![1
The supremum of first eigenvalues of
conformally covariant operators
in a conformal class
bernd ammann and pierre jammes
Abstract
Let (M, g) be a compact Riemannian manifold of dimension ≥ 3. We show that
there is a metric g̃ conformal to g and of volume 1 such that the first positive
eigenvalue of the conformal Laplacian with respect to g̃ is arbitrarily large.
A similar statement is proven for the first positive eigenvalue of the Dirac
operator on a spin manifold of dimension ≥ 2.
1.1 Introduction
The goal of this article is to prove the following theorems.
Theorem 1.1.1 Let (M, g0, χ) be compact Riemannian spin manifold of
dimension n ≥ 2. For any metric g in the conformal class [g0], we denote
the first positive eigenvalue of the Dirac operator on (M, g, χ) by λ+
1 (Dg).
Then
sup
g∈[g0]
λ+
1 (Dg)Vol(M, g)1/n
= ∞.
Theorem 1.1.2 Let (M, g0, χ) be compact Riemannian manifold of dimension
n ≥ 3. For any metric g in the conformal class [g0], we denote the first positive
eigenvalue of the conformal Laplacian Lg := g + n−2
4(n−1)
Scalg (also called
Yamabe operator) on (M, g, χ) by λ+
1 (Lg). Then
sup
g∈[g0]
λ+
1 (Lg)Vol(M, g)2/n
= ∞.
The Dirac operator and the conformal Laplacian belong to a large fam-
ily of operators, defined in details in subsection 1.2.3. These operators are
1](https://image.slidesharecdn.com/89481-250318145131-15e66e01/85/Variational-Problems-in-Differential-Geometry-1st-Edition-Professor-Roger-Bielawski-21-320.jpg)
![2 B. Ammann and P. Jammes
called conformally covariant elliptic operators of order k and of bidegree
((n − k)/2, (n + k)/2), acting on manifolds (M, g) of dimension n k. In
our definition we also claim formal self-adjointness.
All such conformally covariant elliptic operators of order k and of bidegree
((n − k)/2, (n + k)/2) share several analytical properties, in particular they are
associated to the non-compact embedding Hk/2
→ L2n/(n−k)
. Often they have
interpretations in conformal geometry. To give an example, we define for a
compact Riemannian manifold (M, g0)
Y(M, [g0]) := inf
g∈[g0]
λ1(Lg)Vol(M, g)2/n
,
where λ1(Lg) is the lowest eigenvalue of Lg. If Y(M, [g0]) 0, then the
solution of the Yamabe problem [29] tells us that the infimum is attained and
the minimizer is a metric of constant scalar curvature. This famous problem
was finally solved by Schoen and Yau using the positive mass theorem.
In a similar way, for n = 2 the Dirac operator is associated to constant-mean-
curvature conformal immersions of the universal covering into R3
. If a Dirac-
operator-analogue of the positive mass theorem holds for a given manifold
(M, g0), then the infimum
inf
g∈[g0]
λ+
1 (Dg)Vol(M, g)1/n
is attained [3]. However, it is still unclear whether such a Dirac-operator-
analogue of the positive mass theorem holds in general.
The Yamabe problem and its Dirac operator analogue, as well as the
analogues for other conformally covariant operators are typically solved by
minimizing an associated variational problem. As the Sobolev embedding
Hk/2
→ L2n/(n−k)
is non-compact, the direct method of the calculus of variation
fails, but perturbation techniques and conformal blow-up techniques typically
work. Hence all these operators share many properties.
However, only few statements can be proven simultaneously for all confor-
mally covariant elliptic operators of order k and of bidegree ((n − k)/2, (n +
k)/2). Some of the operators are bounded from below (e.g. the Yamabe and
the Paneitz operator), whereas others are not (e.g. the Dirac operator). Some
of them admit a maximum principle, others do not. Some of them act on func-
tions, others on sections of vector bundles. The associated Sobolev space Hk/2
has non-integer order if k is odd, hence it is not the natural domain of a dif-
ferential operator. For Dirac operators, the spin structure has to be considered
in order to derive a statement as Theorem 1.1.1 for n = 2. Because of these
differences, most analytical properties have to be proven for each operator
separately.](https://image.slidesharecdn.com/89481-250318145131-15e66e01/85/Variational-Problems-in-Differential-Geometry-1st-Edition-Professor-Roger-Bielawski-22-320.jpg)
![The supremum of first eigenvalues 3
We consider it hence as remarkable that the proof of our Theorems 1.1.1
and 1.1.2 can be extended to all such operators. Our proof only uses some few
properties of the operators, defined axiomatically in 1.2.3. More exactly we
prove the following.
Theorem 1.1.3 Let Pg be a conformally covariant elliptic operator of order
k, of bidegree ((n − k)/2, (n + k)/2) acting on manifolds of dimension n k.
We also assume that Pg is invertible on Sn−1
× R (see Definition 1.2.4). Let
(M, g0) be compact Riemannian manifold. In the case that Pg depends on
the spin structure, we assume that M is oriented and is equipped with a spin
structure. For any metric g in the conformal class [g0], we denote the first
positive eigenvalue of Pg by λ+
1 (Pg). Then
sup
g∈[g0]
λ+
1 (Pg)Vol(M, g)k/n
= ∞.
The interest in this result is motivated by three questions. At first, as already
mentioned above the infimum
inf
g∈[g0]
λ+
1 (Dg)Vol(M, g)1/n
reflects a rich geometrical structure [3], [4], [5], [7], [8], similarly for the
conformal Laplacian. It seems natural to study the supremum as well.
The second motivation comes from comparing Theorem 1.1.3 to results
about some other differential operators. For the Hodge Laplacian
g
p acting
on p-forms, we have supg∈[g0] λ1(
g
p)Vol(M, g)2/n
= +∞ for n ≥ 4 and 2 ≤
p ≤ n − 2 ([19]). On the other hand, for the Laplacian g
acting on functions,
we have
sup
g∈[g0]
λk(g
)Vol(M, g)2/n
+∞
(the case k = 1 is proven in [20] and the general case in [27]). See [25] for a
synthetic presentation of this subject.
The essential idea in our proof is to construct metrics with longer and longer
cylindrical parts. We will call this an asymptotically cylindrical blowup. Such
metrics are also called Pinocchio metrics in [2, 6]. In [2, 6] the behavior of Dirac
eigenvalues on such metrics has already been studied partially, but the present
article has much stronger results. To extend these existing results provides the
third motivation.
Acknowledgments We thank B. Colbois, M. Dahl, and E. Humbert for
many related discussions. We thank R. Gover for some helpful comments on
conformally covariant operators, and for several references. The first author](https://image.slidesharecdn.com/89481-250318145131-15e66e01/85/Variational-Problems-in-Differential-Geometry-1st-Edition-Professor-Roger-Bielawski-23-320.jpg)
![The supremum of first eigenvalues 5
where r is the distance to 0. Then u is a (strong) solution of (1.1) on . The
same result holds for sections of vector bundles over relatively compact open
subset of Riemannian manifolds.
Proof We show that u is a weak solution of (1.1) in the distributional sense, and
then it follows from standard regularity theory, that it is also a strong solution.
This means that we have to show that for any given compactly supported smooth
test function ψ : → R we have
uP∗
ψ =
f ψ.
Let η : → [0, 1] be a test function that is identically 1 on B0(ε), has
support in B0(2ε), and with |∇m
η| ≤ Cm/εm
. It follows that
sup |P∗
(ηψ)| ≤ C(P, , ψ)ε−k
,
on B0(2ε) B0(ε) and sup |P∗
(ηψ)| ≤ C(P, , ψ) on B0(ε) and hence
uP∗
(ηψ)
≤ Cε−k
B0(2ε)B0(ε)
|u| + C
B0(ε)
|u|
≤ C
B0(2ε)B0(ε)
|u|r−k
+ C
B0(ε)
|u| → 0.
(1.3)
We conclude
uP∗
ψ =
uP∗
(ηψ) +
uP∗
((1 − η)ψ)
=
uP∗
(ηψ)
→0
+
(Pu)(1 − η)ψ
→
f ψ
(1.4)
for ε → 0. Hence the lemma follows.
Condition (1.2) is obviously satisfied if
|u|r−k
∞. It is also satisfied if
|u|2
r−k
∞ and k ≤ n, (1.5)
as in this case
B0(2ε)B0(ε)
|u|r−k
2
≤
|u|2
r−k
B0(2ε)B0(ε)
r−k
≤C
.](https://image.slidesharecdn.com/89481-250318145131-15e66e01/85/Variational-Problems-in-Differential-Geometry-1st-Edition-Professor-Roger-Bielawski-25-320.jpg)
![6 B. Ammann and P. Jammes
1.2.3 Conformally covariant elliptic operators
In this subsection we present a class of certain conformally covariant elliptic
operators. Many important geometric operators are in this class, in particular
the conformal Laplacian, the Paneitz operator, the Dirac operator, see also
[18, 21, 22] for more examples. Readers who are only interested in the Dirac
operator, the Conformal Laplacian or the Paneitz operator, can skip this part
and continue with section 1.3.
Such a conformally covariant operator is not just one single differential oper-
ator, but a procedure how to associate to an n-dimensional Riemannian manifold
(M, g) (potentially with some additional structure) a differential operator Pg
of order k acting on a vector bundle. The important fact is that if g2 = f 2
g1,
then one claims
Pg2
= f − n+k
2 Pg1
f
n−k
2 . (1.6)
One also expresses this by saying that P has bidegree ((n − k)/2, (n + k)/2).
The sense of this equation is apparent if Pg is an operator from C∞
(M)
to C∞
(M). If Pg acts on a vector bundle or if some additional structure (as
e.g. spin structure) is used for defining it, then a rigorous and careful defini-
tion needs more attention. The language of categories provides a good formal
framework [30]. The concept of conformally covariant elliptic operators is
already used by many authors, but we do not know of a reference where a
formal definition is carried out that fits to our context. (See [26] for a similar
categorial approach that includes some of the operators presented here.) Often
an intuitive definition is used. The intuitive definition is obviously sufficient if
one deals with operators acting on functions, such as the conformal Laplacian
or the Paneitz operator. However to properly state Theorem 1.1.3 we need the
following definition.
Let Riemn
(resp. Riemspinn
) be the category n-dimensional Riemannian
manifolds (resp. n-dimensional Riemannian manifolds with orientation and
spin structure). Morphisms from (M1, g1) to (M2, g2) are conformal embed-
dings (M1, g1) → (M2, g2) (resp. conformal embeddings preserving orienta-
tion and spin structure).
Let Laplacen
k (resp. Diracn
k ) be the category whose objects are
{(M, g), Vg, Pg}
where (M, g) in an object of Riemn
(resp. Riemspinn
), where Vg is a vector
bundle with a scalar product on the fibers, where Pg : (Vg) → (Vg) is an
elliptic formally self-adjoint differential operator of order k.](https://image.slidesharecdn.com/89481-250318145131-15e66e01/85/Variational-Problems-in-Differential-Geometry-1st-Edition-Professor-Roger-Bielawski-26-320.jpg)
![8 B. Ammann and P. Jammes
1.2.4 Invertibility on Sn−1
× R
Let P be a conformally covariant elliptic operator of order k and of bide-
gree ((n − k)/2, (n + k)/2). For (M, g) = Sn−1
× R, the operator Pg is a
self-adjoint operator Hk
⊂ L2
→ L2
(see Lemma 1.3.1 and the comments
thereafter).
Definition 1.2.4 We say that P is invertible on Sn−1
× R if Pg is an invertible
operator Hk
→ L2
where g is the standard product metric on Sn−1
× R. In
order words there is a constant σ 0 such that the spectrum of Pg : Hk (Vg) →
L2 (Vg) is contained in (−∞, −σ] ∪ [σ, ∞) for any g ∈ U. In the following,
the largest such σ will be called σP .
We conjecture that any conformally covariant elliptic operator of order k
and of bidegree ((n − k)/2, (n + k)/2) with k n is invertible on Sn−1
× R.
1.2.5 Examples
Example 1: The Conformal Laplacian
Let
Lg := g +
n − 2
4(n − 1)
Scalg,
be the conformal Laplacian. It acts on functions on a Riemannian manifold
(M, g), i.e. Vg is the trivial real line bundle R. Let ι : (M1, g1) → (M2, g2)
be a conformal embedding. Then we can choose κ := Id : ι∗
Vg2
→ Vg1
and
formula (1.7) holds for k = 2 (see e.g. [15, Section 1.J]). All coefficients of
Lg depend continuously on g in the C2
-topology. Hence L is a conformally
covariant elliptic operator of order 2 and of bidegree ((n − 2)/2, (n + 2)/2).
The scalar curvature of Sn−1
× R is (n − 1)(n − 2). The spectrum of Lg on
Sn−1
× R of Lg coincides with the essential spectrum of Lg and is [σL, ∞) with
σL := (n − 2)2
/4. Hence L is invertible on Sn−1
× R if (and only if) n 2.
Example 2: The Paneitz operator
Let (M, g) be a smooth, compact Riemannian manifold of dimension n ≥ 5.
The Paneitz operator Pg is given by
Pgu = (g)2
u − divg(Ag du) +
n − 4
2
Qgu
where
Ag :=
(n − 2)2
+ 4
2(n − 1)(n − 2)
Scalgg −
4
n − 2
Ricg,
Qg =
1
2(n − 1)
gScalg +
n3
− 4n2
+ 16n − 16
8(n − 1)2(n − 2)2
Scal2
g −
2
(n − 2)2
|Ricg|2
.](https://image.slidesharecdn.com/89481-250318145131-15e66e01/85/Variational-Problems-in-Differential-Geometry-1st-Edition-Professor-Roger-Bielawski-28-320.jpg)
![The supremum of first eigenvalues 9
This operator was defined by Paneitz [32] in the case n = 4, and it was general-
ized by Branson in [17] to arbitrary dimensions ≥ 4. We also refer to Theorem
1.21 of the overview article [16]. The explicit formula presented above can
be found e.g. in [23]. The coefficients of Pg depend continuously on g in the
C4
-topology
As in the previous example we can choose for κ the identity, and then the
Paneitz operator Pg is a conformally covariant elliptic operator of order 4 and
of bidegree ((n − 4)/2, (n + 4)/2).
On Sn−1
× R one calculates
Ag =
(n − 4)n
2
Id + 4πR 0
where πR is the projection to vectors parallel to R.
Qg =
(n − 4)n2
8
.
We conclude
σP = Q =
(n − 4)n2
8
and P is invertible on Sn−1
× R if (and only if) n 4.
Examples 3: The Dirac operator.
Let g̃ = f 2
g. Let gM resp. g̃M be the spinor bundle of (M, g) resp.
(M, g̃). Then there is a fiberwise isomorphism β
g
g̃ : gM → g̃M, preserving
the norm such that
Dg̃ ◦ β
g
g̃ (ϕ) = f − n+1
2 β
g
g̃ ◦ Dg f
n−1
2 ϕ ,
see [24, 14] for details. Furthermore, the cocycle conditions
β
g
g̃ ◦ βg̃
g = Id and βĝ
g ◦ β
g̃
ĝ ◦ β
g
g̃ = Id
hold for conformal metrics g, g̃ and ĝ. We will hence use the map β
g
g̃ to identify
gM with g̃M. Hence we simply get
Dg̃ϕ = f − n+1
2 ◦ Dg f
n−1
2 ϕ . (1.8)
All coefficients of Dg depend continuously on g in the C1
-topology. Hence
D is a conformally covariant elliptic operator of order 1 and of bidegree
((n − 1)/2, (n + 1)/2).
The Dirac operator on Sn−1
× R can be decomposed in a part Dvert deriving
along Sn−1
and a part Dhor deriving along R, Dg = Dvert + Dhor, see [1] or [2].](https://image.slidesharecdn.com/89481-250318145131-15e66e01/85/Variational-Problems-in-Differential-Geometry-1st-Edition-Professor-Roger-Bielawski-29-320.jpg)
![10 B. Ammann and P. Jammes
Locally
Dvert =
n−1
i=1
ei · ∇ei
for a local frame (e1, . . . , en−1) of Sn−1
. Here · denotes the Clifford multi-
plication T M ⊗ gM → gM. Furthermore Dhor = ∂t · ∇∂t
, where t ∈ R is
the standard coordinate of R. The operators Dvert and Dhor anticommute. For
n ≥ 3, the spectrum of Dvert coincides with the spectrum of the Dirac operator
on Sn−1
, we cite [12] and obtain
specDvert =
±
n − 1
2
+ k | k ∈ N0
.
The operator (Dhor)2
is the ordinary Laplacian on R and hence has spectrum
[0, ∞). Together this implies that the spectrum of the Dirac operator on Sn−1
×
R is the set (−∞, −σD] ∪ [σD, ∞) with σD = n−1
2
.
In the case n = 2 these statements are only correct if the circle Sn−1
= S1
carries the spin structure induced from the ball. Only this spin structure extends
to the conformal compactification that is given by adding one point at infinity
for each end. For this reason, we will understand in the whole article that all
circles S1
should be equipped with this bounding spin structure. The exten-
sion of the spin structure is essential in order to have a spinor bundle on the
compactification. The methods used in our proof use this extension implicitly.
Hence D is invertible on Sn−1
× R if (and only if) n 1.
Most techniques used in the literature on estimating eigenvalues of the
Dirac operators do not use the spin structure and hence these techniques cannot
provide a proof in the case n = 2.
Example 4: The Rarita-Schwinger operator and many other Fegan type
operators are conformally covariant elliptic operators of order 1 and of bide-
gree ((n − 1)/2, (n + 1)/2). See [21] and in the work of T. Branson for more
information.
Example 5: Assume that (M, g) is a Riemannian spin manifold that carries
a vector bundle W → M with metric and metric connection. Then there is a
natural first order operator (gM ⊗ W) → (gM ⊗ W), the Dirac opera-
tor twisted by W. This operator has similar properties as conformally covariant
elliptic operators of order 1 and of bidegree ((n − 1)/2, (n + 1)/2). The meth-
ods of our article can be easily adapted in order to show that Theorem 1.1.3
is also true for this twisted Dirac operator. However, twisted Dirac operators
are not “conformally covariant elliptic operators” in the above sense. They
could have been included in this class by replacing the category Riemspinn
by](https://image.slidesharecdn.com/89481-250318145131-15e66e01/85/Variational-Problems-in-Differential-Geometry-1st-Edition-Professor-Roger-Bielawski-30-320.jpg)
![The supremum of first eigenvalues 11
Figure 1.1 Asymptotically cylindrical metrics gL (alias Pinocchio metrics) with
growing nose length L.
a category of Riemannian spin manifolds with twisting bundles. In order not to
overload the formalism we chose not to present these larger categories.
The same discussion applies to the spinc
-Dirac operator of a spinc
-manifold.
1.3 Asymptotically cylindrical blowups
1.3.1 Convention
From now on we suppose that Pg is a conformally covariant elliptic operator of
order k, of bidegree ((n − k)/2, (n + k)/2), acting on manifolds of dimension
n and invertible on Sn−1
× R.
1.3.2 Definition of the metrics
Let g0 be a Riemannian metric on a compact manifold M. We can suppose
that the injectivity radius in a fixed point y ∈ M is larger than 1. The geodesic
distance from y to x is denoted by d(x, y).
We choose a smooth function F∞ : M {y} → [1, ∞) such such that
F∞(x) = 1 if d(x, y) ≥ 1, F∞(x) ≤ 2 if d(x, y) ≥ 1/2 and such that F∞(x) =
d(x, y)−1
if d(x, y) ∈ (0, 1/2]. Then for L ≥ 1 we define FL to be a smooth
positive function on M, depending only on d(x, y), such that FL(x) = F∞(x)
if d(x, y) ≥ e−L
and FL(x) ≤ d(x, y)−1
= F∞(x) if d(x, y) ≤ e−L
.
For any L ≥ 1 or L = ∞ set gL := F2
Lg0. The metric g∞ is a complete
metric on M∞.
The family of metrics (gL) is called an asymptotically cylindrical blowup,
in the literature it is denoted as a family of Pinocchio metrics [6], see also
Figure 1.1.
1.3.3 Eigenvalues and basic properties on (M, gL)
For the P-operator associated to (M, gL), L ∈ {0} ∪ [1, ∞) (or more exactly
its self-adjoint extension) we simply write PL instead of PgL
. As M is compact
the spectrum of PL is discrete.](https://image.slidesharecdn.com/89481-250318145131-15e66e01/85/Variational-Problems-in-Differential-Geometry-1st-Edition-Professor-Roger-Bielawski-31-320.jpg)
![12 B. Ammann and P. Jammes
We will denote the spectrum of PL in the following way
. . . ≤ λ−
1 (PL) 0 = 0 . . . = 0 λ+
1 (PL) ≤ λ+
2 (PL) ≤ . . . ,
where each eigenvalue appears with the multiplicity corresponding to the
dimension of the eigenspace. The zeros might appear on this list or not, depend-
ing on whether PL is invertible or not. The spectrum might be entirely positive
(for example the conformal Laplacian Yg on the sphere) in which case λ−
1 (PL)
is not defined. Similarly, λ+
1 (PL) is not defined if the spectrum of (PL) is
negative.
1.3.4 Analytical facts about (M∞, g∞)
The analysis of non-compact manifolds as (M∞, g∞) is more complicated than
in the compact case. Nevertheless (M∞, g∞) is an asymptotically cylindrical
manifold, and for such manifolds an extensive literature is available. One pos-
sible approach would be Melrose’s b-calculus [31]: our cylindrical manifold is
such a b-manifold, but for simplicity and self-containedness we avoid this the-
ory. We will need some few properties that we will summarize in the following
proposition.
We assume in the whole section that P is a conformally covariant elliptic
operator that is invertible on Sn−1
× R, and we write P∞ := Pg∞
for the operator
acting on sections of the bundle V over (M∞, g∞).
