![1
Infinite Planar Graphs with Non-negative
Combinatorial Curvature
Bobo Hua and Yanhui Su
Abstract
In this chapter, we survey some results on infinite planar graphs with non-
negative combinatorial curvature, related to the total curvature, the number of
vertices with positive curvature and the automorphism group.
1.1 Introduction
The combinatorial curvature for planar graphs was introduced by Nevanlinna,
Stone, Gromov, and Ishida [Nev70, Sto76, Gro87, Ish90] respectively, which
resembles the Gaussian curvature for smooth surfaces. Many interesting geo-
metric and combinatorial results have been obtained under such curvature
conditions since then (see, e.g., [Ż97, Woe98, Hig01, BP01, HJL02, LPZ02,
HS03, SY04, RBK05, BP06, DM07, CC08, Zha08, Che09, Kel10, KP11,
Kel11, Oh17, Ghi17]).
Let (V, E) be a (possibly infinite) locally finite, undirected simple graph
with the set of vertices V and the set of edges E. It is called planar if it can be
topologically embedded into the sphere S2 or the plane R2, where we distin-
guish S2 with R2 while they are identified in the theory of finite planar graphs.
We write G = (V, E, F) for the cellular complex structure of a planar graph
induced by the embedding where F is the set of faces, i.e., connected compo-
nents of the complement of the embedding image of the graph (V, E) in S2 or
R2. We say that a planar graph G is a planar tessellation if the following hold
(see, e.g., [Kel11]):
(i) Every face is homeomorphic to a disc whose boundary consists of finitely
many edges of the graph.
1](https://image.slidesharecdn.com/17589716-250609015023-bc326dbe/85/Analysis-And-Geometry-On-Graphs-And-Manifolds-London-Mathematical-Society-Lecture-Note-Series-Series-Number-461-1st-Edition-Matthias-Keller-20-320.jpg)
![Infinite Planar Graphs with Non-negative Curvature 3
with same facial degree, and glue these polygons along the common edges. The
ambient space equipped with the induced metric constructed above is called
the regular polyhedral surface of G, denoted by S(G). In the following, we
always call it the polyhedral surface for the sake of brevity. For a planar graph
G, the combinatorial curvature at the vertex is defined as
(x) = 1 −
deg(x)
2
+
σ∈F:x≺σ
1
deg(σ)
, x ∈ V. (1.1.1)
In this chapter, we mean by the curvature of a planar graph the combinato-
rial curvature of it for simplicity. It turns out that the curvature of a planar
graph is given by the generalized Gaussian curvature of the polyhedral surface
S(G) up to some normalization. Note that for the polyhedral surface S(G) it
is locally isometric to a flat domain in R2 near any interior point of an edge
or a face, while it might be non-smooth near the vertices. As a metric surface,
the generalized Gaussian curvature K of S(G) vanishes at smooth points and
can be regarded as a measure concentrated on the isolated singularities, i.e.,
on vertices. One can show that the mass of the generalized Gaussian curva-
ture at each vertex x is given by K(x) = 2π − x , where x denotes the
total angle at x in the metric space S(G) (see [Ale05]). Moreover, by direct
computation one has K(x) = 2π(x), where the curvature (x) is defined in
(1.1.1). Hence, one can show that a planar graph G has non-negative curvature
if and only if the polyhedral surface S(G) is a generalized convex surface in the
sense of Alexandrov (see [BGP92, BBI01, HJL15]). Furthermore, the polyhe-
dral surface S(G) can be isometrically embedded into R3 as a boundary of a
compact or non-compact convex polyhedron by Alexandrov’s embedding the-
orem ([Ale05]); see Figure 1.2 for an embedded image of S(G) of the planar
graph G in Figure 1.1.
Figure 1.2 The isometric embedding of S(G) of the planar graph G in Figure 1.1](https://image.slidesharecdn.com/17589716-250609015023-bc326dbe/85/Analysis-And-Geometry-On-Graphs-And-Manifolds-London-Mathematical-Society-Lecture-Note-Series-Series-Number-461-1st-Edition-Matthias-Keller-22-320.jpg)
![4 Bobo Hua and Yanhui Su
In this chapter, we study planar graphs with non-negative curvature. We
introduce two classes of planar graphs with positive or non-negative curvature
as follows:
● PC0 := {G : (x) 0, ∀x ∈ V } is the class of planar graphs with
positive curvature everywhere.
● PC≥0 := {G : (x) ≥ 0, ∀x ∈ V } is the class of planar graphs with
non-negative curvature everywhere.
We review some known results on the class PC0. Stone [Sto76] first proved
a Myers-type theorem: a planar graph with the curvature bounded below uni-
formly by a positive constant is a finite graph. Higuchi proposed a stronger
conjecture that any G ∈ PC0 is a finite graph (see [Hig01, Conjecture 3.2]).
This is certainly wrong for smooth surfaces since there are many non-compact
convex surfaces in R3, which have positive curvature everywhere. However,
for a planar graph it is hopefully true by the combinatorial restriction of reg-
ular polygons as its faces. DeVos and Mohar [DM07] proved the conjecture
by showing a generalized Gauss–Bonnet formula (see [SY04] for the case of
cubic graphs).
For any finite planar graph G ∈ PC≥0, in particular any G ∈ PC0,
by Alexandrov’s embedding theorem its polyhedral surface S(G) can be iso-
metrically embedded into R3 as a boundary of a convex polyhedron (see,
e.g., [Ale05]). From this point of view, we obtain many examples for the
class PC0, e.g., the 1-skeletons of 5 Planotic solids, 13 Archimedean solids,
and 92 Johnson solids. Any of them has regular Euclidean polygons as its
faces in its embedded image in R3. Note that these are all examples of
planar graphs in PC0 whose faces of the embedded image in R3 are reg-
ular polygons (see [Joh66, Zal67]). Besides these, the class PC0 contains
many other examples, such as an example of 138 vertices constructed by
Réti, Bitay, and Kosztolányi [RBK05], examples of 208 vertices by Nichol-
son and Sneddon [NS11], Ghidelli [Ghi17], and Oldridge [Old17], which
cannot be realized as the boundary of a convex polyhedron whose faces are
regular polygons. In fact, although any face of G ∈ PC0 is isometric
to a regular polygon in S(G), it may split into several pieces of non-
coplanar faces in the embedded image of S(G) as the boundary of a convex
polyhedron in R3.
There are two special families of graphs in PC0 called prisms and
antiprisms, both consisting of infinite many examples (see, e.g., [DM07]).
Besides them, DeVos and Mohar [DM07] proved that there are only finitely
many graphs in PC0 and proposed the following problem to find out the
largest graph among them.](https://image.slidesharecdn.com/17589716-250609015023-bc326dbe/85/Analysis-And-Geometry-On-Graphs-And-Manifolds-London-Mathematical-Society-Lecture-Note-Series-Series-Number-461-1st-Edition-Matthias-Keller-23-320.jpg)
![Infinite Planar Graphs with Non-negative Curvature 5
Problem 1.1.1 ([DM07]) What is the number
CS2 := max
G=(V,E,F)
V ,
where the maximum is taken over graphs in PC0, which are not prisms or
antiprisms, and V denotes the cardinality of V ?
On the one hand, as some examples of 208 vertices in PC0 have been
constructed in [NS11, Ghi17, Old17], we have the lower bound estimate that
CS2 ≥ 208. On the other hand, DeVos and Mohar [DM07] initiated to use
the discharging methods to obtain the upper bound estimate CS2 ≤ 3444. The
discharging methods were adopted in the proof of the four-colour theorem in
the literature (see [AH77, RSST97]). The upper bound was later improved to
CS2 ≤ 380 by Oh [Oh17]. By a delicate argument, Ghidelli [Ghi17] showed
that CS2 ≤ 208, which completely solves DeVos and Mohar’s problem that
CS2 = 208.
Next, we consider the class of planar graphs with non-negative curvature,
i.e., PC≥0, which turns out to be much larger than PC0 and contains many
interesting examples. The class of PC0 consists of essentially finite many
examples, while the class PC≥0 contains infinitely many examples of differ-
ent combinatorial types. A fullerene is a finite cubic planar graph whose faces
are either pentagon or hexagon. There are plenty of examples of fullerenes
which are important in the real-world applications, to cite a few examples
[KHO+85, Thu98, BD97, BGM12, BE17a, BE17b]. Note that any fullerene
is a planar graph with non-negative curvature. As shown by Thurston [Thu98],
the number of combinatorial types of fullerenes with N hexagons grows as
N9 as N → ∞. Besides these examples of finite graphs, there are plenty of
examples of infinite graphs. Any planar tiling with regular polygons as tiles
(see, e.g., [GS89, Gal09]) is in the class PC≥0. Note that there are infinitely
many such planar tilings, for which only a few examples with symmetry can
be classified. These motivate us to investigate the general structure of planar
graphs in the class PC≥0.
1.2 Total Curvature of Planar Graphs with Non-negative
Curvature
For a smooth non-compact surface with absolutely integrable Gaussian curva-
ture, its total curvature encodes the global geometric information of the space,
e.g., the boundary at infinity (see [SST03]). For example, the total curvature of
a convex surface in R3 describes the apex angle of the cone at infinity of the](https://image.slidesharecdn.com/17589716-250609015023-bc326dbe/85/Analysis-And-Geometry-On-Graphs-And-Manifolds-London-Mathematical-Society-Lecture-Note-Series-Series-Number-461-1st-Edition-Matthias-Keller-24-320.jpg)
![6 Bobo Hua and Yanhui Su
surface, which is useful to study global geometric and analytic properties of
the surface, such as harmonic functions and heat kernels, following [CM97b,
Xu14]. For planar graphs with non-negative curvature G, we denote by
(G) :=
x∈V
(x)
the total curvature of G whenever the summation converges absolutely. In case
of finite graphs, the Gauss–Bonnet theorem reads as (see, e.g., [DM07])
(G) = 2. (1.2.1)
For an infinite planar graph G ∈ PC≥0, the Cohn-Vossen type theorem, proven
by [DM07, Theorem 1.3] or [Che09, Theorem 1.6], yields that
(G) ≤ 1. (1.2.2)
This means that for any infinite G ∈ PC≥0, the total curvature of G satisfies
0 ≤
x∈V
(x) ≤ 1.
In this section, we study all possible values of total curvature of infinite planar
graphs with non-negative curvature, i.e., the following set
{(G) : G infinite,G ∈ PC≥0}. (1.2.3)
As is well known in Riemannian geometry that for any real number 0 ≤
a ≤ 2π, there is a convex surface whose total curvature is given by a. Hence,
the above set for non-compact convex surfaces turns out to be an interval in
the continuous setting. However, combinatorial structure of planar graphs with
non-negative curvature gives us more information and restrictions for the set
(1.2.3).
For any G = (V, E, F) ∈ PC≥0, we denote by
TG := {v ∈ V : (x) 0} (1.2.4)
the set of vertices with positive curvature, and by
DG := sup
σ∈F
deg(σ) (1.2.5)
the maximal facial degree of G. Chen and Chen [CC08, Che09] proved an
interesting result that the set of vertices with positive curvature in a planar
graph with non-negative curvature is a finite set. Hence, the supremum in
(1.2.5) is in fact the maximum.](https://image.slidesharecdn.com/17589716-250609015023-bc326dbe/85/Analysis-And-Geometry-On-Graphs-And-Manifolds-London-Mathematical-Society-Lecture-Note-Series-Series-Number-461-1st-Edition-Matthias-Keller-25-320.jpg)
![Infinite Planar Graphs with Non-negative Curvature 7
Theorem 1.2.1 (Chen and Chen) For any G ∈ PC≥0, TG is a finite set.
This result makes our combinatorial setting distinguished from the
Riemannian setting. Note that there are many non-compact convex surfaces
with positive curvature everywhere, e.g., the elliptic paraboloid, i.e., the
revolution surface of the graph y = x2 with respect to the z axis in R3.
Moreover, if the maximal facial degree DG of G ∈ PC≥0 is at least 43,
then G has rather special structure, analogous to the prisms or antiprisms in
the finite case (see [HJL15] or Theorem 1.3.2 in this chapter). In that case,
one gets (G) = 1. Hence, for our purposes to understand the set (1.2.3), it
suffices to consider planar graphs G with DG ≤ 42. Note that there are finitely
many vertex patterns, consisting of faces of degree at most 42, with positive
curvature (see Table 1.1 in the Appendix). Then one is ready to see that the set
(1.2.3) is a discrete subset in [0, 1] (see, e.g., [HS17b, Proposition 2.3]).
T. Réti [HL16, Conjecture 2.1] was motivated to determine the following
value
τ1 := inf
(G) : G ∈ PC≥0, (G) 0
,
which is called the first gap of total curvature for infinite planar graphs in the
class of PC≥0. He suggested that τ1 = 1
6 and the minimum is attained by the
graph consisting of a pentagon and infinitely many hexagons, which is a kind
of infinite fullerene (see Figure 1.1). In [HS17a], we give an answer to Réti’s
problem.
Theorem 1.2.2 (Theorem 1.3 in [HS17a])
τ1 =
1
12
.
A planar graph G ∈ PC≥0 satisfies (G) = 1
12 if and only if the polyhedral
surface S(G) is isometric to either
(a) a cone with the apex angle θ = 2 arcsin 11
12 , or
(b) a ‘frustum’ with a hendecagon base (see Figure 1.3).
The proof strategy is straightforward and involves tedious case studies. For
a vertex with positive curvature, if the curvature of the vertex is less than 1
12 ,
then we try to find some nearby vertices with positive curvature such that the
sum of these curvatures is at least 1
12 and prove the results case by case. Note
that there are examples of graphs in PC≥0 whose total curvature attains the
first gap 1
12 (see Figure 1.4 and [HS17a] for more examples). Although graph
structures of infinite planar graphs attaining the first gap of total curvature
could be as complicated as planar tilings (see [HS17a]) we are able to classify](https://image.slidesharecdn.com/17589716-250609015023-bc326dbe/85/Analysis-And-Geometry-On-Graphs-And-Manifolds-London-Mathematical-Society-Lecture-Note-Series-Series-Number-461-1st-Edition-Matthias-Keller-26-320.jpg)
![8 Bobo Hua and Yanhui Su
Figure 1.3 A ‘frustum’ with a hendecagon base
A A
B
B
Figure 1.4 This is an example of total curvature 1
12 , where the half lines with
same labels, A or B, are identified
metric structures of polyhedral surfaces for such planar graphs in the above
theorem.
Inspired by Réti’s question, it will be interesting to know other values
in the set (1.2.3). Using Chen and Chen’s result, Theorem 1.2.1, and the
Gauss–Bonnet theorem for compact subsets with boundary, we are able to
determine all possible total curvatures in the class PC≥0.
Theorem 1.2.3 (Theorem 1.1 in [HS17b]) The set of all values of total curva-
ture of infinite planar graphs with non-negative curvature (1.2.3) is given by
i
12
: 0 ≤ i ≤ 12, i ∈ Z
.
As a corollary, we also obtain that τ1 = 1
12 , which provides an alternative
proof to Réti’s problem. Moreover, as the part of the theorem, one may con-
struct planar graphs with non-negative curvature whose total curvatures attain](https://image.slidesharecdn.com/17589716-250609015023-bc326dbe/85/Analysis-And-Geometry-On-Graphs-And-Manifolds-London-Mathematical-Society-Lecture-Note-Series-Series-Number-461-1st-Edition-Matthias-Keller-27-320.jpg)
![Infinite Planar Graphs with Non-negative Curvature 9
all values listed above (see [HS17b]). We sketch the proof of the theorem as
follows: by Theorem 1.2.1, we know that TG is a finite set. We choose a suf-
ficiently large compact subset K ⊂ S(G), homeomorphic to a closed disc,
such that it contains TG and consists of faces in F. Note that the vertices on the
boundary of K have vanishing curvature, so that their patterns appear in the list
of 17 possible patterns in Table 1.2 in the Appendix. By some combinatorial
restrictions, one can further exclude several patterns from the list and conclude
that any vertex on the boundary is incident to a triangle, a square, a hexagon,
an octagon, or a dodecagon. Then using the Gauss–Bonnet formula on K, we
may prove the theorem. Similar proof strategies apply to the problems on the
total curvature of a planar graph with boundary, i.e., a graph embedded into
the disc or a half plane (see [HS17b]).
Although we crucially use the finiteness structure of TG in the proof of
Theorem 1.2.3, we don’t know much about the structure of the subset TG
which still lies in a black box. By a byproduct of the proof of Theorem
1.2.2, we can show that for G ∈ PC≥0, the induced subgraph on TG has at
most 14 connected components. It was conjecturally at most 12 (see [HS17a,
Conjecture 5.2]).
1.3 The Vertices of Positive Curvature in Planar Graphs
with Non-negative Curvature
In this section, we survey some results on the set of vertices with positive
curvature in planar graphs with non-negative curvature. For any finite (infinite
resp.) G ∈ PC≥0, Alexandrov’s embedding theorem [Ale05] yields that an
isometric embedding of the polyhedral surface S(G) into R3 as a boundary of
a compact (non-compact resp.) convex polyhedron. The set TG serves as the
set of the vertices/corners of the convex polyhedron, so that much geometric
information of the polyhedron is contained in TG. We are interested in the
structure of the set TG.
By the solution to DeVos and Mohar’s problem [Ghi17], besides the prisms
and antiprisms the largest number of vertices in a finite graph in PC0 is 208.
We would like to study analogous problems for planar graphs in PC≥0. We
define some analogues to prisms and antiprisms in the class PC≥0.
Definition 1.3.1 We call a planar graph G = (V, E, F) ∈ PC≥0 a prism-like
graph if either
(1) G is an infinite graph and DG ≥ 43, where DG is defined in (1.2.5), or
(2) G is a finite graph and there are at least two faces with facial degree at
least 43.](https://image.slidesharecdn.com/17589716-250609015023-bc326dbe/85/Analysis-And-Geometry-On-Graphs-And-Manifolds-London-Mathematical-Society-Lecture-Note-Series-Series-Number-461-1st-Edition-Matthias-Keller-28-320.jpg)
![10 Bobo Hua and Yanhui Su
σ
Figure 1.5 A half flat-cylinder in R3
By dividing hexagons into triangles, one may assume that there is no
hexagon in G. Note that ‘prism-like’ graphs have rather special structures
which can be completely determined by the following theorems. For any face
σ, we denote by
∂σ := {x ∈ V : x ≺ σ}
the vertex boundary of σ.
Theorem 1.3.2 ([HJL15]) Let G = (V, E, F) be an infinite planar graph with
non-negative curvature and DG ≥ 43. Then there is only one face σ of degree
at least 43. Suppose that there is no hexagonal face. Then the set of faces F
consists of σ, triangles or squares. Moreover,
F = σ ∪ (∪∞
i=1Li ),
where Li , i ≥ 1, are sets of faces of the same type (triangle or square) which
composite a band, i.e., an annulus, and is defined inductively: Li is the next
layer attaching to the previous layer Li−1 with L0 = {σ}. S(G) is isomet-
ric to the boundary of a half flat-cylinder in R3 (see Figure 1.5). Moreover,
(G) = 1.
Theorem 1.3.3 ([HS18]) Let G = (V, E, F) be a finite prism-like graph.
Then there are exactly two disjoint faces σ1 and σ2 of same facial degree at
least 43. Suppose that there is no hexagonal face. Then the set of faces F
consists of σ1 and σ2, triangles, or squares. Moreover,
F = σ1 ∪ (∪M
i=1Li ) ∪ σ2,
where M ≥ 1, and Li , 1 ≤ i ≤ M, are defined similarly as in Theorem 1.3.2.
S(G) is isometric to the boundary of a cylinder barrel in R3 (see Figure 1.6).
The following problem was proposed in [HL16] as an analogue to DeVos
and Mohar’s problem.](https://image.slidesharecdn.com/17589716-250609015023-bc326dbe/85/Analysis-And-Geometry-On-Graphs-And-Manifolds-London-Mathematical-Society-Lecture-Note-Series-Series-Number-461-1st-Edition-Matthias-Keller-29-320.jpg)
![Infinite Planar Graphs with Non-negative Curvature 11
σ1
σ2
Figure 1.6 A cylinder barrel in R3
Problem 1.3.4 What are the numbers
KS2 := max
finite G
TG, KR2 := max
infinite G
TG,
where the maxima are taken over finite and infinite graphs in PC≥0 which are
not prism-like graphs respectively?
In [HS18], we prove the following theorem.
Theorem 1.3.5 ([HS18])
KR2 = 132.
Moreover, a graph in this class attains the maximum if and only if its
polyhedral surface contains 12 disjoint hendecagons.
On the one hand, we give the upper bound KR2 ≤ 132 by the discharging
methods initiated by [DM07, Oh17, Ghi17] for the case PC0. The curvature
at vertices of a planar graph G can be regarded as the charge concentrated
on vertices. The discharging method is to re-distribute the charge on vertices,
via transferring the charge on vertices with large curvature to vertices with
curvature less than 1
132 , such that the terminal charge on involved vertices in
TG after the distribution process is uniformly bounded below by 1
132 . Then the
estimate of vertices in TG follows from the Cohn–Vossen-type theorem (1.2.2).
On the other hand, we can construct an example possessing 132 vertices of
positive curvature (see [HS18]). This completely answers the second part of
Problem 1.3.4.
For the first part of Problem 1.3.4, one can construct a family of infinitely
many examples of finite graphs in PC≥0 with arbitrarily large number of ver-
tices of positive curvature, see Example 3.3 in [HS18], which are not prism-like
graphs. Hence
KS2 = ∞.](https://image.slidesharecdn.com/17589716-250609015023-bc326dbe/85/Analysis-And-Geometry-On-Graphs-And-Manifolds-London-Mathematical-Society-Lecture-Note-Series-Series-Number-461-1st-Edition-Matthias-Keller-30-320.jpg)
![12 Bobo Hua and Yanhui Su
However, since the argument in the proof of Theorem 1.3.5 is local, we can
prove the similar results for the modified quantity,
KS2 , for finite graphs in
PC≥0.
Theorem 1.3.6 ([HS18]) Let
KS2 := max
finite G
TG,
where the maximum is taken over finite graphs in PC≥0 whose maximal facial
degree are less than 132. Then
KS2 = 264.
Moreover, a graph in this class attains the maximum if and only if its
polyhedral surface contains 24 disjoint hendecagons, i.e., 11-gons.
