A note on isoperimetric inequalities of Gromov hyperbolic manifolds and graphs
Alvaro Martinez-Perez Jose M. Rodriguez
math.DG

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Álvaro Martínez-Pérez]Álvaro Martínez-Pérez






José M. Rodríguez]José M. Rodríguez





















In this paper we study the relationship of hyperbolicity and (Cheeger) isoperimetric inequality in the context of Riemannian manifolds and graphs.
We characterize the hyperbolic manifolds and graphs (with bounded local geometry) verifying this isoperimetric inequality, in terms of their Gromov boundary improving similar results from a previous work.
In particular, we prove that having a pole is a necessary condition and, therefore, it can be removed as hypothesis.



A note on isoperimetric inequalities of Gromov hyperbolic manifolds and graphs
    [
    December 30, 2023
==============================================================================






2010 AMS Subject Classification numbers: Primary 53C21, 53C23; Secondary 58C40.


Keywords: Cheeger isoperimetric constant; Gromov hyperbolicity; bounded local geometry; pole.



Author names and affiliations:



Álvaro Martínez-Pérez,
Facultad CC. Sociales de Talavera,
Avda. Real Fbrica de Seda, s/n. 45600 Talavera de la Reina, Toledo, Spain,


alvaro.martinezperez@uclm.es

ORCID 0000-0002-1344-6189



José M. Rodríguez (corresponding author),
Departamento de Matemáticas, Universidad Carlos III de Madrid,
Avenida de la Universidad 30, 28911 Leganés, Madrid, Spain, Phone number 34 91 624 9098,

jomaro@math.uc3m.es

ORCID 0000-0003-2851-7442













§ INTRODUCTION


Isoperimetric inequalities are of interest in pure and applied mathematics (see, e.g., <cit.>, <cit.>).
There are close connections between isoperimetric inequality and some conformal invariants of Riemannian manifolds and graphs, namely
Poincaré-Sobolev inequalities,
the bottom of the spectrum of the Laplace-Beltrami operator, the exponent of convergence,
and the Hausdorff dimensions of the sets of both bounded geodesics and escaping geodesics in a negatively curved surface
(see <cit.>, <cit.>, <cit.>, <cit.>, <cit.>, <cit.>, <cit.>, <cit.>, <cit.>, <cit.>, <cit.>, <cit.>).
The Cheeger isoperimetric inequality is closely related to the project of Ancona
on the space of positive harmonic functions of Gromov-hyperbolic manifolds and graphs (<cit.>, <cit.> and <cit.>).
In fact, in the study of the Laplace operator on a hyperbolic manifold or graph X,
Ancona obtained in those three last papers interesting results, under the additional assumption that the bottom of the spectrum of the Laplace spectrum ł_1(X) is positive.
The well-known Cheeger inequality ł_1(X) ≥1/4 h(X)^2, where h(X) is the isoperimetric constant of X,
guarantees that ł_1(X)>0 when h(X) > 0 (see <cit.> for a converse inequality).
Hence, the results of this paper are useful in order to apply these Ancona's results.


There is a natural connection between hyperbolicity and Cheeger isoperimetric inequality. In fact, one of the alternative definitions of Gromov hyperbolicity uses some kind of isoperimetric inequality (see <cit.>, <cit.>).


Cao proved in <cit.> that hyperbolicity with an extra hypothesis on the Gromov boundary implies (Cheeger) isoperimetric inequality
(an extra hypothesis is necessary, since there exist hyperbolic graphs without isoperimetric inequality, as the Cayley graph of the group ).



In <cit.> we studied the relationship of hyperbolicity and Cheeger isoperimetric inequality in the context of Riemannian manifolds and graphs
with bounded local geometry.





Given any Riemannian n-manifold M, the Cheeger isoperimetric constant of M is defined as

    h(M) = inf_A _n-1(∂ A)/_n(A) ,

where A ranges over all non-empty bounded open subsets of M,
and _k(B) denotes the k-dimensional Riemannian volume of the set B.

Given any graph =(V,E)=(V(),E()), let us consider the natural length metric d_ Γ where every edge has length 1.
For any graph Γ, any vertex v∈ V and any k∈ℕ, let S(v,k):={w∈ V   |   d_ Γ(v,w)=k }.
As usual, we denote by B(v,k) and B̅(v,k) the open and closed balls, respectively.