Proposition 1.3.1 P∞ extends to a bounded operator from
Hk(M∞,g∞)(V ) → L2(M∞,g∞)(V )
and it satisfies the following regularity estimate
(∇∞
)s
u L2(M∞,g∞) ≤ C( u L2(M∞,g∞) + P∞u L2(M∞,g∞)) (1.9)
for all u ∈ Hk(M∞,g∞)(V ) and all s ∈ {0, 1, . . . , k}. The operator
P∞ : Hk(M∞,g∞)(V ) → L2(M∞,g∞)(V )
is self-adjoint in the sense of an operator in L2(M∞,g∞)(V ).
The proof of the proposition will be sketched in the appendix.
Proposition 1.3.2 The essential spectrum of P∞ coincides with the essen-
tial spectrum of the P-operator on the standard cylinder Sn−1
× R. Thus the
essential spectrum of P∞ is contained in (−∞, −σP ] ∪ [σP , ∞).](https://image.slidesharecdn.com/89481-250318145131-15e66e01/85/Variational-Problems-in-Differential-Geometry-1st-Edition-Professor-Roger-Bielawski-32-320.jpg)
![The supremum of first eigenvalues 13
This proposition follows from the characterization of the essential spectrum
in terms of Weyl sequences, a well-known technique which is for example
carried out and well explained in [13].
The second proposition states that the spectrum of P∞ in the interval
(−σP , σP ) is discrete as well. Eigenvalues of P∞ in this interval will be called
small eigenvalues of P∞. Similarly to above we use the notation λ±
j (P∞) for
the small eigenvalues of P∞.
1.3.5 The kernel
Having recalled these well-known facts we will now study the kernel of con-
formally covariant operators.
If g and g̃ = f 2
are conformal metrics on a compact manifold M, then
ϕ → f − n−k
2 ϕ
obviously defines an isomorphism from ker Pg to ker Pg̃. It is less obvious that
a similar statement holds if we compare g0 and g∞ defined before:
Proposition 1.3.3 The map
ker P0 → ker P∞
ϕ0 → ϕ∞ = F
− n−k
2
∞ ϕ0
is an isomorphism of vector spaces.
Proof Suppose ϕ0 ∈ ker P0. Using standard regularity results it is clear that
sup |ϕ0| ∞. Then
M∞
|ϕ∞|2
dvg∞
≤
MBy (1/2)
|ϕ∞|2
dvg∞
+ sup |ϕ0|2
By (1/2)
F−(n−k)
∞ dvg∞
≤ 2k
MBy (1/2)
|ϕ0|2
dvg0
+ sup |ϕ0|2
ωn−1
1/2
0
rn−1
rk
dr ∞.
(1.10)
Here we used that up to lower order terms dvg∞
coincides with the product
measure of the standard measure on the sphere with the measure d(log r) =
1
r
dr. Furthermore, formula (1.6) implies P∞ϕ∞ = 0. Hence the map is well-
defined. In order to show that it is an isomorphism we show that the obvious
inverse ϕ∞ → ϕ0 := F
n−k
2
∞ ϕ∞ is well defined. To see this we start with an
L2
-section in the kernel of P∞.](https://image.slidesharecdn.com/89481-250318145131-15e66e01/85/Variational-Problems-in-Differential-Geometry-1st-Edition-Professor-Roger-Bielawski-33-320.jpg)
![14 B. Ammann and P. Jammes
We calculate
M
Fk
∞|ϕ0|2
dvg0
=
M∞
|ϕ∞|2
dvg∞
.
Using again (1.6) we see that this section satisfies P0ϕ0 = 0 on M {y}. Hence
condition (1.5) is satisfied, and together with the removal of singularity lemma
(Lemma 1.2.1) one obtains that the inverse map is well defined. The proposition
follows.
1.4 Proof of the main theorem
1.4.1 Stronger version of the main theorem
We will now show the following theorem.
Theorem 1.4.1 Let P be a conformally covariant elliptic operator of order
k, of bidegree ((n − k)/2, (n + k)/2), on manifolds of dimension n k. We
assume that P is invertible on Sn−1
× R.
If lim infL→∞ |λ±
j (PL)| σP , then
λ±
j (PL) → λ±
j (P∞) ∈ (−σP , σP ) for L → ∞.
In the case Spec(Pg0
) ⊂ (0, ∞) the theorem only makes a statement about
λ+
j , and conversely in the case that Spec(Pg0
) ⊂ (−∞, 0) it only makes a
statement about λ−
j .
Obviously this theorem implies Theorem 1.1.3.
1.4.2 The supremum part of the proof of Theorem 1.4.1
At first we prove that
lim sup
L→∞
(λ+
j (PL)) ≤ λ+
j (P∞). (1.11)
Let ϕ1, . . . , ϕj be sequence of L2
-orthonormal eigenvectors of P∞ to
eigenvalues λ+
1 (P∞), . . . , λ+
j (P∞) ∈ [−λ̄, λ̄], λ̄ σP . We choose a cut-off
function χ : M → [0, 1] with χ(x) = 1 for − log(d(x, y)) ≤ T , χ(y) = 0
for − log(d(x, y)) ≥ 2T , and |(∇∞
)s
χ|g∞
≤ Cs/T s
for all s ∈ {0, . . . , k}.
Let ϕ be a linear combination of the eigenvectors ϕ1, . . . , ϕj . From Propo-
sition 1.3.1 we see that
(∇∞
)s
ϕ L2(M∞,g∞) ≤ C ϕ L2(M∞,g∞)](https://image.slidesharecdn.com/89481-250318145131-15e66e01/85/Variational-Problems-in-Differential-Geometry-1st-Edition-Professor-Roger-Bielawski-34-320.jpg)
![The supremum of first eigenvalues 15
where C only depends on (M∞, g∞). Hence for sufficiently large T
P∞(χϕ) − χP∞ϕ L2(M∞,g∞) ≤ kC/T ϕ L2(M∞,g∞) ≤ 2kC/T χϕ L2(M∞,g∞)
as χϕ L2(M∞,g∞) → ϕ L2(M∞,g∞) for T → ∞. The section χϕ can be inter-
preted as a section on (M, gL) if L 2T , and on the support of χϕ
we have gL = g∞ and P∞(χϕ) = PL(χϕ). Hence standard Rayleigh quo-
tient arguments imply that if P∞ has m eigenvalues (counted with mul-
tiplicity) in the interval [a, b] then PL has m eigenvalues in the interval
[a − 2kC/T, b + 2kC/T ]. Taking the limit T → ∞ we obtain (1.11).
By exchanging some obvious signs we obtain similarly
lim sup
L→∞
(−λ−
j (PL)) ≤ −λ−
j (P∞). (1.12)
1.4.3 The infimum part of the proof of Theorem 1.4.1
We now prove
lim inf
L→∞
(±λ±
j (PL)) ≥ ±λ±
j (P∞). (1.13)
We assume that we have a sequence Li → ∞, and that for each i we have a
system of orthogonal eigenvectors ϕi,1, . . . , ϕi,m of PLi
, i.e. PLi
ϕi, = λi,ϕi,
for ∈ {1, . . . , m}. Furthermore we suppose that λi, → λ̄ ∈ (−σP , σP ) for
∈ {1, . . . , m}.
Then
ψi, :=
FLi
F∞
n−k
2
ϕi,
satisfies
P∞ψi, = hi,ψi, with hi, :=
FLi
F∞
k
λi,.
Furthermore
ψi,
2
L2(M∞,g∞) =
M
FLi
F∞
−k
|ϕi,|2
dvgLi
≤ sup
M
|ϕi,|2
M
FLi
F∞
−k
dvgLi
Because of
M
FLi
F∞
−k
dvgL
≤ C
rn−1−k
dr ∞](https://image.slidesharecdn.com/89481-250318145131-15e66e01/85/Variational-Problems-in-Differential-Geometry-1st-Edition-Professor-Roger-Bielawski-35-320.jpg)
![Appendix A Analysis on (M∞, g∞)
The aim of this appendix is to sketch how to prove Proposition 1.3.1. All
properties in this appendix are well-known to experts, but explicit references
are not evident to find. Thus this summary might be helpful to the reader.
The geometry of (M∞, g∞) is asymptotically cylindrical. The metric g∞
is even a b-metric in the sense of Melrose [31], but to keep the presentation
simple, we avoid the b-calculus.
If (r, γ ) ∈ R+
× Sn−1
denote polar normal coordinates with respect to the
metric g0, and if we set t := − log r, then (t, γ ) defines a diffeomorphism α :
B
(M,g0)
y (1/2) {y} → [log 2, ∞) × Sn−1
such that (α−1
)∗
g∞ = dt2
+ ht for a
family of metrics such that (α−1
)∗
g∞, all of its derivatives, its curvature, and all
derivatives of the curvature tend to the standard metric on the cylinder, and the
speed of the convergence is majorised by a multiple of et
. Thus the continuity
of the coefficients property implies, that P∞ extends to a bounded operator
from Hk(M∞,g∞)(V ) → L2(M∞,g∞)(V ).
The formal self-adjointness of P∞ implies that
M∞
ψ, P∞ϕ =
M∞
P∞ψ, ϕ (A.18)
holds for ϕ, ψ ∈ c(V ) and as c(V ) is dense in Hk
, property (A.18) follows
all Hk
-sections ϕ, ψ.
To show Proposition 1.3.1 it remains to prove the regularity estimate and
then to verify that the adjoint of P∞ : Hk(M∞,g∞)(V ) → L2(M∞,g∞)(V ) has
domain Hk(M∞,g∞)(V ).
For proving the regularity estimate we need the following local estimate.
Lemma A.1 Let K be a compact subset of a Riemannian manifold (U, g).
Let P be an elliptic differential operator on U of order k ≥ 1. Then there is a
19](https://image.slidesharecdn.com/89481-250318145131-15e66e01/85/Variational-Problems-in-Differential-Geometry-1st-Edition-Professor-Roger-Bielawski-39-320.jpg)
![20 B. Ammann and P. Jammes
constant C = C(U, K, P, g) such that
u Hk(K,g) ≤ C
u L2(U,g) + Pu L2(U,g)
. (A.19)
Here the Hk
(K, g)-norm is defined via the Levi-Civita connection for g.
This estimate holds uniformly in an ε-neighborhood of P and g in the
following sense. Assume that P̃ is another differential operator, and that the
C0
-norm of the coeffcients of P̃ − P is at most ε, where ε is small. Also
assume that g̃ is ε-close to g in the Ck
-topology. Then the estimate (A.19)
holds for P̃ instead of P and for g̃ instead of g and again for a constant
C = C(U, K, P, g, ε).
Proof of the lemma. We cover the compact set K by a finite number of
coordinate neighborhoods U1, . . . , Um. We choose open sets Vi ⊂ Ui such that
the closure of Vi is compact in Ui and such that K ⊂ V1 ∪ . . . ∪ Vm. One can
choose compact sets Ki ⊂ Vi such that K = K1 ∪ . . . ∪ Km. To prove (A.19)
it is sufficient to prove u Hk(Ki ,g) ≤ C( u L2(Vi ,g) + Pu L2(Vi ,g)) for any i.
We write this inequality in coordinates. As the closure of Vi is a compactum
in Ui, the transition to coordinates changes the above inequality only by a
constant. The operator P, written in a coordinate chart is again elliptic.
We have thus reduced the prove of (A.19) to the prove of the special case
that U and K are open subsets of flat Rn
.
The proof of this special case is explained in detail for example in in [33,
Corollary III 1.5]. The idea is to construct a parametrix for P, i.e. a pseudodif-
ferential operator of order −k such that S1 := QP − Id and S2 := PQ − Id are
infinitely smoothing operators. Thus Q is bounded from L2
(U) to the Sobolev
space Hk
(U), in particular Q(P(u)) Hk ≤ C P(u) L2 . Smoothing operators
map the Sobolev space L2
continuously to Hk
. We obtain
u Hk(K) ≤ u Hk(U) ≤ Q(P(u)) Hk(U) + S1(u) Hk(U)
≤ C
P(u) L2(U) + u L2(U)
.
See also [28, III §3] for a good presentation on how to construct and work with
such a parametrix.
To see the uniformicity, one verifies that
u Hk(K,g̃)
u Hk(K,g)
− 1
≤ C g̃ − g Ck ≤ Cε
and
P̃(u) L2(U)
P(u) L2(U)
− 1
≤ Cε u Hk(U).
The unformicity statement thus follows.](https://image.slidesharecdn.com/89481-250318145131-15e66e01/85/Variational-Problems-in-Differential-Geometry-1st-Edition-Professor-Roger-Bielawski-40-320.jpg)
![The supremum of first eigenvalues 21
Proof of the regularity estimate in Proposition 1.3.1. We write M∞ as MB ∪
([0, ∞) × Sn−1
), such that the metric g∞ is asymptotic (in the C∞
-sense) to the
standard cylindrical metric. The metric g∞ restricted to [R − 1, R + 2] × Sn−1
then converges in the Ck
-topology to the cylindrical metric dt2
+ σn−1
on
[0, 3] × Sn−1
for R → ∞. As the coefficients of Pg depend continuously on
the metric, the P-operators on [R − 1, R + 2] × Sn−1
is in an ε-neighborhood
of P, for R ≥ R0 = R0(ε). Applying the preceding lemma for K = [R, R +
1] × Sn−1
and U = (R − 1, R + 2) × Sn−1
we obtain
∇s
u L2([R,R+1]×Sn−1,g∞) ≤ C
u L2((R−1,R+2)×Sn−1,g∞)
+ P∞u L2((R−1,R+2)×Sn−1,g∞)
. (A.20)
Similarly, applying the lemma to K = MB ∪ ([0, R0] × Sn−1
) and U =
MB ∪ ([0, R0 + 1) × Sn−1
) gives
∇s
u L2(MB ∪([0,R0]×Sn−1),g∞) ≤ C
u L2(MB ∪([0,R0+1)×Sn−1),g∞)
+ P∞u L2(MB ∪([0,R0+1)×Sn−1),g∞)
. (A.21)
Taking the sum of estimate (A.21), of estimate (A.20) for R = R0, again
estimate (A.20) but for R = R0 + 1, and so for all R ∈ {R0 + 2, R0 + 3, . . .}
we obtain (1.9), with a larger constant C.
Now we study the domain D of the adjoint of
P∞ : Hk(M∞,g∞)(V ) → L2(M∞,g∞)(V ).
By definition a section ϕ : L2(M∞,g∞)(V ) is in D if and only if
Hk(M∞,g∞)(V ) u →
M∞
P∞u, ϕ (A.22)
is bounded as a map from L2
to R. For ϕ ∈ Hk(M∞,g∞)(V ) we know that
P∞ϕ is L2
and thus property (A.18) directly implies this boundedness. Thus
Hk(M∞,g∞)(V ) ⊂ D.
Conversely assume the boundedness of (A.22). Then there is a v ∈
L2(M∞,g∞)(V ) such that
M∞
u, v =
M∞
P∞u, ϕ, or in other words P∞ϕ =
v holds weakly. Standard regularity theory implies
ϕ ∈ Hk(M∞,g∞)(V ).
We obtain Hk(M∞,g∞)(V ) = D, and thus the self-adjointness of P∞ follows.
Proposition 1.3.1 is thus shown.](https://image.slidesharecdn.com/89481-250318145131-15e66e01/85/Variational-Problems-in-Differential-Geometry-1st-Edition-Professor-Roger-Bielawski-41-320.jpg)
![22 B. Ammann and P. Jammes
References
[1] B. Ammann, The Dirac Operator on Collapsing Circle Bundles, Sém. Th. Spec.
Géom Inst. Fourier Grenoble 16 (1998), 33–42.
[2] B. Ammann, Spin-Strukturen und das Spektrum des Dirac-Operators, Ph.D. thesis,
University of Freiburg, Germany, 1998, Shaker-Verlag Aachen 1998, ISBN 3-
8265-4282-7.
[3] , The smallest Dirac eigenvalue in a spin-conformal class and cmc-
immersions, Comm. Anal. Geom. 17 (2009), 429–479.
[4] , A spin-conformal lower bound of the first positive Dirac eigenvalue, Diff.
Geom. Appl. 18 (2003), 21–32.
[5] , A variational problem in conformal spin geometry, Habilitationsschrift,
Universität Hamburg, 2003.
[6] B. Ammann and C. Bär, Dirac eigenvalues and total scalar curvature, J. Geom.
Phys. 33 (2000), 229–234.
[7] B. Ammann and E. Humbert, The first conformal Dirac eigenvalue on 2-
dimensional tori, J. Geom. Phys. 56 (2006), 623–642.
[8] B. Ammann, E. Humbert, and B. Morel, Mass endomorphism and spinorial Yam-
abe type problems, Comm. Anal. Geom. 14 (2006), 163–182.
[9] B. Ammann, A. D. Ionescu, and V. Nistor, Sobolev spaces on Lie manifolds and
regularity for polyhedral domains, Doc. Math. 11 (2006), 161–206.
[10] B. Ammann, R. Lauter, and V. Nistor, On the geometry of Riemannian manifolds
with a Lie structure at infinity, Int. J. Math. Math. Sci. (2004), 161–193.
[11] , Pseudodifferential operators on manifolds with a Lie structure at infinity,
Ann. of Math. 165 (2007), 717–747.
[12] C. Bär, The Dirac operator on space forms of positive curvature, J. Math. Soc.
Japan 48 (1996), 69–83.
[13] C. Bär, The Dirac operator on hyperbolic manifolds of finite volume, J. Differ.
Geom. 54 (2000), 439–488.
[14] H. Baum, Spin-Strukturen und Dirac-Operatoren über pseudoriemannschen Man-
nigfaltigkeiten, Teubner Verlag, 1981.
[15] A. L. Besse, Einstein manifolds, Ergebnisse der Mathematik und ihrer Grenzgebi-
ete, 3. Folge, no. 10, Springer-Verlag, 1987.
[16] T. P. Branson, Differential operators canonically associated to a conformal struc-
ture, Math. Scand. 57 (1985), 293–345.
[17] , Group representations arising from Lorentz conformal geometry, J. Funct.
Anal. 74 (1987), no. 2, 199–291.
[18] , Second order conformal covariants, Proc. Amer. Math. Soc. 126 (1998),
1031–1042.
[19] B. Colbois and A. El Soufi, Eigenvalues of the Laplacian acting on p-forms and
metric conformal deformations, Proc. of Am. Math. Soc. 134 (2006), 715–721.
[20] A. El Soufi and S. Ilias, Immersions minimales, première valeur propre du laplacien
et volume conforme, Math. Ann. 275 (1986), 257–267.
[21] H. D. Fegan, Conformally invariant first order differential operators., Quart. J.
Math. Oxford, II. series 27 (1976), 371–378.
[22] R. Gover and L. J. Peterson, Conformally invariant powers of the Laplacian,
Q-curvature, and tractor calculus, Comm. Math. Phys. 235 (2003), 339–378.](https://image.slidesharecdn.com/89481-250318145131-15e66e01/85/Variational-Problems-in-Differential-Geometry-1st-Edition-Professor-Roger-Bielawski-42-320.jpg)
![The supremum of first eigenvalues 23
[23] E. Hebey and F. Robert, Coercivity and Struwe’s compactness for Paneitz type
operators with constant coefficients, Calc. Var. Partial Differential Equations 13
(2001), 491–517.
[24] N. Hitchin, Harmonic spinors, Adv. Math. 14 (1974), 1–55.
[25] P. Jammes, Extrema de valeurs propres dans une classe conforme, Sémin. Théor.
Spectr. Géom. 24 (2007), 23–42.
[26] I. Kolář, P. W. Michor, and J. Slovák, Natural operations in differential geometry,
Springer-Verlag, Berlin, 1993.
[27] N. Korevaar, Upper bounds for eigenvalues of conformal metrics, J. Differ. Geom.
37 (1993), 73–93.
[28] H.-B. Lawson and M.-L. Michelsohn, Spin Geometry, Princeton University Press,
1989.
[29] J. M. Lee and T. H. Parker. The Yamabe problem. Bull. Am. Math. Soc., New Ser.
17 (1987), 37–91.
[30] S. Mac Lane, Categories for the working mathematician, Graduate Texts in Math-
ematics, vol. 5, Springer-Verlag, New York, 1998.
[31] R. B. Melrose, The Atiyah-Patodi-Singer index theorem, Research Notes in Math-
ematics, vol. 4, A K Peters Ltd., Wellesley, MA, 1993.
[32] S. M. Paneitz, A quartic conformally covariant differential operator for arbitrary
pseudo-Riemannian manifolds, Preprint 1983, published in SIGMA 4 (2008).
[33] M. E. Taylor, M. E., Pseudodifferential operators, Princeton University Press,
1981.
Authors’ addresses:
Bernd Ammann
Facultät für Mathematik
Universität Regensburg
93040 Regensburg
Germany
bernd.ammann@mathematik.uni-regensburg.de
Pierre Jammes
Laboratoire J.-A. Diendonné, Université Nice-Sophia Antipolis,
Parc Valrose, F-06108 Nice Cedex02, France
pjammes@unice.fr](https://image.slidesharecdn.com/89481-250318145131-15e66e01/85/Variational-Problems-in-Differential-Geometry-1st-Edition-Professor-Roger-Bielawski-43-320.jpg)
![2
K-Destabilizing test configurations with
smooth central fiber
claudio arezzo, alberto della vedova, and
gabriele la nave
Abstract
In this note we point out a simple application of a result by the authors in
[2]. We show how to construct many families of strictly K-unstable polarized
manifolds, destabilized by test configurations with smooth central fiber. The
effect of resolving singularities of the central fiber of a given test configuration
is studied, providing many new examples of manifolds which do not admit
Kähler constant scalar curvature metrics in some classes.
2.1 Introduction
In this note we want to speculate about the following Conjecture due to Tian-
Yau-Donaldson ([23], [24], [25], [7]):
Conjecture 2.1.1 A polarized manifold (M, A) admits a Kähler metric of
constant scalar curvature in the class c1(A) if and only if it is K-polystable.