1.4 Automorphism Groups of Planar Graphs with
Non-negative Curvature
In this section, we study automorphism groups of planar graphs with non-
negative curvature. The automorphism groups of planar graphs have been
extensively studied in the literature (see, e.g., [Man71, Bab75, CBGS08,
SS98]). First, we introduce several definitions of isomorphisms on planar
graphs.
Definition 1.4.1 Let G1 = (V1, E1, F1) and G2 = (V2, E2, F2) be two planar
graphs.
(1) G1 and G2 are said to be graph-isomorphic if there is a graph isomorphism
between (V1, E1) and (V2, E2), i.e., R : V1 → V2 such that for any v, w ∈
V, v ∼ w if and only if R(v) ∼ R(w).
(2) G1 and G2 are said to be cell-isomorphic if there is a cellular isomorphism
H = (HV , HE , HF ) between (V1, E1, F1) and (V2, E2, F2) in the sense
of cell complexes, i.e., three bijections HV : V1 → V2, HE : E1 → E2
and HF : F1 → F2 preserving the incidence relations, that is, for any
v ∈ V, e ∈ E, σ ∈ F, v ≺ e if and only if HV (v) ≺ HE (e) and e ≺ σ if
and only if HE (e) ≺ HF (σ).
(3) G1 and G2 are said to be metric-isomorphic if there is an isometric map in
the sense of metric spaces L : S(G1) → S(G2), such that the restriction
map L is cell-isomorphic between (V1, E1, F1) and (V2, E2, F2).
For a planar graph G, a graph (cellular, metric resp.) isomorphism from G to
itself is called a graph (cellular, metric resp.) automorphism of G. We denote](https://image.slidesharecdn.com/17589716-250609015023-bc326dbe/85/Analysis-And-Geometry-On-Graphs-And-Manifolds-London-Mathematical-Society-Lecture-Note-Series-Series-Number-461-1st-Edition-Matthias-Keller-31-320.jpg)
![Infinite Planar Graphs with Non-negative Curvature 13
by Aut(G), (
Aut(G), L(G) resp.) the group of graph (cellular, metric resp.)
automorphisms of a planar graph G. By the standard identification,
L(G) ≤
Aut(G) ≤ Aut(G),
where ≤ indicates that the former can be embedded as a sub-group of the latter.
By our definition of polyhedral surfaces, it is easy to see that
L(G) ∼
=
Aut(G).
Moreover, by the results in [Whi33, Moh88] for a 3-connected planar graph
G, any graph automorphism R of G can be uniquely realized as a cellu-
lar automorphism H such that HV = R, which is called the associated
cellular automorphism of R. This implies that for a 3-connected planar
graph G,
L(G) ∼
=
Aut(G) ∼
= Aut(G).
In [HS18], we prove that the cellular automorphism group of a planar graph
in PC≥0 with positive total curvature is a finite group.
Theorem 1.4.2 Let G = (V, E, F) be a planar graph with non-negative cur-
vature and positive total curvature. Then the automorphism group of G is finite.
Set a := TG and b := maxv∈TG deg(v). We have the following:
(1) If DG ≤ 42,
Aut(G) divides a!b!.
(2) If DG 42, then
Aut(G) divides
2DG, G is infinite,
4DG, G is finite.
By combining the estimates of the size of TG in Theorems 1.3.5 and
1.3.6 with the above result, we obtain the estimates for the orders of cellular
automorphism groups:
(1) If G is infinite, then
Aut(G) ≤
132! × 5!, for DG ≤ 42,
2DG, for DG 42.
(2) If G is finite, then
Aut(G) ≤
264! × 5!, for DG ≤ 42,
4DG, for DG 42.](https://image.slidesharecdn.com/17589716-250609015023-bc326dbe/85/Analysis-And-Geometry-On-Graphs-And-Manifolds-London-Mathematical-Society-Lecture-Note-Series-Series-Number-461-1st-Edition-Matthias-Keller-32-320.jpg)
![14 Bobo Hua and Yanhui Su
1.5 Analysis on Planar Graphs with Non-negative Curvature
In this section, we study analysis problems on planar graphs with non-negative
curvature. For any G ∈ PC≥0, it is easy to see that for any x ∈ V,
3 ≤ deg(x) ≤ 6.
For any x ∈ V and r 0, we denote by
Br (x) := {y ∈ V : d(y, x) ≤ r}
the ball of radius r centred at x. We introduce the definitions of the volume-
doubling property and the Poincaré inequality on graphs.
Definition 1.5.1
(DV ) A graph G = (V, E) is called satisfying the volume-doubling property
DV (C) for constant C 0 if for all x ∈ V and all r 0:
B2r (x) ≤ CBr (x).
(P) A graph G is called satisfying the Poincaré inequality P(C) for a
constant C 0 if for any functions f on V , x0 ∈ V and r 0,
x∈Br (x0)
| f (x) − fB|2
≤ Cr2
x,y∈B2r (x0),y∼x
( f (y) − f (x))2
,
where
fB :=
1
Br (x0)
x∈Br (x0)
f (x).
The first author, Jost and Liu [HJL15], proved these properties for graphs in
the class PC≥0.
Theorem 1.5.2 ([HJL15]) For any G ∈ PC≥0, the volume-doubling prop-
erty DV (C1) and the Poincaré inequality P(C2) hold for some C1, C2 0.
Moreover, there is a constant C such that
Br (x) ≤ Cr2
, ∀x ∈ V,r 0.
The general principle hidden in this result dates back to [CSC95], in which
they showed that the volume-doubling property and the Poincaré inequality
are quasi-isometric invariants (see [SC04, Woe00] for definitions). Since the
planar graph G with bounded facial degree is properly embedded into the reg-
ular polygonal surface S(G), they are in fact quasi-isometric to each other.
For convex surfaces, even more general Alexandrov spaces with non-negative
curvature, the volume-doubling property follows from the Bishop–Gromov](https://image.slidesharecdn.com/17589716-250609015023-bc326dbe/85/Analysis-And-Geometry-On-Graphs-And-Manifolds-London-Mathematical-Society-Lecture-Note-Series-Series-Number-461-1st-Edition-Matthias-Keller-33-320.jpg)
![Infinite Planar Graphs with Non-negative Curvature 15
volume comparison [BBI01] and the Poincaré inequality is obtained by
[KMS01, Hua09].
For a graph G = (V, E), the Laplace operator , called Laplacian for short,
is defined as follows for any function f on V :
f (x) =
1
deg(x)
y∈V :y∼x
( f (y) − f (x)), ∀x ∈ V.
A function f is called harmonic (subharmonic, superharmonic resp.) on ⊂
V if f ≡ 0 ( f ≥ 0, f ≤ 0 resp.) on . As a corollary of Theorem 1.5.2,
the simple random walk on a planar graph with non-negative curvature is
recurrent, i.e., any positive superharmonic function is constant, which follows
from the assertion that the volume growth is at most quadratic. Moreover,
by Moser iteration, Delmotte [Del97] proved the elliptic Harnack inequality
on graphs under the assumptions of the volume-doubling property and the
Poincaré inequality. In particular, Theorem 1.5.2 implies that for G ∈ PC≥0
and any positive harmonic function f on B2r (p) ⊂ V, r 0, we have
max
Br (p)
f ≤ C min
Br (p)
f, (1.5.1)
where C is a constant independent of r. For any k 0, we denote by
Hk
(G) := { f : V → R| f ≡ 0, | f (x)| ≤ C(1 + d(x, p))k
,
for some p ∈ V, C 0}
the space of harmonic functions on V of polynomial growth whose growth
order are less than or equal to k. The Harnack inequality (1.5.1) yields that
dim Hk
(G) = 1, for some k 1.
In fact, the combination of volume-doubling property and the Poincaré
inequality turns out to be equivalent to the parabolic Harnack inequalities
[Del99]. Furthermore, one can prove the finite-dimensional property of the
space of harmonic functions of polynomial growth with growth rate bounded
above, following Colding, Minicozzi, and Li [CM97a, CM98a, CM98b, Li97,
STW00].
Theorem 1.5.3 ([HJL15, HJ15]) For any G ∈ PC≥0,
dim Hk
(G) ≤ Ck, ∀k ≥ 1,
where C is a universal constant.](https://image.slidesharecdn.com/17589716-250609015023-bc326dbe/85/Analysis-And-Geometry-On-Graphs-And-Manifolds-London-Mathematical-Society-Lecture-Note-Series-Series-Number-461-1st-Edition-Matthias-Keller-34-320.jpg)
![16 Bobo Hua and Yanhui Su
Table 1.1 The patterns of a vertex with positive curvature
Patterns (x)
(3, 3, k) 3 ≤ k 1/6 + 1/k
(3, 4, k) 4 ≤ k 1/12 + 1/k
(3, 5, k) 5 ≤ k 1/30 + 1/k
(3, 6, k) 6 ≤ k 1/k
(3, 7, k) 7 ≤ k ≤ 41 1/k − 1/42
(3, 8, k) 8 ≤ k ≤ 23 1/k − 1/24
(3, 9, k) 9 ≤ k ≤ 17 1/k − 1/18
(3, 10, k) 10 ≤ k ≤ 14 1/k − 1/15
(3, 11, k) 11 ≤ k ≤ 13 1/k − 5/66
(4, 4, k) 4 ≤ k 1/k
(4, 5, k) 5 ≤ k ≤ 19 1/k − 1/20
(4, 6, k) 6 ≤ k ≤ 11 1/k − 1/12
(4, 7, k) 7 ≤ k ≤ 9 1/k − 3/28
(5, 5, k) 5 ≤ k ≤ 9 1/k − 1/10
(5, 6, k) 6 ≤ k ≤ 7 1/k − 2/15
(3, 3, 3, k) 3 ≤ k 1/k
(3, 3, 4, k) 4 ≤ k ≤ 11 1/k − 1/12
(3, 3, 5, k) 5 ≤ k ≤ 7 1/k − 2/15
(3, 4, 4, k) 4 ≤ k ≤ 5 1/k − 1/6
(3, 3, 3, 3, k) 3 ≤ k ≤ 5 1/k − 1/6
Table 1.2 The patterns of a vertex with vanishing curvature
(3, 7, 42), (3, 8, 24), (3, 9, 18), (3, 10, 15), (3, 12, 12),
(4, 5, 20), (4, 6, 12), (4, 8, 8), (5, 5, 10), (6, 6, 6),
(3, 3, 4, 12), (3, 3, 6, 6), (3, 4, 4, 6), (4, 4, 4, 4), (3, 3, 3, 3, 6),
(3, 3, 3, 4, 4), (3, 3, 3, 3, 3, 3).
1.6 Appendix
Table 1.1 lists all possible patterns of a vertex with positive curvature (see
[DM07, CC08]); Table 1.2 lists all possible patterns of a vertex with vanishing
curvature (see [GS89, CC08]).
Acknowledgements
B. H. is supported by NSFC (China) under grant nos. 11831004 and 11826031.
Y. S. is supported by NSFC (China) under grant no. 11771083 and NSF of
Fujian Province through grants 2017J01556 and 2016J01013.](https://image.slidesharecdn.com/17589716-250609015023-bc326dbe/85/Analysis-And-Geometry-On-Graphs-And-Manifolds-London-Mathematical-Society-Lecture-Note-Series-Series-Number-461-1st-Edition-Matthias-Keller-35-320.jpg)
![Infinite Planar Graphs with Non-negative Curvature 17
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![2
Curvature Calculations for Antitrees
David Cushing, Shiping Liu, Florentin Münch, and Norbert Peyerimhoff
Abstract
In this chapter we prove that antitrees with suitable growth properties are
examples of infinite graphs exhibiting strictly positive curvature in various con-
texts: in the normalized and non-normalized Bakry–Émery setting as well as in
the Ollivier–Ricci curvature case. We also show that these graphs do not have
global positive lower curvature bounds, which one would expect in view of
discrete analogues of the Bonnet–Myers theorem. The proofs in the different
settings require different techniques.
2.1 Introduction and Results
The main protagonists in this chapter are antitrees. While these exam-
ples had been studied already in 1988, they were given the name antitree
in talks by Radoslaw Wojciechowsi around 2010. A proper definition of
antitrees, in their most general form, appeared first in [19]. Like in the
case of a tree, the vertices of an antitree are partitioned in generations Vi ,
with the first generation V1 called its root set. While trees are connected
graphs with as few connections as possible between subsequent genera-
tions, antitrees have the maximal number of connections. More precisely,
antitrees are simple (i.e., no loops and no multiple edges), connected graphs
such that
(i) any root vertex x ∈ V1 is connected to all vertices in V2, and no vertices
in Vk, k ≥ 3,
(ii) any vertex x ∈ Vk, k ≥ 2, is connected to all vertices in Vk−1 and Vk+1,
and no vertices in Vl, |k − l| ≥ 2.
21](https://image.slidesharecdn.com/17589716-250609015023-bc326dbe/85/Analysis-And-Geometry-On-Graphs-And-Manifolds-London-Mathematical-Society-Lecture-Note-Series-Series-Number-461-1st-Edition-Matthias-Keller-40-320.jpg)
![22 David Cushing, et al.
Note that this definition allows for the possibility of edges between ver-
tices of the same generation. We will refer to such edges as spherical
edges. Edges between vertices of different generations are called radial
edges. Any radial or spherical edge incident to a vertex in V1 is called
radial or spherical root-edge, respectively. All other edges are called inner
edges.
Antitrees are particularly interesting examples with regard to stochastic
completeness. Section 2.2, provided by Radoslaw Wojciechowki, gives a more
in-depth look at the history of antitrees. In this chapter, we investigate cur-
vature properties of antitrees. Relations between curvature asymptotics and
stochastic completeness were investigated recently in [17] in the Bakry–Émery
setting and in [22] in the Ollivier–Ricci curvature setting.
For our curvature considerations, we consider only antitrees where the
induced subgraph of any one generation Vk is complete, i.e., any two ver-
tices in the same generation are neighbours. For any given finite or infinite
sequence (ak)1≤k≤N , N ∈ N ∪ {∞}, the corresponding unique such antitree
with |Vk| = ak for all 1 ≤ k ≤ N is denoted by AT ((ak)). Note that in the case
of a finite antitree, that is N ∞, (ii) has to be understood in the case k = N
that any vertex x ∈ VN is connected to all vertices in VN−1. Later in this intro-
duction, we will only present results for infinite antitrees, but, since curvature
is a local notion, we need only investigate curvatures of suitable finite antitrees
for the proofs.
Figure 2.1 The antitree AT ((2, 3, 5))](https://image.slidesharecdn.com/17589716-250609015023-bc326dbe/85/Analysis-And-Geometry-On-Graphs-And-Manifolds-London-Mathematical-Society-Lecture-Note-Series-Series-Number-461-1st-Edition-Matthias-Keller-41-320.jpg)
![Curvature Calculations for Antitrees 23
Two particular curvature notions on graphs have been studied actively in
recent years:
● Bakry–Émery curvature taking values on the vertices and based on
Bochner’s formula with respect to a suitable graph Laplacian,
● Ollivier–Ricci curvature taking values on the edges and based on optimal
transport of lazy random walks.
Basic graph theoretical notions are introduced in Section 2.3.1 and precise
definitions of these curvature concepts are given in Sections 2.3.2 and 2.3.3,
respectively.
For both curvature notions there are graph theoretical analogues of the
fundamental Bonnet–Myers theorem for Riemannian manifolds with strictly
positive Ricci curvature bounded away from zero.
Let us first consider Bakry–Émery curvature. Generally, on a combinatorial
graph G = (V, E) with vertex set V and edge set E, the graph Laplacian on
functions f : V → R is of the form
f (x) =
1
μ(x)
y∼x
( f (y) − f (x)), (2.1.1)
with a vertex measure μ : V → (0, ∞). In this chapter, we consider two
specific choices of vertex measures:
● μ ≡ 1, which we refer to as the non-normalized case,
● μ(x) = dx (the vertex degree of x ∈ V ), which we refer to as the normalized
case.
The corresponding discrete Bonnet–Myers theorems in both settings are as
follows.
Theorem 2.1.1 (see [21]) Let G = (V, E) be a connected graph satisfying
C D(K, ∞) for some K 0 in the non-normalized case and dx ≤ D for all
x ∈ V and some finite D. Then G is a finite graph and, furthermore,
diam(G) ≤
2D
K
.
Theorem 2.1.2 (see [21]) Let G = (V, E) be a connected graph satisfying
C D(K, ∞) for some K 0 in the normalized case (possibly of unbounded
vertex degree). Then G is a finite graph and, furthermore,
diam(G) ≤
2
K
.](https://image.slidesharecdn.com/17589716-250609015023-bc326dbe/85/Analysis-And-Geometry-On-Graphs-And-Manifolds-London-Mathematical-Society-Lecture-Note-Series-Series-Number-461-1st-Edition-Matthias-Keller-42-320.jpg)
![24 David Cushing, et al.
Ollivier–Ricci curvature depends upon an idleness parameter p ∈ [0, 1]
describing the laziness of the associated random walk. Here, the discrete
Bonnet–Myers theorem takes the following form.
Theorem 2.1.3 (see [23]) Let G = (V, E) be a connected graph satisfying
κp(x, y) ≥ K 0 for all x ∼ y and a fixed idleness p ∈ [0, 1]. Then G is a
finite graph and, furthermore,
diam(G) ≤
2(1 − p)
K
. (2.1.2)
These results give rise to the following natural questions:
● Do there exist examples of infinite connected graphs with strictly positive
curvature? (That is, relaxing the condition of a uniform strictly positive
lower curvature bound.)
● In the non-normalized case, does there exist an infinite connected graphs
satisfying C D(K, ∞) for K 0 of unbounded vertex degree?
This chapter provides a positive answer to the first question. In fact, we show
that antitrees AT ((ak)) with suitable growth properties of the infinite sequence
(ak) have strictly positive curvature for all curvature notions mentioned above.
More precisely, we have the following in the Bakry–Émery curvature case.
Theorem 2.1.4 In both the normalized and non-normalized setting, the infinite
antitree AT ((k)) satisfies C D(Kx , ∞, x) for all vertices x with a family of
constants Kx 0 depending only on the generation of x. Furthermore,
lim inf
k→∞, x∈Vk
Kx = 0.
Remark 2.1.5 In fact, the method of proof relies on some Maple calculations
which can be extended to also provide the following results (without going into
the details):
(i) Linear growth: The same curvature results hold true for the infinite
antitrees AT ((1 + (k − 1)t)) with arbitrary t ∈ N.
(ii) Exponential growth: The same curvature results hold true for the infinite
antitree AT ((2k−1)) in the normalized case and fail to satisfy C D(0, ∞)
in the non-normalized case.
Due to Bakry–Émery curvature being a local property, in order to calcu-
late the curvatures KG,x (∞) of vertices x in the first two generations of
G = AT ((2k−1)) as defined later in (2.3.1), it is sufficient to consider the](https://image.slidesharecdn.com/17589716-250609015023-bc326dbe/85/Analysis-And-Geometry-On-Graphs-And-Manifolds-London-Mathematical-Society-Lecture-Note-Series-Series-Number-461-1st-Edition-Matthias-Keller-43-320.jpg)
![Curvature Calculations for Antitrees 25
0.078
0.417
0.078
0.58
0.269
0.269
0.269
0.269
0.269
0.269
0.269
0.269
0.58
0.58
0.58
Figure 2.2 Normalized curvature KG,x (∞)
0.5
–1.157
–1.157
0.595
–1
–1
–1
–1
–1
–1
–1
–1
0.595
0.595
0.595
Figure 2.3 Non-normalized curvature KG,x (∞)
graph presented in Figures 2.2 and 2.3 (spherical edges of 2-spheres around
a vertex do not contribute to the curvature, see [7]). These figures are in
agreement with the statements in Remark 2.1.5(ii).](https://image.slidesharecdn.com/17589716-250609015023-bc326dbe/85/Analysis-And-Geometry-On-Graphs-And-Manifolds-London-Mathematical-Society-Lecture-Note-Series-Series-Number-461-1st-Edition-Matthias-Keller-44-320.jpg)
![26 David Cushing, et al.
Now we consider Ollivier–Ricci curvature. Here our main result is the
following.
Theorem 2.1.6 Let G = AT ((ak)) be an infinite antitree with 1 = a1 and
ak+1 ≥ ak for all k ∈ N and x, y be neighbouring vertices in G.
● Radial root edges: If x ∈ V1 and y ∈ V2:
κp(x, y) =
⎧
⎨
⎩
a2−1
a2+a3
+ a2+2a3+1
a2+a−3 p, if p ∈ 0, 1
a2+a3+1 ,
a2+1
a2+a3
(1 − p), if p ∈ 1
a2+a3+1 , 1 .
● Radial edges: If x ∈ Vk and y ∈ Vk+1, k ≥ 2, p ∈ [0, 1]:
κp(x, y) =
2ak + ak+1 − 1
ak + ak+1 + ak+2 − 1
−
2ak−1 + ak − 1
ak−1 + ak + ak+1 − 1
(1 − p).
● Spherical edges: If x, y ∈ Vk, x = y, k ≥ 2:
κp(x, y) =
⎧
⎨
⎩
ak−1+ak+ak+1−2
ak−1+ak+ak+1−1 + ak−1+ak+ak+1
ak−1+ak+ak+1−1 p, if p ∈ 0, 1
ak−1+ak+ak+1
,
ak−1+ak+ak+1
ak−1+ak+ak+1−1 (1 − p), if p ∈ 1
ak−1+ak+ak+1
, 1 .
Let us consider special cases.
Corollary 2.1.7 (Linear growth) Let G = AT ((1+(k−1)t)), t ∈ N arbitrary.
Then
κ0(x, y) =
⎧
⎪
⎪
⎨
⎪
⎪
⎩
t
3t+2 for x ∈ V1, y ∈ V2,
6t2
(3kt+2)(3kt+2−3t) for x ∈ Vk, y ∈ Vk+1,
1 − 1
3kt+2−3t for x, y ∈ Vk, x = y, k ≥ 2.
In particular, κ0 of radial edges decays asymptotically like 2
3k2 as k → ∞.
Corollary 2.1.8 (Exponential growth) We have for G = AT ((rk−1), r ∈ N:
κ0(x, y) =
⎧
⎪
⎪
⎨
⎪
⎪
⎩
r−1
r(r+1) for x ∈ V1, y ∈ V2,
(r−1)2(r+1)rk−2
(rk+rk−1+rk−2−1)(rk+1+rk+rk−1−1)
for x ∈ Vk, y ∈ Vk+1,
1 − 1
rk+rk−1+rk−2−1
for x, y ∈ Vk, x = y, k ≥ 2.
