![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/89481-250507100606-de65b034/85/Variational-Problems-in-Differential-Geometry-1st-Edition-Professor-Roger-Bielawski-20-320.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/89481-250507100606-de65b034/85/Variational-Problems-in-Differential-Geometry-1st-Edition-Professor-Roger-Bielawski-21-320.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/89481-250507100606-de65b034/85/Variational-Problems-in-Differential-Geometry-1st-Edition-Professor-Roger-Bielawski-22-320.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/89481-250507100606-de65b034/85/Variational-Problems-in-Differential-Geometry-1st-Edition-Professor-Roger-Bielawski-24-320.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/89481-250507100606-de65b034/85/Variational-Problems-in-Differential-Geometry-1st-Edition-Professor-Roger-Bielawski-25-320.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/89481-250507100606-de65b034/85/Variational-Problems-in-Differential-Geometry-1st-Edition-Professor-Roger-Bielawski-27-320.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/89481-250507100606-de65b034/85/Variational-Problems-in-Differential-Geometry-1st-Edition-Professor-Roger-Bielawski-28-320.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/89481-250507100606-de65b034/85/Variational-Problems-in-Differential-Geometry-1st-Edition-Professor-Roger-Bielawski-29-320.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/89481-250507100606-de65b034/85/Variational-Problems-in-Differential-Geometry-1st-Edition-Professor-Roger-Bielawski-30-320.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/89481-250507100606-de65b034/85/Variational-Problems-in-Differential-Geometry-1st-Edition-Professor-Roger-Bielawski-31-320.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/89481-250507100606-de65b034/85/Variational-Problems-in-Differential-Geometry-1st-Edition-Professor-Roger-Bielawski-32-320.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/89481-250507100606-de65b034/85/Variational-Problems-in-Differential-Geometry-1st-Edition-Professor-Roger-Bielawski-33-320.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/89481-250507100606-de65b034/85/Variational-Problems-in-Differential-Geometry-1st-Edition-Professor-Roger-Bielawski-34-320.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/89481-250507100606-de65b034/85/Variational-Problems-in-Differential-Geometry-1st-Edition-Professor-Roger-Bielawski-38-320.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/89481-250507100606-de65b034/85/Variational-Problems-in-Differential-Geometry-1st-Edition-Professor-Roger-Bielawski-39-320.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/89481-250507100606-de65b034/85/Variational-Problems-in-Differential-Geometry-1st-Edition-Professor-Roger-Bielawski-40-320.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/89481-250507100606-de65b034/85/Variational-Problems-in-Differential-Geometry-1st-Edition-Professor-Roger-Bielawski-41-320.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/89481-250507100606-de65b034/85/Variational-Problems-in-Differential-Geometry-1st-Edition-Professor-Roger-Bielawski-42-320.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/89481-250507100606-de65b034/85/Variational-Problems-in-Differential-Geometry-1st-Edition-Professor-Roger-Bielawski-43-320.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/89481-250507100606-de65b034/85/Variational-Problems-in-Differential-Geometry-1st-Edition-Professor-Roger-Bielawski-44-320.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/89481-250507100606-de65b034/85/Variational-Problems-in-Differential-Geometry-1st-Edition-Professor-Roger-Bielawski-45-320.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/89481-250507100606-de65b034/85/Variational-Problems-in-Differential-Geometry-1st-Edition-Professor-Roger-Bielawski-46-320.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/89481-250507100606-de65b034/85/Variational-Problems-in-Differential-Geometry-1st-Edition-Professor-Roger-Bielawski-47-320.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/89481-250507100606-de65b034/85/Variational-Problems-in-Differential-Geometry-1st-Edition-Professor-Roger-Bielawski-48-320.jpg)
![NEW INSIGHTS. See
THINGS THAT BOUNCE.
THINGS THAT FALL.
THINGS THAT FLOAT.
THINGS THAT FLY.
THINGS THAT ROLL.
THINGS THAT STICK.
THE NEW JERSEY SHORELINE. Environmental Films. 18 min., sd.,
color, 16 mm. Appl. author: Thomas P. Grazulis. ©
Environmental Films, Inc.; 1Aug70; MP2O8O5.
THE NEW PARENT LEARNS TO LIVE WITH BABY (Filmstrip) Mt.
San Jacinto College. 42 fr., color, 35 mm. (Marriage and the
family) By Beverlie Burgard. © Mt. San Jacinto College a.a.d.o.
Mt. San Jacinto Junior College District (in notice: Mt. San Jacinto
College); 1Jan68; JP12529.
A NEW PLACE TO LIVE. See
[DAILY LIVING]
NEW ROOM. See
MY THREE SONS.
A NEW SHAPE FOR FREDDY. See
STORIES ABOUT SHAPES.
NEW ZEALAND: PROSPERITY BASED ON TRADE (Filmstrip)
McGraw-Hill Book Co. Made by American Broadcasting Co.
Merchandising, Jules Power International Productions McGraw-
Hill. 46 fr., color, 35 mm. (South Pacific geography series) ©](https://image.slidesharecdn.com/89481-250507100606-de65b034/85/Variational-Problems-in-Differential-Geometry-1st-Edition-Professor-Roger-Bielawski-52-320.jpg)
![American Broadcasting Co. Merchandising, Inc. McGraw-Hill,
Inc.; 30Dec68; JP12469.
A NEWCOMER COMES TO TOWN (Filmstrip) Troll Associates. 34 fr.,
color, 35 mm. (Practicing good citizenship) Author, Bertha
Sickels; illustrator, Jacqueline Blair. © Troll Associates; 6Feb70;
A166407.
A NICE PLACE TO VISIT. See
THE MONKEES. 33.
NIGHT OF.
For titles beginning with Night of See THE WILD, WILD WEST.
NIKO, BOY OF GREECE. ACI Productions. 21 min., sd., color, 16
mm. Appl. author: Stelios Roccos. © ACI Productions, Inc.;
19Feb68; MP20606.
1970 LOW SPEED CAR CRASH TESTS. Insurance Institute for
Highway Safety. 34 min., sd., color, 16 mm. Appl. author: Albert
Benjamin Kelley. © Insurance Institute for Highway Safety;
24Aug70; MP20785.
[1971 BARRACUDA FEATURES] (Filmstrip) Chrysler Corp. Made by
Ross Roy, Inc. 50 fr., color, 35 mm. © Chrysler Corp.; 17Sep70;
JP12548.
1971 CHRYSLER FEATURES. See FOR ALL THE LIVING YOU DO,
CHRYSLER COMES THROUGH.
1971 CHRYSLER-PLYMOUTH FURY COMING THROUGH BIG
(Filmstrip) Chrysler Corp. Made by Ross Roy, Inc. 49 fr., color, 35
mm. Appl. ti.: 1971 Fury features. © Chrysler Corp.; 17Sep70;
JP12540.
[1971 DART/DEMON PRODUCT FILM] (Filmstrip) Chrysler Corp.
Made by Ross Roy, Inc. 67 fr., color, 35 mm. © Chrysler Corp.;
3Sep70; JP12541.](https://image.slidesharecdn.com/89481-250507100606-de65b034/85/Variational-Problems-in-Differential-Geometry-1st-Edition-Professor-Roger-Bielawski-53-320.jpg)
![1971 DODGE TRUCKS: MAXIVALUE (Filmstrip) Chrysler Corp.
Made by Ross Roy, Inc. 76 fr., color, 35 mm. © Chrysler Corp.;
30Sep70; JP12551.