The notion of K-stability will be recalled below. For the moment it suffices to
say, loosely speaking, that a polarized manifold, or more generally a polarized
variety (V, A), is K-stable if and only if any special degeneration or test
configuration of (V, A) has an associated non positive weight, called Futaki
invariant and that this is zero only for the product configuration, i.e. the trivial
degeneration.
We do not even attempt to give a survey of results about Conjecture 2.1.1, but
as far as the results of this note are concerned, it is important to recall the reader
that Tian [24], Donaldson [7], Stoppa [22], using the results in [3] and [4], and
Mabuchi [17] have proved the sufficiency part of the Conjecture. Destabilizing
a polarizing manifold then implies non existence results of Kähler constant
scalar curvature metrics in the corresponding classes.
24](https://image.slidesharecdn.com/89481-250318145131-15e66e01/85/Variational-Problems-in-Differential-Geometry-1st-Edition-Professor-Roger-Bielawski-44-320.jpg)
![K-Destabilizing test configurations 25
One of the main problems in this subject is that under a special degeneration
a smooth manifold often becomes very singular, in fact just a polarized scheme
in general. This makes all the analytic tool available at present very difficult to
use.
Hence one naturally asks which type of singularities must be introduced to
make the least effort to destabilize a smooth manifold without cscK metrics.
The aim of this note is to provide a large class of examples of special
degenerations with positive Futaki invariant and smooth limit. In fact we want to
provide a “machine” which associates to any special degeneration of a polarized
normal variety (V, A) with positive Futaki invariant a special degeneration for
a polarized manifold (M̃, Ã) with smooth central fiber and still positive Futaki
invariant.
To the best of our knowledge, before this work the only known examples of
special degeneration with non negative Futaki invariant and smooth central fiber
are the celebrated example of Mukai-Umemura’s Fano threefold ([18]) used
by Tian in [24] to exhibit the first examples of Fano manifolds with discrete
automorphism group and no Kähler-Einstein metrics (other Fano manifolds
with these properties have been then produced in [1]). In this case there exist
non trivial special degenerations with smooth limit and zero Futaki invariant
(hence violating the definition of K-stability). It then falls in the borderline
case, making this example extremely interesting and delicate. We stress that
our “machine” does not work in this borderline case, because a priori the Futaki
invariant of the new test configuration is certainly small (by [2]) but we cannot
control its sign.
To state our result more precisely we now recall the relevant definitions:
Definition 2.1.2 Let (V, A) be a n-dimensional polarized variety or scheme.
Given a one-parameter subgroup ρ : C∗
→ Aut(V ) with a linearization on A
and denoted by w(V, A) the weight of the C∗
-action induced on
top
H0
(V, A),
we have the following asymptotic expansions as k 0:
h0
(V, Ak
) = a0kn
+ a1kn−1
+ O(kn−2
) (2.1)
w(V, Ak
) = b0kn+1
+ b1kn
+ O(kn−1
) (2.2)
The (normalized) Futaki invariant of the action is the rational number
F(V, A, ρ) =
b1
a0
−
b0 a1
a2
0
.
Definition 2.1.3 A test configuration (X, L) for a polarized variety (V, A)
consists of a scheme X endowed with a C∗
-action that linearizes on a line
bundle L over X, and a flat C∗
-equivariant map f : X → C (where C has the](https://image.slidesharecdn.com/89481-250318145131-15e66e01/85/Variational-Problems-in-Differential-Geometry-1st-Edition-Professor-Roger-Bielawski-45-320.jpg)
![26 C. Arezzo, A. Della Vedova, and Gabriele La Nave
usual weight one C∗
-action) such that L|f −1(0) is ample on f −1
(0) and we have
(f −1
(1), L|f −1(1)) (V, Ar
) for some r 0.
When (V, A) has a C∗
-action ρ : C∗
→ Aut(V ), a test configuration where
X = V × C and C∗
acts on X diagonally through ρ is called product configu-
ration.
Given a test configuration (X, L) we will denote by F(X, L) the Futaki
invariant of the C∗
-action induced on the central fiber (f −1
(0), L|f −1(0)).
If (X, L) is a product configuration as above, clearly we have F(X, L) =
F(V, A, ρ).
Definition 2.1.4 The polarized manifold (M, A) is K-stable if for each test
configuration for (M, A) the Futaki invariant of the induced action on the central
fiber (f −1
(0), L|f −1(0)) is less than or equal to zero, with equality if and only if
we have a product configuration.
A test configuration (X, L) is called destabilizing if the Futaki invariant of
the induced action on (f −1
(0), L|f −1(0)) is greater than zero.
Test configurations for an embedded variety V ⊂ PN
endowed with the hyper-
plane polarization A can be constructed as follows. Given a one-parameter
subgroup ρ : C∗
→ GL(N + 1), which induces an obvious diagonal C∗
-action
on PN
× C, it clear that the subscheme
X =
(z, t) ∈ PN × C | t = 0, (ρ(t−1)z, t) ∈ V
⊂ PN
× C,
is invariant and projects equivariantly on C. Thus considering the relatively
ample polarization L induced by the hyperplane bundle gives test configuration
for (V, A). On the other hand, given a test configuration (X, L) for a polarized
variety (V, A), the relative projective embedding given by Lr
, with r sufficiently
large, realizes X as above (see details in [21]).
We can now describe our “machine”: consider a test configuration (X, L)
for a polarized normal variety (V, A) with F(X, L) 0. Up to raise L to a
suitable power – which does not affect the Futaki invariant – we can suppose
being in the situation above with X ⊂ PN
× C invariantly, and L induced by the
hyperplane bundle of PN
. At this point we consider the central fiber X0 ⊂ PN
,
which is invariant with respect to ρ, and we apply the (equivariant) resolution of
singularities [14, Corollary 3.22 and Proposition 3.9.1]. Thus there is a smooth
manifold P̃ acted on by C∗
and an equivariant map
β : P̃ → PN
which factorizes through a sequence of blow-ups, such that the strict transform
X̃0 of X0 is invariant and smooth. The key observation is that the strict transform
X̃1 of the fiber X1 ⊂ X degenerate to X0 under the given C∗
action on P̃ , thus it](https://image.slidesharecdn.com/89481-250318145131-15e66e01/85/Variational-Problems-in-Differential-Geometry-1st-Edition-Professor-Roger-Bielawski-46-320.jpg)
![K-Destabilizing test configurations 27
must be smooth. This gives an invariant family X̃ ⊂ P × C and an equivariant
birational morphism
π : X̃ → X.
Some comments are in order:
1 all the fibers of X̃ are smooth, but π is never a resolution of singularities of
X (except the trivial case when the central fiber of X was already smooth)
since it fails to be an isomorphism on the smooth locus of X;
2 L̃ = π∗
L is not a relatively ample line bundle any more, but just a big and nef
one. It is not then even clear what it means to compute its Futaki invariant;
3 the fiber over the generic point of C of the new (big and nef) test configuration
(X̃, L̃) is different from V ;
4 the family X̃ is not unique since the resolution β it is not.
The issue raised at point (2) was addressed in [2] and it was proved that the
following natural (topological) definition makes the Futaki invariant a continu-
ous function around big and nef points in the Kähler cone. We will give simple
self-contained proofs in the cases of smooth manifolds and varieties with just
normal singularities in Section 2.
Definition 2.1.5 Let V be a projective variety or scheme endowed with a C∗
-
action and let B be a big and nef line bundle on V . Choosing a linearization of
the action on B gives a C∗
-representation on
dim V
j=0 Hj
(V, Bk
)(−1)j
(here the
E−1
denotes the dual of E). We set w(V, Bk
) = tr Ak, where Ak is the generator
of that representation. As k → +∞ we have the following expansion
w(V, Bk
)
χ(V, Bk)
= F0k + F1 + O(k−1
),
and we define
F(V, B) = F1
to be the Donaldson–Futaki invariant of the chosen action on (V, B)
The existence of the expansion involved in definition above follows from the
standard fact that χ(V, Bk
) is a polynomial of degree dim V , whose proof (see
for example [11]) can be easily adapted to show that w(V, Bk
) is a polynomial
of degree at most dim V + 1.](https://image.slidesharecdn.com/89481-250318145131-15e66e01/85/Variational-Problems-in-Differential-Geometry-1st-Edition-Professor-Roger-Bielawski-47-320.jpg)
![28 C. Arezzo, A. Della Vedova, and Gabriele La Nave
The key technical Theorem proved in [2] is then the following:
Theorem 2.1.6 Let B, A be linearized line bunldes on a scheme V acted on
by C∗
. Suppose that B is big and nef and A ample. We have
F(V, Br
⊗ A) = F(V, B) + O
1
r
, as r → ∞.
Having established a good continuity property of the Futaki invariant up to these
boundary point, we need to address the question of the effect of a resolution
of singularities of the central fiber. This is a particular case of the following
non trivial extension of previous analysis by Ross and Thomas [21] which was
proved in [2] where the general case of birational morphisms has been studied:
Theorem 2.1.7 Given a test configuration f : (X, L) → C as above, let f
:
(X
, L
) → C be another flat equivariant family with X
normal and let β :
(X
, L
) → (X, L) be a C∗
-equivariant birational morphism such that f
=
f ◦ β and L
= β∗
L. Then we have
F(X
, L
) ≥ F(X, L),
with strict inequality if and only if the support of β∗(OX )/OX has codimension
one.
The proof of these results uses some heavy algebraic machinery, yet their proof
when (V, A) or the central fiber of (X, L) have only normal singularities (a
case largely studied) is quite simple and we give it in Section 2.
The Corollary of Theorem 2.1.6 and Theorem 2.1.7 we want to point out in
this note is then the following:
Theorem 2.1.8 Let (X, L) be a test configuration for the polarized normal
variety (V, A) with positive Futaki invariant. Let moreover (X̃, L̃) be a (big
and nef) test configuration obtained from (X, L) as above and let (M̃, B̃) be
the smooth (big and nef) fiber over the point 1 ∈ C. Let R be any relatively
ample line bundle over X̃.
Then (X̃, L̃r
⊗ R) is a test configuration for (M̃, B̃r
⊗ R|M̃ ) with following
properties:
1 smooth central fiber;
2 positive Futaki invariant for r sufficiently large.
In particular M̃ does not admit a constant scalar curvature Kähler metric in
any class of the form c1(B̃r
⊗ R|M̃ ), with r large enough.](https://image.slidesharecdn.com/89481-250318145131-15e66e01/85/Variational-Problems-in-Differential-Geometry-1st-Edition-Professor-Roger-Bielawski-48-320.jpg)
![K-Destabilizing test configurations 29
While this Theorem clearly follows from Theorems 2.1.6 and Theorem 2.1.7,
but for the specific case of central fiber with normal singularities it follows
from the much simpler Proposition 2.2.1 and Theorem 2.2.3.
The range of applicability of the above theorem is very large. We go through
the steps of the resolution of singularities in an explicit example by Ding-
Tian [6] of a complex orbifold of dimension 2. In this simple example explicit
calculations are easy to perform, yet we point out that the final example is
somehow trivial since it ends on a product test configuration. On the other
hand abundance of similar examples even in dimension 2 can be obtained by
the reader as an exercise using the results of Jeffres [12] and Nagakawa [19],
in which cases we loose an explicit description of the resulting destabilized
manifold, but we get new nontrivial examples. In fact in higher dimensions one
can use the approach described in this note to test also the Arezzo-Pacard blow
up theorems [3] [4], when the resolution of singularities requires a blow up of
a scheme of positive dimension.
2.2 The case of normal singularities
In this section we give simple proofs of the continuity of the Futaki invariant
at boundary points for smooth manifolds or varieties with normal singularities.
More general results of this type have been proved in [2] but we want to stress
that under these assumptions proofs become much easier.
The fundamental continuity property we will need, and proved in Corollary
2.1.6, can be stated in the following form for smooth bases:
Proposition 2.2.1 Let A, L be respectively an ample and a big and nef line
bundle on a smooth projective manifold M. For every C∗
-action on M that
linearizes to A and L, as r → +∞ we have
F(M, Lr
⊗ A) = F(M, L) + O
1
r
.
Proof The result is a simple application of the equivariant Riemann-Roch
Theorem. We present here the details of the calculations involved, since we
could not find precise references for them.
Fix an hermitian metrics on A that is invariant with respect to the action of
S1
⊂ C∗
and suppose that the curvature ω is a Kähler metric. Since L is nef,
for each r 0 we can choose an invariant metric on L whose curvature ηr
satisfy rηr + ω 0. In other words rηr + ω is a Kähler form which coincides
with the curvature of the induced hermitian metric on the line bundle Lr
⊗ A.](https://image.slidesharecdn.com/89481-250318145131-15e66e01/85/Variational-Problems-in-Differential-Geometry-1st-Edition-Professor-Roger-Bielawski-49-320.jpg)
Variational Problems in Differential Geometry 1st Edition Professor Roger Bielawski
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- 7. LONDON MATHEMATICAL SOCIETY LECTURE NOTE SERIES Managing Editor: Professor M. Reid, Mathematics Institute, University of Warwick, Coventry CV4 7AL, United Kingdom The titles below are available from booksellers, or from Cambridge University Press at http://www.cambridge.org/mathematics 337 Methods in Banach space theory, J.M.F. CASTILLO & W.B. JOHNSON (eds) 338 Surveys in geometry and number theory, N. YOUNG (ed) 339 Groups St Andrews 2005 I, C.M. CAMPBELL, M.R. QUICK, E.F. ROBERTSON & G.C. SMITH (eds) 340 Groups St Andrews 2005 II, C.M. CAMPBELL, M.R. QUICK, E.F. ROBERTSON & G.C. SMITH (eds) 341 Ranks of elliptic curves and random matrix theory, J.B. CONREY, D.W. FARMER, F. MEZZADRI & N.C. SNAITH (eds) 342 Elliptic cohomology, H.R. MILLER & D.C. RAVENEL (eds) 343 Algebraic cycles and motives I, J. NAGEL & C. PETERS (eds) 344 Algebraic cycles and motives II, J. NAGEL & C. PETERS (eds) 345 Algebraic and analytic geometry, A. NEEMAN 346 Surveys in combinatorics 2007, A. HILTON & J. TALBOT (eds) 347 Surveys in contemporary mathematics, N. YOUNG & Y. CHOI (eds) 348 Transcendental dynamics and complex analysis, P.J. RIPPON & G.M. STALLARD (eds) 349 Model theory with applications to algebra and analysis I, Z. CHATZIDAKIS, D. MACPHERSON, A. PILLAY & A. WILKIE (eds) 350 Model theory with applications to algebra and analysis II, Z. CHATZIDAKIS, D. MACPHERSON, A. PILLAY & A. WILKIE (eds) 351 Finite von Neumann algebras and masas, A.M. SINCLAIR & R.R. SMITH 352 Number theory and polynomials, J. MCKEE & C. SMYTH (eds) 353 Trends in stochastic analysis, J. BLATH, P. MÖRTERS & M. SCHEUTZOW (eds) 354 Groups and analysis, K. TENT (ed) 355 Non-equilibrium statistical mechanics and turbulence, J. CARDY, G. FALKOVICH & K. GAWEDZKI 356 Elliptic curves and big Galois representations, D. DELBOURGO 357 Algebraic theory of differential equations, M.A.H. MACCALLUM & A.V. MIKHAILOV (eds) 358 Geometric and cohomological methods in group theory, M.R. BRIDSON, P.H. KROPHOLLER & I.J. LEARY (eds) 359 Moduli spaces and vector bundles, L. BRAMBILA-PAZ, S.B. BRADLOW, O. GARCÍA-PRADA & S. RAMANAN (eds) 360 Zariski geometries, B. ZILBER 361 Words: Notes on verbal width in groups, D. SEGAL 362 Differential tensor algebras and their module categories, R. BAUTISTA, L. SALMERÓN & R. ZUAZUA 363 Foundations of computational mathematics, Hong Kong 2008, F. CUCKER, A. PINKUS & M.J. TODD (eds) 364 Partial differential equations and fluid mechanics, J.C. ROBINSON & J.L. RODRIGO (eds) 365 Surveys in combinatorics 2009, S. HUCZYNSKA, J.D. MITCHELL & C.M. RONEY-DOUGAL (eds) 366 Highly oscillatory problems, B. ENGQUIST, A. FOKAS, E. HAIRER & A. ISERLES (eds) 367 Random matrices: High dimensional phenomena, G. BLOWER 368 Geometry of Riemann surfaces, F.P. GARDINER, G. GONZÁLEZ-DIEZ & C. KOUROUNIOTIS (eds) 369 Epidemics and rumours in complex networks, M. DRAIEF & L. MASSOULIÉ 370 Theory of p-adic distributions, S. ALBEVERIO, A.YU. KHRENNIKOV & V.M. SHELKOVICH 371 Conformal fractals, F. PRZYTYCKI & M. URBAŃSKI 372 Moonshine: The first quarter century and beyond, J. LEPOWSKY, J. MCKAY & M.P. TUITE (eds) 373 Smoothness, regularity, and complete intersection, J. MAJADAS & A. G. RODICIO 374 Geometric analysis of hyperbolic differential equations: An introduction, S. ALINHAC 375 Triangulated categories, T. HOLM, P. JØRGENSEN & R. ROUQUIER (eds) 376 Permutation patterns, S. LINTON, N. RUŠKUC & V. VATTER (eds) 377 An introduction to Galois cohomology and its applications, G. BERHUY 378 Probability and mathematical genetics, N. H. BINGHAM & C. M. GOLDIE (eds) 379 Finite and algorithmic model theory, J. ESPARZA, C. MICHAUX & C. STEINHORN (eds) 380 Real and complex singularities, M. MANOEL, M.C. ROMERO FUSTER & C.T.C WALL (eds) 381 Symmetries and integrability of difference equations, D. LEVI, P. OLVER, Z. THOMOVA & P. WINTERNITZ (eds) 382 Forcing with random variables and proof complexity, J. KRAJÍČEK 383 Motivic integration and its interactions with model theory and non-Archimedean geometry I, R. CLUCKERS, J. NICAISE & J. SEBAG (eds) 384 Motivic integration and its interactions with model theory and non-Archimedean geometry II, R. CLUCKERS, J. NICAISE & J. SEBAG (eds) 385 Entropy of hidden Markov processes and connections to dynamical systems, B. MARCUS, K. PETERSEN & T. WEISSMAN (eds) 386 Independence-friendly logic, A.L. MANN, G. SANDU & M. SEVENSTER 387 Groups St Andrews 2009 in Bath I, C.M. CAMPBELL et al (eds) 388 Groups St Andrews 2009 in Bath II, C.M. CAMPBELL et al (eds) 389 Random fields on the sphere, D. MARINUCCI & G. PECCATI 390 Localization in periodic potentials, D.E. PELINOVSKY 391 Fusion systems in algebra and topology M. ASCHBACHER, R. KESSAR & B. OLIVER 392 Surveys in combinatorics 2011, R. CHAPMAN (ed) 393 Non-abelian fundamental groups and Iwasawa theory, J. COATES et al (eds) 394 Variational problems in differential geometry, R. BIELAWSKI, K. HOUSTON & M. SPEIGHT (eds)
- 9. London Mathematical Society Lecture Note Series: 394 Variational Problems in Differential Geometry University of Leeds 2009 Edited by R. BIELAWSKI K. HOUSTON J.M. SPEIGHT University of Leeds
- 10. cambridge university press Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo, Delhi, Tokyo, Mexico City Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521282741 C Cambridge University Press 2012 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2012 Printed in the United Kingdom at the University Press, Cambridge A catalogue record for this publication is available from the British Library Library of Congress Cataloguing in Publication data Variational problems in differential geometry : University of Leeds, 2009 / edited by R. Bielawski, K. Houston, J.M. Speight. p. cm. – (London Mathematical Society lecture note series ; 394) Includes bibliographical references. ISBN 978-0-521-28274-1 (pbk.) 1. Geometry, Differential – Congresses. I. Bielawski, R. II. Houston, Kevin, 1968– III. Speight, J. M. (J. Martin) IV. Title. V. Series. QA641.V37 2012 516.3 6 – dc23 2011027490 ISBN 978-0-521-28274-1 Paperback Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.