In particular, κ0 of radial edges decays asymptotically like 1
rk as k → ∞.
Remark 2.1.9 Note that for any finite sequence (ak)1≤k≤N , N ≥ 2, with 1 =
a1 and ak+1 ≥ ak for all 1 ≤ k ≤ N, we can find a large enough aN+1 ≥ aN
such that κ0(x, y) 0 for x ∈ VN−1 and y ∈ VN .](https://image.slidesharecdn.com/17589716-250609015023-bc326dbe/85/Analysis-And-Geometry-On-Graphs-And-Manifolds-London-Mathematical-Society-Lecture-Note-Series-Series-Number-461-1st-Edition-Matthias-Keller-45-320.jpg)
![Curvature Calculations for Antitrees 27
The chapter is organised as follows: We start with some historical comments
on antitrees in Section 2.2 which was provided by Radosław Wojciechowski.
Section 2.3 introduces the readers into Bakry–Émery curvature and Ollivier–
Ricci curvature. The following two Sections 2.4 and 2.5 present the concrete
curvature investigations in both settings. Appendices A, B, and C provide the
Maple code used for the results in Section 2.4.
2.2 A (Partial) History of Antitrees
To our knowledge, the first known appearance of an antitree is the case of
|Sr | = r + 1 in the article of Dodziuk and Karp [8]. They study the nor-
malized Laplacian and give conditions for transience of the simple random
walk in terms of r r where r is the distance to a vertex. It appears in [8,
Example 2.5] as a case of a transient graph with bottom of the spectrum 0
whose Green’s function decays like 1/r. The same antitree appears in the
article of Weber [24]. Weber extends the result of Dodziuk and Mathai [9]
concerning the stochastic completeness of the semigroup associated with the
non-normalized Laplacian . Indeed, Dodziuk/Mathai prove stochastic com-
pleteness in the case of bounded vertex degree. Weber improves this result to
give stochastic completeness in the case of r ≥ K for some constant K. The
antitree mentioned above is then given as an example of a graph whose vertex
degree is unbounded but which satisfies r ≥ K (see [24, Figure 1, p. 156]).
The general case of antitrees with arbitrary spherical growth |Sr | = f (r) where
f is any natural number-valued function is considered in [25, Example 4.11].
There it is shown that antitrees are stochastically complete if and only if
r
r
k=0 f (k)
f (r) f (r + 1)
= ∞.
This is used to give a counter-example to a direct analogue to Grigor’yan’s
result for stochastic completeness of manifolds (see [13]). Indeed,
Grigor’yan’s result says that any stochastically incomplete manifold must
have super-exponential volume growth while the result above gives stochas-
tically incomplete graphs which have only polynomial volume growth when
the combinatorial graph metric is used. These examples give the smallest such
examples in the combinatorial graph metric by a result of Huang, Grigor’yan
and Masamune [12, Theorem 1.4], where the example (and name) of antitrees
also appears. This might be the first time in print that the name is used and
they refer to them as the ‘antitree of Wojciechowski’. A proper definition with
the name of antitree first appears in [19, Definition 6.3]. Here the result on](https://image.slidesharecdn.com/17589716-250609015023-bc326dbe/85/Analysis-And-Geometry-On-Graphs-And-Manifolds-London-Mathematical-Society-Lecture-Note-Series-Series-Number-461-1st-Edition-Matthias-Keller-46-320.jpg)
![28 David Cushing, et al.
stochastic completeness is generalized to all weakly spherically symmetric
graphs of which the antitrees are but an example. Furthermore, it is shown
that the non-normalized Laplacian on any such stochastically incomplete
antitree has positive bottom of the spectrum (see [19, Corollary 6.6]). This
gives a counter-example to a direct analogue to a theorem of Brooks [5] which
states that the bottom of the spectrum of the Laplacian on any manifold with
sub-exponential volume growth is zero. This sparked an interest in applying
intrinsic metrics as defined by Frank, Lenz, and Wingert in [10] to study the
question involving volume growth on graphs of unbounded vertex degree. In
particular, the analogue to Grigor’yan’s theorem was first proven in [11] (see
also [18] for an analytic proof) while the analogue to Brooks’ theorem was
shown in [16]. Since then, antitrees appear in a variety of places. Their spec-
tral theory is thoroughly analysed by Breuer and Keller in [4]. Here it should
be noted that the spectrum consists mainly of eigenvalues with compactly sup-
ported eigenfunctions and a further spectral component which can be singular
continuous in certain cases. Antitrees are also used as a counterexample to a
conjecture presented by Golenia and Schumacher in [14] concerning the defi-
ciency indices of the adjacency matrix (see [15]). They are also used to show
the utility of the new bottom of the spectrum estimate for a Cheeger constant
involving intrinsic metrics in [1].
2.3 Definitions and Notations
2.3.1 Basic Graph Theoretical Notations
Let G = (V, E) be a locally finite connected simple combinatorial graph
(that is, no loops and no multiple edges) with vertex set V and edge set E.
For any x, y ∈ V we write x ∼ y if {x, y} ∈ E. The degree of a vertex
x ∈ V is denoted by dx . Let d : V × V → N ∪ {0} be the combinatorial
distance function, i.e., d(x, y) is the length of the shortest path from x to y.
For x ∈ V , the combinatorial spheres and balls of radius r ≥ 0 around x are
denoted by
Sr (x) = {y ∈ V | d(x, y) = r},
Br (x) = {y ∈ V | d(x, y) ≤ r},
respectively. The diameter of G is defined as
diam(G) = sup{d(x, y) | x, y ∈ V } ∈ N ∪ {0, ∞}.](https://image.slidesharecdn.com/17589716-250609015023-bc326dbe/85/Analysis-And-Geometry-On-Graphs-And-Manifolds-London-Mathematical-Society-Lecture-Note-Series-Series-Number-461-1st-Edition-Matthias-Keller-47-320.jpg)
![Curvature Calculations for Antitrees 29
2.3.2 Bakry–Émery Curvature
As mentioned before, this curvature notion is rooted on Bochner’s formula
using a Laplacian operator leading to the curvature-dimension inequality (CD-
inequality for short). This approach was pursued by Bakry–Émery [2] via
an elegant -calculus and leads to a substitute of the lower Ricci curvature
bound of the underlying space for much more general settings. (Some further
information on the Bochner approach can be found, e.g., in [7, Remark 1.3].)
Recall definition (2.1.1) of the normalized (μ(x) = dx ) and non-normalized
Laplacian (μ ≡ 1) from the Introduction. Such a choice of Laplacian leads to
the following operator for all f, g : V → R:
( f, g)(x) =
1
2
( ( f g) − f g − g f )(x)
=
1
2μ(x)
y∼x
( f (y) − f (x))(g(y) − g(x)).
For simplicity, we always write ( f ) := ( f, f ). Iterating , we can define
another operator 2, given by
2( f, g)(x) =
1
2
( ( f, g) − ( f, g) − (g, f ))(x).
Again, we abbreviate 2( f ) = 2( f, f ). The Bakry–Émery curvature is
defined via these operators in the following way.
Definition 2.3.1 Let K ∈ R and N ∈ (0, ∞].
(i) The pointwise curvature dimension condition C D(K, N, x) for x ∈ V is
defined by
2( f )(x) ≥ K( f )(x) +
1
N
( f )2
(x), for any f : V → R.
(ii) The global curvature dimension condition C D(K, N) holds if and only if
C D(K, N, x) holds for any x ∈ V .
(iii) For any x ∈ V , we define
KG,x (N) := sup{K ∈ R | C D(K, N, x)}. (2.3.1)
In this chapter, we are only concerned with ∞-curvature, that is, N = ∞.
Following [7, Prop. 2.1], the condition C D(K, ∞, x) is equivalent to
2(x) ≥ K(x), (2.3.2)
where 2(x) and (x) are symmetric matrices of the corresponding quadratic
forms evaluated at x ∈ V . Since only local information needs to be taken](https://image.slidesharecdn.com/17589716-250609015023-bc326dbe/85/Analysis-And-Geometry-On-Graphs-And-Manifolds-London-Mathematical-Society-Lecture-Note-Series-Series-Number-461-1st-Edition-Matthias-Keller-48-320.jpg)
![30 David Cushing, et al.
into account, they are of size |B2(x)| × |B2(x)| and |B1(x)| × |B1(x)|, respec-
tively, and to make sense of (2.3.2) the smaller size matrix must be padded
with 0 entries. For more information in the non-normalized case, see [7, Sec-
tions 2.1–2.3]. The entries of these matrices in the general weighted case are
explicitly given in [7, Section 12]. (Note that for the context of this chapter, the
edge weights w : E → [0, ∞) take only values 0, 1 and reflect adjacency of
vertices and the vertex measure μ : V → (0, ∞) will only correspond to the
normalized and non-normalized cases.)
The main tool to prove strictly positive curvature is [7, Corollary 2.7], that
is, the following properties are equivalent:
● 2(x) is positive semi-definite with one-dimensional kernel,
● KG,x (∞) 0.
[7, Corollary 2.7] covers only the non-normalized case, but one can easily
check that the equivalence holds also in the setting of general vertex measures.
2.3.3 Ollivier–Ricci Curvature
As mentioned before, Ollivier–Ricci curvature is based on optimal trans-
port. Ollivier–Ricci curvature was introduced in [23]. A fundamental con-
cept in optimal transport is the Wasserstein distance between probability
measures.
Definition 2.3.2 Let G = (V, E) be a locally finite graph. Let μ1, μ2 be two
probability measures on V . The Wasserstein distance W1(μ1, μ2) between μ1
and μ2 is defined as
W1(μ1, μ2) = inf
π
y∈V
x∈V
d(x, y)π(x, y), (2.3.3)
where the infimum runs over all transportation plans π : V × V → [0, 1]
satisfying
μ1(x) =
y∈V
π(x, y), μ2(y) =
x∈V
π(x, y).
The transportation plan π moves a mass distribution given by μ1 into a mass
distribution given by μ2, and W1(μ1, μ2) is a measure for the minimal effort
which is required for such a transition.
If π attains the infimum in (2.3.3) we call it an optimal transport plan
transporting μ1 to μ2.](https://image.slidesharecdn.com/17589716-250609015023-bc326dbe/85/Analysis-And-Geometry-On-Graphs-And-Manifolds-London-Mathematical-Society-Lecture-Note-Series-Series-Number-461-1st-Edition-Matthias-Keller-49-320.jpg)
![Curvature Calculations for Antitrees 31
We define the following probability distributions μx for any x ∈ V, p ∈
[0, 1]:
μ
p
x (z) =
⎧
⎪
⎪
⎨
⎪
⎪
⎩
p, if z = x,
1−p
dx
, if z ∼ x,
0, otherwise.
Definition 2.3.3 The p-Ollivier–Ricci curvature on an edge x ∼ y in G =
(V, E) is
κp(x, y) = 1 − W1(μ
p
x , μ
p
y ),
where p ∈ [0, 1] is called the idleness.
The Ollivier–Ricci curvature introduced by Lin–Lu–Yau in [20] is defined
as
κLLY (x, y) = lim
p→1
κp(x, y)
1 − p
.
A fundamental concept in the optimal transport theory and vital to our work
is Kantorovich duality. First we recall the notion of 1-Lipschitz functions and
then state Kantorovich duality.
Definition 2.3.4 Let G = (V, E) be a locally finite graph, φ : V → R. We
say that φ is 1-Lipschitz if
|φ(x) − φ(y)| ≤ d(x, y)
for all x, y ∈ V. Let 1–Lip denote the set of all 1–Lipschitz functions.
Note that, by triangle inequality, φ is 1-Lipschitz iff |φ(x) − φ(y)| ≤ 1 for
all pairs x ∼ y.
Theorem 2.3.5 (Kantorovich duality) Let G = (V, E) be a locally finite
graph. Let μ1, μ2 be two probability measures on V . Then
W1(μ1, μ2) = sup
φ:V →R
φ∈1–Lip
x∈V
φ(x)(μ1(x) − μ2(x)).
If φ ∈ 1-Lip attains the supremum we call it an optimal Kantorovich
potential transporting μ1 to μ2.
The following result on some properties of p → κp(x, y) for x ∼ y and its
consequences was useful in our curvature considerations.
Theorem 2.3.6 (see [3]) Let G = (V, E) be a locally finite graph. Let x, y ∈
V with x ∼ y. Then the function p → κp(x, y) is concave and piecewise](https://image.slidesharecdn.com/17589716-250609015023-bc326dbe/85/Analysis-And-Geometry-On-Graphs-And-Manifolds-London-Mathematical-Society-Lecture-Note-Series-Series-Number-461-1st-Edition-Matthias-Keller-50-320.jpg)
![32 David Cushing, et al.
linear over [0, 1] with at most 3 linear parts. Furthermore κp(x, y) is linear
on the intervals
0,
1
lcm(dx , dy) + 1
and
1
max(dx , dy) + 1
, 1
.
Thus, if we have the further condition dx = dy, then κp(x, y) has at most two
linear parts.
2.4 Bakry–Émery Curvature of Antitrees
Let us first introduce some notation and a useful general fact (Lemma 2.4.1).
The identity matrix of size d is denoted by Idd and the all-zero and all-one
matrix of size d1 × d2 is denoted by 0d1,d2 and Jd1,d2 , respectively. Moreover,
if d1 = d2, we use the notation Jd1 = Jd1,d1 , and if d2 = 1, we use the notation
1d1 for the all-one column vector of size d1. Moreover, the standard base of
column vectors in RN is denoted by e1, . . . , eN .
Lemma 2.4.1 Let d1, . . . , dr ∈ N and A = (Ai j )1≤i, j≤r be a symmetric
matrix, where the Ai j are block matrices of size di × dj with Aji = A
i j .
Assume that there exist constants αi , βi ∈ R and γi j = γji ∈ R such that, for
1 ≤ i, j ≤ r, j = i,
Aii = αi Iddi + βi Jdi
and
Ai j = γi j Jdi ,dj .
Let Ared = (ai j )1≤i, j≤r be the r × r-matrix given by ai j = 1
di
Ai j 1dj , i.e., for
i = j,
aii = αi di + βi d2
i ,
ai j = γi j di dj .
For any vector w = (w1, . . . , wr ) ∈ Rr let
w := (w11
d1
, . . . , wr 1
dr
)
∈ Rd
with d =
r
j=1 dj . Then we have the following two facts:
(a) For every di ≥ 2, the (di − 1)-dimensional space
Ei =
⎧
⎨
⎩
di
j=1
cj ej+d |
di
j=1
cj = 0
⎫
⎬
⎭
with d =
i−1
j=1 dj consists of eigenvectors to the eigenvalue αi .](https://image.slidesharecdn.com/17589716-250609015023-bc326dbe/85/Analysis-And-Geometry-On-Graphs-And-Manifolds-London-Mathematical-Society-Lecture-Note-Series-Series-Number-461-1st-Edition-Matthias-Keller-51-320.jpg)
![34 David Cushing, et al.
we have in both the normalized and non-normalized case:
KG,x (∞) 0. (2.4.2)
Proof. In this proof, we will keep the values a, b, c, d, e general as long
as possible and only specify them towards the end of the proof. Let G =
AT ((a, b, c, d, e)), 1 ≤ a ≤ b c ≤ d ≤ e and x ∈ V3. To cover
simultaneously both the normalized and non-normalized setting, we choose
− =
μ(x)
μ(y−)
− 1, + =
μ(x)
μ(y+)
− 1,
where y− ∈ V2 and y+ ∈ V4. (Note that μ(z) depends only the generation of
z.) Using the results in [7, Section 12], a tedious but straightforward calculation
shows the following: The matrix A = 4μ(x)22(x) is of the following block
structure A = (Ai j )1≤i, j≤6 where the blocks correspond to an ordering of
B2(x) into the vertex sets {x}, V3{x}, V4, V2, V5, V1:
A11 = dx (dx + 3) + 3b− + 3d+,
A12 = (−(dx + 3) + b− + d+)J1,c−1,
A13 = (−(dx + 3 + e) − (2 + c + e)+)J1,d,
A14 = (−(dx + 3 + a) − (2 + a + c)−)J1,b,
A15 = (d + d+)J1,e,
A16 = (b + b−)J1,a,
A22 = (3(dx + 1) + b− + d+)Idc−1 − 2Jc−1,
A23 = −(2 + 2+)Jc−1,d,
A24 = −(2 + 2−)Jc−1,b,
A25 = 0c−1,e,
A26 = 0c−1,a,
A33 = (−b + 3c + 3d + 3e + (3c + 4d + 3e)+)Idd − (2 + 4+)Jd,
A34 = 2Jd,b,
A35 = −(2 + 2+)Jd,e,
A36 = 0d,a,
A44 = (3a + 3b + 3c − d + (3a + 4b + 3c)−)Idb − (2 + 4−)Jb,
A45 = 0b,e,
A46 = −(2 + 2−)Jb,a,
A55 = (d + d+)Ide,
A56 = 0e,a,
A66 = (b + b−)Ida.](https://image.slidesharecdn.com/17589716-250609015023-bc326dbe/85/Analysis-And-Geometry-On-Graphs-And-Manifolds-London-Mathematical-Society-Lecture-Note-Series-Series-Number-461-1st-Edition-Matthias-Keller-53-320.jpg)
![This ebook is for the use of anyone anywhere in the United States
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under the terms of the Project Gutenberg License included with this
ebook or online at www.gutenberg.org. If you are not located in the
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you are located before using this eBook.