[1971 ENGINEERING FEATURES] (Filmstrip) Chrysler Corp. Made
by Ross Roy, Inc. 58 fr., color, 35 mm. © Chrysler Corp.;
17Sep70; JP12545.
1971 FURY FEATURES. See
1971 CHRYSLER-PLYMOUTH FURY COMING THROUGH BIG.
[1971 MARKET ORIENTATION FILM] (Filmstrip) Chrysler Corp.
Made by Ross Roy, Inc. 77 fr., color, 35 mm. © Chrysler Corp.;
23Sep70; JP12537.
[1971 OPTIONS] (Filmstrip) Chrysler Corp. Made by Ross Roy, Inc.
56 fr., color, 35 mm. © Chrysler Corp.; 17Sep70; JP12542.
[1971 VALIANT/DUSTER FEATURES] (Filmstrip) Chrysler Corp.
Made by Ross Roy, Inc. 56 fr., color, 35 mm. © Chrysler Corp.;
10Sep70; JP12546.
[1971 VALIANT/DUSTER FEATURES] (Filmstrip) Chrysler Corp.
Made by Ross Roy, Inc. 66 fr., color, 35 mm. © Chrysler Corp.;
10Sep70; JP12444.
NO HARVEST FOR THE REAPER. See
WHAT HARVEST FOR THE REAPER.
NO SPACE LIKE HOME. Terrytoons. 5 min., sd., bw, 16 mm.
(Astronut) © Terrytoons, a division of CBS Films, Inc.; 21Jul65;
LP38431.
NO TROUBLE AT ALL. See
THE HIGH CHAPARRAL.
NO-TURN SPEED BROIL. Westinghouse Electric Corp. Made by Jam
Handy Productions, div. of T.T.P. Corp. 8 min., sd., color, 16 mm.](https://image.slidesharecdn.com/89481-250507100606-de65b034/85/Variational-Problems-in-Differential-Geometry-1st-Edition-Professor-Roger-Bielawski-54-320.jpg)
![NUMERATION SYSTEMS (Filmstrip) No. 760. Educational
Projections Corp. 43 fr., color, 35 mm. (Mathematics, level 6) By
Donovan H. Lichtenberg Charles W. Engel. © Educational
Projections Corp.; 11Nov69; JP12671.
NURSE'S AIDE. See
[CAREER DEVELOPMENT: SELF-EVALUATION]
NUTRITION: I'LL TRADE YOU MY COOKIE FOR AN ORANGE
(Filmstrip) McGraw-Hill Book Co. Made by McGraw-Hill Films. 91
fr., color, 35 mm. (Learning to learn series) With guide. ©
McGraw-Hill, Inc.; 30Dec69 (in notice: 1968); JP12494.](https://image.slidesharecdn.com/89481-250507100606-de65b034/85/Variational-Problems-in-Differential-Geometry-1st-Edition-Professor-Roger-Bielawski-57-320.jpg)
![OFFICE MACHINES. See
HOW TO PREPARE A STENCIL USING A TYPEWRITER.
HOW TO USE THE ILLUMINATED DRAWING BOARD.
OH, FREEDOM! New York Times Co. Arno Press. Made by
Rediscovery Productions. 26 min., sd., color, 16 mm. © New York
Times Co. Arno Press; 1Aug70; LP38193.
OHM'S LAW (Filmstrip) Mt. San Jacinto College. 38 fr., color, 35
mm. (Automotive technology, M-2B) Planned written by John
Schuster. © Mt. San Jacinto College a.a.d.o. Mt. San Jacinto
Junior College District (in notice: Mt. San Jacinto College);
1Jan69; JP12510.
OLD FRIENDS. See
BONANZA.
OLD WITCH GOES TO THE BALL. See
TELL ME A STORY.
ON A CLEAR DAY YOU CAN SEE FOREVER. Paramount Pictures
Corp. 129 min., sd., color, 35 mm. Panavision. Based on the
musical play by Alan Jay Lerner. © Paramount Pictures Corp.;
8Jun70; LP38128.
ON MY WAY TO THE CRUSADES I MET A GIRL WHO. Warner
Bros.-Seven Arts. 93 min., sd., color, 35 mm. A Julia film
production. © Warner Bros.-Seven Arts, Inc.; 1Sep69 (in notice:
1967); LP38135.
ON THE JOB BENEFITS. See
[KEEPING A JOB]
ON THE LAM WITH THE HAM. Terrytoons. 5 min., sd., bw, 16
mm. (Deputy Dawg) Appl. ti.: On the lam with ham. ©](https://image.slidesharecdn.com/89481-250507100606-de65b034/85/Variational-Problems-in-Differential-Geometry-1st-Edition-Professor-Roger-Bielawski-59-320.jpg)
![Terrytoons, a division of CBS Films, Inc.; 19Aug64 (in notice:
1963); LP38368.
ON WRITING CHEMICAL EQUATIONS. Pt. 1. John Wiley Sons. 4
min., si., color, Super 8 mm. Loop film. Appl. author: Wendell H.
Slabaugh. © John Wiley Sons, Inc.; 18Jul69; MP20560.
ON WRITING CHEMICAL EQUATIONS. REDOX. Pt. 2. John Wiley
Sons. 4 min., si., color, Super 8 mm. Loop film. Appl. author:
Wendell H. Slabaugh. © John Wiley Sons, Inc.; 18Jul69;
MP20553.
ONCE AND FUTURE KING. See
CAMELOT.
THE ONE BETWEEN. Indiana Public Health Foundation. Made by
Indiana State University Audio-Visual Center. 20 min., sd., color,
16 mm. Produced in cooperation with I.S.U. Dept. of Health
Safety Vigo County School Corp. © Indiana Public Health
Foundation, Inc.; 20Nov70; MU8249.
ONE BIG OCEAN. Reinald Werrenrath, Jr. Made by Journal Films.
10 min., sd., color, 16 mm. © Reinald Werrenrath, Jr.; 1Apr70;
MP20824.
ONE DAY'S GROWTH. See
[REMINGTON ELECTRIC SHAVER TELEVISION COMMERCIALS]
ONE SMALL STEP FOR MAN. See
THE BOLD ONES.
ONLY PEOPLE PRODUCE PROFITS. Caltex Petroleum Corp. Made
by Visualscope. 10 min., sd., color, Super 8 mm. Loop film. ©
Caltex Petroleum Corp.; 13Nov70; MU825O.
OPEN WIDE. Terrytoons. 5 min., sd., bw, 16 mm. (Deputy Dawg)
© Terrytoons, a division of CBS Films, Inc.; 19Aug64 (in notice:](https://image.slidesharecdn.com/89481-250507100606-de65b034/85/Variational-Problems-in-Differential-Geometry-1st-Edition-Professor-Roger-Bielawski-60-320.jpg)
![1963); LP38381.
OPENING A CHECKING ACCOUNT. See
[HANDLING FINANCES]
OPENING A SAVINGS ACCOUNT. See
[HANDLING FINANCES]
OPENING NEW DOORS (Filmstrip) Troll Associates. 36 fr., color, 35
mm. (Developing good work and study habits) Author, Brian
James; illustrator, Gordon Hart. © Troll Associates; 6Feb70;
A166387.
OPERA WITH HENRY BUTLER. Learning Co. of America. 26 min.,
sd., color, 16 mm. © Learning Co. of America, division of
Columbia Pictures Industries, Inc.; 2Nov70; LP38518.