The combinatorial Cheeger isoperimetric constant of Γ is defined to be

    h(Γ) = inf_A |∂ A|/|A| ,

where A ranges over all non-empty finite subsets of vertices in Γ,
∂ A = {v ∈Γ  |   d_ Γ(v,A) = 1} and |A| denotes the cardinality of A.


A Riemannian manifold or graph X satisfies the (Cheeger) isoperimetric inequality if h(X)>0, since in this case

    _n(A) ≤ h(X)^-1_n-1(∂ A) ,

for every bounded open set A ⊆ X if X is a Riemannian n-manifold, and

    |A| ≤ h(X)^-1 |∂ A| ,

for every finite set A ⊆ V(X) if X is a graph.

Along the paper, we just consider manifolds and graphs X which are connected.
This is not a loss of generality, since if X has connected components {X_j}, then h(X)=inf_j h(X_j).

Let (X,d_X) and (Y,d_Y)  be two metric spaces. A map f: X⟶ Y is said to be
an (α, β)-quasi-isometric embedding, with constants α≥ 1,β≥ 0, if for every x, y∈ X:

    α^-1d_X(x,y)-β≤ d_ Y(f(x),f(y))≤α  d_X(x,y)+β.

The function f is ε-full if
for each y ∈ Y there exists x∈ X with d_Y(f(x),y)≤ε.

A map f: X⟶ Y is said to be
a quasi-isometry, if there exist constants α≥ 1,β,ε≥ 0 such that f is an ε-full
(α, β)-quasi-isometric embedding.
Two metric spaces X and Y are quasi-isometric if there exists
a quasi-isometry f:X⟶ Y.
One can check that to be quasi-isometric is an equivalence relation.


A graph Γ is said to be μ-uniform if each vertex p of V has at most μ neighbors, i.e.,

    sup{|N(p)|  |    p∈ V()}≤μ.

If a graph Γ is μ-uniform for some constant μ we say that Γ is uniform
or that it has bounded local geometry.

A Riemannian n-manifold M has bounded local geometry if there exist positive constants r,c, such that for every x ∈ M there is a diffeomorphism F : B(x,r) →^n
with

    1/c d(x_1,x_2) ≤ F(x_1) - F(x_2) ≤ c d(x_1,x_2)

for every x_1,x_2 ∈ B(x,r).

The injectivity radius inj(x) of x∈ M is defined as the supremum of those r>0 such that B(x,r) is simply connected or, equivalently,
as half the infimum of the lengths of the (homotopically non-trivial) loops based at x.
The injectivity radius inj(M) of M is the infimum over x∈ M of inj(x).

 
If M has positive injectivity radius and a lower bound on its Ricci curvature, then M has bounded local geometry <cit.>.


A celebrated theorem of Kanai in <cit.> states that quasi-isometries preserve isoperimetric inequalities
between Riemannian manifolds and graphs with bounded local geometry.
This result also holds with weaker hypotheses in the context of Riemann surfaces <cit.>, <cit.>.




Let X be a metric space. Fix a base point o∈ X and for
x,x'∈ X let

    (x|x')_o=1/2(d(x,o)+d(x',o)-d(x,x')).

The number (x|x')_o is non-negative and it is called the Gromov product of x,x' with respect to o.

 A metric space X is (Gromov)
hyperbolic if it satisfies the δ-inequality

    (x|y)_o≥min{(x|z)_o,(z|y)_o}-δ
 for some δ≥
0, for every base point o∈ X and all x,y,z ∈ X.


We denote by (̣X) the sharp hyperbolicity constant of X:

    (̣X)= sup{min{(x|z)_o,(z|y)_o} - (x|y)_o  |    x,y,z,o ∈ X }.

Hence, X is hyperbolic if and only if (̣X)<∞.

The theory of Gromov hyperbolic spaces was introduced by M. Gromov for
the study of finitely generated groups (see <cit.>).
The concept of Gromov hyperbolicity grasps the essence of negatively curved
spaces like the classical hyperbolic space, Riemannian manifolds of
negative sectional curvature bounded away from 0, and of discrete spaces like trees
and the Cayley graphs of many finitely generated groups. It is remarkable
that a simple concept leads to such a rich
general theory (see <cit.>).
This theory has been developed from a geometric point of view to the extent of making
hyperbolic spaces an important class of metric spaces to be studied on their
own (see, e.g., <cit.>). In the last years, Gromov hyperbolicity
has been intensely studied in graphs (see, e.g.,
<cit.> and the references therein).
Gromov hyperbolicity, specially in graphs, has found applications in different areas such as phylogenetics (see
<cit.>), real networks (see <cit.>) or the secure transmission
of information and virus propagation on networks (see <cit.>).