Title: Kurze Aufsätze
Author: Annette Kolb
Release date: November 21, 2013 [eBook #44251]
Most recently updated: October 23, 2024
Language: German
Credits: Produced by Jens Sadowski
*** START OF THE PROJECT GUTENBERG EBOOK KURZE AUFSÄTZE
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Analysis And Geometry On Graphs And Manifolds London Mathematical Society Lecture Note Series Series Number 461 1st Edition Matthias Keller
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- 6. L O N D O N M AT H E M AT I C A L S O C I E T Y L E C T U R E N OT E S E R I E S Managing Editor: Professor Endre Süli, Mathematical Institute, University of Oxford, Woodstock Road, Oxford OX2 6GG, United Kingdom The titles below are available from booksellers, or from Cambridge University Press at www.cambridge.org/mathematics 353 Trends in stochastic analysis, J. BLATH, P. MÖRTERS & M. SCHEUTZOW (eds) 354 Groups and analysis, K. TENT (ed) 355 Non-equilibrium statistical mechanics and turbulence, J. CARDY, G. FALKOVICH & K. GAWEDZKI 356 Elliptic curves and big Galois representations, D. DELBOURGO 357 Algebraic theory of differential equations, M.A.H. MACCALLUM & A.V. MIKHAILOV (eds) 358 Geometric and cohomological methods in group theory, M.R. BRIDSON, P.H. KROPHOLLER & I.J. LEARY (eds) 359 Moduli spaces and vector bundles, L. BRAMBILA-PAZ, S.B. BRADLOW, O. GARCÍA-PRADA & S. RAMANAN (eds) 360 Zariski geometries, B. ZILBER 361 Words: Notes on verbal width in groups, D. SEGAL 362 Differential tensor algebras and their module categories, R. BAUTISTA, L. SALMERÓN & R. ZUAZUA 363 Foundations of computational mathematics, Hong Kong 2008, F. CUCKER, A. PINKUS & M.J. TODD (eds) 364 Partial differential equations and fluid mechanics, J.C. ROBINSON & J.L. RODRIGO (eds) 365 Surveys in combinatorics 2009, S. HUCZYNSKA, J.D. MITCHELL & C.M. RONEY-DOUGAL (eds) 366 Highly oscillatory problems, B. ENGQUIST, A. FOKAS, E. HAIRER & A. ISERLES (eds) 367 Random matrices: High dimensional phenomena, G. BLOWER 368 Geometry of Riemann surfaces, F.P. GARDINER, G. GONZÁLEZ-DIEZ & C. KOUROUNIOTIS (eds) 369 Epidemics and rumours in complex networks, M. DRAIEF & L. MASSOULIÉ 370 Theory of p-adic distributions, S. ALBEVERIO, A.YU. KHRENNIKOV & V.M. SHELKOVICH 371 Conformal fractals, F. PRZYTYCKI & M. URBAŃSKI 372 Moonshine: The first quarter century and beyond, J. LEPOWSKY, J. MCKAY & M.P. TUITE (eds) 373 Smoothness, regularity and complete intersection, J. MAJADAS & A. G. RODICIO 374 Geometric analysis of hyperbolic differential equations: An introduction, S. ALINHAC 375 Triangulated categories, T. HOLM, P. JØRGENSEN & R. ROUQUIER (eds) 376 Permutation patterns, S. LINTON, N. RUŠKUC & V. VATTER (eds) 377 An introduction to Galois cohomology and its applications, G. BERHUY 378 Probability and mathematical genetics, N. H. BINGHAM & C. M. GOLDIE (eds) 379 Finite and algorithmic model theory, J. ESPARZA, C. MICHAUX & C. STEINHORN (eds) 380 Real and complex singularities, M. MANOEL, M.C. ROMERO FUSTER & C.T.C WALL (eds) 381 Symmetries and integrability of difference equations, D. LEVI, P. OLVER, Z. THOMOVA & P. WINTERNITZ (eds) 382 Forcing with random variables and proof complexity, J. KRAJÍČEK 383 Motivic integration and its interactions with model theory and non-Archimedean geometry I, R. CLUCKERS, J. NICAISE & J. SEBAG (eds) 384 Motivic integration and its interactions with model theory and non-Archimedean geometry II, R. CLUCKERS, J. NICAISE & J. SEBAG (eds) 385 Entropy of hidden Markov processes and connections to dynamical systems, B. MARCUS, K. PETERSEN & T. WEISSMAN (eds) 386 Independence-friendly logic, A.L. MANN, G. SANDU & M. SEVENSTER 387 Groups St Andrews 2009 in Bath I, C.M. CAMPBELL et al (eds) 388 Groups St Andrews 2009 in Bath II, C.M. CAMPBELL et al (eds) 389 Random fields on the sphere, D. MARINUCCI & G. PECCATI 390 Localization in periodic potentials, D.E. PELINOVSKY 391 Fusion systems in algebra and topology, M. ASCHBACHER, R. KESSAR & B. OLIVER 392 Surveys in combinatorics 2011, R. CHAPMAN (ed) 393 Non-abelian fundamental groups and Iwasawa theory, J. COATES et al (eds) 394 Variational problems in differential geometry, R. BIELAWSKI, K. HOUSTON & M. SPEIGHT (eds) 395 How groups grow, A. MANN 396 Arithmetic differential operators over the p-adic integers, C.C. RALPH & S.R. SIMANCA 397 Hyperbolic geometry and applications in quantum chaos and cosmology, J. BOLTE & F. STEINER (eds) 398 Mathematical models in contact mechanics, M. SOFONEA & A. MATEI 399 Circuit double cover of graphs, C.-Q. ZHANG 400 Dense sphere packings: a blueprint for formal proofs, T. HALES 401 A double Hall algebra approach to affine quantum Schur–Weyl theory, B. DENG, J. DU & Q. FU 402 Mathematical aspects of fluid mechanics, J.C. ROBINSON, J.L. RODRIGO & W. SADOWSKI (eds)
- 7. 403 Foundations of computational mathematics, Budapest 2011, F. CUCKER, T. KRICK, A. PINKUS & A. SZANTO (eds) 404 Operator methods for boundary value problems, S. HASSI, H.S.V. DE SNOO & F.H. SZAFRANIEC (eds) 405 Torsors, étale homotopy and applications to rational points, A.N. SKOROBOGATOV (ed) 406 Appalachian set theory, J. CUMMINGS & E. SCHIMMERLING (eds) 407 The maximal subgroups of the low-dimensional finite classical groups, J.N. BRAY, D.F. HOLT & C.M. RONEY-DOUGAL 408 Complexity science: the Warwick master’s course, R. BALL, V. KOLOKOLTSOV & R.S. MACKAY (eds) 409 Surveys in combinatorics 2013, S.R. BLACKBURN, S. GERKE & M. WILDON (eds) 410 Representation theory and harmonic analysis of wreath products of finite groups, T. CECCHERINI-SILBERSTEIN, F. SCARABOTTI & F. TOLLI 411 Moduli spaces, L. BRAMBILA-PAZ, O. GARCÍA-PRADA, P. NEWSTEAD & R.P. THOMAS (eds) 412 Automorphisms and equivalence relations in topological dynamics, D.B. ELLIS & R. ELLIS 413 Optimal transportation, Y. OLLIVIER, H. PAJOT & C. VILLANI (eds) 414 Automorphic forms and Galois representations I, F. DIAMOND, P.L. KASSAEI & M. KIM (eds) 415 Automorphic forms and Galois representations II, F. DIAMOND, P.L. KASSAEI & M. KIM (eds) 416 Reversibility in dynamics and group theory, A.G. O’FARRELL & I. SHORT 417 Recent advances in algebraic geometry, C.D. HACON, M. MUSTAŢǍ & M. POPA (eds) 418 The Bloch–Kato conjecture for the Riemann zeta function, J. COATES, A. RAGHURAM, A. SAIKIA & R. SUJATHA (eds) 419 The Cauchy problem for non-Lipschitz semi-linear parabolic partial differential equations, J.C. MEYER & D.J. NEEDHAM 420 Arithmetic and geometry, L. DIEULEFAIT et al (eds) 421 O-minimality and Diophantine geometry, G.O. JONES & A.J. WILKIE (eds) 422 Groups St Andrews 2013, C.M. CAMPBELL et al (eds) 423 Inequalities for graph eigenvalues, Z. STANIĆ 424 Surveys in combinatorics 2015, A. CZUMAJ et al (eds) 425 Geometry, topology and dynamics in negative curvature, C.S. ARAVINDA, F.T. FARRELL & J.-F. LAFONT (eds) 426 Lectures on the theory of water waves, T. BRIDGES, M. GROVES & D. NICHOLLS (eds) 427 Recent advances in Hodge theory, M. KERR & G. PEARLSTEIN (eds) 428 Geometry in a Fréchet context, C.T.J. DODSON, G. GALANIS & E. VASSILIOU 429 Sheaves and functions modulo p, L. TAELMAN 430 Recent progress in the theory of the Euler and Navier–Stokes equations, J.C. ROBINSON, J.L. RODRIGO, W. SADOWSKI & A. VIDAL-LÓPEZ (eds) 431 Harmonic and subharmonic function theory on the real hyperbolic ball, M. STOLL 432 Topics in graph automorphisms and reconstruction (2nd Edition), J. LAURI & R. SCAPELLATO 433 Regular and irregular holonomic D-modules, M. KASHIWARA & P. SCHAPIRA 434 Analytic semigroups and semilinear initial boundary value problems (2nd Edition), K. TAIRA 435 Graded rings and graded Grothendieck groups, R. HAZRAT 436 Groups, graphs and random walks, T. CECCHERINI-SILBERSTEIN, M. SALVATORI & E. SAVA-HUSS (eds) 437 Dynamics and analytic number theory, D. BADZIAHIN, A. GORODNIK & N. PEYERIMHOFF (eds) 438 Random walks and heat kernels on graphs, M.T. BARLOW 439 Evolution equations, K. AMMARI & S. GERBI (eds) 440 Surveys in combinatorics 2017, A. CLAESSON et al (eds) 441 Polynomials and the mod 2 Steenrod algebra I, G. WALKER & R.M.W. WOOD 442 Polynomials and the mod 2 Steenrod algebra II, G. WALKER & R.M.W. WOOD 443 Asymptotic analysis in general relativity, T. DAUDÉ, D. HÄFNER & J.-P. NICOLAS (eds) 444 Geometric and cohomological group theory, P.H. KROPHOLLER, I.J. LEARY, C. MARTÍNEZ-PÉREZ & B.E.A. NUCINKIS (eds) 445 Introduction to hidden semi-Markov models, J. VAN DER HOEK & R.J. ELLIOTT 446 Advances in two-dimensional homotopy and combinatorial group theory, W. METZLER & S. ROSEBROCK (eds) 447 New directions in locally compact groups, P.-E. CAPRACE & N. MONOD (eds) 448 Synthetic differential topology, M.C. BUNGE, F. GAGO & A.M. SAN LUIS 449 Permutation groups and cartesian decompositions, C.E. PRAEGER & C. SCHNEIDER 450 Partial differential equations arising from physics and geometry, M. BEN AYED et al (eds) 451 Topological methods in group theory, N. BROADDUS, M. DAVIS, J.-F. LAFONT & I. ORTIZ (eds) 452 Partial differential equations in fluid mechanics, C.L. FEFFERMAN, J.C. ROBINSON & J.L. RODRIGO (eds) 453 Stochastic stability of differential equations in abstract spaces, K. LIU 454 Beyond hyperbolicity, M. HAGEN, R. WEBB & H. WILTON (eds) 455 Groups St Andrews 2017 in Birmingham, C.M. CAMPBELL et al (eds) 456 Surveys in combinatorics 2019, A. LO, R. MYCROFT, G. PERARNAU & A. TREGLOWN (eds) 457 Shimura varieties, T. HAINES & M. HARRIS (eds) 458 Integrable systems and algebraic geometry I, R. DONAGI & T. SHASKA (eds) 459 Integrable systems and algebraic geometry II, R. DONAGI & T. SHASKA (eds) 460 Wigner-type theorems for Hilbert Grassmannians, M. PANKOV 461 Analysis and Geometry on Graphs and Manifolds M. KELLER, D. LENZ & R.K. WOJCIECHOWSKI
- 8. London Mathematical Society Lecture Note Series: 461 Analysis and Geometry on Graphs and Manifolds MATTHIAS KELLER The University of Potsdam DANIEL LENZ Friedrich-Schiller University of Jena RADOSLAW K. WOJCIECHOWSKI Graduate Center and York College of the City University of New York
- 9. University Printing House, Cambridge CB2 8BS, United Kingdom One Liberty Plaza, 20th Floor, New York, NY 10006, USA 477 Williamstown Road, Port Melbourne, VIC 3207, Australia 314–321, 3rd Floor, Plot 3, Splendor Forum, Jasola District Centre, New Delhi – 110025, India 79 Anson Road, #06–04/06, Singapore 079906 Cambridge University Press is part of the University of Cambridge. It furthers the University’s mission by disseminating knowledge in the pursuit of education, learning, and research at the highest international levels of excellence. www.cambridge.org Information on this title: www.cambridge.org/9781108713184 DOI: 10.1017/9781108615259 c Cambridge University Press 2020 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 2020 Printed and bound in Great Britain by Clays Ltd, Elcograf S.p.A. A catalogue record for this publication is available from the British Library. Library of Congress Cataloging-in-Publication Data Names: Analysis and Geometry on Graphs and Manifolds (2017 : Potsdam, Germany), author. | Keller, Matthias, editor. | Lenz, Daniel, editor. | Wojciechowski, Radoslaw K., editor. Title: Analysis and geometry on graphs and manifolds / edited by Matthias Keller, Daniel Lenz, Radoslaw K. Wojciechowski. Description: Cambridge ; New York, NY : Cambridge University Press, 2020. | Series: London Mathematical Society lecture note series | Includes bibliographical references. Identifiers: LCCN 2020005418 | ISBN 9781108713184 (paperback) | ISBN 9781108615259 (epub) Subjects: LCSH: Geometric analysis – Congresses. | Graph theory – Congresses. | Manifolds (Mathematics) – Congresses. Classification: LCC QA360 .A63 2017 | DDC 515/.15–dc23 LC record available at https://lccn.loc.gov/2020005418 ISBN 978-1-108-71318-4 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 vii Preface xi 1 Infinite Planar Graphs with Non-negative Combinatorial Curvature 1 Bobo Hua and Yanhui Su 2 Curvature Calculations for Antitrees 21 David Cushing, Shiping Liu, Florentin Münch, and Norbert Peyer- imhoff 3 Gromov–Lawson Tunnels with Estimates 55 Józef Dodziuk 4 Norm Convergence of the Resolvent for Wild Perturbations 66 Colette Anné and Olaf Post 5 Manifolds with Ricci Curvature in the Kato Class: Heat Kernel Bounds and Applications 76 Christian Rose and Peter Stollmann 6 Multiple Boundary Representations of λ-Harmonic Functions on Trees 95 Massimo A. Picardello and Wolfgang Woess 7 Internal DLA on Sierpinski Gasket Graphs 126 Joe P. Chen, Wilfried Huss, Ecaterina Sava-Huss, and Alexander Teplyaev 8 Universal Lower Bounds for Laplacians on Weighted Graphs 156 D. Lenz and P. Stollmann 9 Critical Hardy Inequalities on Manifolds and Graphs 172 Matthias Keller, Yehuda Pinchover, and Felix Pogorzelski v
- 11. vi Contents 10 Neumann Domains on Graphs and Manifolds 203 Lior Alon, Ram Band, Michael Bersudsky, and Sebastian Egger 11 On the Existence and Uniqueness of Self-Adjoint Realizations of Discrete (Magnetic) Schrödinger Operators 250 Marcel Schmidt 12 Box Spaces: Geometry of Finite Quotients 328 Ana Khukhro and Alain Valette 13 Ramanujan Graphs and Digraphs 344 Ori Parzanchevski 14 From Partial Differential Equations to Groups 368 Andrzej Zuk 15 Spectral Properties of Limit-Periodic Operators 382 David Damanik and Jake Fillman 16 Uniform Existence of the IDS on Lattices and Groups 445 C. Schumacher, F. Schwarzenberger, and I. Veselić
- 12. Contributors Lior Alon Department of Mathematics, Technion–Israel Institute of Technology, Haifa 32000, Israel Colette Anné Laboratoire de Mathématiques Jean Leray, CNRS–Université de Nantes, Fac- ulté des Sciences, BP 92208, 44322 Nantes, France Ram Band Department of Mathematics, Technion–Israel Institute of Technology, Haifa 32000, Israel Michael Bersudsky Department of Mathematics, Technion–Israel Institute of Technology, Haifa 32000, Israel Joe P. Chen Department of Mathematics, Colgate University, 13 Oak Drive, Hamilton NY 13346, USA David Cushing Department of Mathematical Sciences, Durham University, Science Laborato- ries, South Road, Durham, England David Damanik Department of Mathematics, Rice University, Houston, TX 77005, USA Józef Dodziuk CUNY Graduate Center, 365 Fifth Avenue, New York, NY, 10016, USA vii
- 13. viii List of Contributors Sebastian Egger Department of Mathematics, Technion–Israel Institute of Technology, Haifa 32000, Israel Jake Fillman Department of Mathematics, Virginia Polytechnic Institute and State Univer- sity, 225 Stanger Street, Blacksburg, VA 24061, USA Bobo Hua School of Mathematical Sciences, LMNS, Fudan University, Shanghai 200433, China Shanghai Center for Mathematical Sciences, Fudan University, Shanghai 200433, China Wilfried Huss Graz University of Technology, Department of Mathematical Structure The- ory, Steyrergasse 30, 8010, Graz, Austria Matthias Keller Institut für Mathematik, Universität Potsdam, 14476 Potsdam, Germany Ana Khukhro Centre for Mathematical Sciences, University of Cambridge, Wilberforce Rd, Cambridge CB3 0WB, United Kingdom Daniel Lenz Mathematisches Institut, Friedrich Schiller Universität Jena, 07743 Jena, Ger- many Shiping Liu School of Mathematical Sciences, University of Science and Technology of China, 96 Jinzhai Road, Hefei 230026, Anhui Province, China Florentin Münch Universität Potsdam, Institut für Mathematik, Campus Golm, Haus 9, Karl- Liebknecht-Straße 24-25, 14476 Potsdam, Germany Ori Parzanchevski The Hebrew University of Jerusalem, Givat Ram. Jerusalem, 9190401, Israel Norbert Peyerimhoff Department of Mathematical Sciences, Durham University, Science Laborato- ries, South Road, Durham, England
- 14. List of Contributors ix Massimo A. Picardello Dipartimento di Matematica, Università di Roma ‘Tor Vergata’, I-00133 Rome, Italy Yehuda Pinchover Department of Mathematics, Technion-Israel Institute of Technology, 3200003 Haifa, Israel Felix Pogorzelski Institut für Mathematik, Universität Leipzig, 04109 Leipzig, Germany Olaf Post Fachbereich 4 – Mathematik, Universität Trier, 54286 Trier, Germany Christian Rose Max-Planck Institute for Mathematics in the Sciences, D-04103 Leipzig, Ger- many Ecaterina Sava-Huss Graz University of Technology, Austria, Institute of Discrete Mathematics, Steyrergasse 30/III, Office ST 03 228, 8010 Graz, Austria Marcel Schmidt Mathematisches Institut, Friedrich Schiller Universität Jena, 07743 Jena, Ger- many Christoph Schumacher Technische Universität Dortmund, Fakultät für Mathematik, Lehrstuhl LSIX, Vogelpothsweg 87, 44227 Dortmund, Germany Fabian Schwarzenberger Universität Leipzig, Institut für Medizinische Informatik, Statistik und Epi- demiologie, Härtelstraße 16-18, 04107 Leipzig, Germany Peter Stollmann Technische Universität Chemnitz, Faculty of Mathematics, D - 09107 Chem- nitz, Germany Yanhui Su College of Mathematics and Computer Science, Fuzhou University, Fuzhou 350116, China
- 15. x List of Contributors Alexander Teplyaev Department of Mathematics, University of Connecticut, Storrs, CT 06269- 1009, USA Alain Valette Institut de Mathématiques, Unimail, 11 Rue Emile Argand, CH-2000 Neuchâ- tel, Switzerland Ivan Veselić Technische Universität Dortmund, Fakultät für Mathematik, Lehrstuhl LSIX, Vogelpothsweg 87, 44227 Dortmund, Germany Wolfgang Woess Institut für Diskrete Mathematik, Technische Universität Graz, Steyrergasse 30, A-8010 Graz, Austria Andrzej Żuk Institut de Mathematiques, Universite Paris 7, 13 rue Albert Einstein, 75013 Paris, France
- 16. Preface This book brings together contributions for the conference ‘Analysis and geometry on graphs and manifolds,’ which took place at the University of Potsdam in Potsdam, Germany, from 31st July to 4th August 2017. The aim of the conference was to bring together leading experts in geometric analysis, in both the discrete and continuous settings. This included researchers working on such diverse models as manifolds, graphs, fractals, groups, and metric mea- sure spaces. The goal was for these researchers to share their expertise and to explore common ground. Each day there was also an extensive afternoon ses- sion which provided time for young researchers to present partial results and early work. The overall theme of the conference and of the contributions contained in this volume is the interplay of geometry, spectral theory, and stochastics. This interplay has a long and fruitful history and can be seen as a driving force behind many developments in modern mathematics. The present volume focuses on the global effects of local properties. This can be explored in both the discrete and continuous settings, and there has been a continual interest in contrasting what happens in these two cases. The main goal of this volume is to give an expository overview of these topics. This is achieved by presenting a mixture of survey chapters which examine the landscape of certain subjects and shorter chapters which focus on specific techniques and problems. We will now briefly comment on the content of the chapters contained in this volume. In doing so, we will also point out some of the connections between the chapters. Curvature is a natural local quantity arising when studying the geometry of a space. It has been thoroughly investigated in the case of Riemannian man- ifolds. Recent years have seen an explosion of research aimed at establishing curvature notions in the discrete setting. Here we present several contributions in this direction. Namely, the chapter by Bobo Hua and Yanhui Su gives an xi
- 17. xii Preface overview of results concerning combinatorial curvature in the case of planar tessellations. In particular, they analyse combinatorial, potential, and spectral theoretic consequences of global lower combinatorial curvature bounds. The chapter by David Cushing, Shiping Liu, Florentin Münch and Norbert Pey- erimhoff calculates both the Bakry–Émery and Ollivier–Ricci curvatures of a class of graphs called ‘antitrees’. Antitrees have recently come of interest as they provide surprising counterexamples to direct analogues of statements from the continuous setting. For general graphs, Bakry–Émery and Ollivier– Ricci are the most commonly appearing curvature notions. As for curvature in the continuous setting, the chapter by Józef Dodziuk describes a procedure for constructing tunnels connecting manifolds of arbitrary dimension and posi- tive scalar curvature while preserving the positivity of the curvature. A general treatment of convergence of operators while glueing and removing subsets in the case of suitable curvature control assumptions for manifolds can be found in the chapter by Colette Anné and Olaf Post. Global geometry is strongly reflected in the properties of a Markov pro- cess exploring the space. An analytic approach to this investigation is provided via the heat equation. The connections between curvature and properties of the heat kernel in the case of unbounded Ricci curvature on manifolds are explored in the chapter by Peter Stollmann and Christian Rose. Another geometric fea- ture explored by the Markov process is that of a boundary at infinity. By construction, this boundary captures global geometric properties of the space. Representations of generalizations of harmonic functions on such boundaries at infinity is the topic of the chapter by Massimo A. Picardello and Wolfgang Woess. Here, the areas of random walks, geometry, and potential theory merge. The trajectories of simple random walks on Sierpinski gasket graphs is the subject of the chapter by Joe P. Chen, Wilfried Huss, Ecaterina Sava-Huss, and Alexander Teplyaev. Spectral geometry deals with the interplay of spectral theory and both local and global geometric properties. In particular, the bottom of the spectrum and properties of ground states have been investigated in both the discrete and continuous settings. In this book, three chapters deal with these issues. Specif- ically, the chapter by Daniel Lenz and Peter Stollmann provides lower bounds for eigenvalues on graphs in terms of the inradius of subsets. Furthermore, the chapter by Matthias Keller, Yehuda Pinchover, and Felix Pogorzelski discusses the Hardy inequality, and the chapter by Lior Alon, Michael Bersudsky, Sebas- tian Egger, and Ram Band deals with Neumann domains of eigenfunctions. These two chapters deal with both graphs and manifolds. The even more fun- damental question of the existence and uniqueness of self-adjoint realizations in the discrete setting is detailed in the chapter by Marcel Schmidt.
- 18. Preface xiii The bottom of the spectrum is also a prominent topic in models investigated in geometric group theory. In particular, the chapter by Ana Khukhro and Alain Valette discusses expanders in the context of box spaces of Cayley graphs. Furthermore, the article by Ori Parzanchevski deals with Ramanujan digraphs, a counterpart to Ramanujan graphs, which are a special class of expanders. Finally, the chapter by Andrzej Żuk deals with the discretization of partial differential equations and connections to automata groups. The spectral theory of Schrödinger operators in the case of a simple geo- metrical setting strongly depends on global features of the potential. This is analogous to how global features for complicated geometries influence the spectral theory of the Laplace–Beltrami operator. The impact of the potential on spectral theory is particularly prominent in the case of potentials gen- erated by random processes or dynamical systems. The chapter by David Damanik and Jake Fillman gives a thorough overview of the spectral the- ory of Schrödinger operators on the one-dimensional lattice with potentials which are periodic or limit-periodic. The convergence of the integrated den- sity of states in the case of random Schrödinger operators over lattices and amenable groups is investigated in the chapter by Christoph Schumacher, Fabian Schwarzenberger, and Ivan Veselić. Acknowledgements. We gratefully acknowledge financial support for the conference provided by the DFG priority programme ‘Geometry at infinity’ SPP 2026, and the National Science Foundation (NSF) grant no. 1707722. Furthermore, the third editor acknowledges financial support provided by PSC- CUNY Awards, jointly funded by the Professional Staff Congress and the City University of New York, and the Collaboration Grant for Mathematicians, funded by the Simons Foundation. Finally, we are grateful for the hospitality and the financial support of the University of Potsdam.