OPERATION OF THE pH METER, BECKMAN MODEL 72. John Wiley
Sons. 4 min., si, color, Super 8 mm. Loop film. Appl. author:
Wendell H. Slabaugh. © John Wiley Sons, Inc.; 18Jul69;
MP20554.
OPPORTUNITY AND THE GOOD LIFE. Aerojet-General Corp. 7
min., sd., color, 16 mm. © Aerojet-General Corp.; 29Jul69;
LP38118.
OPTOMETRIC ASSISTANT. See
[CAREER DEVELOPMENT: SELF-EVALUATION]
ORAL MEDICATIONS. See
ADMINISTRATION OF MEDICATIONS.
ORAL REPORTS: HOW TO TALK OUT LOUD (Filmstrip) Troll
Associates. 33 fr., color, 35 mm. (Developing good work and
study habits) Author, Anne Brent; illustrator, Herbert Leopold. ©
Troll Associates; 6Feb70; A166391.](https://image.slidesharecdn.com/89481-250507100606-de65b034/85/Variational-Problems-in-Differential-Geometry-1st-Edition-Professor-Roger-Bielawski-61-320.jpg)
![OTHER WOMAN. See
MY THREE SONS.
OTROS TRABAJADORES DE LA COMUNIDAD. See
[SPANISH PROGRAM]
OUR CHILDREN'S HERITAGE (Filmstrip) Cooper Films Records. 6
filmstrips, color, 35 mm. With Teacher's guide, 31 p. Contents:
Peter and the wolf. 32 fr.--Toads and diamonds. 29 fr.--The ugly
duckling. 42 fr.--The town mouse and the country mouse. 26 fr.--
Pinocchio. 31 fr.--Silly Joe. 28 fr. Illustrated by Carroll E. Spinney.
© Cooper Films Records, Inc.; 17Aug70 (1969 in notices on
filmstrips); A182384.
OUR COMMUNITY. See
WHAT IS A COMMUNITY.
OUR FRIEND THE ROBIN. Troll Associates. 4 min., si., color, Super
8 mm. Loop film. © Troll Associates; 16Jan70; MP20680.
OUR HERITAGE FROM ANCIENT ROME (Filmstrip) Harcourt, Brace
World. 2 filmstrips (pt. 1, 92 fr.; pt. 2, 117 fr.), color, 35 mm.
(Adventure in literature) Appl. author: Guidance Associates of
Pleasantville. © Harcourt, Brace World, Inc.; 19Dec69;
JP12421.
OUR HERITAGE FROM THE PAST. See
ARAB WORLD AND ISLAM.
THE BYZANTINE EMPIRE.
LIFE IN ANCIENT EGYPT.
LIFE IN THE DARK AGES.
LIFE IN THE MIDDLE AGES.](https://image.slidesharecdn.com/89481-250507100606-de65b034/85/Variational-Problems-in-Differential-Geometry-1st-Edition-Professor-Roger-Bielawski-63-320.jpg)
![PAPER SHAPES. See
PHONO-VIEWER PROGRAM, ART SERIES 2/DISCOVERING
TECHNIQUES.
PARADE. See
[FOCUS ON SELF-DEVELOPMENT, STAGE ONE: AWARENESS]
PARAGRAPHS: HENRY LEARNS SOMETHING NEW (Filmstrip) Troll
Associates. 35 fr., color, 35 mm. (New adventures in language)
Author, Edward McCullough; illustrator, Dodie O'Keefe. © Troll
Associates; 10Jan70; A166443.
PARTS OF SPEECH: UP AND AWAY IN A FLYING BOAT (Filmstrip)
Troll Associates. 30 fr., color, 35 mm. (New adventures in
language) Author, Edward McCullough; illustrator, Karen Tureck.
© Troll Associates; 14Jan70; A166432.
PARTY PLANNING. Dart Industries. 13 min., sd., color, Super 8
mm. (Stop and go learning, session 4) © Dart Industries, Inc.;
1Oct69; MP20619.
THE PASSION OF ANNA. A. B. Svensk Filmindustri. Released by
United Artists Corp. 99 min., sd., color, 35 mm. © A. B. Svensk
Filmindustri; 28Mar70 (in notice: 1969); LP38433.
EL PATITO FEO. See
[SPANISH PROGRAM]
EL PATITO VALIENTE Y EL VIENTO FRIO DEL NORTE (Filmstrip) S-
132. Educational Projections Corp. 34 fr., color, 35 mm. Traducido
del ingles por Carlos Rivera. © Educational Projections Corp.;
23Dec68; JP12383.
PATRIOTIC POETRY BY AMERICAN WRITERS. See
A VISUAL ANTHOLOGY OF POETRY.](https://image.slidesharecdn.com/89481-250507100606-de65b034/85/Variational-Problems-in-Differential-Geometry-1st-Edition-Professor-Roger-Bielawski-66-320.jpg)
![PAUL CEZANNE (Filmstrip) Films Slides. 15 fr., color, 35 mm. ©
Films Slides; 1Dec61; JP12700.
PEACE. Todd N. Tuckey. 6 min., si., color, 16 mm. © Todd N.
Tuckey; 31Aug70; MU8231.
PEACH PLUCKIN' KANGAROO. Terrytoons. 5 min., sd., bw, 16
mm. (Deputy Dawg) © Terrytoons, a division of CBS Films, Inc.;
19Aug64 (in notice: 1963); LP38345.
PECOS BILL. See
AMERICAN FOLKLORE.
PECOS BILL AND LIGHTNING (Filmstrip) Troll Associates. 43 fr.,
color, 35 mm. (American folk heroes and tall tales) Illustrator,
Ettie de Laczay. © Troll Associates; 8Jan70; A166483.
PECOS BILL AND THE LONG LASSO, See
TELL ME A STORY.
PEEK-A-BOO. See
[CHAS. PFIZER CO. TELEVISION COMMERCIALS]
PEGASUS THE WINGED HORSE (Filmstrip) Troll Associates. 44 fr.,
color, 35 mm. (Myths and legends of ancient Greece) Illustrator,
Regina Fisher. © Troll Associates; 6Feb70; A166490.
PEOPLE AGAINST ORTEGA. See
THE BOLD ONES.
PEOPLE PLEASERS (Filmstrip) Chrysler Corp. Made by Ross Roy,
Inc. 59 fr., color, 35 mm. © Chrysler Corp.; 17Sep70; JP12538.
PEOPLE SOUP. Pangloss Productions. Released by Columbia
Pictures Corp. 11 min., sd., color, 35 mm. © Pangloss
Productions, Inc.; 1Apr70 (in notice: 1969); MP20551.](https://image.slidesharecdn.com/89481-250507100606-de65b034/85/Variational-Problems-in-Differential-Geometry-1st-Edition-Professor-Roger-Bielawski-68-320.jpg)
![PERCENT (Filmstrip) No. 769. Educational Projections Corp. 39 fr.,
color, 35 mm. (Mathematics, level 6) By Donovan R. Lichtenberg
Charles W. Engel. © Educational Projections Corp.; 27Feb70;
JP12680.
PERCEPTION. Appleton-Century-Crofts. 1 reel, sd., color, 16 mm.
(Analysis of behavior) Appl. authors: Robert Johnson Michael
Ball. © Meredith Corp.; 5Nov70; MP20952.
PERCHING BIRDS, LARGEST FAMILY OF BIRDS. Troll Associates. 4
min., si., color, Super 8 mm. Loop film. © Troll Associates;
16Jan70; MP20682.
PEREGRINE FALCON. See
THE WONDERFUL WORLD OF DISNEY.