- 11. Contents List of contributors page viii Preface xi 1 The supremum of first eigenvalues of conformally covariant operators in a conformal class 1 Bernd Ammann and Pierre Jammes 1.1 Introduction 1 1.2 Preliminaries 4 1.3 Asymptotically cylindrical blowups 11 1.4 Proof of the main theorem 14 Appendix A Analysis on (M∞, g∞) 19 References 22 2 K-Destabilizing test configurations with smooth central fiber 24 Claudio Arezzo, Alberto Della Vedova, and Gabriele La Nave 2.1 Introduction 24 2.2 The case of normal singularities 29 2.3 Proof of Theorem 2.1.8 and examples 32 References 34 3 Explicit constructions of Ricci solitons 37 Paul Baird 3.1 Introduction 37 3.2 Solitons from a dynamical system 40 3.3 Reduction of the equations to a 2-dimensional system 44 3.4 Higher dimensional Ricci solitons via projection 48 3.5 The 4-dimensional geometry Nil4 50 References 55 v
- 12. vi Contents 4 Open Iwasawa cells and applications to surface theory 56 Josef F. Dorfmeister 4.1 Introduction 56 4.2 Basic notation and the Birkhoff decomposition 58 4.3 Iwasawa decomposition 59 4.4 Iwasawa decomposition via Birkhoff decomposition 60 4.5 A function defining the open Iwasawa cells 62 4.6 Applications to surface theory 64 References 66 5 Multiplier ideal sheaves and geometric problems 68 Akito Futaki and Yuji Sano 5.1 Introduction 68 5.2 An overview of multiplier ideal sheaves 72 5.3 Direct relationships between multiplier ideal sheaves and the obstruction F 83 References 90 6 Multisymplectic formalism and the covariant phase space 94 Frédéric Hélein 6.1 The multisymplectic formalism 95 6.2 The covariant phase space 110 6.3 Geometric quantization 117 References 123 7 Nonnegative curvature on disk bundles 127 Lorenz J. Schwachhöfer 7.1 Introduction 127 7.2 Normal homogeneous metrics and Cheeger deformations 128 7.3 Homogeneous metrics of nonnegative curvature 130 7.4 Collar metrics of nonnegative curvature 131 7.5 Bundles with normal homogeneous collar 132 7.6 Cohomogeneity one manifolds 139 References 140 8 Morse theory and stable pairs 142 Richard A. Wentworth and Graeme Wilkin 8.1 Introduction 142 8.2 Stable pairs 146 8.3 Morse theory 154 8.4 Cohomology of moduli spaces 174 References 180
- 13. Contents vii 9 Manifolds with k-positive Ricci curvature 182 Jon Wolfson 9.1 Introduction 182 9.2 Manifolds with k-positive Ricci curvature 183 9.3 Fill radius and an approach to Conjecture 1 192 9.4 The fundamental group and fill radius bounds 198 References 200
- 14. Contributors Bernd Ammann Facultät für Mathematik, Universität Regensburg, 93040 Regensburg, Germany Pierre Jammes Laboratoire J.-A. Dieudonné, Université Nice – Sophia Antipolis, Parc Valrose, F-06108 NICE Cedex 02, France Claudio Arezzo Abdus Salam International Center for Theoretical Physics, Strada Costiera 11, Trieste (Italy) and Dipartimento di Matematica, Università di Parma, Parco Area delle Scienze 53/A, Parma, Italy Alberto Della Vedova Fine Hall, Princeton University, Princeton, NJ 08544 and Dipartimento di Matematica, Università di Parma, Parco Area delle Scienze 53/A, Parma, Italy Gabriele La Nave Department of Mathematics, Yeshiva University, 500 West 185 Street, New York, NY, USA Paul Baird Département de Mathématiques, Université de Bretagne Occidentale, 6 Avenue Le Gorgeu – CS 93837, 29238 Brest, France Josef F. Dorfmeister Fakultät für Mathematik, Technische Universität München, Boltzmannstr. 3, D-85747 Garching, Germany viii
- 15. List of contributors ix Akito Futaki Department of Mathematics, Tokyo Institute of Technology, 2-12-1, O-okayama, Meguro, Tokyo 152-8551, Japan Yuji Sano Department of Mathematics, Kyushu University, 6-10-1, Hakozaki, Higashiku, Fukuoka-city, Fukuoka 812-8581 Japan Frédéric Hélein Institut de Mathématiques de Jussieu, UMR CNRS 7586, Université Denis Diderot Paris 7, 175 rue du Chevaleret, 75013 Paris, France Lorenz J. Schwachhöfer Fakultät für Mathematik, Technische Universität Dortmund, Vogelpothsweg 87, 44221 Dortmund, Germany Richard A. Wentworth Department of Mathematics, University of Maryland, College Park, MD 20742, USA Graeme Wilkin Department of Mathematics, University of Colorado, Boulder, CO 80309, USA Jon Wolfson Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA
- 17. Preface The workshop Variational Problems in Differential Geometry was held at the University of Leeds from March 30 to April 2nd, 2009. The aim of the meeting was to bring together researchers working on disparate geometric problems, all of which admit a variational formulation. Among the topics discussed were recent developments in harmonic maps and morphisms, minimal and CMC surfaces, extremal Kähler metrics, the Yam- abe functional, Hamiltonian variational problems, and topics related to gauge theory and to the Ricci flow. The meeting incorporated a special session in honour of John C. Wood, on the occasion of his 60th birthday, to celebrate his seminal contributions to the theory of harmonic maps and morphisms. The following mathematicians gave one-hour talks: Bernd Ammann, Clau- dio Arezzo, Paul Baird, Olivier Biquard, Christoph Boehm, Francis Burstall, Josef Dorfmeister, Akito Futaki, Mark Haskins, Frederic Helein, Nicolaos Kapouleas, Mario Micallef, Frank Pacard, Simon Salamon, Lorenz Schwach- hoefer, Peter Topping, Richard Wentworth, and Jon Wolfson. There were about 50 participants from the UK, US, Japan and several Euro- pean countries. The schedule allowed plenty of opportunities for discussion and interaction between official talks and made for a successful and stimulat- ing meeting. The workshop was financially supported by the London Mathematical Soci- ety, the Engineering and Physical Sciences Research Council of Great Britain and the School of Mathematics, University of Leeds. The articles presented in this volume represent the whole spectrum of the subject. The supremum of first eigenvalues of conformally covariant operators in a conformal class by Ammann and Jammes is concerned with the first eigenvalues of the Yamabe operator, the Dirac operator, and more general conformally xi
- 18. xii Preface covariant elliptic operators on compact Riemannian manifolds. It is well known that the infimum of the first eigenvalue in a given conformal class reflects a rich geometric structure. In this article, the authors study the supremum of the first eigenvalue and show that, for a very general class of operators, this supremum is infinite. The article, K-Destabilizing test configurations with smooth central fiber by Arezzo, Della Vedova, and La Nave is concerned with the famous Tian- Yau-Donaldson conjecture about existence of constant scalar curvature Kähler metrics. They construct many new families of K-unstable manifolds, and, consequently, many new examples of manifolds which do not admit Kähler constant scalar curvature metrics in some cohomology classes. As has been now understood, a very natural extension of Einstein metrics are the Ricci solitons. These are the subject of Paul Baird’s article Explicit constructions of Ricci solitons, in which he does precisely that: he constructs many explicit examples, including some in the more exotic geometries Sol3, Nil3, and Nil4. Josef Dorfmeister is concerned with a more classical topic: that of constant mean curvature and Willmore surfaces. In recent years, many new examples of such surfaces were constructed using loop groups. The method relies on finding “Iwasawa-like” decompositions of loop groups and the article Open Iwasawa cells in twisted loop groups and some applications to harmonic maps discusses such decompositions and their singularities. The currently extremely important notions of K-stability and K- polystability are the topic of the paper by Futaki and Sano Multiplier ideal sheaves and geometric problems. This is an expository article giving state-of- the-art presentation of the powerful method of multiplier ideal sheaves and their applications to Kähler-Einstein and Sasaki-Einstein geometries. Multisymplectic formalism and the covariant phase space by Frédéric Hélein takes us outside Riemannian geometry. The author presents an alternative (in fact, two of them) to the Feynman integral as a foundation of quantum field theory. Lorenz Schwachhöfer’s Nonnegative curvature on disk bundles is a survey of the glueing method used to construct Riemannian manifolds with nonnegative sectional curvature - one of the classical problems in geometry. Morse theory and stable pairs by Wentworth and Wilkin introduces new techniques to compute equivariant cohomology of certain natural moduli spaces. The main ingredient is a version of Morse-Atiyah-Bott theory adapted to singular infinite dimensional spaces. The final article, Manifolds with k-positive Ricci curvature, by Jon Wolf- son, is a survey of results and conjectures about Riemannian n-manifolds with
- 19. Preface xiii k-positive Ricci curvature. These interpolate between positive scalar curva- ture (n-positive Ricci curvature) and positive Ricci curvature (1-positive Ricci curvature), and the author shows how the results about k-positive Ricci curva- ture, 1 k n, also interpolate, or should do, between what is known about manifolds satisfying those two classical notions of positivity. We would like to extend our thanks to our colleague John Wood for his help and assistance in preparing these proceedings. R. Bielawski K. Houston J.M. Speight Leeds, UK
- 21. 1 The supremum of first eigenvalues of conformally covariant operators in a conformal class bernd ammann and pierre jammes Abstract Let (M, g) be a compact Riemannian manifold of dimension ≥ 3. We show that there is a metric g̃ conformal to g and of volume 1 such that the first positive eigenvalue of the conformal Laplacian with respect to g̃ is arbitrarily large. A similar statement is proven for the first positive eigenvalue of the Dirac operator on a spin manifold of dimension ≥ 2. 1.1 Introduction The goal of this article is to prove the following theorems. Theorem 1.1.1 Let (M, g0, χ) be compact Riemannian spin manifold of dimension n ≥ 2. For any metric g in the conformal class [g0], we denote the first positive eigenvalue of the Dirac operator on (M, g, χ) by λ+ 1 (Dg). Then sup g∈[g0] λ+ 1 (Dg)Vol(M, g)1/n = ∞. Theorem 1.1.2 Let (M, g0, χ) be compact Riemannian manifold of dimension n ≥ 3. For any metric g in the conformal class [g0], we denote the first positive eigenvalue of the conformal Laplacian Lg := g + n−2 4(n−1) Scalg (also called Yamabe operator) on (M, g, χ) by λ+ 1 (Lg). Then sup g∈[g0] λ+ 1 (Lg)Vol(M, g)2/n = ∞. The Dirac operator and the conformal Laplacian belong to a large fam- ily of operators, defined in details in subsection 1.2.3. These operators are 1
- 22. 2 B. Ammann and P. Jammes called conformally covariant elliptic operators of order k and of bidegree ((n − k)/2, (n + k)/2), acting on manifolds (M, g) of dimension n k. In our definition we also claim formal self-adjointness. All such conformally covariant elliptic operators of order k and of bidegree ((n − k)/2, (n + k)/2) share several analytical properties, in particular they are associated to the non-compact embedding Hk/2 → L2n/(n−k) . Often they have interpretations in conformal geometry. To give an example, we define for a compact Riemannian manifold (M, g0) Y(M, [g0]) := inf g∈[g0] λ1(Lg)Vol(M, g)2/n , where λ1(Lg) is the lowest eigenvalue of Lg. If Y(M, [g0]) 0, then the solution of the Yamabe problem [29] tells us that the infimum is attained and the minimizer is a metric of constant scalar curvature. This famous problem was finally solved by Schoen and Yau using the positive mass theorem. In a similar way, for n = 2 the Dirac operator is associated to constant-mean- curvature conformal immersions of the universal covering into R3 . If a Dirac- operator-analogue of the positive mass theorem holds for a given manifold (M, g0), then the infimum inf g∈[g0] λ+ 1 (Dg)Vol(M, g)1/n is attained [3]. However, it is still unclear whether such a Dirac-operator- analogue of the positive mass theorem holds in general. The Yamabe problem and its Dirac operator analogue, as well as the analogues for other conformally covariant operators are typically solved by minimizing an associated variational problem. As the Sobolev embedding Hk/2 → L2n/(n−k) is non-compact, the direct method of the calculus of variation fails, but perturbation techniques and conformal blow-up techniques typically work. Hence all these operators share many properties. However, only few statements can be proven simultaneously for all confor- mally covariant elliptic operators of order k and of bidegree ((n − k)/2, (n + k)/2). Some of the operators are bounded from below (e.g. the Yamabe and the Paneitz operator), whereas others are not (e.g. the Dirac operator). Some of them admit a maximum principle, others do not. Some of them act on func- tions, others on sections of vector bundles. The associated Sobolev space Hk/2 has non-integer order if k is odd, hence it is not the natural domain of a dif- ferential operator. For Dirac operators, the spin structure has to be considered in order to derive a statement as Theorem 1.1.1 for n = 2. Because of these differences, most analytical properties have to be proven for each operator separately.
- 23. The supremum of first eigenvalues 3 We consider it hence as remarkable that the proof of our Theorems 1.1.1 and 1.1.2 can be extended to all such operators. Our proof only uses some few properties of the operators, defined axiomatically in 1.2.3. More exactly we prove the following. Theorem 1.1.3 Let Pg be a conformally covariant elliptic operator of order k, of bidegree ((n − k)/2, (n + k)/2) acting on manifolds of dimension n k. We also assume that Pg is invertible on Sn−1 × R (see Definition 1.2.4). Let (M, g0) be compact Riemannian manifold. In the case that Pg depends on the spin structure, we assume that M is oriented and is equipped with a spin structure. For any metric g in the conformal class [g0], we denote the first positive eigenvalue of Pg by λ+ 1 (Pg). Then sup g∈[g0] λ+ 1 (Pg)Vol(M, g)k/n = ∞. The interest in this result is motivated by three questions. At first, as already mentioned above the infimum inf g∈[g0] λ+ 1 (Dg)Vol(M, g)1/n reflects a rich geometrical structure [3], [4], [5], [7], [8], similarly for the conformal Laplacian. It seems natural to study the supremum as well. The second motivation comes from comparing Theorem 1.1.3 to results about some other differential operators. For the Hodge Laplacian g p acting on p-forms, we have supg∈[g0] λ1( g p)Vol(M, g)2/n = +∞ for n ≥ 4 and 2 ≤ p ≤ n − 2 ([19]). On the other hand, for the Laplacian g acting on functions, we have sup g∈[g0] λk(g )Vol(M, g)2/n +∞ (the case k = 1 is proven in [20] and the general case in [27]). See [25] for a synthetic presentation of this subject. The essential idea in our proof is to construct metrics with longer and longer cylindrical parts. We will call this an asymptotically cylindrical blowup. Such metrics are also called Pinocchio metrics in [2, 6]. In [2, 6] the behavior of Dirac eigenvalues on such metrics has already been studied partially, but the present article has much stronger results. To extend these existing results provides the third motivation. Acknowledgments We thank B. Colbois, M. Dahl, and E. Humbert for many related discussions. We thank R. Gover for some helpful comments on conformally covariant operators, and for several references. The first author
- 24. 4 B. Ammann and P. Jammes wants to thank the Albert Einstein institute at Potsdam-Golm for its very kind hospitality which enabled to write the article. 1.2 Preliminaries 1.2.1 Notations In this article By(r) denotes the ball of radius r around y, Sy(r) = ∂By(r) its boundary. The standard sphere S0(1) ⊂ Rn in Rn is denoted by Sn−1 , its volume is ωn−1. For the volume element of (M, g) we use the notation dvg . In our article, (V ) (resp. c(V )) always denotes the set of all smooth sections (resp. all compactly supported smooth sections) of the vector bundle V → M. For sections u of V → M over a Riemannian manifold (M, g) the Sobolev norms L2 and Hs , s ∈ N, are defined as u 2 L2(M,g) := M |u|2 dvg u 2 Hs (M,g) := u 2 L2(M,g) + ∇u 2 L2(M,g) + · · · + ∇s u 2 L2(M,g). The vector bundle V will be suppressed in the notation. If M and g are clear from the context, we write just L2 and Hs . The completion of {u ∈ (V ) | u Hs (M,g) ∞} with respect to the Hs (M, g)-norm is denoted by Hs (M,g)(V ), or if (M, g) or V is clear from the context, we alternatively write Hs (V ) or Hs (M, g) for Hs (M,g)(V ). The same definitions are used for L2 instead of Hs . And similarly Ck(M,g)(V ) = Ck (V ) = Ck (M, g) is the set of all Ck -sections, k ∈ N ∪ {∞}. 1.2.2 Removal of singularities In the proof we will use the following removal of singularities lemma. Lemma 1.2.1 (Removal of singularities lemma) Let be a bounded open subset of Rn containing 0. Let P be an elliptic differential operator of order k on , f ∈ C∞ (), and let u ∈ C∞ ( {0}) be a solution of Pu = f (1.1) on {0} with lim ε→0 B0(2ε)−B0(ε) |u|r−k = 0 and lim ε→0 B0(ε) |u| = 0 (1.2)
- 25. The supremum of first eigenvalues 5 where r is the distance to 0. Then u is a (strong) solution of (1.1) on . The same result holds for sections of vector bundles over relatively compact open subset of Riemannian manifolds. Proof We show that u is a weak solution of (1.1) in the distributional sense, and then it follows from standard regularity theory, that it is also a strong solution. This means that we have to show that for any given compactly supported smooth test function ψ : → R we have uP∗ ψ = f ψ. Let η : → [0, 1] be a test function that is identically 1 on B0(ε), has support in B0(2ε), and with |∇m η| ≤ Cm/εm . It follows that sup |P∗ (ηψ)| ≤ C(P, , ψ)ε−k , on B0(2ε) B0(ε) and sup |P∗ (ηψ)| ≤ C(P, , ψ) on B0(ε) and hence uP∗ (ηψ) ≤ Cε−k B0(2ε)B0(ε) |u| + C B0(ε) |u| ≤ C B0(2ε)B0(ε) |u|r−k + C B0(ε) |u| → 0. (1.3) We conclude uP∗ ψ = uP∗ (ηψ) + uP∗ ((1 − η)ψ) = uP∗ (ηψ) →0 + (Pu)(1 − η)ψ → f ψ (1.4) for ε → 0. Hence the lemma follows. Condition (1.2) is obviously satisfied if |u|r−k ∞. It is also satisfied if |u|2 r−k ∞ and k ≤ n, (1.5) as in this case B0(2ε)B0(ε) |u|r−k 2 ≤ |u|2 r−k B0(2ε)B0(ε) r−k ≤C .
- 26. 6 B. Ammann and P. Jammes 1.2.3 Conformally covariant elliptic operators In this subsection we present a class of certain conformally covariant elliptic operators. Many important geometric operators are in this class, in particular the conformal Laplacian, the Paneitz operator, the Dirac operator, see also [18, 21, 22] for more examples. Readers who are only interested in the Dirac operator, the Conformal Laplacian or the Paneitz operator, can skip this part and continue with section 1.3. Such a conformally covariant operator is not just one single differential oper- ator, but a procedure how to associate to an n-dimensional Riemannian manifold (M, g) (potentially with some additional structure) a differential operator Pg of order k acting on a vector bundle. The important fact is that if g2 = f 2 g1, then one claims Pg2 = f − n+k 2 Pg1 f n−k 2 . (1.6) One also expresses this by saying that P has bidegree ((n − k)/2, (n + k)/2). The sense of this equation is apparent if Pg is an operator from C∞ (M) to C∞ (M). If Pg acts on a vector bundle or if some additional structure (as e.g. spin structure) is used for defining it, then a rigorous and careful defini- tion needs more attention. The language of categories provides a good formal framework [30]. The concept of conformally covariant elliptic operators is already used by many authors, but we do not know of a reference where a formal definition is carried out that fits to our context. (See [26] for a similar categorial approach that includes some of the operators presented here.) Often an intuitive definition is used. The intuitive definition is obviously sufficient if one deals with operators acting on functions, such as the conformal Laplacian or the Paneitz operator. However to properly state Theorem 1.1.3 we need the following definition. Let Riemn (resp. Riemspinn ) be the category n-dimensional Riemannian manifolds (resp. n-dimensional Riemannian manifolds with orientation and spin structure). Morphisms from (M1, g1) to (M2, g2) are conformal embed- dings (M1, g1) → (M2, g2) (resp. conformal embeddings preserving orienta- tion and spin structure). Let Laplacen k (resp. Diracn k ) be the category whose objects are {(M, g), Vg, Pg} where (M, g) in an object of Riemn (resp. Riemspinn ), where Vg is a vector bundle with a scalar product on the fibers, where Pg : (Vg) → (Vg) is an elliptic formally self-adjoint differential operator of order k.
- 27. The supremum of first eigenvalues 7 A morphism (ι, κ) from {(M1, g1), Vg1 , Pg1 } to {(M2, g2), Vg2 , Pg2 } consists of a conformal embedding ι : (M1, g1) → (M2, g2) (preserving orientation and spin structure in the case of Diracn k ) together with a fiber isomorphism κ : ι∗ Vg2 → Vg1 preserving fiberwise length, such that Pg1 and Pg2 sat- isfy the conformal covariance property (1.6). For stating this property pre- cisely, let f 0 be defined by ι∗ g2 = f 2 g1, and let κ∗ : (Vg2 ) → (Vg1 ), κ∗(ϕ) = κ ◦ ϕ ◦ ι. Then the conformal covariance property is κ∗Pg2 = f − n+k 2 Pg1 f n−k 2 κ∗. (1.7) In the following the maps κ and ι will often be evident from the context and then will be omitted. The transformation formula (1.7) then simplifies to (1.6). Definition 1.2.2 A conformally covariant elliptic operator of order k and of bidegree ((n − k)/2, (n + k)/2) is a contravariant functor from Riemn (resp. Riemspinn ) to Laplacen k (resp. Diracn k ), mapping (M, g) to (M, g, Vg, Pg) in such a way that the coefficients are continuous in the Ck -topology of metrics (see below). To shorten notation, we just write Pg or P for this functor. It remains to explain the Ck -continuity of the coefficients. For Riemannian metrics g, g1, g2 defined on a compact set K ⊂ M we set d g Ck(K) (g1, g2) := max t=0,...,k (∇g)t (g1 − g2) C0(K). For a fixed background metric g, the relation d g Ck(K) ( · , · ) defines a distance function on the space of metrics on K. The topology induced by dg is inde- pendent of this background metric and it is called the Ck -topology of metrics on K. Definition 1.2.3 We say that the coefficients of P are continuous in the Ck - topology of metrics if for any metric g on a manifold M, and for any compact subset K ⊂ M there is a neighborhood U of g|K in the Ck -topology of met- rics on K, such that for all metrics g̃, g̃|K ∈ U, there is an isomorphism of vector bundles κ̂ : Vg|K → Vg̃|K over the identity of K with induced map κ̂∗ : (Vg|K ) → (Vg̃|K ) with the property that the coefficients of the differ- ential operator Pg − (κ̂∗)−1 Pg̃κ̂∗ depend continuously on g̃ (with respect to the Ck -topology of metrics).