- 20. 1 Infinite Planar Graphs with Non-negative Combinatorial Curvature Bobo Hua and Yanhui Su Abstract In this chapter, we survey some results on infinite planar graphs with non- negative combinatorial curvature, related to the total curvature, the number of vertices with positive curvature and the automorphism group. 1.1 Introduction The combinatorial curvature for planar graphs was introduced by Nevanlinna, Stone, Gromov, and Ishida [Nev70, Sto76, Gro87, Ish90] respectively, which resembles the Gaussian curvature for smooth surfaces. Many interesting geo- metric and combinatorial results have been obtained under such curvature conditions since then (see, e.g., [Ż97, Woe98, Hig01, BP01, HJL02, LPZ02, HS03, SY04, RBK05, BP06, DM07, CC08, Zha08, Che09, Kel10, KP11, Kel11, Oh17, Ghi17]). Let (V, E) be a (possibly infinite) locally finite, undirected simple graph with the set of vertices V and the set of edges E. It is called planar if it can be topologically embedded into the sphere S2 or the plane R2, where we distin- guish S2 with R2 while they are identified in the theory of finite planar graphs. We write G = (V, E, F) for the cellular complex structure of a planar graph induced by the embedding where F is the set of faces, i.e., connected compo- nents of the complement of the embedding image of the graph (V, E) in S2 or R2. We say that a planar graph G is a planar tessellation if the following hold (see, e.g., [Kel11]): (i) Every face is homeomorphic to a disc whose boundary consists of finitely many edges of the graph. 1
- 21. 2 Bobo Hua and Yanhui Su Figure 1.1 A planar graph G consists of a pentagon and infinitely many hexagons (ii) Every edge is contained in exactly two different faces. (iii) For any two faces whose closures have non-empty intersection, the intersection is either a vertex or an edge. In this chapter, we only consider planar tessellations (see Figure 1.1 for an example) and call them planar graphs for the sake of simplicity. For a planar tessellation, it is finite (infinite resp.) if and only if it embeds into S2 (R2 resp.). We say that a vertex x is incident to an edge e, denoted by x ≺ e, (similarly, an edge e is incident to a face σ, denoted by e ≺ σ; or a vertex x is incident to a face σ, denoted by x ≺ σ) if the former is a subset of the closure of the latter. Two vertices x and y are called ‘neighbours’ if there is an edge e such that x ≺ e and y ≺ e, in this case denoted by x ∼ y. We denote by deg(x) the degree of a vertex x, i.e., the number of neighbours of a vertex x, and by deg(σ) the degree of a face σ, i.e., the number of edges incident to a face σ (equivalently, the number of vertices incident to σ). We always assume that for any vertex x and face σ, deg(x) ≥ 3, deg(σ) ≥ 3. We denote by (deg(σ1), deg(σ2), · · · , deg(σN )) the pattern of a vertex x where N = deg(x), {σi }N i=1 are the faces which x is incident to, and deg(σ1) ≤ deg(σ2) ≤ · · · ≤ deg(σN ). Given a planar graph G = (V, E, F), one may canonically endow its ambi- ent space S2 or R2 with a piecewise flat metric as follows: assign each edge length one, replace each face by a regular Euclidean polygon of side length one
- 22. Infinite Planar Graphs with Non-negative Curvature 3 with same facial degree, and glue these polygons along the common edges. The ambient space equipped with the induced metric constructed above is called the regular polyhedral surface of G, denoted by S(G). In the following, we always call it the polyhedral surface for the sake of brevity. For a planar graph G, the combinatorial curvature at the vertex is defined as (x) = 1 − deg(x) 2 + σ∈F:x≺σ 1 deg(σ) , x ∈ V. (1.1.1) In this chapter, we mean by the curvature of a planar graph the combinato- rial curvature of it for simplicity. It turns out that the curvature of a planar graph is given by the generalized Gaussian curvature of the polyhedral surface S(G) up to some normalization. Note that for the polyhedral surface S(G) it is locally isometric to a flat domain in R2 near any interior point of an edge or a face, while it might be non-smooth near the vertices. As a metric surface, the generalized Gaussian curvature K of S(G) vanishes at smooth points and can be regarded as a measure concentrated on the isolated singularities, i.e., on vertices. One can show that the mass of the generalized Gaussian curva- ture at each vertex x is given by K(x) = 2π − x , where x denotes the total angle at x in the metric space S(G) (see [Ale05]). Moreover, by direct computation one has K(x) = 2π(x), where the curvature (x) is defined in (1.1.1). Hence, one can show that a planar graph G has non-negative curvature if and only if the polyhedral surface S(G) is a generalized convex surface in the sense of Alexandrov (see [BGP92, BBI01, HJL15]). Furthermore, the polyhe- dral surface S(G) can be isometrically embedded into R3 as a boundary of a compact or non-compact convex polyhedron by Alexandrov’s embedding the- orem ([Ale05]); see Figure 1.2 for an embedded image of S(G) of the planar graph G in Figure 1.1. Figure 1.2 The isometric embedding of S(G) of the planar graph G in Figure 1.1
- 23. 4 Bobo Hua and Yanhui Su In this chapter, we study planar graphs with non-negative curvature. We introduce two classes of planar graphs with positive or non-negative curvature as follows: ● PC0 := {G : (x) 0, ∀x ∈ V } is the class of planar graphs with positive curvature everywhere. ● PC≥0 := {G : (x) ≥ 0, ∀x ∈ V } is the class of planar graphs with non-negative curvature everywhere. We review some known results on the class PC0. Stone [Sto76] first proved a Myers-type theorem: a planar graph with the curvature bounded below uni- formly by a positive constant is a finite graph. Higuchi proposed a stronger conjecture that any G ∈ PC0 is a finite graph (see [Hig01, Conjecture 3.2]). This is certainly wrong for smooth surfaces since there are many non-compact convex surfaces in R3, which have positive curvature everywhere. However, for a planar graph it is hopefully true by the combinatorial restriction of reg- ular polygons as its faces. DeVos and Mohar [DM07] proved the conjecture by showing a generalized Gauss–Bonnet formula (see [SY04] for the case of cubic graphs). For any finite planar graph G ∈ PC≥0, in particular any G ∈ PC0, by Alexandrov’s embedding theorem its polyhedral surface S(G) can be iso- metrically embedded into R3 as a boundary of a convex polyhedron (see, e.g., [Ale05]). From this point of view, we obtain many examples for the class PC0, e.g., the 1-skeletons of 5 Planotic solids, 13 Archimedean solids, and 92 Johnson solids. Any of them has regular Euclidean polygons as its faces in its embedded image in R3. Note that these are all examples of planar graphs in PC0 whose faces of the embedded image in R3 are reg- ular polygons (see [Joh66, Zal67]). Besides these, the class PC0 contains many other examples, such as an example of 138 vertices constructed by Réti, Bitay, and Kosztolányi [RBK05], examples of 208 vertices by Nichol- son and Sneddon [NS11], Ghidelli [Ghi17], and Oldridge [Old17], which cannot be realized as the boundary of a convex polyhedron whose faces are regular polygons. In fact, although any face of G ∈ PC0 is isometric to a regular polygon in S(G), it may split into several pieces of non- coplanar faces in the embedded image of S(G) as the boundary of a convex polyhedron in R3. There are two special families of graphs in PC0 called prisms and antiprisms, both consisting of infinite many examples (see, e.g., [DM07]). Besides them, DeVos and Mohar [DM07] proved that there are only finitely many graphs in PC0 and proposed the following problem to find out the largest graph among them.
- 24. Infinite Planar Graphs with Non-negative Curvature 5 Problem 1.1.1 ([DM07]) What is the number CS2 := max G=(V,E,F) V , where the maximum is taken over graphs in PC0, which are not prisms or antiprisms, and V denotes the cardinality of V ? On the one hand, as some examples of 208 vertices in PC0 have been constructed in [NS11, Ghi17, Old17], we have the lower bound estimate that CS2 ≥ 208. On the other hand, DeVos and Mohar [DM07] initiated to use the discharging methods to obtain the upper bound estimate CS2 ≤ 3444. The discharging methods were adopted in the proof of the four-colour theorem in the literature (see [AH77, RSST97]). The upper bound was later improved to CS2 ≤ 380 by Oh [Oh17]. By a delicate argument, Ghidelli [Ghi17] showed that CS2 ≤ 208, which completely solves DeVos and Mohar’s problem that CS2 = 208. Next, we consider the class of planar graphs with non-negative curvature, i.e., PC≥0, which turns out to be much larger than PC0 and contains many interesting examples. The class of PC0 consists of essentially finite many examples, while the class PC≥0 contains infinitely many examples of differ- ent combinatorial types. A fullerene is a finite cubic planar graph whose faces are either pentagon or hexagon. There are plenty of examples of fullerenes which are important in the real-world applications, to cite a few examples [KHO+85, Thu98, BD97, BGM12, BE17a, BE17b]. Note that any fullerene is a planar graph with non-negative curvature. As shown by Thurston [Thu98], the number of combinatorial types of fullerenes with N hexagons grows as N9 as N → ∞. Besides these examples of finite graphs, there are plenty of examples of infinite graphs. Any planar tiling with regular polygons as tiles (see, e.g., [GS89, Gal09]) is in the class PC≥0. Note that there are infinitely many such planar tilings, for which only a few examples with symmetry can be classified. These motivate us to investigate the general structure of planar graphs in the class PC≥0. 1.2 Total Curvature of Planar Graphs with Non-negative Curvature For a smooth non-compact surface with absolutely integrable Gaussian curva- ture, its total curvature encodes the global geometric information of the space, e.g., the boundary at infinity (see [SST03]). For example, the total curvature of a convex surface in R3 describes the apex angle of the cone at infinity of the
- 25. 6 Bobo Hua and Yanhui Su surface, which is useful to study global geometric and analytic properties of the surface, such as harmonic functions and heat kernels, following [CM97b, Xu14]. For planar graphs with non-negative curvature G, we denote by (G) := x∈V (x) the total curvature of G whenever the summation converges absolutely. In case of finite graphs, the Gauss–Bonnet theorem reads as (see, e.g., [DM07]) (G) = 2. (1.2.1) For an infinite planar graph G ∈ PC≥0, the Cohn-Vossen type theorem, proven by [DM07, Theorem 1.3] or [Che09, Theorem 1.6], yields that (G) ≤ 1. (1.2.2) This means that for any infinite G ∈ PC≥0, the total curvature of G satisfies 0 ≤ x∈V (x) ≤ 1. In this section, we study all possible values of total curvature of infinite planar graphs with non-negative curvature, i.e., the following set {(G) : G infinite,G ∈ PC≥0}. (1.2.3) As is well known in Riemannian geometry that for any real number 0 ≤ a ≤ 2π, there is a convex surface whose total curvature is given by a. Hence, the above set for non-compact convex surfaces turns out to be an interval in the continuous setting. However, combinatorial structure of planar graphs with non-negative curvature gives us more information and restrictions for the set (1.2.3). For any G = (V, E, F) ∈ PC≥0, we denote by TG := {v ∈ V : (x) 0} (1.2.4) the set of vertices with positive curvature, and by DG := sup σ∈F deg(σ) (1.2.5) the maximal facial degree of G. Chen and Chen [CC08, Che09] proved an interesting result that the set of vertices with positive curvature in a planar graph with non-negative curvature is a finite set. Hence, the supremum in (1.2.5) is in fact the maximum.
- 26. Infinite Planar Graphs with Non-negative Curvature 7 Theorem 1.2.1 (Chen and Chen) For any G ∈ PC≥0, TG is a finite set. This result makes our combinatorial setting distinguished from the Riemannian setting. Note that there are many non-compact convex surfaces with positive curvature everywhere, e.g., the elliptic paraboloid, i.e., the revolution surface of the graph y = x2 with respect to the z axis in R3. Moreover, if the maximal facial degree DG of G ∈ PC≥0 is at least 43, then G has rather special structure, analogous to the prisms or antiprisms in the finite case (see [HJL15] or Theorem 1.3.2 in this chapter). In that case, one gets (G) = 1. Hence, for our purposes to understand the set (1.2.3), it suffices to consider planar graphs G with DG ≤ 42. Note that there are finitely many vertex patterns, consisting of faces of degree at most 42, with positive curvature (see Table 1.1 in the Appendix). Then one is ready to see that the set (1.2.3) is a discrete subset in [0, 1] (see, e.g., [HS17b, Proposition 2.3]). T. Réti [HL16, Conjecture 2.1] was motivated to determine the following value τ1 := inf (G) : G ∈ PC≥0, (G) 0 , which is called the first gap of total curvature for infinite planar graphs in the class of PC≥0. He suggested that τ1 = 1 6 and the minimum is attained by the graph consisting of a pentagon and infinitely many hexagons, which is a kind of infinite fullerene (see Figure 1.1). In [HS17a], we give an answer to Réti’s problem. Theorem 1.2.2 (Theorem 1.3 in [HS17a]) τ1 = 1 12 . A planar graph G ∈ PC≥0 satisfies (G) = 1 12 if and only if the polyhedral surface S(G) is isometric to either (a) a cone with the apex angle θ = 2 arcsin 11 12 , or (b) a ‘frustum’ with a hendecagon base (see Figure 1.3). The proof strategy is straightforward and involves tedious case studies. For a vertex with positive curvature, if the curvature of the vertex is less than 1 12 , then we try to find some nearby vertices with positive curvature such that the sum of these curvatures is at least 1 12 and prove the results case by case. Note that there are examples of graphs in PC≥0 whose total curvature attains the first gap 1 12 (see Figure 1.4 and [HS17a] for more examples). Although graph structures of infinite planar graphs attaining the first gap of total curvature could be as complicated as planar tilings (see [HS17a]) we are able to classify
- 27. 8 Bobo Hua and Yanhui Su Figure 1.3 A ‘frustum’ with a hendecagon base A A B B Figure 1.4 This is an example of total curvature 1 12 , where the half lines with same labels, A or B, are identified metric structures of polyhedral surfaces for such planar graphs in the above theorem. Inspired by Réti’s question, it will be interesting to know other values in the set (1.2.3). Using Chen and Chen’s result, Theorem 1.2.1, and the Gauss–Bonnet theorem for compact subsets with boundary, we are able to determine all possible total curvatures in the class PC≥0. Theorem 1.2.3 (Theorem 1.1 in [HS17b]) The set of all values of total curva- ture of infinite planar graphs with non-negative curvature (1.2.3) is given by i 12 : 0 ≤ i ≤ 12, i ∈ Z . As a corollary, we also obtain that τ1 = 1 12 , which provides an alternative proof to Réti’s problem. Moreover, as the part of the theorem, one may con- struct planar graphs with non-negative curvature whose total curvatures attain
- 28. Infinite Planar Graphs with Non-negative Curvature 9 all values listed above (see [HS17b]). We sketch the proof of the theorem as follows: by Theorem 1.2.1, we know that TG is a finite set. We choose a suf- ficiently large compact subset K ⊂ S(G), homeomorphic to a closed disc, such that it contains TG and consists of faces in F. Note that the vertices on the boundary of K have vanishing curvature, so that their patterns appear in the list of 17 possible patterns in Table 1.2 in the Appendix. By some combinatorial restrictions, one can further exclude several patterns from the list and conclude that any vertex on the boundary is incident to a triangle, a square, a hexagon, an octagon, or a dodecagon. Then using the Gauss–Bonnet formula on K, we may prove the theorem. Similar proof strategies apply to the problems on the total curvature of a planar graph with boundary, i.e., a graph embedded into the disc or a half plane (see [HS17b]). Although we crucially use the finiteness structure of TG in the proof of Theorem 1.2.3, we don’t know much about the structure of the subset TG which still lies in a black box. By a byproduct of the proof of Theorem 1.2.2, we can show that for G ∈ PC≥0, the induced subgraph on TG has at most 14 connected components. It was conjecturally at most 12 (see [HS17a, Conjecture 5.2]). 1.3 The Vertices of Positive Curvature in Planar Graphs with Non-negative Curvature In this section, we survey some results on the set of vertices with positive curvature in planar graphs with non-negative curvature. For any finite (infinite resp.) G ∈ PC≥0, Alexandrov’s embedding theorem [Ale05] yields that an isometric embedding of the polyhedral surface S(G) into R3 as a boundary of a compact (non-compact resp.) convex polyhedron. The set TG serves as the set of the vertices/corners of the convex polyhedron, so that much geometric information of the polyhedron is contained in TG. We are interested in the structure of the set TG. By the solution to DeVos and Mohar’s problem [Ghi17], besides the prisms and antiprisms the largest number of vertices in a finite graph in PC0 is 208. We would like to study analogous problems for planar graphs in PC≥0. We define some analogues to prisms and antiprisms in the class PC≥0. Definition 1.3.1 We call a planar graph G = (V, E, F) ∈ PC≥0 a prism-like graph if either (1) G is an infinite graph and DG ≥ 43, where DG is defined in (1.2.5), or (2) G is a finite graph and there are at least two faces with facial degree at least 43.
- 29. 10 Bobo Hua and Yanhui Su σ Figure 1.5 A half flat-cylinder in R3 By dividing hexagons into triangles, one may assume that there is no hexagon in G. Note that ‘prism-like’ graphs have rather special structures which can be completely determined by the following theorems. For any face σ, we denote by ∂σ := {x ∈ V : x ≺ σ} the vertex boundary of σ. Theorem 1.3.2 ([HJL15]) Let G = (V, E, F) be an infinite planar graph with non-negative curvature and DG ≥ 43. Then there is only one face σ of degree at least 43. Suppose that there is no hexagonal face. Then the set of faces F consists of σ, triangles or squares. Moreover, F = σ ∪ (∪∞ i=1Li ), where Li , i ≥ 1, are sets of faces of the same type (triangle or square) which composite a band, i.e., an annulus, and is defined inductively: Li is the next layer attaching to the previous layer Li−1 with L0 = {σ}. S(G) is isomet- ric to the boundary of a half flat-cylinder in R3 (see Figure 1.5). Moreover, (G) = 1. Theorem 1.3.3 ([HS18]) Let G = (V, E, F) be a finite prism-like graph. Then there are exactly two disjoint faces σ1 and σ2 of same facial degree at least 43. Suppose that there is no hexagonal face. Then the set of faces F consists of σ1 and σ2, triangles, or squares. Moreover, F = σ1 ∪ (∪M i=1Li ) ∪ σ2, where M ≥ 1, and Li , 1 ≤ i ≤ M, are defined similarly as in Theorem 1.3.2. S(G) is isometric to the boundary of a cylinder barrel in R3 (see Figure 1.6). The following problem was proposed in [HL16] as an analogue to DeVos and Mohar’s problem.
- 30. Infinite Planar Graphs with Non-negative Curvature 11 σ1 σ2 Figure 1.6 A cylinder barrel in R3 Problem 1.3.4 What are the numbers KS2 := max finite G TG, KR2 := max infinite G TG, where the maxima are taken over finite and infinite graphs in PC≥0 which are not prism-like graphs respectively? In [HS18], we prove the following theorem. Theorem 1.3.5 ([HS18]) KR2 = 132. Moreover, a graph in this class attains the maximum if and only if its polyhedral surface contains 12 disjoint hendecagons. On the one hand, we give the upper bound KR2 ≤ 132 by the discharging methods initiated by [DM07, Oh17, Ghi17] for the case PC0. The curvature at vertices of a planar graph G can be regarded as the charge concentrated on vertices. The discharging method is to re-distribute the charge on vertices, via transferring the charge on vertices with large curvature to vertices with curvature less than 1 132 , such that the terminal charge on involved vertices in TG after the distribution process is uniformly bounded below by 1 132 . Then the estimate of vertices in TG follows from the Cohn–Vossen-type theorem (1.2.2). On the other hand, we can construct an example possessing 132 vertices of positive curvature (see [HS18]). This completely answers the second part of Problem 1.3.4. For the first part of Problem 1.3.4, one can construct a family of infinitely many examples of finite graphs in PC≥0 with arbitrarily large number of ver- tices of positive curvature, see Example 3.3 in [HS18], which are not prism-like graphs. Hence KS2 = ∞.
- 31. 12 Bobo Hua and Yanhui Su However, since the argument in the proof of Theorem 1.3.5 is local, we can prove the similar results for the modified quantity, KS2 , for finite graphs in PC≥0. Theorem 1.3.6 ([HS18]) Let KS2 := max finite G TG, where the maximum is taken over finite graphs in PC≥0 whose maximal facial degree are less than 132. Then KS2 = 264. Moreover, a graph in this class attains the maximum if and only if its polyhedral surface contains 24 disjoint hendecagons, i.e., 11-gons. 1.4 Automorphism Groups of Planar Graphs with Non-negative Curvature In this section, we study automorphism groups of planar graphs with non- negative curvature. The automorphism groups of planar graphs have been extensively studied in the literature (see, e.g., [Man71, Bab75, CBGS08, SS98]). First, we introduce several definitions of isomorphisms on planar graphs. Definition 1.4.1 Let G1 = (V1, E1, F1) and G2 = (V2, E2, F2) be two planar graphs. (1) G1 and G2 are said to be graph-isomorphic if there is a graph isomorphism between (V1, E1) and (V2, E2), i.e., R : V1 → V2 such that for any v, w ∈ V, v ∼ w if and only if R(v) ∼ R(w). (2) G1 and G2 are said to be cell-isomorphic if there is a cellular isomorphism H = (HV , HE , HF ) between (V1, E1, F1) and (V2, E2, F2) in the sense of cell complexes, i.e., three bijections HV : V1 → V2, HE : E1 → E2 and HF : F1 → F2 preserving the incidence relations, that is, for any v ∈ V, e ∈ E, σ ∈ F, v ≺ e if and only if HV (v) ≺ HE (e) and e ≺ σ if and only if HE (e) ≺ HF (σ). (3) G1 and G2 are said to be metric-isomorphic if there is an isometric map in the sense of metric spaces L : S(G1) → S(G2), such that the restriction map L is cell-isomorphic between (V1, E1, F1) and (V2, E2, F2). For a planar graph G, a graph (cellular, metric resp.) isomorphism from G to itself is called a graph (cellular, metric resp.) automorphism of G. We denote
- 32. Infinite Planar Graphs with Non-negative Curvature 13 by Aut(G), ( Aut(G), L(G) resp.) the group of graph (cellular, metric resp.) automorphisms of a planar graph G. By the standard identification, L(G) ≤ Aut(G) ≤ Aut(G), where ≤ indicates that the former can be embedded as a sub-group of the latter. By our definition of polyhedral surfaces, it is easy to see that L(G) ∼ = Aut(G). Moreover, by the results in [Whi33, Moh88] for a 3-connected planar graph G, any graph automorphism R of G can be uniquely realized as a cellu- lar automorphism H such that HV = R, which is called the associated cellular automorphism of R. This implies that for a 3-connected planar graph G, L(G) ∼ = Aut(G) ∼ = Aut(G). In [HS18], we prove that the cellular automorphism group of a planar graph in PC≥0 with positive total curvature is a finite group. Theorem 1.4.2 Let G = (V, E, F) be a planar graph with non-negative cur- vature and positive total curvature. Then the automorphism group of G is finite. Set a := TG and b := maxv∈TG deg(v). We have the following: (1) If DG ≤ 42, Aut(G) divides a!b!. (2) If DG 42, then Aut(G) divides 2DG, G is infinite, 4DG, G is finite. By combining the estimates of the size of TG in Theorems 1.3.5 and 1.3.6 with the above result, we obtain the estimates for the orders of cellular automorphism groups: (1) If G is infinite, then Aut(G) ≤ 132! × 5!, for DG ≤ 42, 2DG, for DG 42. (2) If G is finite, then Aut(G) ≤ 264! × 5!, for DG ≤ 42, 4DG, for DG 42.