PERFECT TEENAGER. See
THE PATTY DUKE SHOW.
PERFORMA PANTY HOSE. See
[PRO-TEL PRODUCTS TELEVISION COMMERCIALS]
PERFORMANCE OF DOWNCOMERS IN DISTILLATION COLUMNS.
Fractionation Research. 15 min., sd., bw, 16 mm. ©
Fractionation Research, Inc.; 18Feb70; MP20595.
PERIODONTAL DISEASE. Teaching Films. 9 min., sd., color, 16 mm.
(Prevention control of dental disease) © Teaching Films, Inc.,
division of A-V Corp.; 26Oct70; MU8258.
PERSEUS AND MEDUSA (Filmstrip) Troll Associates. 43 fr., color, 35
mm. (Myths and legends of ancient Greece) Illustrator, Regina
Fisher. © Troll Associates; 6Feb70; A166488.
PERSONAL DEVELOPMENT. See
GETTING LOST.](https://image.slidesharecdn.com/89481-250507100606-de65b034/85/Variational-Problems-in-Differential-Geometry-1st-Edition-Professor-Roger-Bielawski-69-320.jpg)
Variational Problems in Differential Geometry 1st Edition Professor Roger Bielawski
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- 8. 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
- 9. 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.
- 10. 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
- 11. 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
- 12. 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
- 13. 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
- 14. 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
- 16. 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
- 17. 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
- 18. 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
- 20. 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
- 21. 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.
- 22. 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
- 23. 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)
- 24. 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 .
- 25. 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.
- 26. 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).
- 27. 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 .
- 28. 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].
- 29. 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
- 30. 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.
- 31. 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 , ∞).
- 32. 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∞.
- 33. 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∞)
- 34. 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 ∞
- 35. 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|
- 36. 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.
- 37. 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.
- 38. 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
- 39. 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.
- 40. 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.
- 41. 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.
- 42. 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
- 43. 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
- 44. 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
- 45. 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
- 46. 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.
- 47. 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.
- 48. 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.
- 49. Another Random Scribd Document with Unrelated Content
- 50. SCHOOLS, FAMILIES, NEIGHBORHOODS. NEITHER PUNCH NOR JUDY. See MARCUS WELBY, M.D. NEON. See HOT DOG. No. 1. NERVES AND HEARTBEAT RATE. Regents of University of Colorado. 4 min., si., color, Super 8 mm. (BSCS single topic inquiry films) Loop film. Appl. author: Biological Sciences Curriculum Study. © Regents of University of Colorado; 1Jul69 (in notice: 1968); MP20746. NETSILIK LIFE (Filmstrip) Education Development Center. 19 fr., color, 35 mm. © Education Development Center, Inc.; 26Jun69; JP12653. THE NETSILIK TODAY (Filmstrip) Education Development Center. 16 fr., bw, 35 mm. With the cooperation of the National Museum of Canada the National Film Board of Canada. © Education Development Center, Inc.; 11Jun67; JP12652. NEVER ON THIRSTY. Mirisch-DePatie-Freleng. Released by United Artists Corp. 7 min., sd., color, 35 mm. © Mirisch-D.F.; 5Aug70; LP38225. THE NEVERGLADES. Terrytoons. 5 min., sd., bw, 16 mm. (Deputy Dawg) © Terrytoons, a division of CBS Films, Inc.; 19Aug64 (in notice: 1963); LP38343. NEW ADVENTURES IN LANGUAGE. See CAPITALIZATION: DON'T HIDE IN TREES, PLEASE. GRAMMAR USAGE: LET'S GIVE A SHOW. INTRODUCING ENGLISH.
- 51. LEARNING TO WRITE LETTERS. MORE PARTS OF SPEECH: THE CASE OF THE MISSING MONEY BAG. MORE SENTENCES: THE HAUNTED HOUSE MYSTERY. NOUNS: WHO'S WHO IN THIS CRAZY ZOO. PARAGRAPHS: HENRY LEARNS SOMETHING NEW. PARTS OF SPEECH: UP AND AWAY IN A FLYING BOAT. PUNCTUATION MARKS: MELVIN MAKES HIS MARK. PUNCTUATION: THERE'S A DRAGON IN MY BACKYARD. SENTENCES: A SILLY DAY WITH CECIL. SPELLING: HOW SPELLING ALMOST RUINED MUDVILLE'S LITTLE LEAGUE. USING REFERENCE MATERIALS. VERBS: A STRANGE NEW BROTHER. NEW DOCTORS. See THE BOLD ONES. NEW ENGLAND'S HERITAGE TRAIL. Paramount Pictures Corp. Made by Winik Films. 2 reels, sd., color, 35 mm. (See the U.S.A.) © Paramount Pictures Corp.; 9Apr69; MP20630. (See also New England's heritage trail; 9Apr69; MP20946) NEW ENGLAND'S HERITAGE TRAIL. Paramount Pictures Corp. Made by Winik Films Corp. 2 reels, sd., color, 35 mm. (See the U.S.A.) © Paramount Pictures Corp.; 9Apr69; MP20946. (See also New England's heritage trail; 9Apr69; MP20630) NEW HOSTESS IN TOWN. See THE HIGH CHAPARRAL.
- 52. NEW INSIGHTS. See THINGS THAT BOUNCE. THINGS THAT FALL. THINGS THAT FLOAT. THINGS THAT FLY. THINGS THAT ROLL. THINGS THAT STICK. THE NEW JERSEY SHORELINE. Environmental Films. 18 min., sd., color, 16 mm. Appl. author: Thomas P. Grazulis. © Environmental Films, Inc.; 1Aug70; MP2O8O5. THE NEW PARENT LEARNS TO LIVE WITH BABY (Filmstrip) Mt. San Jacinto College. 42 fr., color, 35 mm. (Marriage and the family) By Beverlie Burgard. © Mt. San Jacinto College a.a.d.o. Mt. San Jacinto Junior College District (in notice: Mt. San Jacinto College); 1Jan68; JP12529. A NEW PLACE TO LIVE. See [DAILY LIVING] NEW ROOM. See MY THREE SONS. A NEW SHAPE FOR FREDDY. See STORIES ABOUT SHAPES. NEW ZEALAND: PROSPERITY BASED ON TRADE (Filmstrip) McGraw-Hill Book Co. Made by American Broadcasting Co. Merchandising, Jules Power International Productions McGraw- Hill. 46 fr., color, 35 mm. (South Pacific geography series) ©
- 53. American Broadcasting Co. Merchandising, Inc. McGraw-Hill, Inc.; 30Dec68; JP12469. A NEWCOMER COMES TO TOWN (Filmstrip) Troll Associates. 34 fr., color, 35 mm. (Practicing good citizenship) Author, Bertha Sickels; illustrator, Jacqueline Blair. © Troll Associates; 6Feb70; A166407. A NICE PLACE TO VISIT. See THE MONKEES. 33. NIGHT OF. For titles beginning with Night of See THE WILD, WILD WEST. NIKO, BOY OF GREECE. ACI Productions. 21 min., sd., color, 16 mm. Appl. author: Stelios Roccos. © ACI Productions, Inc.; 19Feb68; MP20606. 1970 LOW SPEED CAR CRASH TESTS. Insurance Institute for Highway Safety. 34 min., sd., color, 16 mm. Appl. author: Albert Benjamin Kelley. © Insurance Institute for Highway Safety; 24Aug70; MP20785. [1971 BARRACUDA FEATURES] (Filmstrip) Chrysler Corp. Made by Ross Roy, Inc. 50 fr., color, 35 mm. © Chrysler Corp.; 17Sep70; JP12548. 1971 CHRYSLER FEATURES. See FOR ALL THE LIVING YOU DO, CHRYSLER COMES THROUGH. 1971 CHRYSLER-PLYMOUTH FURY COMING THROUGH BIG (Filmstrip) Chrysler Corp. Made by Ross Roy, Inc. 49 fr., color, 35 mm. Appl. ti.: 1971 Fury features. © Chrysler Corp.; 17Sep70; JP12540. [1971 DART/DEMON PRODUCT FILM] (Filmstrip) Chrysler Corp. Made by Ross Roy, Inc. 67 fr., color, 35 mm. © Chrysler Corp.; 3Sep70; JP12541.