- 28. 8 B. Ammann and P. Jammes 1.2.4 Invertibility on Sn−1 × R Let P be a conformally covariant elliptic operator of order k and of bide- gree ((n − k)/2, (n + k)/2). For (M, g) = Sn−1 × R, the operator Pg is a self-adjoint operator Hk ⊂ L2 → L2 (see Lemma 1.3.1 and the comments thereafter). Definition 1.2.4 We say that P is invertible on Sn−1 × R if Pg is an invertible operator Hk → L2 where g is the standard product metric on Sn−1 × R. In order words there is a constant σ 0 such that the spectrum of Pg : Hk (Vg) → L2 (Vg) is contained in (−∞, −σ] ∪ [σ, ∞) for any g ∈ U. In the following, the largest such σ will be called σP . We conjecture that any conformally covariant elliptic operator of order k and of bidegree ((n − k)/2, (n + k)/2) with k n is invertible on Sn−1 × R. 1.2.5 Examples Example 1: The Conformal Laplacian Let Lg := g + n − 2 4(n − 1) Scalg, be the conformal Laplacian. It acts on functions on a Riemannian manifold (M, g), i.e. Vg is the trivial real line bundle R. Let ι : (M1, g1) → (M2, g2) be a conformal embedding. Then we can choose κ := Id : ι∗ Vg2 → Vg1 and formula (1.7) holds for k = 2 (see e.g. [15, Section 1.J]). All coefficients of Lg depend continuously on g in the C2 -topology. Hence L is a conformally covariant elliptic operator of order 2 and of bidegree ((n − 2)/2, (n + 2)/2). The scalar curvature of Sn−1 × R is (n − 1)(n − 2). The spectrum of Lg on Sn−1 × R of Lg coincides with the essential spectrum of Lg and is [σL, ∞) with σL := (n − 2)2 /4. Hence L is invertible on Sn−1 × R if (and only if) n 2. Example 2: The Paneitz operator Let (M, g) be a smooth, compact Riemannian manifold of dimension n ≥ 5. The Paneitz operator Pg is given by Pgu = (g)2 u − divg(Ag du) + n − 4 2 Qgu where Ag := (n − 2)2 + 4 2(n − 1)(n − 2) Scalgg − 4 n − 2 Ricg, Qg = 1 2(n − 1) gScalg + n3 − 4n2 + 16n − 16 8(n − 1)2(n − 2)2 Scal2 g − 2 (n − 2)2 |Ricg|2 .
- 29. The supremum of first eigenvalues 9 This operator was defined by Paneitz [32] in the case n = 4, and it was general- ized by Branson in [17] to arbitrary dimensions ≥ 4. We also refer to Theorem 1.21 of the overview article [16]. The explicit formula presented above can be found e.g. in [23]. The coefficients of Pg depend continuously on g in the C4 -topology As in the previous example we can choose for κ the identity, and then the Paneitz operator Pg is a conformally covariant elliptic operator of order 4 and of bidegree ((n − 4)/2, (n + 4)/2). On Sn−1 × R one calculates Ag = (n − 4)n 2 Id + 4πR 0 where πR is the projection to vectors parallel to R. Qg = (n − 4)n2 8 . We conclude σP = Q = (n − 4)n2 8 and P is invertible on Sn−1 × R if (and only if) n 4. Examples 3: The Dirac operator. Let g̃ = f 2 g. Let gM resp. g̃M be the spinor bundle of (M, g) resp. (M, g̃). Then there is a fiberwise isomorphism β g g̃ : gM → g̃M, preserving the norm such that Dg̃ ◦ β g g̃ (ϕ) = f − n+1 2 β g g̃ ◦ Dg f n−1 2 ϕ , see [24, 14] for details. Furthermore, the cocycle conditions β g g̃ ◦ βg̃ g = Id and βĝ g ◦ β g̃ ĝ ◦ β g g̃ = Id hold for conformal metrics g, g̃ and ĝ. We will hence use the map β g g̃ to identify gM with g̃M. Hence we simply get Dg̃ϕ = f − n+1 2 ◦ Dg f n−1 2 ϕ . (1.8) All coefficients of Dg depend continuously on g in the C1 -topology. Hence D is a conformally covariant elliptic operator of order 1 and of bidegree ((n − 1)/2, (n + 1)/2). The Dirac operator on Sn−1 × R can be decomposed in a part Dvert deriving along Sn−1 and a part Dhor deriving along R, Dg = Dvert + Dhor, see [1] or [2].
- 30. 10 B. Ammann and P. Jammes Locally Dvert = n−1 i=1 ei · ∇ei for a local frame (e1, . . . , en−1) of Sn−1 . Here · denotes the Clifford multi- plication T M ⊗ gM → gM. Furthermore Dhor = ∂t · ∇∂t , where t ∈ R is the standard coordinate of R. The operators Dvert and Dhor anticommute. For n ≥ 3, the spectrum of Dvert coincides with the spectrum of the Dirac operator on Sn−1 , we cite [12] and obtain specDvert = ± n − 1 2 + k | k ∈ N0 . The operator (Dhor)2 is the ordinary Laplacian on R and hence has spectrum [0, ∞). Together this implies that the spectrum of the Dirac operator on Sn−1 × R is the set (−∞, −σD] ∪ [σD, ∞) with σD = n−1 2 . In the case n = 2 these statements are only correct if the circle Sn−1 = S1 carries the spin structure induced from the ball. Only this spin structure extends to the conformal compactification that is given by adding one point at infinity for each end. For this reason, we will understand in the whole article that all circles S1 should be equipped with this bounding spin structure. The exten- sion of the spin structure is essential in order to have a spinor bundle on the compactification. The methods used in our proof use this extension implicitly. Hence D is invertible on Sn−1 × R if (and only if) n 1. Most techniques used in the literature on estimating eigenvalues of the Dirac operators do not use the spin structure and hence these techniques cannot provide a proof in the case n = 2. Example 4: The Rarita-Schwinger operator and many other Fegan type operators are conformally covariant elliptic operators of order 1 and of bide- gree ((n − 1)/2, (n + 1)/2). See [21] and in the work of T. Branson for more information. Example 5: Assume that (M, g) is a Riemannian spin manifold that carries a vector bundle W → M with metric and metric connection. Then there is a natural first order operator (gM ⊗ W) → (gM ⊗ W), the Dirac opera- tor twisted by W. This operator has similar properties as conformally covariant elliptic operators of order 1 and of bidegree ((n − 1)/2, (n + 1)/2). The meth- ods of our article can be easily adapted in order to show that Theorem 1.1.3 is also true for this twisted Dirac operator. However, twisted Dirac operators are not “conformally covariant elliptic operators” in the above sense. They could have been included in this class by replacing the category Riemspinn by
- 31. The supremum of first eigenvalues 11 Figure 1.1 Asymptotically cylindrical metrics gL (alias Pinocchio metrics) with growing nose length L. a category of Riemannian spin manifolds with twisting bundles. In order not to overload the formalism we chose not to present these larger categories. The same discussion applies to the spinc -Dirac operator of a spinc -manifold. 1.3 Asymptotically cylindrical blowups 1.3.1 Convention From now on we suppose that Pg is a conformally covariant elliptic operator of order k, of bidegree ((n − k)/2, (n + k)/2), acting on manifolds of dimension n and invertible on Sn−1 × R. 1.3.2 Definition of the metrics Let g0 be a Riemannian metric on a compact manifold M. We can suppose that the injectivity radius in a fixed point y ∈ M is larger than 1. The geodesic distance from y to x is denoted by d(x, y). We choose a smooth function F∞ : M {y} → [1, ∞) such such that F∞(x) = 1 if d(x, y) ≥ 1, F∞(x) ≤ 2 if d(x, y) ≥ 1/2 and such that F∞(x) = d(x, y)−1 if d(x, y) ∈ (0, 1/2]. Then for L ≥ 1 we define FL to be a smooth positive function on M, depending only on d(x, y), such that FL(x) = F∞(x) if d(x, y) ≥ e−L and FL(x) ≤ d(x, y)−1 = F∞(x) if d(x, y) ≤ e−L . For any L ≥ 1 or L = ∞ set gL := F2 Lg0. The metric g∞ is a complete metric on M∞. The family of metrics (gL) is called an asymptotically cylindrical blowup, in the literature it is denoted as a family of Pinocchio metrics [6], see also Figure 1.1. 1.3.3 Eigenvalues and basic properties on (M, gL) For the P-operator associated to (M, gL), L ∈ {0} ∪ [1, ∞) (or more exactly its self-adjoint extension) we simply write PL instead of PgL . As M is compact the spectrum of PL is discrete.
- 32. 12 B. Ammann and P. Jammes We will denote the spectrum of PL in the following way . . . ≤ λ− 1 (PL) 0 = 0 . . . = 0 λ+ 1 (PL) ≤ λ+ 2 (PL) ≤ . . . , where each eigenvalue appears with the multiplicity corresponding to the dimension of the eigenspace. The zeros might appear on this list or not, depend- ing on whether PL is invertible or not. The spectrum might be entirely positive (for example the conformal Laplacian Yg on the sphere) in which case λ− 1 (PL) is not defined. Similarly, λ+ 1 (PL) is not defined if the spectrum of (PL) is negative. 1.3.4 Analytical facts about (M∞, g∞) The analysis of non-compact manifolds as (M∞, g∞) is more complicated than in the compact case. Nevertheless (M∞, g∞) is an asymptotically cylindrical manifold, and for such manifolds an extensive literature is available. One pos- sible approach would be Melrose’s b-calculus [31]: our cylindrical manifold is such a b-manifold, but for simplicity and self-containedness we avoid this the- ory. We will need some few properties that we will summarize in the following proposition. We assume in the whole section that P is a conformally covariant elliptic operator that is invertible on Sn−1 × R, and we write P∞ := Pg∞ for the operator acting on sections of the bundle V over (M∞, g∞). Proposition 1.3.1 P∞ extends to a bounded operator from Hk(M∞,g∞)(V ) → L2(M∞,g∞)(V ) and it satisfies the following regularity estimate (∇∞ )s u L2(M∞,g∞) ≤ C( u L2(M∞,g∞) + P∞u L2(M∞,g∞)) (1.9) for all u ∈ Hk(M∞,g∞)(V ) and all s ∈ {0, 1, . . . , k}. The operator P∞ : Hk(M∞,g∞)(V ) → L2(M∞,g∞)(V ) is self-adjoint in the sense of an operator in L2(M∞,g∞)(V ). The proof of the proposition will be sketched in the appendix. Proposition 1.3.2 The essential spectrum of P∞ coincides with the essen- tial spectrum of the P-operator on the standard cylinder Sn−1 × R. Thus the essential spectrum of P∞ is contained in (−∞, −σP ] ∪ [σP , ∞).
- 33. The supremum of first eigenvalues 13 This proposition follows from the characterization of the essential spectrum in terms of Weyl sequences, a well-known technique which is for example carried out and well explained in [13]. The second proposition states that the spectrum of P∞ in the interval (−σP , σP ) is discrete as well. Eigenvalues of P∞ in this interval will be called small eigenvalues of P∞. Similarly to above we use the notation λ± j (P∞) for the small eigenvalues of P∞. 1.3.5 The kernel Having recalled these well-known facts we will now study the kernel of con- formally covariant operators. If g and g̃ = f 2 are conformal metrics on a compact manifold M, then ϕ → f − n−k 2 ϕ obviously defines an isomorphism from ker Pg to ker Pg̃. It is less obvious that a similar statement holds if we compare g0 and g∞ defined before: Proposition 1.3.3 The map ker P0 → ker P∞ ϕ0 → ϕ∞ = F − n−k 2 ∞ ϕ0 is an isomorphism of vector spaces. Proof Suppose ϕ0 ∈ ker P0. Using standard regularity results it is clear that sup |ϕ0| ∞. Then M∞ |ϕ∞|2 dvg∞ ≤ MBy (1/2) |ϕ∞|2 dvg∞ + sup |ϕ0|2 By (1/2) F−(n−k) ∞ dvg∞ ≤ 2k MBy (1/2) |ϕ0|2 dvg0 + sup |ϕ0|2 ωn−1 1/2 0 rn−1 rk dr ∞. (1.10) Here we used that up to lower order terms dvg∞ coincides with the product measure of the standard measure on the sphere with the measure d(log r) = 1 r dr. Furthermore, formula (1.6) implies P∞ϕ∞ = 0. Hence the map is well- defined. In order to show that it is an isomorphism we show that the obvious inverse ϕ∞ → ϕ0 := F n−k 2 ∞ ϕ∞ is well defined. To see this we start with an L2 -section in the kernel of P∞.
- 34. 14 B. Ammann and P. Jammes We calculate M Fk ∞|ϕ0|2 dvg0 = M∞ |ϕ∞|2 dvg∞ . Using again (1.6) we see that this section satisfies P0ϕ0 = 0 on M {y}. Hence condition (1.5) is satisfied, and together with the removal of singularity lemma (Lemma 1.2.1) one obtains that the inverse map is well defined. The proposition follows. 1.4 Proof of the main theorem 1.4.1 Stronger version of the main theorem We will now show the following theorem. Theorem 1.4.1 Let P be a conformally covariant elliptic operator of order k, of bidegree ((n − k)/2, (n + k)/2), on manifolds of dimension n k. We assume that P is invertible on Sn−1 × R. If lim infL→∞ |λ± j (PL)| σP , then λ± j (PL) → λ± j (P∞) ∈ (−σP , σP ) for L → ∞. In the case Spec(Pg0 ) ⊂ (0, ∞) the theorem only makes a statement about λ+ j , and conversely in the case that Spec(Pg0 ) ⊂ (−∞, 0) it only makes a statement about λ− j . Obviously this theorem implies Theorem 1.1.3. 1.4.2 The supremum part of the proof of Theorem 1.4.1 At first we prove that lim sup L→∞ (λ+ j (PL)) ≤ λ+ j (P∞). (1.11) Let ϕ1, . . . , ϕj be sequence of L2 -orthonormal eigenvectors of P∞ to eigenvalues λ+ 1 (P∞), . . . , λ+ j (P∞) ∈ [−λ̄, λ̄], λ̄ σP . We choose a cut-off function χ : M → [0, 1] with χ(x) = 1 for − log(d(x, y)) ≤ T , χ(y) = 0 for − log(d(x, y)) ≥ 2T , and |(∇∞ )s χ|g∞ ≤ Cs/T s for all s ∈ {0, . . . , k}. Let ϕ be a linear combination of the eigenvectors ϕ1, . . . , ϕj . From Propo- sition 1.3.1 we see that (∇∞ )s ϕ L2(M∞,g∞) ≤ C ϕ L2(M∞,g∞)
- 35. The supremum of first eigenvalues 15 where C only depends on (M∞, g∞). Hence for sufficiently large T P∞(χϕ) − χP∞ϕ L2(M∞,g∞) ≤ kC/T ϕ L2(M∞,g∞) ≤ 2kC/T χϕ L2(M∞,g∞) as χϕ L2(M∞,g∞) → ϕ L2(M∞,g∞) for T → ∞. The section χϕ can be inter- preted as a section on (M, gL) if L 2T , and on the support of χϕ we have gL = g∞ and P∞(χϕ) = PL(χϕ). Hence standard Rayleigh quo- tient arguments imply that if P∞ has m eigenvalues (counted with mul- tiplicity) in the interval [a, b] then PL has m eigenvalues in the interval [a − 2kC/T, b + 2kC/T ]. Taking the limit T → ∞ we obtain (1.11). By exchanging some obvious signs we obtain similarly lim sup L→∞ (−λ− j (PL)) ≤ −λ− j (P∞). (1.12) 1.4.3 The infimum part of the proof of Theorem 1.4.1 We now prove lim inf L→∞ (±λ± j (PL)) ≥ ±λ± j (P∞). (1.13) We assume that we have a sequence Li → ∞, and that for each i we have a system of orthogonal eigenvectors ϕi,1, . . . , ϕi,m of PLi , i.e. PLi ϕi, = λi,ϕi, for ∈ {1, . . . , m}. Furthermore we suppose that λi, → λ̄ ∈ (−σP , σP ) for ∈ {1, . . . , m}. Then ψi, := FLi F∞ n−k 2 ϕi, satisfies P∞ψi, = hi,ψi, with hi, := FLi F∞ k λi,. Furthermore ψi, 2 L2(M∞,g∞) = M FLi F∞ −k |ϕi,|2 dvgLi ≤ sup M |ϕi,|2 M FLi F∞ −k dvgLi Because of M FLi F∞ −k dvgL ≤ C rn−1−k dr ∞
- 36. 16 B. Ammann and P. Jammes (for n k) the norm ψi, L2(M∞,g∞) is finite as well, and we can renormalize such that ψi, L2(M∞,g∞) = 1. Lemma 1.4.2 For any δ 0 and any ∈ {0, . . . , m} the sequence ψi, Ck+1(MBy (δ),g∞) i is bounded. Proof of the lemma. After removing finitely many i, we can assume that λi ≤ 2λ̄ and e−Li δ/2. Hence FL = F∞ and hi = λi on M By(δ/2). Because of MBy (δ/2) |(P∞)s ψi|2 dvg∞ ≤ (2λ̄)2s MBy (δ/2) |ψi|2 dvg∞ ≤ (2λ̄)2s we obtain boundedness of ψi in the Sobolev space Hsk (M By(3δ/4), g∞), and hence, for sufficiently large s boundedness in Ck+1 (M By(δ), g∞). The lemma is proved. Hence after passing to a subsequence ψi, converges in Ck,α (M By(δ), g∞) to a solution ψ̄ of P∞ψ̄ = λ̄ψ̄. By taking a diagonal sequence, one can obtain convergence in Ck,α loc (M∞) of ψi, to ψ̄. It remains to prove that ψ̄1, . . . ,ψ̄m are linearly independent, in particular that any ψ̄ = 0. For this we use the following lemma. Lemma 1.4.3 For any ε 0 there is δ0 and i0 such that ψi, L2(By (δ0),g∞) ≤ ε ψi, L2(M∞,g∞) for all i ≥ i0 and all ∈ {0, . . . , m}. In particular, ψi, L2(MBy (δ0),g∞) ≥ (1 − ε) ψi, L2(M∞,g∞). Proof of the lemma. Because of Proposition 1.3.1 and P∞ψi, L2(M∞,g∞) ≤ |λ̄| ψi, L2(M∞,g∞) = |λ̄|
- 37. The supremum of first eigenvalues 17 we get (∇∞ )s ψi, L2(M∞,g∞) ≤ C for all s ∈ {0, . . . , k}. Let χ be a cut-off function as in Subsection 1.4.2 with T = − log δ. Hence P∞ (1 − χ)ψi, − (1 − χ)P∞(ψi,) L2(M∞,g∞) ≤ C T = C − log δ . (1.14) On the other hand (By(δ) {y}, g∞) converges for suitable choices of base points for δ → 0 to Sn−1 × (0, ∞) in the C∞ -topology of Riemannian man- ifolds with base points. Hence there is a function τ(δ) converging to 0 such that P∞ (1 − χ)ψi, L2(M∞,g∞) ≥ (σp − τ(δ)) (1 − χ)ψi, L2(M∞,g∞). (1.15) Using the obvious relation (1 − χ)P∞(ψi,) L2(M∞,g∞) ≤ |λi,| (1 − χ)ψi, L2(M∞,g∞) we obtain with (1.14) and (1.15) ψi, L2(By (δ2),g∞) ≤ (1 − χ)ψi, L2(M∞,g∞) ≤ C | log δ|(σP − τ(δ) − |λi,|) . The right hand side is smaller than ε for i sufficiently large and δ suffi- ciently small. The main statement of the lemma then follows for δ0 := δ2 . The Minkowski inequality yields. ψi, L2(MBy (δ2),g∞) ≥ 1 − ψi, L2(By (δ2),g∞) ≥ 1 − ε. The convergence in C1 (M By(δ0)) implies strong convergence in L2 (M By(δ0), g∞) of ψi, to ψ̄. Hence ψ̄ L2(MBy (δ0),g∞) ≥ 1 − ε, and thus ψ̄ L2(M∞,g∞) = 1. The orthogonality of these sections is pro- vided by the following lemma, and the inequality (1.13) then follows immediately. Lemma 1.4.4 The sections ψ̄1, . . . , ψ̄m are orthogonal.
- 38. 18 B. Ammann and P. Jammes Proof of the lemma. The sections ϕi,1, . . . , ϕi, are orthogonal. For any fixed δ0 (given by the previous lemma), it follows for sufficiently large i that MBy (δ0) ψi,, ψi, ˜ dvg∞ = MBy (δ0) ϕi,, ϕi, ˜ dvgLi = By (δ0) ϕi,, ϕi, ˜ dvgLi = By (δ0) FLi F∞ k ≤1 ψi,, ψi, ˜ dvg∞ ≤ ε2 (1.16) Because of strong L2 convergence on M By(δ0) this implies MBy (δ0) ψ̄, ψ̄˜ dvg∞ ≤ ε2 (1.17) for ˜ = , and hence in the limit ε → 0 (and δ0 → 0) we get the orthogonality of ψ̄1, . . . , ψ̄m.