- 33. 14 Bobo Hua and Yanhui Su 1.5 Analysis on Planar Graphs with Non-negative Curvature In this section, we study analysis problems on planar graphs with non-negative curvature. For any G ∈ PC≥0, it is easy to see that for any x ∈ V, 3 ≤ deg(x) ≤ 6. For any x ∈ V and r 0, we denote by Br (x) := {y ∈ V : d(y, x) ≤ r} the ball of radius r centred at x. We introduce the definitions of the volume- doubling property and the Poincaré inequality on graphs. Definition 1.5.1 (DV ) A graph G = (V, E) is called satisfying the volume-doubling property DV (C) for constant C 0 if for all x ∈ V and all r 0: B2r (x) ≤ CBr (x). (P) A graph G is called satisfying the Poincaré inequality P(C) for a constant C 0 if for any functions f on V , x0 ∈ V and r 0, x∈Br (x0) | f (x) − fB|2 ≤ Cr2 x,y∈B2r (x0),y∼x ( f (y) − f (x))2 , where fB := 1 Br (x0) x∈Br (x0) f (x). The first author, Jost and Liu [HJL15], proved these properties for graphs in the class PC≥0. Theorem 1.5.2 ([HJL15]) For any G ∈ PC≥0, the volume-doubling prop- erty DV (C1) and the Poincaré inequality P(C2) hold for some C1, C2 0. Moreover, there is a constant C such that Br (x) ≤ Cr2 , ∀x ∈ V,r 0. The general principle hidden in this result dates back to [CSC95], in which they showed that the volume-doubling property and the Poincaré inequality are quasi-isometric invariants (see [SC04, Woe00] for definitions). Since the planar graph G with bounded facial degree is properly embedded into the reg- ular polygonal surface S(G), they are in fact quasi-isometric to each other. For convex surfaces, even more general Alexandrov spaces with non-negative curvature, the volume-doubling property follows from the Bishop–Gromov
- 34. Infinite Planar Graphs with Non-negative Curvature 15 volume comparison [BBI01] and the Poincaré inequality is obtained by [KMS01, Hua09]. For a graph G = (V, E), the Laplace operator , called Laplacian for short, is defined as follows for any function f on V : f (x) = 1 deg(x) y∈V :y∼x ( f (y) − f (x)), ∀x ∈ V. A function f is called harmonic (subharmonic, superharmonic resp.) on ⊂ V if f ≡ 0 ( f ≥ 0, f ≤ 0 resp.) on . As a corollary of Theorem 1.5.2, the simple random walk on a planar graph with non-negative curvature is recurrent, i.e., any positive superharmonic function is constant, which follows from the assertion that the volume growth is at most quadratic. Moreover, by Moser iteration, Delmotte [Del97] proved the elliptic Harnack inequality on graphs under the assumptions of the volume-doubling property and the Poincaré inequality. In particular, Theorem 1.5.2 implies that for G ∈ PC≥0 and any positive harmonic function f on B2r (p) ⊂ V, r 0, we have max Br (p) f ≤ C min Br (p) f, (1.5.1) where C is a constant independent of r. For any k 0, we denote by Hk (G) := { f : V → R| f ≡ 0, | f (x)| ≤ C(1 + d(x, p))k , for some p ∈ V, C 0} the space of harmonic functions on V of polynomial growth whose growth order are less than or equal to k. The Harnack inequality (1.5.1) yields that dim Hk (G) = 1, for some k 1. In fact, the combination of volume-doubling property and the Poincaré inequality turns out to be equivalent to the parabolic Harnack inequalities [Del99]. Furthermore, one can prove the finite-dimensional property of the space of harmonic functions of polynomial growth with growth rate bounded above, following Colding, Minicozzi, and Li [CM97a, CM98a, CM98b, Li97, STW00]. Theorem 1.5.3 ([HJL15, HJ15]) For any G ∈ PC≥0, dim Hk (G) ≤ Ck, ∀k ≥ 1, where C is a universal constant.
- 35. 16 Bobo Hua and Yanhui Su Table 1.1 The patterns of a vertex with positive curvature Patterns (x) (3, 3, k) 3 ≤ k 1/6 + 1/k (3, 4, k) 4 ≤ k 1/12 + 1/k (3, 5, k) 5 ≤ k 1/30 + 1/k (3, 6, k) 6 ≤ k 1/k (3, 7, k) 7 ≤ k ≤ 41 1/k − 1/42 (3, 8, k) 8 ≤ k ≤ 23 1/k − 1/24 (3, 9, k) 9 ≤ k ≤ 17 1/k − 1/18 (3, 10, k) 10 ≤ k ≤ 14 1/k − 1/15 (3, 11, k) 11 ≤ k ≤ 13 1/k − 5/66 (4, 4, k) 4 ≤ k 1/k (4, 5, k) 5 ≤ k ≤ 19 1/k − 1/20 (4, 6, k) 6 ≤ k ≤ 11 1/k − 1/12 (4, 7, k) 7 ≤ k ≤ 9 1/k − 3/28 (5, 5, k) 5 ≤ k ≤ 9 1/k − 1/10 (5, 6, k) 6 ≤ k ≤ 7 1/k − 2/15 (3, 3, 3, k) 3 ≤ k 1/k (3, 3, 4, k) 4 ≤ k ≤ 11 1/k − 1/12 (3, 3, 5, k) 5 ≤ k ≤ 7 1/k − 2/15 (3, 4, 4, k) 4 ≤ k ≤ 5 1/k − 1/6 (3, 3, 3, 3, k) 3 ≤ k ≤ 5 1/k − 1/6 Table 1.2 The patterns of a vertex with vanishing curvature (3, 7, 42), (3, 8, 24), (3, 9, 18), (3, 10, 15), (3, 12, 12), (4, 5, 20), (4, 6, 12), (4, 8, 8), (5, 5, 10), (6, 6, 6), (3, 3, 4, 12), (3, 3, 6, 6), (3, 4, 4, 6), (4, 4, 4, 4), (3, 3, 3, 3, 6), (3, 3, 3, 4, 4), (3, 3, 3, 3, 3, 3). 1.6 Appendix Table 1.1 lists all possible patterns of a vertex with positive curvature (see [DM07, CC08]); Table 1.2 lists all possible patterns of a vertex with vanishing curvature (see [GS89, CC08]). Acknowledgements B. H. is supported by NSFC (China) under grant nos. 11831004 and 11826031. Y. S. is supported by NSFC (China) under grant no. 11771083 and NSF of Fujian Province through grants 2017J01556 and 2016J01013.
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- 40. 2 Curvature Calculations for Antitrees David Cushing, Shiping Liu, Florentin Münch, and Norbert Peyerimhoff Abstract In this chapter we prove that antitrees with suitable growth properties are examples of infinite graphs exhibiting strictly positive curvature in various con- texts: in the normalized and non-normalized Bakry–Émery setting as well as in the Ollivier–Ricci curvature case. We also show that these graphs do not have global positive lower curvature bounds, which one would expect in view of discrete analogues of the Bonnet–Myers theorem. The proofs in the different settings require different techniques. 2.1 Introduction and Results The main protagonists in this chapter are antitrees. While these exam- ples had been studied already in 1988, they were given the name antitree in talks by Radoslaw Wojciechowsi around 2010. A proper definition of antitrees, in their most general form, appeared first in [19]. Like in the case of a tree, the vertices of an antitree are partitioned in generations Vi , with the first generation V1 called its root set. While trees are connected graphs with as few connections as possible between subsequent genera- tions, antitrees have the maximal number of connections. More precisely, antitrees are simple (i.e., no loops and no multiple edges), connected graphs such that (i) any root vertex x ∈ V1 is connected to all vertices in V2, and no vertices in Vk, k ≥ 3, (ii) any vertex x ∈ Vk, k ≥ 2, is connected to all vertices in Vk−1 and Vk+1, and no vertices in Vl, |k − l| ≥ 2. 21
- 41. 22 David Cushing, et al. Note that this definition allows for the possibility of edges between ver- tices of the same generation. We will refer to such edges as spherical edges. Edges between vertices of different generations are called radial edges. Any radial or spherical edge incident to a vertex in V1 is called radial or spherical root-edge, respectively. All other edges are called inner edges. Antitrees are particularly interesting examples with regard to stochastic completeness. Section 2.2, provided by Radoslaw Wojciechowki, gives a more in-depth look at the history of antitrees. In this chapter, we investigate cur- vature properties of antitrees. Relations between curvature asymptotics and stochastic completeness were investigated recently in [17] in the Bakry–Émery setting and in [22] in the Ollivier–Ricci curvature setting. For our curvature considerations, we consider only antitrees where the induced subgraph of any one generation Vk is complete, i.e., any two ver- tices in the same generation are neighbours. For any given finite or infinite sequence (ak)1≤k≤N , N ∈ N ∪ {∞}, the corresponding unique such antitree with |Vk| = ak for all 1 ≤ k ≤ N is denoted by AT ((ak)). Note that in the case of a finite antitree, that is N ∞, (ii) has to be understood in the case k = N that any vertex x ∈ VN is connected to all vertices in VN−1. Later in this intro- duction, we will only present results for infinite antitrees, but, since curvature is a local notion, we need only investigate curvatures of suitable finite antitrees for the proofs. Figure 2.1 The antitree AT ((2, 3, 5))
- 42. Curvature Calculations for Antitrees 23 Two particular curvature notions on graphs have been studied actively in recent years: ● Bakry–Émery curvature taking values on the vertices and based on Bochner’s formula with respect to a suitable graph Laplacian, ● Ollivier–Ricci curvature taking values on the edges and based on optimal transport of lazy random walks. Basic graph theoretical notions are introduced in Section 2.3.1 and precise definitions of these curvature concepts are given in Sections 2.3.2 and 2.3.3, respectively. For both curvature notions there are graph theoretical analogues of the fundamental Bonnet–Myers theorem for Riemannian manifolds with strictly positive Ricci curvature bounded away from zero. Let us first consider Bakry–Émery curvature. Generally, on a combinatorial graph G = (V, E) with vertex set V and edge set E, the graph Laplacian on functions f : V → R is of the form f (x) = 1 μ(x) y∼x ( f (y) − f (x)), (2.1.1) with a vertex measure μ : V → (0, ∞). In this chapter, we consider two specific choices of vertex measures: ● μ ≡ 1, which we refer to as the non-normalized case, ● μ(x) = dx (the vertex degree of x ∈ V ), which we refer to as the normalized case. The corresponding discrete Bonnet–Myers theorems in both settings are as follows. Theorem 2.1.1 (see [21]) Let G = (V, E) be a connected graph satisfying C D(K, ∞) for some K 0 in the non-normalized case and dx ≤ D for all x ∈ V and some finite D. Then G is a finite graph and, furthermore, diam(G) ≤ 2D K . Theorem 2.1.2 (see [21]) Let G = (V, E) be a connected graph satisfying C D(K, ∞) for some K 0 in the normalized case (possibly of unbounded vertex degree). Then G is a finite graph and, furthermore, diam(G) ≤ 2 K .
- 43. 24 David Cushing, et al. Ollivier–Ricci curvature depends upon an idleness parameter p ∈ [0, 1] describing the laziness of the associated random walk. Here, the discrete Bonnet–Myers theorem takes the following form. Theorem 2.1.3 (see [23]) Let G = (V, E) be a connected graph satisfying κp(x, y) ≥ K 0 for all x ∼ y and a fixed idleness p ∈ [0, 1]. Then G is a finite graph and, furthermore, diam(G) ≤ 2(1 − p) K . (2.1.2) These results give rise to the following natural questions: ● Do there exist examples of infinite connected graphs with strictly positive curvature? (That is, relaxing the condition of a uniform strictly positive lower curvature bound.) ● In the non-normalized case, does there exist an infinite connected graphs satisfying C D(K, ∞) for K 0 of unbounded vertex degree? This chapter provides a positive answer to the first question. In fact, we show that antitrees AT ((ak)) with suitable growth properties of the infinite sequence (ak) have strictly positive curvature for all curvature notions mentioned above. More precisely, we have the following in the Bakry–Émery curvature case. Theorem 2.1.4 In both the normalized and non-normalized setting, the infinite antitree AT ((k)) satisfies C D(Kx , ∞, x) for all vertices x with a family of constants Kx 0 depending only on the generation of x. Furthermore, lim inf k→∞, x∈Vk Kx = 0. Remark 2.1.5 In fact, the method of proof relies on some Maple calculations which can be extended to also provide the following results (without going into the details): (i) Linear growth: The same curvature results hold true for the infinite antitrees AT ((1 + (k − 1)t)) with arbitrary t ∈ N. (ii) Exponential growth: The same curvature results hold true for the infinite antitree AT ((2k−1)) in the normalized case and fail to satisfy C D(0, ∞) in the non-normalized case. Due to Bakry–Émery curvature being a local property, in order to calcu- late the curvatures KG,x (∞) of vertices x in the first two generations of G = AT ((2k−1)) as defined later in (2.3.1), it is sufficient to consider the
- 44. Curvature Calculations for Antitrees 25 0.078 0.417 0.078 0.58 0.269 0.269 0.269 0.269 0.269 0.269 0.269 0.269 0.58 0.58 0.58 Figure 2.2 Normalized curvature KG,x (∞) 0.5 –1.157 –1.157 0.595 –1 –1 –1 –1 –1 –1 –1 –1 0.595 0.595 0.595 Figure 2.3 Non-normalized curvature KG,x (∞) graph presented in Figures 2.2 and 2.3 (spherical edges of 2-spheres around a vertex do not contribute to the curvature, see [7]). These figures are in agreement with the statements in Remark 2.1.5(ii).
- 45. 26 David Cushing, et al. Now we consider Ollivier–Ricci curvature. Here our main result is the following. Theorem 2.1.6 Let G = AT ((ak)) be an infinite antitree with 1 = a1 and ak+1 ≥ ak for all k ∈ N and x, y be neighbouring vertices in G. ● Radial root edges: If x ∈ V1 and y ∈ V2: κp(x, y) = ⎧ ⎨ ⎩ a2−1 a2+a3 + a2+2a3+1 a2+a−3 p, if p ∈ 0, 1 a2+a3+1 , a2+1 a2+a3 (1 − p), if p ∈ 1 a2+a3+1 , 1 . ● Radial edges: If x ∈ Vk and y ∈ Vk+1, k ≥ 2, p ∈ [0, 1]: κp(x, y) = 2ak + ak+1 − 1 ak + ak+1 + ak+2 − 1 − 2ak−1 + ak − 1 ak−1 + ak + ak+1 − 1 (1 − p). ● Spherical edges: If x, y ∈ Vk, x = y, k ≥ 2: κp(x, y) = ⎧ ⎨ ⎩ ak−1+ak+ak+1−2 ak−1+ak+ak+1−1 + ak−1+ak+ak+1 ak−1+ak+ak+1−1 p, if p ∈ 0, 1 ak−1+ak+ak+1 , ak−1+ak+ak+1 ak−1+ak+ak+1−1 (1 − p), if p ∈ 1 ak−1+ak+ak+1 , 1 . Let us consider special cases. Corollary 2.1.7 (Linear growth) Let G = AT ((1+(k−1)t)), t ∈ N arbitrary. Then κ0(x, y) = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ t 3t+2 for x ∈ V1, y ∈ V2, 6t2 (3kt+2)(3kt+2−3t) for x ∈ Vk, y ∈ Vk+1, 1 − 1 3kt+2−3t for x, y ∈ Vk, x = y, k ≥ 2. In particular, κ0 of radial edges decays asymptotically like 2 3k2 as k → ∞. Corollary 2.1.8 (Exponential growth) We have for G = AT ((rk−1), r ∈ N: κ0(x, y) = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ r−1 r(r+1) for x ∈ V1, y ∈ V2, (r−1)2(r+1)rk−2 (rk+rk−1+rk−2−1)(rk+1+rk+rk−1−1) for x ∈ Vk, y ∈ Vk+1, 1 − 1 rk+rk−1+rk−2−1 for x, y ∈ Vk, x = y, k ≥ 2. In particular, κ0 of radial edges decays asymptotically like 1 rk as k → ∞. Remark 2.1.9 Note that for any finite sequence (ak)1≤k≤N , N ≥ 2, with 1 = a1 and ak+1 ≥ ak for all 1 ≤ k ≤ N, we can find a large enough aN+1 ≥ aN such that κ0(x, y) 0 for x ∈ VN−1 and y ∈ VN .
- 46. Curvature Calculations for Antitrees 27 The chapter is organised as follows: We start with some historical comments on antitrees in Section 2.2 which was provided by Radosław Wojciechowski. Section 2.3 introduces the readers into Bakry–Émery curvature and Ollivier– Ricci curvature. The following two Sections 2.4 and 2.5 present the concrete curvature investigations in both settings. Appendices A, B, and C provide the Maple code used for the results in Section 2.4. 2.2 A (Partial) History of Antitrees To our knowledge, the first known appearance of an antitree is the case of |Sr | = r + 1 in the article of Dodziuk and Karp [8]. They study the nor- malized Laplacian and give conditions for transience of the simple random walk in terms of r r where r is the distance to a vertex. It appears in [8, Example 2.5] as a case of a transient graph with bottom of the spectrum 0 whose Green’s function decays like 1/r. The same antitree appears in the article of Weber [24]. Weber extends the result of Dodziuk and Mathai [9] concerning the stochastic completeness of the semigroup associated with the non-normalized Laplacian . Indeed, Dodziuk/Mathai prove stochastic com- pleteness in the case of bounded vertex degree. Weber improves this result to give stochastic completeness in the case of r ≥ K for some constant K. The antitree mentioned above is then given as an example of a graph whose vertex degree is unbounded but which satisfies r ≥ K (see [24, Figure 1, p. 156]). The general case of antitrees with arbitrary spherical growth |Sr | = f (r) where f is any natural number-valued function is considered in [25, Example 4.11]. There it is shown that antitrees are stochastically complete if and only if r r k=0 f (k) f (r) f (r + 1) = ∞. This is used to give a counter-example to a direct analogue to Grigor’yan’s result for stochastic completeness of manifolds (see [13]). Indeed, Grigor’yan’s result says that any stochastically incomplete manifold must have super-exponential volume growth while the result above gives stochas- tically incomplete graphs which have only polynomial volume growth when the combinatorial graph metric is used. These examples give the smallest such examples in the combinatorial graph metric by a result of Huang, Grigor’yan and Masamune [12, Theorem 1.4], where the example (and name) of antitrees also appears. This might be the first time in print that the name is used and they refer to them as the ‘antitree of Wojciechowski’. A proper definition with the name of antitree first appears in [19, Definition 6.3]. Here the result on
- 47. 28 David Cushing, et al. stochastic completeness is generalized to all weakly spherically symmetric graphs of which the antitrees are but an example. Furthermore, it is shown that the non-normalized Laplacian on any such stochastically incomplete antitree has positive bottom of the spectrum (see [19, Corollary 6.6]). This gives a counter-example to a direct analogue to a theorem of Brooks [5] which states that the bottom of the spectrum of the Laplacian on any manifold with sub-exponential volume growth is zero. This sparked an interest in applying intrinsic metrics as defined by Frank, Lenz, and Wingert in [10] to study the question involving volume growth on graphs of unbounded vertex degree. In particular, the analogue to Grigor’yan’s theorem was first proven in [11] (see also [18] for an analytic proof) while the analogue to Brooks’ theorem was shown in [16]. Since then, antitrees appear in a variety of places. Their spec- tral theory is thoroughly analysed by Breuer and Keller in [4]. Here it should be noted that the spectrum consists mainly of eigenvalues with compactly sup- ported eigenfunctions and a further spectral component which can be singular continuous in certain cases. Antitrees are also used as a counterexample to a conjecture presented by Golenia and Schumacher in [14] concerning the defi- ciency indices of the adjacency matrix (see [15]). They are also used to show the utility of the new bottom of the spectrum estimate for a Cheeger constant involving intrinsic metrics in [1]. 2.3 Definitions and Notations 2.3.1 Basic Graph Theoretical Notations Let G = (V, E) be a locally finite connected simple combinatorial graph (that is, no loops and no multiple edges) with vertex set V and edge set E. For any x, y ∈ V we write x ∼ y if {x, y} ∈ E. The degree of a vertex x ∈ V is denoted by dx . Let d : V × V → N ∪ {0} be the combinatorial distance function, i.e., d(x, y) is the length of the shortest path from x to y. For x ∈ V , the combinatorial spheres and balls of radius r ≥ 0 around x are denoted by Sr (x) = {y ∈ V | d(x, y) = r}, Br (x) = {y ∈ V | d(x, y) ≤ r}, respectively. The diameter of G is defined as diam(G) = sup{d(x, y) | x, y ∈ V } ∈ N ∪ {0, ∞}.