- 54. 1971 DODGE TRUCKS: MAXIVALUE (Filmstrip) Chrysler Corp. Made by Ross Roy, Inc. 76 fr., color, 35 mm. © Chrysler Corp.; 30Sep70; JP12551. [1971 ENGINEERING FEATURES] (Filmstrip) Chrysler Corp. Made by Ross Roy, Inc. 58 fr., color, 35 mm. © Chrysler Corp.; 17Sep70; JP12545. 1971 FURY FEATURES. See 1971 CHRYSLER-PLYMOUTH FURY COMING THROUGH BIG. [1971 MARKET ORIENTATION FILM] (Filmstrip) Chrysler Corp. Made by Ross Roy, Inc. 77 fr., color, 35 mm. © Chrysler Corp.; 23Sep70; JP12537. [1971 OPTIONS] (Filmstrip) Chrysler Corp. Made by Ross Roy, Inc. 56 fr., color, 35 mm. © Chrysler Corp.; 17Sep70; JP12542. [1971 VALIANT/DUSTER FEATURES] (Filmstrip) Chrysler Corp. Made by Ross Roy, Inc. 56 fr., color, 35 mm. © Chrysler Corp.; 10Sep70; JP12546. [1971 VALIANT/DUSTER FEATURES] (Filmstrip) Chrysler Corp. Made by Ross Roy, Inc. 66 fr., color, 35 mm. © Chrysler Corp.; 10Sep70; JP12444. NO HARVEST FOR THE REAPER. See WHAT HARVEST FOR THE REAPER. NO SPACE LIKE HOME. Terrytoons. 5 min., sd., bw, 16 mm. (Astronut) © Terrytoons, a division of CBS Films, Inc.; 21Jul65; LP38431. NO TROUBLE AT ALL. See THE HIGH CHAPARRAL. NO-TURN SPEED BROIL. Westinghouse Electric Corp. Made by Jam Handy Productions, div. of T.T.P. Corp. 8 min., sd., color, 16 mm.
- 55. © Westinghouse Electric Corp.; 6Jul70; MU8206. NOBODY WANTS A FAT JOCKEY. See MARCUS WELBY, M.D. NORTH AMERICAN DESERTS (Filmstrip) Educational Filmstrips. 6 filmstrips, color, 35 mm. With manual, 26 p. Contents: Face of the desert. 57 fr.--Life in the desert. 57 fr.--The Chihuahuan Desert. 58 fr.--The Sonoran Desert. 58 fr.--The Mojave Desert. 58 fr.--The Great Basin and the Painted Desert. 58 fr. Written photographed by Sidney Mary Lee Nolan. © Educational Filmstrips; 1Apr70; JP12333. NOSE. See GRIMM BROTHERS' FAVORITES. NOUNS AND PRONOUNS (Filmstrip) 652. Educational Projections Corp. 36 fr., color, 35 mm. © Educational Projections Corp.; 1Feb68; JP12415. NOUNS: WHO'S WHO IN THIS CRAZY ZOO? (Filmstrip) Troll Associates. 30 fr., color, 35 mm. (New adventures in language) Author, Edward McCullough; illustrator, Dodie O'Keefe. © Troll Associates; 14Jan70; A166430. NOW. see THE PANTHERS. NUESTRA FAMILIA EN CASA (Filmstrip) S-109. Educational Projections Corp. 29 fr., color, 35 mm. © Educational Projections Corp.; 20Sep68; JP12377. NUESTRO CUMPLEANOS (Filmstrip) S-158. Educational Projections Corp. 29 fr., color, 35 mm. © Educational Projections Corp.; 18Apr69; JP12386. NUMBER PATTERNS. See
- 56. MATHEMATICS, LEVEL 2. NUMBER SENTENCES. See MATHEMATICS, LEVEL 2. MATHEMATICS, LEVEL 3. NUMBER THEORY (Filmstrip) No. 742. Educational Projections Corp. 44 fr., color, 35 mm. (Mathematics, level 4) By Donovan R. Lichtenberg Charles. W. Engel. © Educational Projections Corp.; 3Feb69; JP12665. NUMBER THEORY (Filmstrip) No. 761. Educational Projections Corp. 49 fr., color, 35 mm. (Mathematics, level 6) By Donovan R. Lichtenberg Charles W. Engel. © Educational Projections Corp.; 9Dec69; JP12672. NUMBERS LESS THAN ZERO (Filmstrip) No. 1251. Popular Science Audio-Visuals. 42 fr., color, 35 mm. (A Mathematics release) With Filmstrip guide, 5 p. © Popular Science Audio-Visuals, Inc.; 2Feb70; A196082. NUMBERS 0-4. See MATHEMATICS, LEVEL 1. NUMBERS 5-9. See MATHEMATICS, LEVEL 1. NUMERATION. See MATHEMATICS, LEVEL 3. NUMERATION SYSTEMS (Filmstrip) No. 736. Educational Projections Corp. 45 fr., color, 35 mm. (Mathematics, level 4) By Donovan R. Lichtenberg Charles W. Engel. © Educational Projections Corp.; 1Nov68; JP12659.
- 57. NUMERATION SYSTEMS (Filmstrip) No. 760. Educational Projections Corp. 43 fr., color, 35 mm. (Mathematics, level 6) By Donovan H. Lichtenberg Charles W. Engel. © Educational Projections Corp.; 11Nov69; JP12671. NURSE'S AIDE. See [CAREER DEVELOPMENT: SELF-EVALUATION] NUTRITION: I'LL TRADE YOU MY COOKIE FOR AN ORANGE (Filmstrip) McGraw-Hill Book Co. Made by McGraw-Hill Films. 91 fr., color, 35 mm. (Learning to learn series) With guide. © McGraw-Hill, Inc.; 30Dec69 (in notice: 1968); JP12494.
- 58. O OBNOXIOUS OBIE. Terrytoons. 5 min., sd., bw, 16 mm. (Deputy Dawg) © Terrytoons, a division of CBS Films, Inc.; 19Aug64; LP38333. O'CASEY SCANDAL. See MY THREE SONS. THE OCEAN: A NEW FRONTIER (Filmstrip) No. 859. Popular Science Audio-Visuals. 43 fr., color, 35 mm. (A Social studies release) With Filmstrip guide, 5 p. © Popular Science Audio- Visuals, Inc.; 5Jan70; A196108. THE OCEAN OF AIR. See ENVIRONMENT OF MAN. OCEANA: AN ISLAND WORLD (Filmstrip) McGraw-Hill Book Co. 47 fr., color, 35 mm. (South Pacific geography series) Produced in collaboration with Authentic Pictures. © McGraw-Hill, Inc.; 30Dec68; JP12467. THE ODYSSEY OF DR. PAP. American Cancer Society. Made by Harry Olesker Productions. 30 min., sd., color, 16 mm. © American Cancer Society, Inc.; 6Jun69; LP38170. OF FOOD AND LAND. See ENVIRONMENT OF MAN. OF LIFE AND DEATH. National Funeral Directors Assn. of the U.S. Made by Alpha Corp. of America. 27 min., sd., color, 16 mm. © National Funeral Directors Assn. of the U.S., Inc.; 17Feb70; LP38123.