- 39. Appendix A Analysis on (M∞, g∞) The aim of this appendix is to sketch how to prove Proposition 1.3.1. All properties in this appendix are well-known to experts, but explicit references are not evident to find. Thus this summary might be helpful to the reader. The geometry of (M∞, g∞) is asymptotically cylindrical. The metric g∞ is even a b-metric in the sense of Melrose [31], but to keep the presentation simple, we avoid the b-calculus. If (r, γ ) ∈ R+ × Sn−1 denote polar normal coordinates with respect to the metric g0, and if we set t := − log r, then (t, γ ) defines a diffeomorphism α : B (M,g0) y (1/2) {y} → [log 2, ∞) × Sn−1 such that (α−1 )∗ g∞ = dt2 + ht for a family of metrics such that (α−1 )∗ g∞, all of its derivatives, its curvature, and all derivatives of the curvature tend to the standard metric on the cylinder, and the speed of the convergence is majorised by a multiple of et . Thus the continuity of the coefficients property implies, that P∞ extends to a bounded operator from Hk(M∞,g∞)(V ) → L2(M∞,g∞)(V ). The formal self-adjointness of P∞ implies that M∞ ψ, P∞ϕ = M∞ P∞ψ, ϕ (A.18) holds for ϕ, ψ ∈ c(V ) and as c(V ) is dense in Hk , property (A.18) follows all Hk -sections ϕ, ψ. To show Proposition 1.3.1 it remains to prove the regularity estimate and then to verify that the adjoint of P∞ : Hk(M∞,g∞)(V ) → L2(M∞,g∞)(V ) has domain Hk(M∞,g∞)(V ). For proving the regularity estimate we need the following local estimate. Lemma A.1 Let K be a compact subset of a Riemannian manifold (U, g). Let P be an elliptic differential operator on U of order k ≥ 1. Then there is a 19
- 40. 20 B. Ammann and P. Jammes constant C = C(U, K, P, g) such that u Hk(K,g) ≤ C u L2(U,g) + Pu L2(U,g) . (A.19) Here the Hk (K, g)-norm is defined via the Levi-Civita connection for g. This estimate holds uniformly in an ε-neighborhood of P and g in the following sense. Assume that P̃ is another differential operator, and that the C0 -norm of the coeffcients of P̃ − P is at most ε, where ε is small. Also assume that g̃ is ε-close to g in the Ck -topology. Then the estimate (A.19) holds for P̃ instead of P and for g̃ instead of g and again for a constant C = C(U, K, P, g, ε). Proof of the lemma. We cover the compact set K by a finite number of coordinate neighborhoods U1, . . . , Um. We choose open sets Vi ⊂ Ui such that the closure of Vi is compact in Ui and such that K ⊂ V1 ∪ . . . ∪ Vm. One can choose compact sets Ki ⊂ Vi such that K = K1 ∪ . . . ∪ Km. To prove (A.19) it is sufficient to prove u Hk(Ki ,g) ≤ C( u L2(Vi ,g) + Pu L2(Vi ,g)) for any i. We write this inequality in coordinates. As the closure of Vi is a compactum in Ui, the transition to coordinates changes the above inequality only by a constant. The operator P, written in a coordinate chart is again elliptic. We have thus reduced the prove of (A.19) to the prove of the special case that U and K are open subsets of flat Rn . The proof of this special case is explained in detail for example in in [33, Corollary III 1.5]. The idea is to construct a parametrix for P, i.e. a pseudodif- ferential operator of order −k such that S1 := QP − Id and S2 := PQ − Id are infinitely smoothing operators. Thus Q is bounded from L2 (U) to the Sobolev space Hk (U), in particular Q(P(u)) Hk ≤ C P(u) L2 . Smoothing operators map the Sobolev space L2 continuously to Hk . We obtain u Hk(K) ≤ u Hk(U) ≤ Q(P(u)) Hk(U) + S1(u) Hk(U) ≤ C P(u) L2(U) + u L2(U) . See also [28, III §3] for a good presentation on how to construct and work with such a parametrix. To see the uniformicity, one verifies that u Hk(K,g̃) u Hk(K,g) − 1 ≤ C g̃ − g Ck ≤ Cε and P̃(u) L2(U) P(u) L2(U) − 1 ≤ Cε u Hk(U). The unformicity statement thus follows.
- 41. The supremum of first eigenvalues 21 Proof of the regularity estimate in Proposition 1.3.1. We write M∞ as MB ∪ ([0, ∞) × Sn−1 ), such that the metric g∞ is asymptotic (in the C∞ -sense) to the standard cylindrical metric. The metric g∞ restricted to [R − 1, R + 2] × Sn−1 then converges in the Ck -topology to the cylindrical metric dt2 + σn−1 on [0, 3] × Sn−1 for R → ∞. As the coefficients of Pg depend continuously on the metric, the P-operators on [R − 1, R + 2] × Sn−1 is in an ε-neighborhood of P, for R ≥ R0 = R0(ε). Applying the preceding lemma for K = [R, R + 1] × Sn−1 and U = (R − 1, R + 2) × Sn−1 we obtain ∇s u L2([R,R+1]×Sn−1,g∞) ≤ C u L2((R−1,R+2)×Sn−1,g∞) + P∞u L2((R−1,R+2)×Sn−1,g∞) . (A.20) Similarly, applying the lemma to K = MB ∪ ([0, R0] × Sn−1 ) and U = MB ∪ ([0, R0 + 1) × Sn−1 ) gives ∇s u L2(MB ∪([0,R0]×Sn−1),g∞) ≤ C u L2(MB ∪([0,R0+1)×Sn−1),g∞) + P∞u L2(MB ∪([0,R0+1)×Sn−1),g∞) . (A.21) Taking the sum of estimate (A.21), of estimate (A.20) for R = R0, again estimate (A.20) but for R = R0 + 1, and so for all R ∈ {R0 + 2, R0 + 3, . . .} we obtain (1.9), with a larger constant C. Now we study the domain D of the adjoint of P∞ : Hk(M∞,g∞)(V ) → L2(M∞,g∞)(V ). By definition a section ϕ : L2(M∞,g∞)(V ) is in D if and only if Hk(M∞,g∞)(V ) u → M∞ P∞u, ϕ (A.22) is bounded as a map from L2 to R. For ϕ ∈ Hk(M∞,g∞)(V ) we know that P∞ϕ is L2 and thus property (A.18) directly implies this boundedness. Thus Hk(M∞,g∞)(V ) ⊂ D. Conversely assume the boundedness of (A.22). Then there is a v ∈ L2(M∞,g∞)(V ) such that M∞ u, v = M∞ P∞u, ϕ, or in other words P∞ϕ = v holds weakly. Standard regularity theory implies ϕ ∈ Hk(M∞,g∞)(V ). We obtain Hk(M∞,g∞)(V ) = D, and thus the self-adjointness of P∞ follows. Proposition 1.3.1 is thus shown.
- 42. 22 B. Ammann and P. Jammes References [1] B. Ammann, The Dirac Operator on Collapsing Circle Bundles, Sém. Th. Spec. Géom Inst. Fourier Grenoble 16 (1998), 33–42. [2] B. Ammann, Spin-Strukturen und das Spektrum des Dirac-Operators, Ph.D. thesis, University of Freiburg, Germany, 1998, Shaker-Verlag Aachen 1998, ISBN 3- 8265-4282-7. [3] , The smallest Dirac eigenvalue in a spin-conformal class and cmc- immersions, Comm. Anal. Geom. 17 (2009), 429–479. [4] , A spin-conformal lower bound of the first positive Dirac eigenvalue, Diff. Geom. Appl. 18 (2003), 21–32. [5] , A variational problem in conformal spin geometry, Habilitationsschrift, Universität Hamburg, 2003. [6] B. Ammann and C. Bär, Dirac eigenvalues and total scalar curvature, J. Geom. Phys. 33 (2000), 229–234. [7] B. Ammann and E. Humbert, The first conformal Dirac eigenvalue on 2- dimensional tori, J. Geom. Phys. 56 (2006), 623–642. [8] B. Ammann, E. Humbert, and B. Morel, Mass endomorphism and spinorial Yam- abe type problems, Comm. Anal. Geom. 14 (2006), 163–182. [9] B. Ammann, A. D. Ionescu, and V. Nistor, Sobolev spaces on Lie manifolds and regularity for polyhedral domains, Doc. Math. 11 (2006), 161–206. [10] B. Ammann, R. Lauter, and V. Nistor, On the geometry of Riemannian manifolds with a Lie structure at infinity, Int. J. Math. Math. Sci. (2004), 161–193. [11] , Pseudodifferential operators on manifolds with a Lie structure at infinity, Ann. of Math. 165 (2007), 717–747. [12] C. Bär, The Dirac operator on space forms of positive curvature, J. Math. Soc. Japan 48 (1996), 69–83. [13] C. Bär, The Dirac operator on hyperbolic manifolds of finite volume, J. Differ. Geom. 54 (2000), 439–488. [14] H. Baum, Spin-Strukturen und Dirac-Operatoren über pseudoriemannschen Man- nigfaltigkeiten, Teubner Verlag, 1981. [15] A. L. Besse, Einstein manifolds, Ergebnisse der Mathematik und ihrer Grenzgebi- ete, 3. Folge, no. 10, Springer-Verlag, 1987. [16] T. P. Branson, Differential operators canonically associated to a conformal struc- ture, Math. Scand. 57 (1985), 293–345. [17] , Group representations arising from Lorentz conformal geometry, J. Funct. Anal. 74 (1987), no. 2, 199–291. [18] , Second order conformal covariants, Proc. Amer. Math. Soc. 126 (1998), 1031–1042. [19] B. Colbois and A. El Soufi, Eigenvalues of the Laplacian acting on p-forms and metric conformal deformations, Proc. of Am. Math. Soc. 134 (2006), 715–721. [20] A. El Soufi and S. Ilias, Immersions minimales, première valeur propre du laplacien et volume conforme, Math. Ann. 275 (1986), 257–267. [21] H. D. Fegan, Conformally invariant first order differential operators., Quart. J. Math. Oxford, II. series 27 (1976), 371–378. [22] R. Gover and L. J. Peterson, Conformally invariant powers of the Laplacian, Q-curvature, and tractor calculus, Comm. Math. Phys. 235 (2003), 339–378.
- 43. The supremum of first eigenvalues 23 [23] E. Hebey and F. Robert, Coercivity and Struwe’s compactness for Paneitz type operators with constant coefficients, Calc. Var. Partial Differential Equations 13 (2001), 491–517. [24] N. Hitchin, Harmonic spinors, Adv. Math. 14 (1974), 1–55. [25] P. Jammes, Extrema de valeurs propres dans une classe conforme, Sémin. Théor. Spectr. Géom. 24 (2007), 23–42. [26] I. Kolář, P. W. Michor, and J. Slovák, Natural operations in differential geometry, Springer-Verlag, Berlin, 1993. [27] N. Korevaar, Upper bounds for eigenvalues of conformal metrics, J. Differ. Geom. 37 (1993), 73–93. [28] H.-B. Lawson and M.-L. Michelsohn, Spin Geometry, Princeton University Press, 1989. [29] J. M. Lee and T. H. Parker. The Yamabe problem. Bull. Am. Math. Soc., New Ser. 17 (1987), 37–91. [30] S. Mac Lane, Categories for the working mathematician, Graduate Texts in Math- ematics, vol. 5, Springer-Verlag, New York, 1998. [31] R. B. Melrose, The Atiyah-Patodi-Singer index theorem, Research Notes in Math- ematics, vol. 4, A K Peters Ltd., Wellesley, MA, 1993. [32] S. M. Paneitz, A quartic conformally covariant differential operator for arbitrary pseudo-Riemannian manifolds, Preprint 1983, published in SIGMA 4 (2008). [33] M. E. Taylor, M. E., Pseudodifferential operators, Princeton University Press, 1981. Authors’ addresses: Bernd Ammann Facultät für Mathematik Universität Regensburg 93040 Regensburg Germany bernd.ammann@mathematik.uni-regensburg.de Pierre Jammes Laboratoire J.-A. Diendonné, Université Nice-Sophia Antipolis, Parc Valrose, F-06108 Nice Cedex02, France pjammes@unice.fr
- 44. 2 K-Destabilizing test configurations with smooth central fiber claudio arezzo, alberto della vedova, and gabriele la nave Abstract In this note we point out a simple application of a result by the authors in [2]. We show how to construct many families of strictly K-unstable polarized manifolds, destabilized by test configurations with smooth central fiber. The effect of resolving singularities of the central fiber of a given test configuration is studied, providing many new examples of manifolds which do not admit Kähler constant scalar curvature metrics in some classes. 2.1 Introduction In this note we want to speculate about the following Conjecture due to Tian- Yau-Donaldson ([23], [24], [25], [7]): Conjecture 2.1.1 A polarized manifold (M, A) admits a Kähler metric of constant scalar curvature in the class c1(A) if and only if it is K-polystable. The notion of K-stability will be recalled below. For the moment it suffices to say, loosely speaking, that a polarized manifold, or more generally a polarized variety (V, A), is K-stable if and only if any special degeneration or test configuration of (V, A) has an associated non positive weight, called Futaki invariant and that this is zero only for the product configuration, i.e. the trivial degeneration. We do not even attempt to give a survey of results about Conjecture 2.1.1, but as far as the results of this note are concerned, it is important to recall the reader that Tian [24], Donaldson [7], Stoppa [22], using the results in [3] and [4], and Mabuchi [17] have proved the sufficiency part of the Conjecture. Destabilizing a polarizing manifold then implies non existence results of Kähler constant scalar curvature metrics in the corresponding classes. 24
- 45. K-Destabilizing test configurations 25 One of the main problems in this subject is that under a special degeneration a smooth manifold often becomes very singular, in fact just a polarized scheme in general. This makes all the analytic tool available at present very difficult to use. Hence one naturally asks which type of singularities must be introduced to make the least effort to destabilize a smooth manifold without cscK metrics. The aim of this note is to provide a large class of examples of special degenerations with positive Futaki invariant and smooth limit. In fact we want to provide a “machine” which associates to any special degeneration of a polarized normal variety (V, A) with positive Futaki invariant a special degeneration for a polarized manifold (M̃, Ã) with smooth central fiber and still positive Futaki invariant. To the best of our knowledge, before this work the only known examples of special degeneration with non negative Futaki invariant and smooth central fiber are the celebrated example of Mukai-Umemura’s Fano threefold ([18]) used by Tian in [24] to exhibit the first examples of Fano manifolds with discrete automorphism group and no Kähler-Einstein metrics (other Fano manifolds with these properties have been then produced in [1]). In this case there exist non trivial special degenerations with smooth limit and zero Futaki invariant (hence violating the definition of K-stability). It then falls in the borderline case, making this example extremely interesting and delicate. We stress that our “machine” does not work in this borderline case, because a priori the Futaki invariant of the new test configuration is certainly small (by [2]) but we cannot control its sign. To state our result more precisely we now recall the relevant definitions: Definition 2.1.2 Let (V, A) be a n-dimensional polarized variety or scheme. Given a one-parameter subgroup ρ : C∗ → Aut(V ) with a linearization on A and denoted by w(V, A) the weight of the C∗ -action induced on top H0 (V, A), we have the following asymptotic expansions as k 0: h0 (V, Ak ) = a0kn + a1kn−1 + O(kn−2 ) (2.1) w(V, Ak ) = b0kn+1 + b1kn + O(kn−1 ) (2.2) The (normalized) Futaki invariant of the action is the rational number F(V, A, ρ) = b1 a0 − b0 a1 a2 0 . Definition 2.1.3 A test configuration (X, L) for a polarized variety (V, A) consists of a scheme X endowed with a C∗ -action that linearizes on a line bundle L over X, and a flat C∗ -equivariant map f : X → C (where C has the
- 46. 26 C. Arezzo, A. Della Vedova, and Gabriele La Nave usual weight one C∗ -action) such that L|f −1(0) is ample on f −1 (0) and we have (f −1 (1), L|f −1(1)) (V, Ar ) for some r 0. When (V, A) has a C∗ -action ρ : C∗ → Aut(V ), a test configuration where X = V × C and C∗ acts on X diagonally through ρ is called product configu- ration. Given a test configuration (X, L) we will denote by F(X, L) the Futaki invariant of the C∗ -action induced on the central fiber (f −1 (0), L|f −1(0)). If (X, L) is a product configuration as above, clearly we have F(X, L) = F(V, A, ρ). Definition 2.1.4 The polarized manifold (M, A) is K-stable if for each test configuration for (M, A) the Futaki invariant of the induced action on the central fiber (f −1 (0), L|f −1(0)) is less than or equal to zero, with equality if and only if we have a product configuration. A test configuration (X, L) is called destabilizing if the Futaki invariant of the induced action on (f −1 (0), L|f −1(0)) is greater than zero. Test configurations for an embedded variety V ⊂ PN endowed with the hyper- plane polarization A can be constructed as follows. Given a one-parameter subgroup ρ : C∗ → GL(N + 1), which induces an obvious diagonal C∗ -action on PN × C, it clear that the subscheme X = (z, t) ∈ PN × C | t = 0, (ρ(t−1)z, t) ∈ V ⊂ PN × C, is invariant and projects equivariantly on C. Thus considering the relatively ample polarization L induced by the hyperplane bundle gives test configuration for (V, A). On the other hand, given a test configuration (X, L) for a polarized variety (V, A), the relative projective embedding given by Lr , with r sufficiently large, realizes X as above (see details in [21]). We can now describe our “machine”: consider a test configuration (X, L) for a polarized normal variety (V, A) with F(X, L) 0. Up to raise L to a suitable power – which does not affect the Futaki invariant – we can suppose being in the situation above with X ⊂ PN × C invariantly, and L induced by the hyperplane bundle of PN . At this point we consider the central fiber X0 ⊂ PN , which is invariant with respect to ρ, and we apply the (equivariant) resolution of singularities [14, Corollary 3.22 and Proposition 3.9.1]. Thus there is a smooth manifold P̃ acted on by C∗ and an equivariant map β : P̃ → PN which factorizes through a sequence of blow-ups, such that the strict transform X̃0 of X0 is invariant and smooth. The key observation is that the strict transform X̃1 of the fiber X1 ⊂ X degenerate to X0 under the given C∗ action on P̃ , thus it
- 47. K-Destabilizing test configurations 27 must be smooth. This gives an invariant family X̃ ⊂ P × C and an equivariant birational morphism π : X̃ → X. Some comments are in order: 1 all the fibers of X̃ are smooth, but π is never a resolution of singularities of X (except the trivial case when the central fiber of X was already smooth) since it fails to be an isomorphism on the smooth locus of X; 2 L̃ = π∗ L is not a relatively ample line bundle any more, but just a big and nef one. It is not then even clear what it means to compute its Futaki invariant; 3 the fiber over the generic point of C of the new (big and nef) test configuration (X̃, L̃) is different from V ; 4 the family X̃ is not unique since the resolution β it is not. The issue raised at point (2) was addressed in [2] and it was proved that the following natural (topological) definition makes the Futaki invariant a continu- ous function around big and nef points in the Kähler cone. We will give simple self-contained proofs in the cases of smooth manifolds and varieties with just normal singularities in Section 2. Definition 2.1.5 Let V be a projective variety or scheme endowed with a C∗ - action and let B be a big and nef line bundle on V . Choosing a linearization of the action on B gives a C∗ -representation on dim V j=0 Hj (V, Bk )(−1)j (here the E−1 denotes the dual of E). We set w(V, Bk ) = tr Ak, where Ak is the generator of that representation. As k → +∞ we have the following expansion w(V, Bk ) χ(V, Bk) = F0k + F1 + O(k−1 ), and we define F(V, B) = F1 to be the Donaldson–Futaki invariant of the chosen action on (V, B) The existence of the expansion involved in definition above follows from the standard fact that χ(V, Bk ) is a polynomial of degree dim V , whose proof (see for example [11]) can be easily adapted to show that w(V, Bk ) is a polynomial of degree at most dim V + 1.
- 48. 28 C. Arezzo, A. Della Vedova, and Gabriele La Nave The key technical Theorem proved in [2] is then the following: Theorem 2.1.6 Let B, A be linearized line bunldes on a scheme V acted on by C∗ . Suppose that B is big and nef and A ample. We have F(V, Br ⊗ A) = F(V, B) + O 1 r , as r → ∞. Having established a good continuity property of the Futaki invariant up to these boundary point, we need to address the question of the effect of a resolution of singularities of the central fiber. This is a particular case of the following non trivial extension of previous analysis by Ross and Thomas [21] which was proved in [2] where the general case of birational morphisms has been studied: Theorem 2.1.7 Given a test configuration f : (X, L) → C as above, let f : (X , L ) → C be another flat equivariant family with X normal and let β : (X , L ) → (X, L) be a C∗ -equivariant birational morphism such that f = f ◦ β and L = β∗ L. Then we have F(X , L ) ≥ F(X, L), with strict inequality if and only if the support of β∗(OX )/OX has codimension one. The proof of these results uses some heavy algebraic machinery, yet their proof when (V, A) or the central fiber of (X, L) have only normal singularities (a case largely studied) is quite simple and we give it in Section 2. The Corollary of Theorem 2.1.6 and Theorem 2.1.7 we want to point out in this note is then the following: Theorem 2.1.8 Let (X, L) be a test configuration for the polarized normal variety (V, A) with positive Futaki invariant. Let moreover (X̃, L̃) be a (big and nef) test configuration obtained from (X, L) as above and let (M̃, B̃) be the smooth (big and nef) fiber over the point 1 ∈ C. Let R be any relatively ample line bundle over X̃. Then (X̃, L̃r ⊗ R) is a test configuration for (M̃, B̃r ⊗ R|M̃ ) with following properties: 1 smooth central fiber; 2 positive Futaki invariant for r sufficiently large. In particular M̃ does not admit a constant scalar curvature Kähler metric in any class of the form c1(B̃r ⊗ R|M̃ ), with r large enough.
- 49. K-Destabilizing test configurations 29 While this Theorem clearly follows from Theorems 2.1.6 and Theorem 2.1.7, but for the specific case of central fiber with normal singularities it follows from the much simpler Proposition 2.2.1 and Theorem 2.2.3. The range of applicability of the above theorem is very large. We go through the steps of the resolution of singularities in an explicit example by Ding- Tian [6] of a complex orbifold of dimension 2. In this simple example explicit calculations are easy to perform, yet we point out that the final example is somehow trivial since it ends on a product test configuration. On the other hand abundance of similar examples even in dimension 2 can be obtained by the reader as an exercise using the results of Jeffres [12] and Nagakawa [19], in which cases we loose an explicit description of the resulting destabilized manifold, but we get new nontrivial examples. In fact in higher dimensions one can use the approach described in this note to test also the Arezzo-Pacard blow up theorems [3] [4], when the resolution of singularities requires a blow up of a scheme of positive dimension. 2.2 The case of normal singularities In this section we give simple proofs of the continuity of the Futaki invariant at boundary points for smooth manifolds or varieties with normal singularities. More general results of this type have been proved in [2] but we want to stress that under these assumptions proofs become much easier. The fundamental continuity property we will need, and proved in Corollary 2.1.6, can be stated in the following form for smooth bases: Proposition 2.2.1 Let A, L be respectively an ample and a big and nef line bundle on a smooth projective manifold M. For every C∗ -action on M that linearizes to A and L, as r → +∞ we have F(M, Lr ⊗ A) = F(M, L) + O 1 r . Proof The result is a simple application of the equivariant Riemann-Roch Theorem. We present here the details of the calculations involved, since we could not find precise references for them. Fix an hermitian metrics on A that is invariant with respect to the action of S1 ⊂ C∗ and suppose that the curvature ω is a Kähler metric. Since L is nef, for each r 0 we can choose an invariant metric on L whose curvature ηr satisfy rηr + ω 0. In other words rηr + ω is a Kähler form which coincides with the curvature of the induced hermitian metric on the line bundle Lr ⊗ A.