- 48. Curvature Calculations for Antitrees 29 2.3.2 Bakry–Émery Curvature As mentioned before, this curvature notion is rooted on Bochner’s formula using a Laplacian operator leading to the curvature-dimension inequality (CD- inequality for short). This approach was pursued by Bakry–Émery [2] via an elegant -calculus and leads to a substitute of the lower Ricci curvature bound of the underlying space for much more general settings. (Some further information on the Bochner approach can be found, e.g., in [7, Remark 1.3].) Recall definition (2.1.1) of the normalized (μ(x) = dx ) and non-normalized Laplacian (μ ≡ 1) from the Introduction. Such a choice of Laplacian leads to the following operator for all f, g : V → R: ( f, g)(x) = 1 2 ( ( f g) − f g − g f )(x) = 1 2μ(x) y∼x ( f (y) − f (x))(g(y) − g(x)). For simplicity, we always write ( f ) := ( f, f ). Iterating , we can define another operator 2, given by 2( f, g)(x) = 1 2 ( ( f, g) − ( f, g) − (g, f ))(x). Again, we abbreviate 2( f ) = 2( f, f ). The Bakry–Émery curvature is defined via these operators in the following way. Definition 2.3.1 Let K ∈ R and N ∈ (0, ∞]. (i) The pointwise curvature dimension condition C D(K, N, x) for x ∈ V is defined by 2( f )(x) ≥ K( f )(x) + 1 N ( f )2 (x), for any f : V → R. (ii) The global curvature dimension condition C D(K, N) holds if and only if C D(K, N, x) holds for any x ∈ V . (iii) For any x ∈ V , we define KG,x (N) := sup{K ∈ R | C D(K, N, x)}. (2.3.1) In this chapter, we are only concerned with ∞-curvature, that is, N = ∞. Following [7, Prop. 2.1], the condition C D(K, ∞, x) is equivalent to 2(x) ≥ K(x), (2.3.2) where 2(x) and (x) are symmetric matrices of the corresponding quadratic forms evaluated at x ∈ V . Since only local information needs to be taken
- 49. 30 David Cushing, et al. into account, they are of size |B2(x)| × |B2(x)| and |B1(x)| × |B1(x)|, respec- tively, and to make sense of (2.3.2) the smaller size matrix must be padded with 0 entries. For more information in the non-normalized case, see [7, Sec- tions 2.1–2.3]. The entries of these matrices in the general weighted case are explicitly given in [7, Section 12]. (Note that for the context of this chapter, the edge weights w : E → [0, ∞) take only values 0, 1 and reflect adjacency of vertices and the vertex measure μ : V → (0, ∞) will only correspond to the normalized and non-normalized cases.) The main tool to prove strictly positive curvature is [7, Corollary 2.7], that is, the following properties are equivalent: ● 2(x) is positive semi-definite with one-dimensional kernel, ● KG,x (∞) 0. [7, Corollary 2.7] covers only the non-normalized case, but one can easily check that the equivalence holds also in the setting of general vertex measures. 2.3.3 Ollivier–Ricci Curvature As mentioned before, Ollivier–Ricci curvature is based on optimal trans- port. Ollivier–Ricci curvature was introduced in [23]. A fundamental con- cept in optimal transport is the Wasserstein distance between probability measures. Definition 2.3.2 Let G = (V, E) be a locally finite graph. Let μ1, μ2 be two probability measures on V . The Wasserstein distance W1(μ1, μ2) between μ1 and μ2 is defined as W1(μ1, μ2) = inf π y∈V x∈V d(x, y)π(x, y), (2.3.3) where the infimum runs over all transportation plans π : V × V → [0, 1] satisfying μ1(x) = y∈V π(x, y), μ2(y) = x∈V π(x, y). The transportation plan π moves a mass distribution given by μ1 into a mass distribution given by μ2, and W1(μ1, μ2) is a measure for the minimal effort which is required for such a transition. If π attains the infimum in (2.3.3) we call it an optimal transport plan transporting μ1 to μ2.
- 50. Curvature Calculations for Antitrees 31 We define the following probability distributions μx for any x ∈ V, p ∈ [0, 1]: μ p x (z) = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ p, if z = x, 1−p dx , if z ∼ x, 0, otherwise. Definition 2.3.3 The p-Ollivier–Ricci curvature on an edge x ∼ y in G = (V, E) is κp(x, y) = 1 − W1(μ p x , μ p y ), where p ∈ [0, 1] is called the idleness. The Ollivier–Ricci curvature introduced by Lin–Lu–Yau in [20] is defined as κLLY (x, y) = lim p→1 κp(x, y) 1 − p . A fundamental concept in the optimal transport theory and vital to our work is Kantorovich duality. First we recall the notion of 1-Lipschitz functions and then state Kantorovich duality. Definition 2.3.4 Let G = (V, E) be a locally finite graph, φ : V → R. We say that φ is 1-Lipschitz if |φ(x) − φ(y)| ≤ d(x, y) for all x, y ∈ V. Let 1–Lip denote the set of all 1–Lipschitz functions. Note that, by triangle inequality, φ is 1-Lipschitz iff |φ(x) − φ(y)| ≤ 1 for all pairs x ∼ y. Theorem 2.3.5 (Kantorovich duality) Let G = (V, E) be a locally finite graph. Let μ1, μ2 be two probability measures on V . Then W1(μ1, μ2) = sup φ:V →R φ∈1–Lip x∈V φ(x)(μ1(x) − μ2(x)). If φ ∈ 1-Lip attains the supremum we call it an optimal Kantorovich potential transporting μ1 to μ2. The following result on some properties of p → κp(x, y) for x ∼ y and its consequences was useful in our curvature considerations. Theorem 2.3.6 (see [3]) Let G = (V, E) be a locally finite graph. Let x, y ∈ V with x ∼ y. Then the function p → κp(x, y) is concave and piecewise
- 51. 32 David Cushing, et al. linear over [0, 1] with at most 3 linear parts. Furthermore κp(x, y) is linear on the intervals 0, 1 lcm(dx , dy) + 1 and 1 max(dx , dy) + 1 , 1 . Thus, if we have the further condition dx = dy, then κp(x, y) has at most two linear parts. 2.4 Bakry–Émery Curvature of Antitrees Let us first introduce some notation and a useful general fact (Lemma 2.4.1). The identity matrix of size d is denoted by Idd and the all-zero and all-one matrix of size d1 × d2 is denoted by 0d1,d2 and Jd1,d2 , respectively. Moreover, if d1 = d2, we use the notation Jd1 = Jd1,d1 , and if d2 = 1, we use the notation 1d1 for the all-one column vector of size d1. Moreover, the standard base of column vectors in RN is denoted by e1, . . . , eN . Lemma 2.4.1 Let d1, . . . , dr ∈ N and A = (Ai j )1≤i, j≤r be a symmetric matrix, where the Ai j are block matrices of size di × dj with Aji = A i j . Assume that there exist constants αi , βi ∈ R and γi j = γji ∈ R such that, for 1 ≤ i, j ≤ r, j = i, Aii = αi Iddi + βi Jdi and Ai j = γi j Jdi ,dj . Let Ared = (ai j )1≤i, j≤r be the r × r-matrix given by ai j = 1 di Ai j 1dj , i.e., for i = j, aii = αi di + βi d2 i , ai j = γi j di dj . For any vector w = (w1, . . . , wr ) ∈ Rr let w := (w11 d1 , . . . , wr 1 dr ) ∈ Rd with d = r j=1 dj . Then we have the following two facts: (a) For every di ≥ 2, the (di − 1)-dimensional space Ei = ⎧ ⎨ ⎩ di j=1 cj ej+d | di j=1 cj = 0 ⎫ ⎬ ⎭ with d = i−1 j=1 dj consists of eigenvectors to the eigenvalue αi .
- 52. Curvature Calculations for Antitrees 33 (b) For any w ∈ Rr , the corresponding vector w is orthogonal to all spaces Ei in (a) and we have w A w = w Ared w. Proof. Choose a vector û = (u1, . . . , ur ) ∈ Rd with u j ∈ Rdj for 1 ≤ j ≤ r. Then we have Au = ⎛ ⎝ r j=1 A1 j u j , . . . , r j=1 Ar j u j ⎞ ⎠ . Assume now that u j = (cj1, . . . , cjdj ) ∈ Rdj satisfies dj k=1 cjk = 0 for all 1 ≤ j ≤ r. (2.4.1) This implies that Jdi ,dj u j = 0 for all 1 ≤ i, j ≤ r and, therefore, r j=1 Ai j u j = Aii ui = αi ui , which proves (a). For the proof of (b), we assume that w ∈ Rr and w ∈ Rd are related as described in the lemma. It is then easy to see that w is orthogonal to any vec- tor u with components satisfying (2.4.1) and, therefore, to all eigenspaces Ei . Moreover, we have w A w = r i, j=1 wi 1 di Ai j wj 1dj = r i, j=1 wi wj ai j = w Aredw. This finishes the proof of (b). Now we start with our Bakry–Émery curvature considerations for antitrees. Due to localness of the Bakry–Émery curvature notion, we only need to consider KG,x (∞) for (i) a vertex x ∈ V3 in the finite antitree AT ((a, b, c, d, e)), (ii) a vertex x ∈ V2 in the finite antitree AT ((b, c, d, e)), and (iii) a vertex x ∈ V1 in the finite antitree AT ((c, d, e)). The relevant results are given in the following theorems. Theorem 2.4.2 Let x ∈ V3 be a vertex of the finite antitree G = AT ((a, b, c, d, e)). If a = n, b = n + 1, c = n + 2, d = n + 3, and e = n + 4,
- 53. 34 David Cushing, et al. we have in both the normalized and non-normalized case: KG,x (∞) 0. (2.4.2) Proof. In this proof, we will keep the values a, b, c, d, e general as long as possible and only specify them towards the end of the proof. Let G = AT ((a, b, c, d, e)), 1 ≤ a ≤ b c ≤ d ≤ e and x ∈ V3. To cover simultaneously both the normalized and non-normalized setting, we choose − = μ(x) μ(y−) − 1, + = μ(x) μ(y+) − 1, where y− ∈ V2 and y+ ∈ V4. (Note that μ(z) depends only the generation of z.) Using the results in [7, Section 12], a tedious but straightforward calculation shows the following: The matrix A = 4μ(x)22(x) is of the following block structure A = (Ai j )1≤i, j≤6 where the blocks correspond to an ordering of B2(x) into the vertex sets {x}, V3{x}, V4, V2, V5, V1: A11 = dx (dx + 3) + 3b− + 3d+, A12 = (−(dx + 3) + b− + d+)J1,c−1, A13 = (−(dx + 3 + e) − (2 + c + e)+)J1,d, A14 = (−(dx + 3 + a) − (2 + a + c)−)J1,b, A15 = (d + d+)J1,e, A16 = (b + b−)J1,a, A22 = (3(dx + 1) + b− + d+)Idc−1 − 2Jc−1, A23 = −(2 + 2+)Jc−1,d, A24 = −(2 + 2−)Jc−1,b, A25 = 0c−1,e, A26 = 0c−1,a, A33 = (−b + 3c + 3d + 3e + (3c + 4d + 3e)+)Idd − (2 + 4+)Jd, A34 = 2Jd,b, A35 = −(2 + 2+)Jd,e, A36 = 0d,a, A44 = (3a + 3b + 3c − d + (3a + 4b + 3c)−)Idb − (2 + 4−)Jb, A45 = 0b,e, A46 = −(2 + 2−)Jb,a, A55 = (d + d+)Ide, A56 = 0e,a, A66 = (b + b−)Ida.
- 54. Curvature Calculations for Antitrees 35 Let Ared be the corresponding reduced symmetric 6 × 6 matrix Ared = (ai j )1≤i, j≤6, as defined in Lemma 2.4.1. Recalling the equivalence at the end of Section 2.3.2, KG,x (∞) 0 is equivalent to A being positive semi-definite and having one-dimensional ker- nel. Lemma 2.4.1 provides the following eigenvalues and multiplicities of A. ● Since −, + −1 and dx = b + c + d − 1, α2 = 3(dx + 1+b− + d+) 0 is a positive eigenvalue of multiplicity c − 2 ≥ 0. ● Note that in both normalized and non-normalized case we have + ≥ b+c+d−1 c+d+e−1 − 1 and α3 = −b + 3c + 3d + 3e + (3c + 4d + 3e)+ ≥ ≥ −b − d + 3c + 4d + 3e c + d + e − 1 (b + c + d − 1) 0 is a positive eigenvalue of multiplicity d − 1 ≥ 1. ● Note that in both normalized and non-normalized case we have − ≥ 0 and α4 = 3a + 3b + 3c − d + (3a + 4b + 3c)− ≥ 3a + 3b + 3c − d 0 if d 3(a + b + c). This eigenvalue has multiplicity b − 1 ≥ 0. ● Since −, + −1, α5 = d + d+ 0 and α6 = b + b− 0 are both positive eigenvalues of multiplicities e − 1 ≥ 1 and a − 1 ≥ 0, respectively. Moreover, it is easily checked that A1a+b+c+d+e = 0. The orthogonal complement of the direct sum of the corresponding eigenspaces Ei and R1a+b+c+d+e is 5-dimensional and given by W = { w | w ∈ W}, where (d1, d2, d3, d4, d5, d6) = (1, c − 1, d, b, e, a) and W := {w ∈ R6 , 6 i=1 wi di = 0}. Under the assumption d 3(a + b + c), KG,x (∞) 0 is then equivalent to A| W being positive definite, which is equivalent to w A w = w Ared w 0 for all w ∈ W{0}. (2.4.3)
- 55. Discovering Diverse Content Through Random Scribd Documents
- 59. The Project Gutenberg eBook of Kurze Aufsätze
- 60. This ebook is for the use of anyone anywhere in the United States and most other parts of the world at no cost and with almost no restrictions whatsoever. You may copy it, give it away or re-use it under the terms of the Project Gutenberg License included with this ebook or online at www.gutenberg.org. If you are not located in the United States, you will have to check the laws of the country where you are located before using this eBook. Title: Kurze Aufsätze Author: Annette Kolb Release date: November 21, 2013 [eBook #44251] Most recently updated: October 23, 2024 Language: German Credits: Produced by Jens Sadowski *** START OF THE PROJECT GUTENBERG EBOOK KURZE AUFSÄTZE ***
- 61. KURZE AUFSÄTZE VON ANNETTE KOLB. MÜNCHEN 1899. ZU BEZIEHEN DURCH ULRICH PUTZE, BRIENNERSTRASSE 8.
- 62. Bruckmann’sche Buch- und Kunstdruckerei, München.
- 63. INHALT. 1.Der Zufall Seite 5 2.Der Frosch 15 3.Adam und Eva 19 4.Le revenant 23 5.L'Oracle 29 6.Herbstlied 33 7.Der Walchensee 35 8.Die Heruntergekommenen 39 9.Skizze 43 10.Das Traumbuch 49 Musikalisches: 11.Eine musikalische Betrachtung 57 12.Nemesis 63 13.Skizze über die Stellung des heutigen Pianisten 67 14.Epilog 75
- 64. DER ZUFALL? Was giebt es unvermeidlicheres, berechneteres und dabei natürlicheres wie den Zufall? Was ist abgefeimter und grausamer oder gütiger? Wir können ihn weder anklagen, noch ihm danken. — Nie können wir ihn überführen, ihm die Maske entreissen und sagen: »Dies hast du gewollt und über mich gebracht.« — Denn die natürlichste Verkettung der Dinge hat es herbeigeführt. Was sollen wir mit diesem raffinierten Zufall anfangen, der unsere Schritte lenkt und doch nur als ein leerer Schleier in unsern Händen bleibt? — Am besten ist es wohl, ihm zu vertrauen; allein man lernt dies nur nach Jahren, und nach geprüften Jahren. Erst treibt es uns, ihn gewaltsam herbeizuführen, unsern Willen dem seinen gegenüberzustellen, und dann erst wird der Zufall so recht feindselig und allmächtig! Was hängt er nicht alles an eine Begegnung? Ob wir eine Minute früher oder später in diese Gasse bogen, mag über eine unbeschreibliche Reihe von Unglückstagen entscheiden — sie von uns abwenden oder über uns bringen. »Es giebt keinen Zufall!« — sagt Schillers Wallenstein. Aber damit sagte er schon zu viel; denn der Zufall entzieht sich uns so fern, dass er nicht einmal diese Behauptung ermöglicht. Als ich in Paris anfing, mit dem Gedanken umzugehen, ich wäre am liebsten wieder zu Hause, erhielten wir eines Tages aus Marseille einen sorgfältig verpackten Schlüssel und einen Brief. Es war ein Angebot, die Wohnung einer Dame zu beziehen, währenddem diese im Süden weilte und ihr schöner Flügel wurde ganz besonders gerühmt, aber wir machten von all dem keinen Gebrauch, denn es kam so vieles dazwischen.
- 65. Da plagte mich eines Morgens ein unverkennbares Heimweh. Wir wohnten in einer jener engen Strassen, die den Himmel versperren und die Menschen zusammendrängen wie auf einem Schiff. Draussen war es regnerisch und schwül, und ich sehnte mich fort; da fühlte ich zufällig unter meinen Fingern den Schlüssel jener Wohnung, und um mich gewaltsam aus der Stimmung zu reissen, in der ich mich befand, machte ich mich zur Stelle auf den Weg nach diesem Hause. — Als ich aber dort die ziemlich hochgelegene Wohnung betrat, lag sie in so rabenschwarzer Nacht, dass ich alsbald wieder hinunterging, um mir bei dem Concierge ein Licht zu verschaffen. Dieser hatte indes seine Loge verlassen, und ohne auf ihn zu warten, zündete ich mir eine Kerze an und eilte wieder hinauf. — Auch nicht ein Schimmer des Tageslichtes drang in diese Räume! Eiserne, verriegelte Läden schlossen es gänzlich ab, und der Lärm von Paris klang da gar seltsam herein, denn öde war es hier! — Als hätte ein Unglück die Bewohner plötzlich vertrieben, so dass sie alles liessen wie es war, nur dem Lichte wehrend, bevor sie flohen. Denn nichts war aufgeräumt. Im ersten Zimmer stand ein blauseidnes Bett aufgeschlagen und bestaubt, vom Baldachin hing eine lange Kordel zerrissen herab. Die Kerze beleuchtete nur immer dürftig eine einzige Stelle, aber im Vorübergehen sah ich Gegenstände verwahrlost herumliegen, zertrümmertes Krystall, zierliche Louis XV.- Möbel und einen offenen Schrank. Es war, als ob hier Diebe gehaust hätten, und als seien sie dann in der Hast über alles davongestiegen. So unheimlich war der Anblick all dieser Zimmer, dass ich, ohne mich länger umzusehen, den Salon suchte, wo der Flügel stehen musste, um dann schleunigst wieder fortzukommen. Ich entdeckte ihn denn auch, zwischen zwei Fenstern stehend und von einer Decke geschützt. Als ich diese zurückschob, hob sich ein Schwarm von vielleicht tausend Flöhen und stieb in gerader Linie auf mich los. Ich fuhr zurück — wahrscheinlich zu rasch — die Kerze verlosch! — Was dies für mich bedeutete, war mir sofort klar. Denn ich hatte im unverantwortlichen und unbegreiflichen Leichtsinn die Zündhölzer unten gelassen. —
- 66. Nie aber würde ich in dieser Finsternis die Hausthüre finden, und wenn ich sie fände, niemals unterscheiden — den Weg zurück wusste ich nicht. Es waren so viele Zimmer gewesen und kein Gang. Alles ineinand geschachtelt, wie es in französischen Wohnungen oft ist. Ich tastete nach dem Schlüssel, aber der Schrecken hatte mir alle Erinnerung benommen. Ich fand ihn nicht mehr. Mit den Händen fuhr ich der Wand entlang bis zum Fenster, allein die Läden mussten einen eigenen Verschluss haben und schnitten mir in die Finger, ohne zu rücken. Behutsam ging ich vorwärts, vielleicht drang doch in irgend eine Kammer ein Schimmer von Licht und war von dort aus ein Zeichen möglich, aber überall war Finsternis und Staubgeruch als läge ich tief unter der Erde. Der Concierge würde den Leuchter kaum vermissen, den ich unter vielen andern aus seiner Loge fortnahm, keinesfalls aber auf mich geraten und die Meinen hatten keine Ahnung wohin ich gegangen war, denn als ich von Hause fortging war ich allein gewesen. — So war zwar meine Rettung lange noch möglich, noch grösser aber die Gefahr, dass ich hier verschlossen und vergessen bliebe. Meine Wanderungen nach der Hausthüre begannen von neuem. Griffe ich sie, so wollte ich dort stehen und rufen. Allein ich fand sie nicht! Es liess sich keine Thüre von der andern erkennen, kein Zimmer, keine Kammer. Einige waren versperrt. Wie in einer Falle irrte ich blind umher und wurde immer unfähiger, mich zu orientieren; denn von den Räumlichkeiten hatte ich die Verhältnisse nicht entnommen, und der Ausgangspunkt war mir längs verloren. So musste ich mich meinem Schicksal ergeben. Die Zeit verging, und wie rings um mich, so war es jetzt auch in meinem Herzen Nacht. Aber statt der Verzweiflung kamen mir da plötzlich Gedanken: Was für einen Sinn hätte denn ein solcher Abschluss? Welche Deutung konnte ich meinem Tode abgewinnen? In meinem Leben konnte ich nichts entdecken, aber dies Leben selbst erschien mir da merkwürdigerweise wie ein arger Schuldbrief, und ich werde wohl nie mehr so tief und ruhig zu denken vermögen, wie in jenem so hoch über der Erde gelegenem Grab!
- 67. Wie spät es geworden sein mochte ahnte ich nicht. Immer wieder begannen meine finsteren Wanderungen, mein Tasten nach Thüren und mein Rufen. Meine eigne Stimme versetzte mich in solche Angst, dass es wie wahnsinnig in meinen Schläfen pochte. Den Hunger sah ich schon als meinen Gefährten, und heiss und blutig drang mir’s nun ins Gehirn. — Und wie betäubt stiess ich zuletzt gegen eine scharfe Kante und empfand etwas Kaltes unter meinen Händen. Daraus schloss ich, dass ich mich wieder in einem Zimmer befand, denn dies fühlte sich wie ein marmorner Tisch. Ich fasste ihn mit der andern Hand: da durchzuckte mich jäh eine wilde, triumphierende Lebensfreude. Was da meine suchenden Finger ergriffen hatten, war — eine Zündholzschachtel! Zitternd fachte ich eines an und starrte jetzt auf ein gespenstiges Wesen, das mit hohlen Augen unvergesslich auf mich blickte. Allein bevor die Angst noch ihre Klammern auf mich legen konnte, gewahrte ich den hohen Spiegel, vor dem ich stand, woran die schmale Marmorplatte angebracht war, an die ich stiess. Lange Kerzen stacken da in Kandelabern, und mechanisch zündete ich sie an; von meinem eignen Bilde keinen Blick verwendend, denn wie von einem Drama war ich hier gefesselt. Das Entsetzen auf meiner Stirne, die trostlose Ergebenheit meiner Züge, die Todesahnung war auf meinem Gesichte geblieben. Obwohl ich mich gerettet wusste, immer starrte ich noch wie eine Verlorene. Was hinter diesen weitgeöffneten Augen vorgegangen war, wusste ich so wohl, der schon wie eingefallene Mund, warum er so bitter geschlossen war, das herabgezogene Kinn, der zurückgehaltene Grimm. — Und dabei war mir’s als erschaute ich das Selbsterlebte nun zum erstenmale. So blieb ich vor dem Spiegel gebannt, bis meine Augen sich verkleinerten und die Farbe, als sei nichts geschehen, sich allmählich wieder einstellte. — Der Raum, in dem ich mich befand, war ein kleines Durchgangszimmer, und die Begebenheit so einfach und natürlich! Es hatte hier jemand eine Schachtel Streichhölzer vergessen. Weiter nichts!