- 59. OFFICE MACHINES. See HOW TO PREPARE A STENCIL USING A TYPEWRITER. HOW TO USE THE ILLUMINATED DRAWING BOARD. OH, FREEDOM! New York Times Co. Arno Press. Made by Rediscovery Productions. 26 min., sd., color, 16 mm. © New York Times Co. Arno Press; 1Aug70; LP38193. OHM'S LAW (Filmstrip) Mt. San Jacinto College. 38 fr., color, 35 mm. (Automotive technology, M-2B) Planned written by John Schuster. © Mt. San Jacinto College a.a.d.o. Mt. San Jacinto Junior College District (in notice: Mt. San Jacinto College); 1Jan69; JP12510. OLD FRIENDS. See BONANZA. OLD WITCH GOES TO THE BALL. See TELL ME A STORY. ON A CLEAR DAY YOU CAN SEE FOREVER. Paramount Pictures Corp. 129 min., sd., color, 35 mm. Panavision. Based on the musical play by Alan Jay Lerner. © Paramount Pictures Corp.; 8Jun70; LP38128. ON MY WAY TO THE CRUSADES I MET A GIRL WHO. Warner Bros.-Seven Arts. 93 min., sd., color, 35 mm. A Julia film production. © Warner Bros.-Seven Arts, Inc.; 1Sep69 (in notice: 1967); LP38135. ON THE JOB BENEFITS. See [KEEPING A JOB] ON THE LAM WITH THE HAM. Terrytoons. 5 min., sd., bw, 16 mm. (Deputy Dawg) Appl. ti.: On the lam with ham. ©
- 60. Terrytoons, a division of CBS Films, Inc.; 19Aug64 (in notice: 1963); LP38368. ON WRITING CHEMICAL EQUATIONS. Pt. 1. John Wiley Sons. 4 min., si., color, Super 8 mm. Loop film. Appl. author: Wendell H. Slabaugh. © John Wiley Sons, Inc.; 18Jul69; MP20560. ON WRITING CHEMICAL EQUATIONS. REDOX. Pt. 2. John Wiley Sons. 4 min., si., color, Super 8 mm. Loop film. Appl. author: Wendell H. Slabaugh. © John Wiley Sons, Inc.; 18Jul69; MP20553. ONCE AND FUTURE KING. See CAMELOT. THE ONE BETWEEN. Indiana Public Health Foundation. Made by Indiana State University Audio-Visual Center. 20 min., sd., color, 16 mm. Produced in cooperation with I.S.U. Dept. of Health Safety Vigo County School Corp. © Indiana Public Health Foundation, Inc.; 20Nov70; MU8249. ONE BIG OCEAN. Reinald Werrenrath, Jr. Made by Journal Films. 10 min., sd., color, 16 mm. © Reinald Werrenrath, Jr.; 1Apr70; MP20824. ONE DAY'S GROWTH. See [REMINGTON ELECTRIC SHAVER TELEVISION COMMERCIALS] ONE SMALL STEP FOR MAN. See THE BOLD ONES. ONLY PEOPLE PRODUCE PROFITS. Caltex Petroleum Corp. Made by Visualscope. 10 min., sd., color, Super 8 mm. Loop film. © Caltex Petroleum Corp.; 13Nov70; MU825O. OPEN WIDE. Terrytoons. 5 min., sd., bw, 16 mm. (Deputy Dawg) © Terrytoons, a division of CBS Films, Inc.; 19Aug64 (in notice:
- 61. 1963); LP38381. OPENING A CHECKING ACCOUNT. See [HANDLING FINANCES] OPENING A SAVINGS ACCOUNT. See [HANDLING FINANCES] OPENING NEW DOORS (Filmstrip) Troll Associates. 36 fr., color, 35 mm. (Developing good work and study habits) Author, Brian James; illustrator, Gordon Hart. © Troll Associates; 6Feb70; A166387. OPERA WITH HENRY BUTLER. Learning Co. of America. 26 min., sd., color, 16 mm. © Learning Co. of America, division of Columbia Pictures Industries, Inc.; 2Nov70; LP38518. OPERATION OF THE pH METER, BECKMAN MODEL 72. John Wiley Sons. 4 min., si, color, Super 8 mm. Loop film. Appl. author: Wendell H. Slabaugh. © John Wiley Sons, Inc.; 18Jul69; MP20554. OPPORTUNITY AND THE GOOD LIFE. Aerojet-General Corp. 7 min., sd., color, 16 mm. © Aerojet-General Corp.; 29Jul69; LP38118. OPTOMETRIC ASSISTANT. See [CAREER DEVELOPMENT: SELF-EVALUATION] ORAL MEDICATIONS. See ADMINISTRATION OF MEDICATIONS. ORAL REPORTS: HOW TO TALK OUT LOUD (Filmstrip) Troll Associates. 33 fr., color, 35 mm. (Developing good work and study habits) Author, Anne Brent; illustrator, Herbert Leopold. © Troll Associates; 6Feb70; A166391.
- 62. ORBIT A LITTLE BIT. Terrytoons. 5 min., sd., bw, 16 mm. (Deputy Dawg) © Terrytoons, a division of CBS Films, Inc.; 19Aug64 (in notice: 1963); LP38339. ORBITAL SHAPES PATHS. Teaching Films. 12 min., sd., color. (Space technology series) © Teaching Films, Inc., subsidiary of A-V Corp.; 21Sep70; MU8237. ORDEAL OF MAJOR GRIGSBY. See THE LAST GRENADE. ORGANIZING FOR SALES (Filmstrip) American Training Academy. 58 fr., color, 35 mm. Original material, research direction by James L. Miller. Appl. states prev. pub. 26Apr67, JP10254. NM: additions revisions. © James L. Miller; 11Feb70; JU12643. THE ORIGIN OF THE ALPHABET. See OUR LITERARY HERITAGE. THE ORIGIN OF WRITING. See OUR LITERARY HERITAGE. OSCAR'S BIRTHDAY PRESENT. Terrytoons. 5 min., sd., bw, 16 mm. (Astronut) © Terrytoons, a division of CBS Films, Inc.; 21Jul65; LP38427. OSCAR'S THINKING CAP. Terrytoons. 5 min., sd., bw, 16 mm. (Astronut) © Terrytoons, a division of CBS Films, Inc.; 21Jul65; LP38424. OSMOSIS (Filmstrip) No. 1565. Popular Science Audio-Visuals. 42 fr., color, 35 mm. (A Biology release) With Filmstrip guide, 5 p. © Popular Science Audio-Visuals, Inc.; 6Oct69; A196087. OTHER SIDE OF THE CHART. See MARCUS WELBY, M.D.