- 50. Another Random Document on Scribd Without Any Related Topics
- 51. “No, I can’t, Brooks. I oughtn’t to. I—we ain’t got much money, you see.” Brooks observed him, frowning intently. At last he concluded that Cal was speaking the truth and not merely exaggerating his poverty in order to escape practice. “That’s different,” he said. “You come with me.” Wondering what was going to befall him now, Cal accompanied the other across the bridge and along the path to East House. He had never been there before. East House was newer than West and larger. It accommodated fourteen fellows to West House’s eight. On the square porch Cal paused but Brooks beckoned him in and led the way up the stairs and into a nicely furnished room on the second floor. There were lots of pictures on the walls, a good deal of comfortable mission furniture with leather upholstery, and several Oriental rugs on the hardwood floor. Altogether the room was a revelation to Cal of what a school study might be if the occupant possessed both money and good taste. “Sit down, won’t you?” said Brooks, pushing a deep-seated chair forward. Cal seated himself, placed his cloth cap over one knee and smoothed it down there, feeling somewhat embarrassed and ill at ease. Brooks went to a closet and in a moment was back with an armful of togs. “Here you are,” he said, dumping the things in Cal’s lap. “Shirt, breeches and stockings. I haven’t any boots, but I guess you can use what you’ve got for awhile. These things aren’t new by any means, but I guess they’ll last the season out. You can get Mrs. Linn to patch that place in the jersey.” “But—but I oughtn’t to take these,” stammered Cal. “Yes, you ought. Now look here, Boland. I don’t want to be nasty, but honestly you haven’t any business to act like this. You’re a new boy, and I guess that explains it, though. At that, Boland, you’ve been here long enough to know things. Haven’t they told you that we don’t shirk duty here at Oak Park? I suppose it’s Sanderson that’s at fault; he’s a good deal of a duffer, to my mind. Tell him so if you
- 52. want to. It’s a shame you West Housers haven’t got another chap for Leader over there.” “Sandy’s all right,” said Cal with a scowl. Brooks smiled. “Well, I’m glad you’ve got that far,” he said. “At least you’ve learned to stand up for your House. But hasn’t Sandy told you that every fellow is expected to take hold and work for his House? That with us it’s House first, School next and self nowhere?” “Ned Brent said something like that,” answered Cal. “Yes, Ned would. Why don’t you do as Ned does, then? You want House to win, don’t you?” “Of course,” answered Cal indignantly. “Well, why don’t you help us then instead of sulking? What if football practice is hard? I know it well enough. I’ve been all through the—the drudgery, just as you are going. It isn’t any harder for us than it is for Hall, though. It isn’t any harder for you than it is for any other new boy. And after you’ve learned you’ll get a whole lot of pleasure out of it.” “But it don’t seem to me,” muttered Cal, “that I ever can learn. I cal’late I’m no good at football.” “That’s none of your business,” said Frank Brooks sharply. “That’s my look-out. If I didn’t think you could be useful to the team do you suppose I’d waste my time on you for a minute?” This hadn’t occurred to Cal and he digested it a moment. Then, “You mean that you think I can learn to play the game?” he asked. “I mean that I think you can be of use to the House Team. That’s enough. If you can be of use it’s your duty to work hard and forget yourself, Boland. Get that idea?” “Yes.” “All right.” Brooks observed him a moment. Then he smiled and thumped him on his back. “You’ll do, Boland. No more nonsense,
- 53. though, if you please. See you this afternoon.” Cal bundled the togs up. “I’ll do what I can,” he said simply, “but—” “But nothing,” laughed Brooks. “You do what you’re told to do as well as you know how and leave the ‘buts’ to me. Glad you came over, Boland. You’ll get on all right.” “I don’t just like to take these things, though,” Cal objected. “Piffle! They’re no use to me. Call it a loan if you like. You can hand them back after the season’s through—if there’s anything left of them! Good bye.” So ended Cal’s mutiny. Half-way through the park he stopped and examined the contents of the bundle. There was a very fair pair of khaki breeches, properly padded on hip and knee, a somewhat threadbare cherry-red jersey with a three-corner tear on one sleeve and what seemed to be a brand-new pair of red stockings. He felt very proud of these new possessions, very proud, too, that Brooks had assured him that in spite of his own misgivings he was really of some use to the team. He made up his mind to buckle down and do the best that was in him, even if, at the end, he was destined to be only an onlooker when the battles raged. And without intentional disrespect to Sandy, he heartily wished that Frank Brooks was leader at West House. A fairly uneventful week followed. He neither heard nor saw anything more of Miss Molly Elizabeth Curtis and he and the rest of the House forgot their misgivings. They talked of her once or twice during the first few days and then, as she didn’t obtrude herself, thought no more about her. Football practice went on six days in the week. They were hard at signal work now, and Cal, playing tackle on the second eleven, had grown interested in his duties. The first game was only a week away and already the air was surcharged with excitement. House boys began to sport their cherry-red and Hall fellows their blue. Football became the subject for conversation
- 54. at every meal and Mrs. Linn, as was her yearly custom, displayed a well-meaning but frightfully ignorant interest in the game. Lessons suffered proportionally as football fever increased and the instructors, notably Mr. Kendall, familiarly known as Grouch, railed and scolded. Only Mr. Fordyce, who taught English and physics, and who was called Fussy, took it philosophically, apparently realizing that in a month or so affairs would be back on their accustomed plane and no one the worse. For once Fussy belied his title. Mr. James, in whose room Cal had his desk, might have been expected to be more lenient with the fellows in their football madness than anyone else, since as physical instructor he had direct charge of the players. But Jim, as he was called, drew a hard and fast line between class-room and playground and so far as he was concerned athletic prowess was no excuse for lack of attention to studies. Several of the boys found this out during the last of October and the first of November, and it was a dull week indeed when someone was not absent from practice on either Hall or House field because Jim believed that a cessation of athletic industry would improve class standing.
- 55. He stopped and examined the contents of the bundle At West House football put Ned’s misfortune out of everyone’s mind, excepting Ned’s and Cal’s. The mystery remained unexplained, but the generally accepted theory, introduced by Ned himself, was that the money had been mislaid and would sooner or later be discovered. Cal appreciated his room-mate’s generosity in seeking by every means to keep suspicion from him, but he hadn’t forgiven Ned for himself suspecting. The breach widened rather than lessened as the days went by, and Cal wasn’t very happy. Rooming with a chap
- 56. to whom you have nothing to say and who has nothing to say to you is an uncomfortable business. Neither Ned nor Cal knowingly gave any evidence of the estrangement, but it didn’t take the other boys long to discover it. At another time it would have occasioned more interest; just now football was the only topic holding anyone’s attention. On the Monday before the first game Frank Brooks finished his experiments and the First Team as it lined up that afternoon was the team that would face the Hall, barring accidents. West House had secured five places. Sandy was at left guard, Dutch at left tackle, Spud at left end, Ned at right half-back and The Fungus at left half- back. That left six places for East House. Brooks played right guard. The quarter-back was Will M’Crae and on him the Houses pinned much of their faith, for besides being a good general he was an exceptionally good punter. Hoop and Cal had drawn places amongst the substitutes, Hoop as a guard and Cal as a tackle. There had never been much sympathy between these two, for Hoop had a passion for saying mean things without really wanting to hurt, and Cal had not forgotten the incident attending his arrival at West House when Hoop had tripped him up on the steps. Dutch and Hoop got along splendidly together as room-mates, for Dutch was good- natured to a degree and paid very little attention to his chum’s gibes. Most anyone could have got on with Dutch Zoller. Being together in the substitute ranks, however, brought Cal and Hoop together a good deal and Cal soon got to liking the other very well and it wasn’t long before he had ceased entertaining any animosity toward Hoop for the little incident on the steps. They walked back to West House together that Monday afternoon after practice was over and discussed their chances of getting into the first game. By this time Cal had cultivated quite a keen interest in football and no one worked harder or took his knocks more cheerfully. “You’re likely to get in before I do,” said Hoop. “Brooks is bound to play every game through, while as for Sandy, although Truesdale will play all around him, he’s no quitter. But Griffin at right guard gets
- 57. hurt easily. When you do get in it will be to replace him, Cal. Dutch doesn’t know how to get hurt, so you needn’t look for his place.” “I suppose we’re bound to get into one of the games, aren’t we?” Cal asked. “Sure. We may get into them all for awhile. You can’t tell. Brooks might lay off part of Saturday’s game so as to save himself up; he would if the game went our way, I guess. Then I’d get my whack at it. I’m crazy to get up against that duffer Williams of the Hall. He always plays high and I’ll bet I can get right through him.” “I cal’late I’d be kind of scared if they did put me in,” said Cal. “Rot! You wouldn’t either! You’d forget all about being scared after the first play. When the other chap is trying to pull you on your nose or walk up your spine you haven’t time to think whether you’re scared or not. Gee, I’ll be sorry when the season’s over!” “What do they do here in winter, Hoop?” “Oh, play hockey a good deal. We had a fine team last winter. I don’t play myself; can’t skate worth a hang; never seemed to be able to learn how. Do you?” “Yes, I learned when I was about eight, I guess. I’ve never played hockey, though. Is it hard?” “Yes, it is. We play basket-ball, too. That’s good fun. West House won the School Championship last year; beat East House and First and Second Hall. I played.” “No wonder you won, then,” laughed Cal. Hoop grinned. “I didn’t mean it that way,” he said. “Sandy’s the bang-up basket- ball player. He’s a dandy center. And Ned’s a cracker-jack, too. I guess you could make good at that if you went in for it, Cal.” “I’d like to try. I’ve seen them play it at home.” “It’s a lot of fun. Hello!”
- 58. They had walked over in advance of the others and now, as they turned the corner of the house, Hoop stopped still and stared. On the top step sat a girl with a brand-new tennis racket in her hands! “Is that her?” growled Hoop in a whisper. “Yes,” answered Cal, “and it looks as though she’d come to play tennis, Hoop.” He grinned. “Maybe you can beg off, though; tell her you’re too tired and—” But Hoop had fled back around the corner. Cal meditated following him, but at that moment Molly turned her head and saw him. “How do you do?” she called. “I’ve been waiting here the longest time!”
- 59. “H CHAPTER XV MOLLY TAKES A HAND ow do,” returned Cal, walking toward her with unflattering deliberation. “I thought you couldn’t come.” “I know, but I feared you’d feel so bad about it,” she laughed, “that I just made them let me. Aren’t you terribly glad to see me?” “Yes,” answered Cal without much enthusiasm. “How—how did you manage?” “Oh, I just kept at it. Aunt Lydia was on my side and she told Aunt Matilda that she guessed you wouldn’t eat me if I was to come over here. I’ve been calling on Mrs. Linn. She’s a dear, isn’t she?” “Er—yes.” He was looking at the racket with strange fascination and Molly, following his glance, smiled brightly and held it out for his inspection. “I bought it this morning. Is it a good one?” “I think so. I don’t know much about tennis rackets. Ned can tell you. He will be here in a moment; the others, too. Did you—do you want to play today?” “Yes, if it isn’t too late. I’ve been here a long time, but I suppose you have all been playing football.” “Yes, we had a pretty stiff practice and I cal’late we’re rather too tired to—” But at this moment the others came around the corner, Hoop, arm in arm with Sandy and Spud, scowling ferociously and evincing a desire to escape. If Cal expected evidences of embarrassment on the part of the girl he was disappointed. She only smiled interestedly.
- 60. “You’ll have to introduce me, Cal,” she whispered. Cal had never done such a thing in his life, but he managed to get through with the task in some manner, Spud, claiming the privileges of former acquaintance, helping him out. “And this,” said Spud finally, “is Mr. Hooper, who has eagerly volunteered to teach you tennis, Miss—er—Curtis, while here in the background, modest youth that he is, hides Mr. Parker. Mr. Parker is our football guide and wishes me to offer his services to you.” Hoop growled something under his breath that didn’t sound especially flattering to Spud, but Clara walked up and shook hands very nicely. Molly bowed and said “How do you do,” or shook hands and said “I’m very glad to meet you” at each presentation, and the boys, grinning, seated themselves on the steps and frankly looked her over. She didn’t seem very formidable with her pink cheeks and blue eyes, and it was difficult to realize that she figuratively held their welfare in the small hands that gripped her tennis racket. “I suppose,” she said to Sandy, “that Cal has told you that I want awfully to learn to play tennis? He said he didn’t play very well but that he thought one of you would find time to show me a little about it. Do you mind my coming over here?” Sandy proved traitor on the spot. “Of course not,” he declared heartily. “I guess any of us will be glad to play with you. I suppose it’s a bit dull over there with just the Old—I mean with just your aunts.” Spud snickered and Sandy frowned at him. “Awfully,” agreed Molly. “I thought it was very nice of Cal to want me to come over here. And I’m glad you don’t mind.” Hoop surreptitiously kicked Cal in the small of the back. “We don’t mind at all,” said Spud. “We’re tickled. I guess there’s time for a lesson now if you start right away. You’d better get your racket and some balls, Hoop.”
- 61. “I’m tired,” muttered Hoop, casting mutinous eyes around the group. “Miss Molly understands that,” said Ned. “She’ll forgive you if you’re not at your best, I’m sure.” But Molly was viewing Hoop doubtfully. “I guess he doesn’t want to,” she said, turning to Sandy. “I’ll come some other time.” “I’ll give you a lesson myself,” declared Sandy, jumping up. “Find my racket for me, will you, Clara? And bring some balls out.” “Why do you call him Clara?” asked Molly as the boy hurried inside on his errand. “Because his name’s Claire,” answered Dutch. “What a funny name for a boy! And what’s yours?” “Dutch.” Molly laughed and went around the group, nodding her head at each in turn. “Spud.” “Just Ned.” “Sandy.” “The Fungus.” “Hooper.” “He means Hoop.” “And you are Cal,” she said, reaching that youth. “Short for Calamity,” explained Spud gravely. “Isn’t he quick?” sneered Hoop, still resentful. “Quickest thing ever,” answered Spud affably. “Lightning is cold molasses beside me. That’s where I get my name, you know,” he
- 62. added, turning to Molly. “Ex-spud-itious.” The boys groaned, but Molly laughed appreciatively. “I suppose,” she said, “I’ll get you all terribly mixed up at first, and I hope you won’t mind.” “We never mind,” declared Dutch quite flippantly for him. He received his reward from Molly in the shape of a smile and for some time after secretly rather fancied himself as a wit. “My name,” she announced, “is Molly. I guess you’d better call me that, if we are going to be friends.” Clara returned with the racket and she and Sandy proceeded to the tennis court, the others politely electing to watch from a distance so as not to embarrass the novice. “She’s a funny one,” observed The Fungus with a grin. “‘If we are going to be friends,’ said she. She knows mighty well we don’t dare be anything else!” “She’s a good sort,” said Spud. “And I guess we might as well make up our minds to enjoying what they call female society after this. Did you see Sandy fall for her on the spot?” “Conceited idiot!” growled Hoop. “I hope he falls into the net and —and—” “Chokes to death,” added Spud helpfully. “Remarks of that sort from you, Hoop, are sadly out of place. You are a—a renegade.” “That’s all right. I didn’t agree to give her tennis lessons.” “Do I really have to take her to watch football?” asked Clara. “Of course you do,” Dutch said severely. “Don’t you want to?” “I suppose so,” answered the boy. “Seems to me,” observed The Fungus, “that our diplomat isn’t on to his job. Are you—diplomating, Ned?” “Sure thing. Diplomacy is brain-work. I’m thinking.”
- 63. “Don’t see why we gave the job to you, then,” muttered Hoop. “What we ought to do is to find where she keeps that pillow-case and go over and nab it.” “Huh,” Dutch grunted, “I’d like to see anyone go prowling around where Miss Matilda would catch him.” “Pshaw, what’s the good of bothering about that old pillow-case?” asked Spud impatiently. “She isn’t going to be mean. She’s just having a little fun with us. Look at Sandy, fellows; isn’t he having one grand good time?” Sandy was toiling valiantly, chasing balls on all sides of the court. Molly’s efforts were ludicrous and pathetic, and for a time she couldn’t get it into her little head that there was any method to the game beside batting the balls back and forth. The supper bell brought welcome relief to her instructor, although he made believe that he simply hated the thought of stopping.
- 64. Sandy was toiling valiantly, chasing balls on all sides of the court “You did finely,” he declared as they returned to the porch. “All you need is a few more lessons.” “That’s silly,” answered Molly promptly. “I know very well that I was just as stupid as stupid! I’m going to buy one of those little blue books with the rules in them the first thing in the morning. Then I’ll know what it’s all about. Thank you very much for teaching me. Good night.”
- 65. “Good night,” said Sandy, and “Good night,” called the others. And Molly, her racket tucked under her arm, took her departure. Sandy subsided on the top step and said “Whew!” very expressively. The rest observed him grinningly. “How now, gallant squire of dames?” asked Spud. “Someone else has got to take her the next time,” responded Sandy with decision. He glanced at Hoop. But that youth was looking the other way and whistling softly. “Beautiful sunset, Hoop,” murmured Spud. Hoop scowled. “Why don’t you draw lots?” he asked. “We will,” said Sandy, “after supper.” They did. He and Spud arranged the slips of paper and in some remarkable fashion the fatal slip fell to Hoop’s portion. “That isn’t fair!” he objected. “You fellows faked!” But they were very stern with him and in the end he accepted the duty with ill grace. There were three more lessons that week and Hoop officiated at two of them, the other being given by Spud. Strangely enough, Hoop, after the first time, became interested in the task and was quite loth to relinquish in Spud’s favor when the third lesson was due. Clara’s duties began on Wednesday. On that afternoon he took Molly in charge and escorted her to the football field, where she occasioned not a little interest on the part of the candidates. It was something new and novel to have a girl in the audience at practice and I fancy some of the boys worked harder than usual in the hope of distinguishing themselves and so winning a glance of approval from Miss Molly. Clara was very patient and instructive. A few weeks before he had had very little football knowledge himself, but he had watched and studied with enthusiasm and was now a very capable instructor. Molly had never seen the game played before, but, while she objected to it at first as being much too rough, it wasn’t long before she was an ardent champion of the House Team. Clara lent her his rule book and she studied it
- 66. diligently during the next week. Some of the questions she asked were a trifle disconcerting; such as “Why don’t they have the field smaller so they won’t have so far to go to make a touchdown?” or “Would it count anything if they threw the ball over that bar instead of kicking it?” She listened avidly to all the football discussions on the steps of West House and declared on Friday that if House didn’t beat Hall she’d never speak to any of them again. That threat must have nerved the House Team to desperation, for on the next afternoon it battled valiantly against Hall and managed to hold its opponent scoreless through thirty-five of the forty minutes of playing time, and had begun to count on a tied game at least when a miserable fumble by The Fungus on the Hall’s forty yard-line turned the fortunes of the day. It was Pete Grow himself who leaked through the House line, gathered up the ball and, protected by hastily formed interference, romped over the line with it for the only score of the game. They failed at goal and a few minutes later House trailed off the field vanquished to the tune of 5—0. House was heart-broken. To have kept Hall at bay through thirty- five minutes of the fiercest sort of battling and then to lose on a fluke was the hardest sort of luck. The Fungus felt the disgrace keenly and looked forlorn and tragic enough to melt a heart of stone. After the first miserable ten minutes succeeding the game his team-mates set themselves generously to work to cheer him up. “Your fault nothing!” scoffed big, good-hearted Westlake, the House center. “Why, any one of us ought to have got that ball. What if you did fumble it? Gee, we all do that. The trouble was that the rest of us weren’t quick enough to make it safe.” “That’s right,” said Ned sadly. “I ought to have had it myself. That chap Pete Grow, though, was through like a streak.” “I guess,” said Dutch, “it’s up to me, when you come right down to facts. I ought never to have let Grow through.” “Never mind whose fault it was,” said Brooks cheerfully. “We’ve just got to get busy this week and get together. It mustn’t happen
- 67. next time, fellows. We’ve got to develop team-play in the next five days or they’ll wipe up the sod with us. After all, we had them at a standstill until that pesky fumble.” Clara and Molly went back to West House silent and sad. But by the time they had reached the porch and Molly had established herself in her accustomed place with her slim back against a pillar the silence gave place to regrets and discussion. Molly was inclined to be indignant with the Hall. “They oughtn’t to have taken advantage of Fungus’s mistake,” she declared. “I don’t think that was very—very sportsmanlike, do you?” But Clara pointed out to her that ethically Hall had not transgressed. “Fumbling’s part of the game,” he said, “and you’ve got to take advantage of everything, Molly. We played a pretty good game, after all, I think.” “We played a wonderful game!” she assented stoutly. “Why, we just put it all over the Hall at first.” Clara smiled at the phrase she used. “Anyhow, I guess we can do better the next time. The trouble today was that we couldn’t get near enough Hall’s goal to try a drop- kick or placement.” “How near would we have to get?” asked Molly. “Oh, about thirty yards, I guess. M’Crae’s a dandy from the thirty yard-line.” “Wasn’t Spud splendid?” she asked. “He just threw those Hall men about like—like straws!” “Spud’s a dandy end,” Clara agreed. “He played all around Smith. I do wish, though, we might have won. Now we’ve got to get both the other games.” “And we will, too,” said Molly, her eyes flashing. “You just wait and see!”