- 68. Es war eben jener blinde und hundertäugige Zufall, jener unberechenbare Stern, der über unser Leben waltet und es erhält oder vernichtet. Den Schlüssel, die Thür und den Weg ins Freie hatte ich nun bald gefunden; wieder hinab in das rege Paris. Die Boulevards schimmerten im Abendrot, und die Knospen der Bäume waren nach dem Regen hold geschwellt.
- 69. DER FROSCH. Ein Frosch sass im nassen Grase, befriedigt und wohl aufgeblähet, denn er hatte eben gespeist, und da ihm das Verschmauste wohl bekam, so fühlte er sich nicht ungeneigt zu philosophieren, zwinkerte behaglich mit seinen feuchten Augen und dachte: »Was ist doch die Welt so seriöse! — und machen sie alle so fatale Mienen, statt das Leben frisch zu nehmen wie es ist! Ich bin zufrieden, und mir geht es gut; auch nehme ich die Dinge wie sie kommen!« Und obwohl er schon zu viel gegessen hatte, schnappte er noch im Übermute nach einer Fliege, die des Weges flog, und verzog dann sein breites Maul zu einem superiorem Lächeln: Es war doch wirklich alles zu dumm! So hockte er froh an des Teiches Rand, blickte in die laue Luft und hiess die Weltordnung gut. Libellen hingen und schwirrten, dicke Waldschnecken schleppten sich fort, ein Vöglein jammerte und eine hagere Katze schlich umher. Alles beobachtete und genoss der Frosch als heitrer Skeptiker und Bon-vivant und plumpste dann wieder in den Teich. Von Tag zu Tag aber gedieh er, zum Verderben zahlloser Mückchen, die enthusiastisch in der Sonne schillerten. — Kein Wunder, wenn sich der Frosch da »hatte« und seine Lebensanschauung sich zu einem immer insolenterem System abrundete! Und unumwölkt floss sein Dasein dahin, denn jeder ist selbst seines Glückes Schmied.
- 70. ADAM UND EVA. Die Nacht senkte sich vor der Vertriebnen Augen, und nach harter Tagesmühe ruhten sie. Trauer umfloss der Gefallenen Antlitz, und ob des Menschengeschlechtes drang eiserne Schwermut auf sie ein. Keine Thräne hatte noch das Weib; es barg und vertiefte sich das Weh der Erde in ihrem Schosse zur Melancholie, und wortescheu verblieb der Mann, als er sich hingewiesen sah an die harte, unbekannte Scholle, an die unerbittliche Sonne und dem süssen Mond; aber der Welt Zukunft und Not starrte in seinem Geist. Dies Paar, ach! war der Atlas! Das Echo seiner Qual durchdrang den hellen Sinn der Griechen, und eine Weltkugel wälzten sie dem GOTTE auf die Schulter, allein ein Menschenpaar ist es gewesen, das einst die Last des Werdens kostete und trug.
- 72. LE REVENANT. Une nuit je crus errer eu rève dans des siècles passés, et je vis des hommes et des femmes dans leur vie journalière. Je vis des enfants joner, un laquais endormi sur un siège, puis des fruits dans une coupe étrange et soudain sur un balcon trempé de pluie une jeune dame enveloppée dans une grande robe rose et une mante noire. Mon esprit alors fut pris d’un vertige! — et sentant mon rève, je voulus m’en soustraire en le secouant; mais lui aussi-tôt, se faisant plus confus, devint si pesant, que le coeur oppressé, je le subis. — Alors je me vis appuyé contre une fenêtre à ogives à la nuit tombante dans une salle. Brusquement tout au fond une porte s’entr’ouvrit, et un chien s’élança, de ces beaux chiens de chasse! il s’arrèta inquiet, les yeux flambants; puis d’un mouvement jeune et violent, fou de vie et de joie, il se retourna, se jeta vers la porte, et frappant le parquet bruyamment de sa queue, il attendit, guetta plutôt, pour s’élancer sur un homme qui entrait. — Lorsque je vis cet homme qui entrait, je sentis mes lèvres trembler de tristesse. L’on eut dit la vie même, et c’était un mort! — Ah! si vous l’aviez vu s’avancer d’un pas rapide en tournant vers sou chien une figure d’un contour si vif et d’une ciselure si étroite, que cette tête si noire se détachait des ténèbres comme une tache blanche, tant elle était ardente! l’illusion, je vous assure, vous eut gagné, tout comme moi: cas la vie affluait dans chacun de ses gestes; ses yeux étaient chargés et lourds comme certaines fleurs, et sur cette figure fougueuse, le regard était préocupé et rentré, comme pour se poser très-loin sur une vision qui revenait toujours, et faisait sourire malgré lui, sa bouche songeuse et cruelle! — La mort, me disai-je, la mort! —
- 73. Je me sentais si chétif près de cet être si beau, pourtant je vivais moi! n’était-ce pas mieux que ce splendide mirage? La mort!? — mais ce mot même tombait vide devant un pareil revenant! Ce fut alors, qu’il marcha droit vers la fenêtre, où je me tenais et que mes yeux purent plonger dans les siens pour, en chercher l’énigme. Mais hélas! qu’ils étaient loins, et comme mon coeur se serra! une grande douleur fit tomber mes paupières qui brûlaient, et je sentis alors s’approcher de moi, et m’envelopper comme l’haleine du Printemps; je crus respirer toutes les aubépines des bois, et sentir un ciel, des sapins, et des ruisseaux clairs: je vis une truite tachetée de rose, et de l’herbe fraîche et mouillée; et une si afreuse nostalgie passa dans mes veines, que j’étendis un bras éploré vers le spectre, dont la vie m’avait ainsi troublé. Mais lui, quoique sa main pesât sur mon épaule, son regard, qui semblait déborder, se détournait toujours. — Et, voulant jeter un cri d’ angoisse, qui ne fut qu’un souffle, je lui dis: «Je suis lá!» et tout mon être passa dans ces pauvres paroles! L’homme tressaillit, et changeant d’attitude, sa main tomba. Mais en ce moment même il y eut un bruit dans la cour, et je le vis se retourner, faire signe à son chien, et sortir. Ni l’un ni l’autre ne m’avaient vu. — Et alors la Nuit se fit plus profonde, et mon coeur plus froid. Seul mon cerveau s’allumait et marcha. Regarde! dit-il à mes yeux devenus fixes de terreur, regarde sous ces ténèbres croissans cette salle inconnue, et vois ces meubles bizarres! Que peuvent ils te rappeler? Rien! sonna-t-il. Puis toutes les roues de mon cerveau s’ébranlérent avec une vitesse infernale, et j’entendis un glas frapper au fond de moi-même: LE REVENANT, C’ÉTAIT MOI! 1893
- 75. L’ORACLE. Elle était grande et laide, une roche informe et nue, qu’elle hit éclairée ou à l’ombre, toujours triste. Un homme s’y égara un soir, mais perdant pied aussitôt il mourut victime, lui fort et pensant, de cette grande chose inerte et brute, et personne ne la montait plus. Elle demenrait à l’ombre le plus souvent des grandes cimes autour, et le soleil ni la lune ne l’aimaient. Seule la neige s’y plaquait lourde et compacte! Or en une nuit de lune et de Vent (le monde déjà était vieux) quelque chose remua au fond du rocher, et l’emplit soudain, comme d’un profond soupir. Ce ne fut qu’un instant! quelques caillons roulèrent et un peu de neige bleuâtre se détacha. Ce fut tout. Mais en cet instant si vague, et d’infinie lourdeur — le rocher subit sa propre tristesse sourdement, comme la plante comme s’éveille l’aloès du fond de sa torpeur, c’est ainsi que sa propre Enigme vint saisir la montagne et lui révéla son Mystère, les liens occultes, qui la liaient aux longs chagrins et aux incurables misères, à tout ce qui est noir ou navrant dans la création. Tout cela l’enveloppa comme d’une Ombre Géante. Et un accord vibra en ce domaine silencieux! Une source s’agita affolée! elle mouta brûlante et profonde jusque à l’ivresse, pour tarir aussitôt. Mais la Terre — si rèveuse en ces nuits de Lune et de Vent tressaillit et appela. Alors des milliers d’ombres se dégagèrent des plis de Ténèbres et s’agitèrent autour du rocher éteint pour saluer l’Idée — le Symbole — l’Oracle enfin qui venait de parler. 1893
- 76. HERBSTLIED. Herbstlich sinkt der Tag nun. Herbstfarb’nes Licht, so sanft wie süsser Ton, Zart wie bedeutsamer Traum, Der uns beglückend streifte in der Flucht. Ach weile, guter Herbst! Dein ist der tönendste Ton im Jahr! Musik der Dämmerung ist deine Stunde, Beruhigte Leidenschaft dein tiefer Blick. Ist Verfall dein Sinn? Oder lächelst du über den Tod? —
- 77. DER WALCHENSEE. Die Berge zogen ihre hohen, sanften Linien in der bleichen Dämmerung. Ahnungsvoll schien jede Senkung, jede Matte, jeder Schatten, und stumm hielten die Tannen hart am Ufer Wacht. Und Luna zog langsam mit ihrem Gefolge weissgeballter Wolken hinter den Spitzen der Berge einher. Kein Sternengefunkel störte noch des Himmels Ruh’! Und wie tief kündete sich da die Nacht, wie fern schien da Aurora, als käme nimmer der frühe Tau, noch die strahlende Sonne zurück. »Ach!« seufzte da eines Menschen Stimme, »käme nimmer der Morgen!« Doch plötzliches Entsetzen fasste ihn alsbald, und starre Angst trieb ihn dem Gestade entlang, war es ihm doch, als hätte er hier Schatten ins Bewusstsein gerufen und aufgescheucht, als sei ihm das verhängnisvolle Wort entfahren, das diesem See und dieser Natur geheimnisvoll zu Grunde lag, und als seufzte nun alles rings um ihn, von jeder Felswand rauschend und vom Strande wiederhallend, ein traumversunkenes und im Traum gefundenes Echo: Ach, käme nimmer der Morgen! Käme nimmer der Morgen!
- 78. DIE HERUNTERGEKOMMENEN. Als die Nacht hereingebrochen war und der kalte Zug durch die Fensterspalten blies, da wurde es auch stille in dem langen Gang, wo die Ahnenbilder hingen unverrückt an der dunklen Wand und die Finsternis über sich ergehen liessen wie über ihre Gräber. Allein die Nachkommen dieser längst verblichnen Leute wohnten noch in dem alten Schloss und fanden keine Ruhe, denn sie wollten und wünschten mit der wilden Kraft, die sie von den Vätern geerbt! Währenddem die Nacht sich immer tiefer senkte, schlief da Keins. Alle hofften, fürchteten und sehnten sich zu sehr in diesen alten Mauern, als dass der Schlaf sich ihnen rettend nähern konnte. Den hielt der Hass und den die Liebe, alle aber hielt der Lebensdrang, die Heftigkeit des Wunsches und die trübe Ahnung des Unerfüllbaren wach. Die Väter hatten so froh genossen und so wilden Auges gelebt! Sie glichen sich alle in Miene und Blick, und Generationen hindurch verzehrten sich die schönsten Frauen in Liebe um dies Haus! Das Glück aber hielt treue Wacht und zog goldene Gitter um seine Günstlinge. Einem breiten glänzenden Strome glich dies Geschlecht, der schimmernd die schönsten Lande durchzieht, Wälder und hohe Gipfel, glänzende Städte und den ganzen Himmel lachend wiederspiegelt. Zöge sich doch mein Herz nicht zusammen, als ich dieses Vergleichs gedenke! Denn nach hundert Jahren erlosch ein Stern: der herrliche Fluss rauschte weiter; da veränderte sich sein Bett. Hoch und furchtbar drangen kahle Felsenwände auf ihn ein, qualvoll türmte sich da das tiefe Gewässer und wütete gegen die hemmende Wand.
- 79. Sein schrecklicher Schall tönte betäubend durch die Welt. Unerbittlich aber verengten sich noch die Thore, und der Fluss brach sich heulend seine Bahn. Als wilder umdunkelter Bach stürzt er im Schatten dahin. —
- 80. SKIZZE. Vor Jahren fiel mir ein Buch in die Hände, dessen Titel ich mich nicht mehr entsinnen kann, es war eine Übersetzung aus dem Griechischen und mit vielen Anmerkungen versehen, wovon eine einen alten Spruch citierte, der mir immer im Gedächtnisse blieb. Die Worte erinnere ich mir nicht, nur den Sinn, und der war folgender. »Nicht der Mann ist die Weisheit, nicht die Frau ist die Liebe: Die Frau ist Weisheit, der Mann ist Liebe. Des scheinbaren Umtausches sich nicht bewusst, sucht der Mann in der Frau seine eigne Liebe, die Frau im Manne ihre Weisheit wieder.« Dieser Spruch schien mir nach und nach so manches Unerklärliche und Unvereinbare, das in jenen Beziehungen nicht zu begleichen schien, schärfer zu beleuchten. Ein »ganzer Mann« wird einer Frau in so entscheidenden Punkten überlegen sein, dass nur die tiefere Weisheit des schwächeren Teils ein Gleichgewicht herzustellen vermag und in jener Weisheit allein die Möglichkeit liegt, den Blick dieses Mannes ganz wiederzuspiegeln. Ist dieser Spiegel getrübt oder zu stürmisch oder zu seicht, so wendet der Blick sich ermüdet ab und sehnt und sucht nach andern Augen, die wieder versprechen und wieder enttäuschen. Umgekehrt sehen wir oft ganz unbedeutende Männer von einem weiblichen Wesen dauernd gefesselt, von dem sie nie Kenntnis gewinnen können, in dem aber die Weisheit verborgen liegt, die sie mit dumpfer Sehnsucht erfüllt. Eine solche Frau, deren innere Entwicklung ihren eigenen Weg zu folgen bestimmt war, sieht oft zu ihrem stillen Befremden einen ihr so fremden Mann so treu an ihrer Seite.
- 81. Was nun mit jener Weisheit in dem alten Spruche gemeint war, ist sicher nicht die Lebensklugheit noch Schärfe oder Kraft des Geistes, denn die wohnen alle dem Manne viel thätiger inne. Sie wird wohl eher dem Meeresspiegel vergleichbar sein, der tiefer und beschaulicher wird, je mehr sich darin versenkt. — Jeder kennt jenes eigentümliche Gefühl, das ihn angesichts der gleichgültigsten Dinge anwandeln kann, ihn zwingt, innezuhalten und Gedanken einzulassen, die von aussen auf ihn einzudrängen scheinen und deren Bewandtnis er noch nicht erfasst. So stand ich einmal auf einem weiten, freien Feld und dachte an die Druiden, wie die Welt in ihnen wiederhallte, in sie drängend wie ein Strom, so dass sie ihr das Rätsel fast entrieten und, von ihrer Ahnung überwältigt, Wahrheiten stammelten — in undurchdringlichen Worten. Da fiel mir — anscheinend schauerlich unzusammenhängend — der Don Juan ein! War etwa hier ein Gegensatz? — War hier etwas, was sich deckte? Ich weiss es nicht. — Aber mit einem Male begriff ich, wie sich der Zauber und die Tragik im Dasein zweier Geschlechter in jener dunklen Gestalt und ihren Opfern sublimieren konnte, und ich begriff den klärenden Schein, den Mozart um sie wob. Trat in diesem Wesen irgend ein verborgenes Gesetz in Kraft und blieb das nie Erreichte auf weit abliegender Bahn und keinem füglichem Gebiet verwiesen? — Lag etwa im Blicke der Veleda jene Ruh’, die Don Juan in jedem schönen Auge suchte, jenem andern Zuge folgend, der die Liebe so unendlich adelt? — Und lag seine eigne Gewalt in seiner eignen Sehnsucht? —
- 82. DAS TRAUMBUCH. Man wirft mir so gerne vor, dass ich nicht schreibe! — Aber erstens! — — — Und zweitens gehört hiezu doch auch eine leidliche Erfindungsgabe, und ich bin nur deshalb so leichtgläubig, weil ich auf das Gegenteil von dem, was man mir sagt, von selbst gar nicht gerate, eine solche Veranlagung ist nicht eben produktiv! Über Gegebenes, Menschen wie Dinge, kann ich lange und eindringlich nachdenken, nur muss ich sie haben! — Aus der Luft greife ich nichts, denn eine unübersteigbare Kluft trennt mich von jener Fähigkeit zu schaffen, die so beglückend und erhebend sein muss und wohl deshalb so selten ist. Die einzige Genugthuung jedoch, welche mir diese endlich errungene Erkenntnis bot, war, dass ich mich frei sprechen konnte von aller Schuld, wenn keine Gedichte und keine Romane aus meiner Feder flossen, denn wie viel besser wusste ich als alle andern, dass ich keine zu stande brachte! Als ich aber hierüber noch nicht im Reinen war und mir die Menschen so manches versicherten, was mich nicht überzeugen konnte und doch sehr verdross — fasste ich einmal einen verzweifelten Plan, den ich auf die äusserste Spitze treiben wollte und einem Mann von Fach zu eröffnen beschloss. Ich liess mich bei ihm melden und erhielt einige Tage darauf ein zierliches Briefchen, worin er mich auf sein Landgut zu einer Unterredung berief. Nun hatte ich nachts bevor, folgenden Traum: Ich, die nie im Leben geritten war, sass plötzlich hoch zu Ross, ritt andern Reitern, die mich beschworen einzuhalten, voran, liess mich dann langsam herabgleiten und stieg die Treppe zu unserm Hause hinauf.
- 83. Dann erwachte ich. — Da jedoch dieser Traum sehr lebhaft in meinem Gedächtnisse haften blieb, so schlug ich in meinem Traumbuch nach, ob eine Deutung darauf stünde und las folgendes: »Unterlasse nicht, was du vorhast!« Mir aber kam diese Weisung wirklich wie gerufen, denn schon lange wollte ich einen recht flagranten Beweis in Händen haben, der mich von meiner Leichtgläubigkeit endgültig kurierte. Derselbe Abend sollte mich ja noch belehren! Dann verliess ich mein Haus und nahm den Zug. Das Wetter war leuchtend, und zuletzt führte mein Weg auf einem schmalen Fusspfad durch ein hohes Kornfeld. Ganz ergriffen hielt ich da inne; denn die Welt war an diesem Tage zu schön, ihr Schein zu unbeschreiblich! Ovid’s Verwandlungen berührten mich mit einemmale als naturgemäss, und mir war, als würde ich selbst zu jenem singenden, summenden Kornfeld, so sehr entzückte mich gerade an dieser Stelle das goldene Leben unserer Erde. Doch nur wenig Schritte trennten mich von der Besitzung, in der meine Autorität hauste, und nun erschien mir mein Plan erst recht in seiner ganzen Unausführbarkeit. Eine Stunde später ging ich denn auch sehr gemessen denselben Weg wieder zurück: Zuerst war der Mann von Fach sehr ernsthaft drei Schritte zurückgewichen und hatte mich angestarrt. — Aber in sein langes herzliches und eindringliches Lachen musste ich am Ende doch einstimmen. Träume! dachte ich nun und wurde nachdenklicher mit jedem Schritt, denn manches schien mir doch recht befremdend auf der Welt. Wie kam es zum Beispiel, dass die Alten, diese klugen, spöttischen Griechen, denen die Wirklichkeit so voll genügte, solche Acht auf ihre Träume hielten, dass die Geschichte selbst sie uns ganz ernsthaft mit Daten und Thatsachen bringt? Vor jedem Schlachtenberichte stehen sie da als Avantgarde, und jeder Feldherr klügelt über den seinen! Nun denke man sich nur einen modernen Geschichtsschreiber Napoleon’s oder Bismarck’s Träume und dann zum Schluss noch
- 84. seine eignen verzeichnend. Und das mit der gebietenden Miene eines Plutarch! Wäre es möglich, dass hier etwas dahintersteckte und es uns verloren ging? Sonst dienen uns doch die Alten so gerne als Vorbild. Wer aber würde sich heutzutage mit derlei befassen? Die eigentliche Bibliothek des Traumbuchs ist die Küche geworden und geschwätziges oder ungebildetes Volk beratschlagen es. Nur ich besass noch eins, kraft jener Erfindungsunfähigkeit, jener Sucht zu glauben, und auf glaubwürdiges zu lauern. Alle Exzesse und Irrtümer stehen da offen. So dachte ich, von dem wogenden Kornfeld nicht länger impressioniert, im Dämmerlichte des sinkenden Tages einhergehend und eignem Grübeln. Da plötzlich unerwartet, ungeahnt — stand vor meinen bestürzten Augen nicht das Gelingen meines Planes — eine andre Erfüllung, die meinen Traum wachrief wie mit einem langgedehnten Ruf, und wie einen kalten Hauch empfand ich meine eigne Blässe.
- 85. MUSIKALISCHES. MOTTO: Wollen wir hoffen? Richard Wagner, X. Band.
- 86. EINE MUSIKALISCHE BETRACHTUNG. Vor einem mit Plakaten reich übersäten Kioske innehaltend, sagte kürzlich einer zu seinem Freunde: »Sieh doch die vielen Konzerte! Bis über die Wände hinaus klettern die Annoncen!« »Das ist schön!« rief der andere. »Da hast du unser liebes kunstsinniges München!« »Ja, da hast du’s!« brummte wieder der eine. Und wie es so geht auf dieser Welt, als sie eine kleine Strecke weiter gegangen waren, fingen sie fürchterlich zu streiten an. In der Hitze jedoch gebieten wir selten über die überzeugenden Worte, selbst wenn wir im Rechte sind, und grad ein Philister hat da oft leichtes Spiel. Hier siegte denn auch der, dem beim Anblick der vielen Plakate das Herz freudiger schlug, und selbstbewusst und heiter kehrte er nach Hause zur Gattin. Aber wie verdrossen ging der andre heim! Fiel ihm doch jetzt erst alles ein, was er im Eifer nicht fand; und wie sicher gestaltet sich nun seine Rede in den dunklen Strassen! Immer feuriger ging er einher, als müsste er Schritt halten mit seinen Gedanken, und sah recht närrisch dabei aus! Hier sei auch mir eine Bemerkung gestattet: Wage ich mich zwar jetzt mit dem Sprüchwort: Kinder und Narren etc. vor, so werde ich allerdings dem Vorwurf grosser Alltäglichkeit nicht entgehen, bringt uns heute doch fast jeder Plato’s finstre Höhle (die Höhle, ach, du lieber Gott, in der wir alle so gemütlich sitzen!), oder citiert jene grosse Neuigkeit von dem grössten Tragiker, nicht wahr, der zugleich etc. . . . . Denn nur in solchen und ähnlichen Reminiscenzen ergehen
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