- 63. OTHER WOMAN. See MY THREE SONS. OTROS TRABAJADORES DE LA COMUNIDAD. See [SPANISH PROGRAM] OUR CHILDREN'S HERITAGE (Filmstrip) Cooper Films Records. 6 filmstrips, color, 35 mm. With Teacher's guide, 31 p. Contents: Peter and the wolf. 32 fr.--Toads and diamonds. 29 fr.--The ugly duckling. 42 fr.--The town mouse and the country mouse. 26 fr.-- Pinocchio. 31 fr.--Silly Joe. 28 fr. Illustrated by Carroll E. Spinney. © Cooper Films Records, Inc.; 17Aug70 (1969 in notices on filmstrips); A182384. OUR COMMUNITY. See WHAT IS A COMMUNITY. OUR FRIEND THE ROBIN. Troll Associates. 4 min., si., color, Super 8 mm. Loop film. © Troll Associates; 16Jan70; MP20680. OUR HERITAGE FROM ANCIENT ROME (Filmstrip) Harcourt, Brace World. 2 filmstrips (pt. 1, 92 fr.; pt. 2, 117 fr.), color, 35 mm. (Adventure in literature) Appl. author: Guidance Associates of Pleasantville. © Harcourt, Brace World, Inc.; 19Dec69; JP12421. OUR HERITAGE FROM THE PAST. See ARAB WORLD AND ISLAM. THE BYZANTINE EMPIRE. LIFE IN ANCIENT EGYPT. LIFE IN THE DARK AGES. LIFE IN THE MIDDLE AGES.
- 64. MESOPOTAMIA, CRADLE OF CIVILIZATION. THE RENAISSANCE. SPLENDOR OF ANCIENT GREECE. WORLD OF ANCIENT ROME. OUR LITERARY HERITAGE (Filmstrip) Educational Filmstrips. 6 filmstrips, color, 35 mm. With filmstrip manual, 24 p. Contents: Pt. 1-6: The origin of writing. 68 fr.--The origin of the alphabet. 69 fr.--Classical books and libraries. 60 fr.--Monasticism and the book. 63 fr.--The book during the Islamic naissance and European Renaissance. 65 fr.--The development and dissemination of printing. 71 fr. By George Suzanne Russell. © Educational Filmstrips; 1Sep70; JP12627. OUR TEACHING TASK (Filmstrip) Broadman Films. 50 fr., color, 35 mm. With manual, 1 v. © Broadman Films; 1Jul70; A173130. OUTPUT ADMITTANCE--HYBRID. T. M. Adams. 4 min., si., color, Super 8 mm. Loop film. © T. M. Adams; 1Sep70; MP20929. OVER THE HILL TO THE MOON. Teaching Films. 9 min., sd., color. (Space technology series) © Teaching Films, Inc., subsidiary of A-V Corp.; 21Sep70; MU8240. OVEREDGING. Pt. 1, job 1: Closing shoulders (Filmstrip) National Knitted Outerwear Assn. 149 fr., color, 35 mm. © National Knitted Outerwear Assn.; 8Aug69; JP12590. THE OWL AND THE PUSSYCAT. Raster Productions. Released by Columbia Pictures. 95 min., sd., color, 35 mm. Panavision. Based on the play by Bill Manhoff. © Raster Productions, Inc.; 1Nov70; LP38197.
- 65. P PAINT WITH BRUSHES. See PHONO-VIEWER PROGRAM, ART SERIES 2/DISCOVERING TECHNIQUES. PAINT YOUR WAGON. Alan Jay Lerner Productions. 164 min., sd., color, 35 mm. Panavision. Produced with Malpaso Co. Based upon the musical play. © Paramount Pictures Corp. Alan Jay Lerner Productions, Inc.; 6Oct69; LP38469. (See also Paint your wagon); 9Oct69; LP38115. PAINT YOUR WAGON. Alan Jay Lerner Productions. Released by Paramount Pictures Corp. 137 min., sd., color, 35 mm. Panavision. Based upon the musical play. © Paramount Pictures Corp. Alan Jay Lerner Productions, Inc.; 9Oct69; LP38115. (See also Paint your wagon); 6Oct69; LP38469. PAINTING (Filmstrip) Mt. San Jacinto College. 46 fr., color, 35 mm. (Automotive technology, N-5) Through the cooperation of California Department of Education. Planned written by Sheldon Abbott Paul Wells. © Mt. San Jacinto College a.a.d.o. Mt. San Jacinto Junior College District (in notice; Mt. San Jacinto College); 1Jan69; JP12527. PAINTING PROBLEMS (Filmstrip) Mt. San Jacinto College. 36 fr., color, 35 mm. (Automotive technology, unit N-6) Planned written by Paul Wells; audio-visual production, A. H. Waterman; art, Pam Weaver. © Mt. San Jacinto College a.a.d.o. Mt. San Jacinto Junior College District (in notice: Mt. San Jacinto College); 1Jan68; JP12524. THE PANTHERS. American Broadcasting Companies. 28 min, sd., color, 16 mm. (Now) A presentation of ABC News. © American Broadcasting Companies, Inc.; 7Apr70; MP20924.
- 66. PAPER SHAPES. See PHONO-VIEWER PROGRAM, ART SERIES 2/DISCOVERING TECHNIQUES. PARADE. See [FOCUS ON SELF-DEVELOPMENT, STAGE ONE: AWARENESS] PARAGRAPHS: HENRY LEARNS SOMETHING NEW (Filmstrip) Troll Associates. 35 fr., color, 35 mm. (New adventures in language) Author, Edward McCullough; illustrator, Dodie O'Keefe. © Troll Associates; 10Jan70; A166443. PARTS OF SPEECH: UP AND AWAY IN A FLYING BOAT (Filmstrip) Troll Associates. 30 fr., color, 35 mm. (New adventures in language) Author, Edward McCullough; illustrator, Karen Tureck. © Troll Associates; 14Jan70; A166432. PARTY PLANNING. Dart Industries. 13 min., sd., color, Super 8 mm. (Stop and go learning, session 4) © Dart Industries, Inc.; 1Oct69; MP20619. THE PASSION OF ANNA. A. B. Svensk Filmindustri. Released by United Artists Corp. 99 min., sd., color, 35 mm. © A. B. Svensk Filmindustri; 28Mar70 (in notice: 1969); LP38433. EL PATITO FEO. See [SPANISH PROGRAM] EL PATITO VALIENTE Y EL VIENTO FRIO DEL NORTE (Filmstrip) S- 132. Educational Projections Corp. 34 fr., color, 35 mm. Traducido del ingles por Carlos Rivera. © Educational Projections Corp.; 23Dec68; JP12383. PATRIOTIC POETRY BY AMERICAN WRITERS. See A VISUAL ANTHOLOGY OF POETRY.
- 67. PATTON. Twentieth Century-Fox Film Corp. 171 min., sd., color, 70 mm. Dimension 150. Based on factual material from Patton: Ordeal and triumph, by Ladislas Farago, A soldier's story, by Omar N. Bradley. © Twentieth Century-Fox Film Corp.; 30Dec69; LP38179. THE PATTY DUKE SHOW. Chrislaw Productions. Canada. Released by United Artists Television. Approx. 27 min. each, sd., bw, 16 mm. © United Artists Television, Inc. The actress. © 26Nov63; LP38237. Are mothers people. © 18Feb64; LP38233. Auld lang syne. © 31Dec63; LP38241. Christmas present. © 24Dec63; LP38240. The con artists. © 25Feb64; LP38234. The elopement. © 1Oct63; LP38236. Horoscope. © 7Jan64; LP38232. The perfect teenager. © 3Mar64; LP38235. The princess Cathy. © 17Dec63; LP38239. The song writers. © 10Dec63; LP38238. PAUL BUNYAN. See AMERICAN FOLKLORE. PAUL BUNYAN AND HIS BLUE OX. See TELL ME A STORY. PAUL BUNYAN AND HIS GREAT BLUE OX (Filmstrip) Troll Associates. 41 fr., color, 35 mm. (American folk heroes and tall tales) Illustrator, Gloria Fletcher. © Troll Associates; 8Jan70; A166481.