- 68. The others came dejectedly home and until supper time they threshed out the day’s battle over and over again, Molly taking a fair share in the debate. The general tone was pessimistic, but Molly refused to entertain the thought of ultimate defeat for a moment. “You’ve just got to win the next two games,” she declared. “And you’re going to, aren’t you, Sandy?” But she had selected the wrong person in Sandy. He shook his head discouragedly. “I’m afraid not,” he answered. “They’ve got team-play, Molly, and we play every man for himself.” “Oh, you and your team-play!” scoffed Spud. “Why can’t we learn team-play as well as they can? You wait until next Saturday.” “Well, I’m through,” muttered The Fungus miserably. “I guess Brooksie will put in Folsom on Monday.” “Folsom!” jeered Dutch. “Folsom can’t begin to play your game; nor Westlake, either. Don’t you be so sore, old man. You couldn’t help it.” “Of course I could have helped it, only—well, if Brooksie keeps me on I’ll bet it won’t happen again. After this I’m going to dig my nails into it!” “Couldn’t you have explained to them that you didn’t mean to drop that ball?” asked Molly earnestly. “That it was just a—a mistake, Fungus?” The laughter that this question produced cleared the atmosphere not a little and by the time the bell had rung West House was a good deal more cheerful and much hungrier. “Isn’t she the limit?” laughed Spud as they went in to the dining- room. “Asking if Fungus couldn’t have explained that it was a mistake!” “She’s a mighty nice kid,” said Dutch. “She nearly yelled herself hoarse this afternoon,” said Cal. “Did you see her, Hoop?”
- 69. “Yes, and once she was jumping up and down like an Indian. I guess she’s the most enthusiastic rooter we’ve got.” “The Obnoxious Kid,” murmured Spud. “Obnoxious nothing!” objected Sandy indignantly. “She’s all right!” And West House agreed to a man.
- 70. T CHAPTER XVI THE PIPPIN CLUB IS FORMED he next afternoon, a warm Sunday, a strange thing happened. West House in full force filed out of the gate, along the road and in through the entrance to the Curtis place. The invitation had been delivered by Molly after church at noon, with Miss Matilda standing watchfully by and Miss Lydia beaming and nodding over her shoulder. It was a momentous event, for nothing of the sort had ever happened before in the history of West House. The boys had attired themselves in their Sunday best and were a very meek and well-behaving group as they mounted the porch and rang the bell. Spud pretended to be the victim of a vast and overwhelming terror and grasped Sandy’s arm convulsively when they heard the bell jangle inside. “I would I were away,” he muttered. “Ah, woe is me!” To their relief it was Molly herself, Molly in a stiffly starched muslin dress, who opened the door to them. They filed decorously in, deposited their hats and caps on the marble table in the hall and right-wheeled into the parlor. There they seated themselves in a circle about the room and felt very awkward and uncomfortable. Molly did her best to set her guests at ease, but the task was a difficult one. The assemblage was like her dress, very stiff and starchy. They discussed morning service, the weather, Spud’s new necktie and the pictures on the walls, and just when things did seem to be thawing out the least little bit there was the sound of footfalls on the stairs and instantly the guests froze into immobility. Entered Miss Matilda followed by Miss Lydia. The guests arose as one man, painfully polite and serious. Miss Matilda motioned them
- 71. back to their seats. Down they sat with a unanimity that suggested previous rehearsals. Miss Matilda announced that she was very glad to see them, and Sandy murmured—well, nobody ever knew what he murmured. But the tone was quite correct and the murmur served the purpose. Miss Lydia, plainly embarrassed, smoothed her black silk gown over her knees and smiled. Conversation proceeded by fits and starts. It went like a trolley car in a crowded street. Just when they thought it was nicely started, with a clear track ahead, it stopped with a bump. Then, after a dismal silence off it started once more with a jerk. Miss Matilda, Molly, Sandy and Spud were the principal conversationalists. Molly supplied subjects, Miss Matilda frowned them aside, Sandy rescued them and Spud babbled. Babbled is the only word for Spud’s efforts. He babbled of the weather and the dust in the streets and Mrs. Linn’s tonsilitis—a mild attack of no importance save as a subject for discourse—and finally of Molly’s tennis. The others looked on in evident and often open- mouthed admiration and awe. Strangely enough it was Spud’s last babble that cleared the conversational track for several blocks, so to speak. “Well, I’m glad she’s doing nicely at it,” said Miss Matilda with a sniff, “though I don’t see why she wants to learn it. In my day young girls didn’t race around hitting rubber balls with snowshoes.” “It’s—it’s a very pleasant game,” suggested Spud, vastly encouraged by his success, “and quite—er—popular nowadays, ma’am.” “Popular! I dare say; most anything that’s silly enough is popular these days, it seems. When I was a girl sewing and embroidery, yes, and plain cooking, were popular.” “Yes’m.” “Well, I don’t say but what this tennis may be good for Molly. I guess most anything that will keep her nose out of books for awhile will be beneficial. And it’s very kind of you young gentlemen to teach her the game.”
- 72. “Not at all, Miss Curtis,” protested Sandy. “I say it is,” responded Miss Matilda firmly. “Boys don’t usually like to have girls about them. I told Molly that when she first asked me to let her go over to your house. She said you were different.” Miss Matilda smiled briefly. “Maybe you are. My experience with boys makes me convinced that they’re all pretty much alike. I haven’t anything especial against them, though they most usually have dirty shoes—” Eight pair of feet crept under eight chairs—“and are noisy. And sometimes they don’t pay much attention to the eighth commandment.” Rapid glances were exchanged between her hearers. Dutch was plainly striving to recall which commandment was which. Miss Matilda arose in her majesty. “Come, Lydia,” she said. Miss Lydia obeyed, casting a final embarrassed smile over the circle. At the door Miss Matilda paused. “I hope you will come again,” she said quite graciously. “It will be very pleasant for my niece. We will be glad to see you any time so long as you behave yourselves.” Exit Miss Matilda and Miss Lydia. Spud drew a long breath that seemed to come from his shoes and glanced about him. “Did anybody speak?” he inquired. Molly giggled softly. The footsteps of the Misses Curtis died away upstairs. “I guess,” said Sandy, looking around for confirmation, “that we’ll have to be going back now.” “Yes,” said Hoop and The Fungus in a breath. “Oh, please!” exclaimed Molly. “Let’s go out to the shed. It’s dandy out there; and there are lots of apples.” “Well—” began Sandy hesitatingly. But Molly had already jumped up and was leading the way. The boys seized their caps from the table and followed her down the steps and around the side of the house. At Molly’s command the sliding door was pushed back and a fervid aroma of apples met them.
- 73. “Now let’s bring some of those boxes over here by the door,” she said, “and sit down. Two of you can have the wheelbarrow.” Presently they were all seated, Spud and Cal on an empty barrel which creaked ominously whenever they stirred, but not before Molly had led the way to the best apples and they had supplied themselves therewith. “I’m awfully fond of apples,” she announced from her seat on a soap box by the door. “Aren’t you?” She addressed Sandy. “Yes,” he answered, “and these are dandies; aren’t they, Ned?” “Great,” Ned agreed with gusto. “I don’t believe we ever knew about these.” There was a moment of deep silence. Then Molly threw back her head with a peal of laughter and the boys, looking silly and sheepish, finally joined in. So far the incident of the midnight adventure in the orchard had not been mentioned between them. But now Spud said: “That was a great joke you played on us that night, Molly.”
- 74. Presently they were all seated “Were you awfully angry when Cal told you?” she asked. “No. Only Sandy. He was sort of peeved.” “I?” said Sandy in surprise. “Not at all.” He frowned at Spud, but that irrepressible young gentleman went on. “I guess you didn’t know, Molly,” he said, “that Ned was appointed a committee of one to—” “Shut up, Spud!” growled Ned. “To get that pillow-case back. Ned’s our diplomat. Whenever he is extra nice to you you must be very careful. That’s his diplomacy. He’s after the pillow-case.” “Oh, I’m not afraid,” answered Molly. “No one knows where it is, you see.” “I do,” said Cal. “Where?” asked Spud. But Molly gave a cry of alarm.
- 75. “Don’t tell him, Cal! Don’t you dare!” “I guess it wouldn’t do him much good if I did,” said Cal. “He couldn’t get it.” “N-no, I suppose not. Perhaps some day if you’re all just awfully nice to me I’ll give it back to you.” “Tell us about it,” said Ned. “How did you fix yourself up that night?” So Molly recounted her adventures, and by leaning forward they could see the rain-spout that she had clambered up and down by. Viewing it was, however, disastrous to Spud and Cal, for an empty barrel set on its side is at best an uncertain seat, and now when they both leaned forward the barrel “took it into its head,” as Spud explained, to lean backward, with a readily imagined result. When they had picked themselves up Ned tried to clap Spud’s pun. “The barrel,” he said, “was merely trying to stave you off.” “Hoop you choke,” responded Spud promptly. After which scintillations Molly went on with her story. “Of course,” said Sandy, when she had finished, “you have a perfect right to keep the pillow-case, Molly—” “Right of capture,” interpolated The Fungus. “But if your aunts ever found it and told Doctor Webster, we’d be in a bad mess. So don’t you think you’d better—er—better—” “No, I don’t,” laughed Molly. “And you don’t need to worry, Sandy, one bit. It’s in a perfectly safe place, and locked up. And just as long as you’re nice to me, and do everything I want you to it will stay there!” Spud groaned. “She has us in her power, fellows.” “Yes, I have,” Molly exulted laughingly. “And I shall make you do anything I want!” “Well, don’t want too much,” said Hoop.
- 76. “I hope,” inquired Spud concernedly, “that our friend Mr. Hooper is properly attentive, Molly? If he doesn’t do what you want him to let us know and we’ll kill him. And Clara, too. Is he quite satisfactory?” “Clara is perfectly dear,” answered Molly. “And so is Hoop.” Hoop tried to look bored but was quite evidently pleased. “So you all are,” she concluded, beaming about her. “We thank you,” said Ned, laying his hand on his waistcoat in the vicinity of his heart. “Any little thing we can do for you—” “I shall never believe in you again, Ned,” said Molly sadly, “after what Spud told me. Whenever you say anything nice I shall think that you’re after my pillow-case.” “In that case—” began Ned, but he was drowned by a chorus of groans. “I shall sneak another apple,” he finished. “Sneak one for me, too,” said Hoop. “Where do these apples grow, Molly?” he asked carelessly. Molly made a face at him. “I know but I sha’n’t tell you,” she answered. “If I did you might all come over here after some and get caught.” “No, I was just thinking about next year,” Hoop assured her. “We never make more than one raid a year.” “You will please,” Spud admonished Cal, “not throw your cores on the club house floor. Toss them out on the gravel. They look much better there.” “Oh, let’s call this a club!” cried Molly eagerly. “Right,” Ned agreed. “The Woodshed Club.” “The Apple Club would be better,” suggested Clara. “No; let me see.” Sandy frowned thoughtfully. “What kind of apples are these we’ve been eating, Molly?” “Newtown Pippins,” answered Molly. “There you have it, then; the Pippin Club!”
- 77. “Dandy,” said The Fungus. “We’re the Nine Pippins.” “And we’ll meet here every Sunday afternoon,” cried Molly, clapping her hands. “And this shall be our club house.” “Um,” observed Ned doubtfully. “Won’t it be a bit coolish in winter?” “The house committee,” said Spud, “must look into the matter of heating the club house. Steam would do.” “We ought to elect officers,” suggested Hoop. “I’ll be president.” “You dry up. Molly’s president.” This from Sandy. “And I’ll be secretary. And Ned shall be—” “This is a mighty funny election,” interrupted The Fungus. “What am I?” “You’re a toadstool,” said Spud severely. “Sit down and subside. I move that Ned be elected something and that I be made treasurer.” “There won’t be anything to treasure, Spud,” said Molly. “Except the apples.” “Oh, we’ll have initiation fees and dues,” responded Spud cheerfully. “Pay up, please. I need the money.” “Better let Ned be treasurer, then,” said Sandy. “He needs the money worse. He’s shy eight dollars.” That aroused Molly’s curiosity and she had to be told of the mysterious disappearance of Ned’s money, first having been sworn to secrecy. “Oh, I’m so sorry, Ned!” said Molly. “And I don’t mind your being a —a diplomat now.” “I’ve always heard,” remarked The Fungus, “that diplomacy was an expensive career.” “I tell you what!” Molly beamed brightly across at Ned. “I’ll think about it tonight when I’m going to sleep and see if I can’t dream where it is, Ned.”
- 78. Spud made a gesture of triumph. “Ned, your money’s as good as found!” he declared. “Well, you needn’t laugh,” Molly protested. “I have found things that way; once I know of. When I was a little girl I lost my doll and I felt just terribly about it. We hunted everywhere for her, mama and nurse and me. And I was so unhappy that I cried myself to sleep after nurse had gone. And in the night I dreamed that she was hidden under the oak chest in the hall!” “The nurse?” Ned asked. “No, my doll, stupid! And when I got up in the morning I went down and looked and there she was! Now, wasn’t that— remarkable?” “It was. But you see, Molly, we haven’t any oak chest in our hall.” “We might get one, though,” said Spud helpfully. Molly pouted. “Oh, if you’re going to make fun of me—” “We’re not,” protested Ned. “It’s a wonderful scheme. You go ahead and dream, Molly, and see what happens.” “Maybe you’d better eat some mince pie or a welsh rarebit or something before you go to bed,” said Sandy, “so you’ll be sure to dream.” “I always dream,” replied Molly. “Every night of my life. And some of them are just beautiful!” “Wish mine were,” said Spud. “Mine are just awful. You and Cal ought to compare symptoms. Cal has a fine time dreaming, don’t you, Cal? Remember the night Ned lost his money you dreamed of thieves?” “Really?” cried Molly. “Then it was thieves that took your money, Ned!” “I guess it was—if the money was taken. I guess, though, that I just mislaid it.”
- 79. “Gee,” said The Fungus admiringly, “you talk of mislaying eight dollars as though it was eight cents! Wish I was rich like that.” “I tell you what you do, Molly,” said Spud. “You dream about sixteen dollars instead of eight, and then when Ned finds it you and I’ll divide the other eight!” “Spud, you’re too silly for anything,” said Molly severely. “He’s a boiled idiot,” Sandy agreed. “We’ve got to be going, fellows. We’ve had a very nice time, Molly.” “Yes, thanks, and we’ll come again,” said The Fungus. “Next Sunday, then,” Molly replied. “Don’t forget. The Pippin Club meets every Sunday afternoon.” “In their club house on—on Apple Avenue,” added Spud. “I move a vote of thanks to the president for her hospitality. All in favor will signify by taking another apple. It is so moved. As treasurer I’ll take two.” “A terrible thought strikes me,” said Dutch as they left the club house. “We’ll probably have apple-sauce for supper!” A groan, loud, prolonged and dismal, arose on the afternoon air. Spud viewed the two pippins in his hands and shook his head over them. “They don’t look as good as they did,” he muttered. “I guess I’ll put them back—in my pocket.” They said good-bye to Molly at the steps and then ambled back to West House, munching as they went.
- 80. “H CHAPTER XVII CAL BUYS A SUIT ouse Eleven: Practice at 3:45 sharp today. No cuts. Brooks, Captain.” “Sounds like business, what?” asked Spud of Cal as he read the notice in School Building Monday morning. “Say, I hope Brooksie won’t take it out on The Fungus for that beastly fumble. Wasn’t that the meanest luck ever? Between you and me, Cal, Fungus ought to have recovered that ball. He had lots of time. It looked like a case of stage-fright. I guess Fungus was so horrified at what he’d done he couldn’t move for a second. But he will make good all right if Brooksie doesn’t take him off today. But I don’t believe he will. Cap has got a whole lot of common-sense. I guess that’s one thing that makes him such a dandy captain.” Spud was right in his surmise. The Fungus went back to his place at left half-back that afternoon just as though there hadn’t been any fumble. The only change made was in the substitution of Folsom for Boyle at full. It was the hardest practice of the season and lasted until it was almost too dark to see the ball with any certainty. Brooks was trying to make his machine run smoother. All the parts were there and they represented plenty of power, but so far the full power hadn’t materialized. A football team is like, we will say, an engine which is rated at twenty horse-power. If the engine runs smoothly it will develop its twenty, but if the parts aren’t assembled just right, if each one isn’t timed exactly with the others, there’s a loss of power and the twenty is perhaps no better than a fifteen. So it was with the House Team. Brooks, who had, as Spud said, a lot of common- sense—and a good deal of football sense added to it—realized that his team represented the best of the material at hand and that if it
- 81. was to develop the power of which it was capable it must be perfectly adjusted. So that afternoon and every other afternoon that week the constant cry was “Get together!” The back-field was the chief offender. Play after play was pulled off—the team had a repertoire of fourteen at this stage—and always someone was too early or too late. Brooks argued and explained and pleaded and scolded. Ned gave way to H. Westlake at right half and Morris took M’Crae’s place at quarter, and still things went wrong. Hoop went into the line for Brooks so that the captain might coach from back of the team. A thing that exasperated Brooks was that over on the Hall gridiron the rival team was running through its signals with all the smoothness that the House eleven lacked. But Rome wasn’t built in a day and Brooks told himself that it was something accomplished if he had only made the fellows understand what was wanted. Perhaps tomorrow or the next day they would put his preaching into practice. It was a very tired group of players and substitutes that trailed back to the gymnasium at dusk. The Hall Team had long since disappeared and they had the gymnasium to themselves. Brooks, attired scantily in a generous bath towel, spoke a few words to his weary team-mates on his way to the shower. “You fellows can play this game the way it ought to be played,” he said, “play it well enough to lick Hall. But you won’t until you can get it into your heads that a football team isn’t made up of eleven fellows each acting for himself but of eleven fellows acting like one. You know your plays but you don’t know how to use them. That’s what the trouble is. Hall hasn’t any better material than we have in spite of the fact that she has more fellows to draw from. But Hall gets together. The line and the ends and the backs work like so many different parts of a watch, and the result is nice smooth football. You fellows in the line are doing pretty well, but the backs aren’t helping you along. Now tomorrow I want to see this team take hold and run through its plays like clock-work. If it doesn’t there’s going to be another victory for Hall on Saturday. I’m doing all I can. Now it’s up to you fellows.”
- 82. Brooks disappeared into the bath and there was a sound of rushing water beyond the canvas curtain. That’s all the sound there was for a minute. Then Brad Miller whistled a tune softly and stole bathward and one after another the rest followed, as many as there was room for, while the balance waited, subdued and chastened. On Tuesday practice was no less vigorous, but Brooks let them off after an hour and a quarter. There was some improvement noticeable. Cal got in at left tackle for a while and did very well; so well that Dutch, relegated to the substitutes, looked distinctly anxious. It was almost supper time when West House reached home. On the steps sat Molly, a red ribbon pinned to the front of her gown in honor of the Houses. Mrs. Linn had been talking to her from the doorway but hurried kitchenward when the boys appeared. “Didn’t see you at practice, Molly,” said Ned, throwing himself down wearily on the steps. “No, I didn’t go today,” answered Molly. “I was teaching Clara tennis.” “What? Well, you must be getting on!” “I don’t play very well, of course, Ned, but I know what you have to do. And that’s what I was showing Clara.” “Oh, I see. Where is he?” Ned looked about him. “He—he went upstairs.” Molly hesitated and looked troubled. “He got hit with a ball.” “How awful!” laughed Spud. “Did it kill him?” “N-no, but it made his nose bleed. It hit him right square on the nose.” “Why, Molly!” said Spud in shocked tones. “Is that the way you treat your opponents? You ought to be playing football instead of tennis.” “I didn’t mean to, Spud. I just hit a ball across and he was leaning over the net quite near and didn’t see it coming. It—it bled horribly.”
- 83. “Well, he will be all right,” Sandy said comfortingly. “Accidents will happen on the best regulated courts.” “Just the same,” observed Spud, “it isn’t considered sportsmanlike to maim your enemy, Molly.” But Molly looked so troubled that Spud stopped his efforts at teasing. “I see you’re wearing the right color, Molly.” “So is Clara,” murmured Ned. “Yes, but if you don’t beat the Hall next Saturday I’m going to wear blue,” she answered. There was a groan of protest at that. “We’re going to win, though,” said Spud sturdily, “aren’t we, Cal?” “I cal’late we’ll put up a good fight,” was the cautious reply. “We’re going to win,” said The Fungus vehemently as he got up. “That’s what we’re going to do. Now I’ll go up and see how Clara’s nose is behaving. I hope it isn’t damaged. It’s a nice little nose.” It wasn’t damaged, but it presented a reddened and swollen appearance when Clara brought it to the supper table a few minutes later. He had to put up with a good deal of ragging from the others. “I shall have to tell Molly to be more careful with you,” said Spud. “You’re not used to the gentle ways of women, Clara.” The incident, however, brought about more trouble for Molly than for her victim, for the following noon, when Cal returned from morning school, Molly called to him from beyond the lilac hedge that separated the two houses. “Hello,” he said as he went over, “what’s the matter with you?” For Molly looked extremely depressed. “They won’t let me go out of the yard today,” she said mournfully. “And Hoop was going to play tennis with me after dinner.” “Why won’t they?” Cal demanded. “Because I told them about Clara’s nose and Aunt Matilda said I was to stay at home until I had learned to be more careful and lady-
- 84. like. And I told her I didn’t mean to do it, too!” “That’s a shame,” said Cal warmly. “It wasn’t your fault.” “Aunt Matilda says I’m harum-scarum,” sighed Molly. “Do you think I am, Cal?” “I—I cal—I guess I don’t just know what that is,” he answered. “How long have you got to stay in the yard?” “I don’t know. All of today, anyhow. Why, what have you done to your coat, Cal Boland?” “That? That’s just a tear,” replied Cal. “Hoop and I were tussling this morning.” “You must have it mended or it will get worse. Haven’t you another suit you can put on?” “Only my Sunday one.” “Then you’d better buy one at once,” she said severely. “That isn’t fit to be seen in, Cal. All the other boys look so nice, too.” Cal viewed as much of his suit as was in sight to him and shook his head ruefully. “I cal’late I’ve got to,” he said. “Seems like I get into a lot of trouble with my clothes. This was a perfectly good suit when I came here.” Molly laughed. “Well, it’s perfectly good for nothing now. Get a dark suit, Cal, won’t you? You’d look so much nicer in dark clothes.” “That’s what Ned said. Dark clothes show dirt, though, don’t they?” “They couldn’t show much more dirt than those do,” replied Molly scornfully. “Just look at them! You ought to be ashamed to be seen in them.” Cal looked a trifle surprised and a little ashamed.
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