- 68. PAUL CEZANNE (Filmstrip) Films Slides. 15 fr., color, 35 mm. © Films Slides; 1Dec61; JP12700. PEACE. Todd N. Tuckey. 6 min., si., color, 16 mm. © Todd N. Tuckey; 31Aug70; MU8231. PEACH PLUCKIN' KANGAROO. Terrytoons. 5 min., sd., bw, 16 mm. (Deputy Dawg) © Terrytoons, a division of CBS Films, Inc.; 19Aug64 (in notice: 1963); LP38345. PECOS BILL. See AMERICAN FOLKLORE. PECOS BILL AND LIGHTNING (Filmstrip) Troll Associates. 43 fr., color, 35 mm. (American folk heroes and tall tales) Illustrator, Ettie de Laczay. © Troll Associates; 8Jan70; A166483. PECOS BILL AND THE LONG LASSO, See TELL ME A STORY. PEEK-A-BOO. See [CHAS. PFIZER CO. TELEVISION COMMERCIALS] PEGASUS THE WINGED HORSE (Filmstrip) Troll Associates. 44 fr., color, 35 mm. (Myths and legends of ancient Greece) Illustrator, Regina Fisher. © Troll Associates; 6Feb70; A166490. PEOPLE AGAINST ORTEGA. See THE BOLD ONES. PEOPLE PLEASERS (Filmstrip) Chrysler Corp. Made by Ross Roy, Inc. 59 fr., color, 35 mm. © Chrysler Corp.; 17Sep70; JP12538. PEOPLE SOUP. Pangloss Productions. Released by Columbia Pictures Corp. 11 min., sd., color, 35 mm. © Pangloss Productions, Inc.; 1Apr70 (in notice: 1969); MP20551.
- 69. PERCENT (Filmstrip) No. 769. Educational Projections Corp. 39 fr., color, 35 mm. (Mathematics, level 6) By Donovan R. Lichtenberg Charles W. Engel. © Educational Projections Corp.; 27Feb70; JP12680. PERCEPTION. Appleton-Century-Crofts. 1 reel, sd., color, 16 mm. (Analysis of behavior) Appl. authors: Robert Johnson Michael Ball. © Meredith Corp.; 5Nov70; MP20952. PERCHING BIRDS, LARGEST FAMILY OF BIRDS. Troll Associates. 4 min., si., color, Super 8 mm. Loop film. © Troll Associates; 16Jan70; MP20682. PEREGRINE FALCON. See THE WONDERFUL WORLD OF DISNEY. PERFECT TEENAGER. See THE PATTY DUKE SHOW. PERFORMA PANTY HOSE. See [PRO-TEL PRODUCTS TELEVISION COMMERCIALS] PERFORMANCE OF DOWNCOMERS IN DISTILLATION COLUMNS. Fractionation Research. 15 min., sd., bw, 16 mm. © Fractionation Research, Inc.; 18Feb70; MP20595. PERIODONTAL DISEASE. Teaching Films. 9 min., sd., color, 16 mm. (Prevention control of dental disease) © Teaching Films, Inc., division of A-V Corp.; 26Oct70; MU8258. PERSEUS AND MEDUSA (Filmstrip) Troll Associates. 43 fr., color, 35 mm. (Myths and legends of ancient Greece) Illustrator, Regina Fisher. © Troll Associates; 6Feb70; A166488. PERSONAL DEVELOPMENT. See GETTING LOST.
- 70. GOING TO SCHOOL. LEARNING TO DO THINGS FOR YOURSELF. LEARNING TO HELP OTHERS. LEARNING TO LISTEN CAREFULLY. WHAT TO DO WHEN YOU VISIT. PERSONALITY IN BUSINESS (Filmstrip) No. 422. Popular Science Audio-Visuals. 41 fr., color, 35 mm. (A Guidance release) With Filmstrip guide, 5 p. © Popular Science Audio-Visuals, Inc.; 5Jan70; A196144. PERU: INCA HERITAGE. Hartley Productions. 18 min., sd., color, 16 mm. Appl. author: Elda Hartley. © Hartley Productions, Inc.; 5Nov70; MP2O983. PETER AND THE WOLF. See OUR CHILDREN'S HERITAGE. PETER PAN. See FAVORITE CHILDREN'S BOOKS. PETS CAN READ. Dade County School B Board. 6 min., sd., color, 16 mm. © Dade County School Board; 25Aug70; MP20874. THE PHARMACIST AND CANCER. American Cancer Society. Made by Campus Film Productions. 22 min., sd., color, 16 mm. © American Cancer Society, Inc.; 16Apr69; MP20790. PHONO-VIEWER PROGRAM, ART SERIES 1/EXPLORING MATERIALS (Filmstrip) General Learning Corp., Early Learning Division. 5 filmstrips (15 fr. each), color, 16 mm. With Kit. Contents: This is finger paint.--This is paint.--These are crayons.-- This is paper.--This is clay. Prepared in cooperation with Binney Smith, Inc.; art consultant: Margaret Johnson; photographer: John Naso; designed by Sara Stein; written by Carol Murdock.
- 71. Appl. author: General Learning Corp., employer for hire. © General Learning Corp.; 31Dec69; A189588-189592. PHONO-VIEWER PROGRAM, ART SERIES 2/DISCOVERING TECHNIQUES (Filmstrip) General Learning Corp., Early Learning Division. 5 filmstrips (15 fr. each), color, 16 mm. With kit. Contents: Paper shapes.--Crayon over, crayon under.--Paint with brushes.--Print with paint.--Putting together. Prepared in cooperation with Binney Smith, Inc.; art consultant: Margaret Johnson; photographer: John Naso; designed by Sara Stein; written by Carol Murdock. Appl. author: General Learning Corp., employer for hire. © General Learning Corp.; 31Dec69; A189583-189587. PHOTORECEPTION AND FLOWERING. Regents of University of Colorado. 4 min., si., color, Super 8 mm. (BSCS single topic inquiry films) Loop film. Appl. author: Biological Sciences Curriculum Study. © Regents of University of Colorado; 1Jul69 (in notice: 1968); MP20755. PHOTOSYNTHESIS: THE BIOCHEMICAL PROCESS. Coronet Instructional Films. 17 min., sd., bw, 16 mm. © Coronet Instructional Films, a division of Esquire, Inc.; 1Apr70; MP20847. THE PHYLA: WHO'S WHO IN THE ANIMAL KINGDOM. Reela Educational Films, a division of Wometco Enterprises. Released by Sterling Movies, Educational Films Division. 17 min., sd., color, 16 mm. Produced in cooperation with University of Miami School of Marine Atmospheric Sciences, Dade County Public Schools Editors of International Oceanographic Foundation Publications. © Reela Educational Films a.a.d.o. Reela Films, a division of Reela Films Laboratories, Inc.; 21Jul70; MP20809. PHYSICAL FITNESS: SLOW DOWN, I CAN'T KEEP UP (Filmstrip) McGraw-Hill Book Co. Made by McGraw-Hill Films. 49 fr., color, 35 mm. (Learning to learn series) With guide. © McGraw-Hill, Inc.; 30Dec69 (in notice: 1968); JP12496. PHYSIOLOGY FILM SERIES. See
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