We want to remark that the main examples of hyperbolic graphs are the trees.
In fact, the hyperbolicity constant of a metric space can be viewed as a measure of
how “tree-like” the space is, since those spaces X with δ(X) = 0 are precisely the metric trees.
This is an interesting subject since, in
many applications, one finds that the borderline between tractable and intractable
cases may be the tree-like degree of the structure to be dealt with
(see, e.g., <cit.>).




In <cit.> we characterized, in terms of their Gromov boundary, the uniform hyperbolic graphs  and a large class of hyperbolic manifolds satisfying isoperimetric inequality (see, respectively, Theorems <ref> and <ref>).
In Theorems <ref>, <ref> and <cit.>
it is used the hypothesis of the existence of a pole
(in fact, although <cit.> apparently uses a different hypothesis, actually, in hyperbolic spaces, it is equivalent
to the existence of a pole).
The hypothesis on Gromov hyperbolicity is natural (we need it in order to deal with the Gromov boundary).
However, although the hypothesis on the pole is technically needed in the proofs, it does not look natural.
The goal of this paper is to remove this hypothesis in the statements of Theorems <ref> and <ref>. Thus, in Theorems <ref> and
<ref> we prove that having a pole is not needed as an hypothesis but it is also a necessary condition of the characterization.












§ SOME PREVIOUS RESULTS


Recall that a geodesic space is a metric space such that for every couple of points there exists a geodesic joining them.

  A geodesic space X has a pole in a point v if there
exists M > 0 such that each point of X lies in an M-neighborhood of some geodesic
ray emanating from v.






If X is a geodesic metric space and x_1,x_2,x_3∈ X, the union
of three geodesics [x_1 x_2], [x_2 x_3] and [x_3 x_1] is a
geodesic triangle that will be denoted by T={x_1,x_2,x_3}
and we will say that x_1,x_2 and x_3 are the vertices of T.
We say that T is $̣- thin if any side ofTis contained in the$̣-neighborhood of the union of the two other sides.
We denote by _̣th(T) the sharp thin constant of T, i.e., _̣th(T):=inf{≥̣0:
  T   is $̣-thin }. The space X is $̣-thin (or satisfies the  Rips condition with
constant$̣) if every geodesic triangle in X is $̣-thin.
We denote by_̣th(X)the sharp thin constant ofX, i.e.,_̣th(X):=sup{_̣th(T):
  T   is a geodesic triangle in X }. It is well-known that a geodesic metric space is hyperbolic if and only if it satisfies the Rips condition for some constant (see, e.g., <cit.>).
In the classical references on this subject (see, e.g., <cit.>)
appear many different definitions of Gromov hyperbolicity, which are equivalent in the sense
that ifXis$̣-hyperbolic with respect to one definition,
then it is '̣-hyperbolic with respect to another definition (for some '̣ related to $̣).


In order to consider a graphGas a geodesic metric space, identify (by an isometry)
any edgeuv∈ E(G)with the interval[0,1]in the real line;
then the edgeuv(considered as a graph with just one edge)
is isometric to the interval[0,1].
Thus, the points inGare the vertices and, also, the points in the interior
of any edge ofG.
In this way, any connected graphGhas a natural distance
defined on its points, induced by taking shortest paths inG,
and we can seeGas a metric graph.



Let us adapt the following definition from <cit.> where we introduce the constantε_0for convenience. Notice that for bounded metric spaces both definitions coincide. Since herein this property will be always applied to compact spaces all the results work as well with the original definition.

 Given a metric space (X,d) and a constant S>1, we say that (X,d) is S-uniformly perfect if there exists some ε_0>0 such that for every x∈ X and every 0<ε≤ε_0 there exist a point y∈ X such that
ε/S <d(x,y) ≤ε. We say that (X,d) is uniformly perfect if there exists some S such that (X,d) is S-uniformly perfect.


§ HYPERBOLIC GRAPHS


Let us recall the concepts of geodesic and sequential boundary of a hyperbolic space and some basic properties. For further information and proofs we refer the reader to <cit.>.





LetXbe a hyperbolic space ando∈ Xa base point.

The relative geodesic boundary ofXwith respect to the base-pointois

    ∂_o^g X := { [γ]    |  γ: [0,∞) → X γ(0) = o},

whereγ_1 ∼γ_2if there exists someK>0such thatd(γ_1(t),γ_2(t))<K, for everyt≥ 0.In fact, the definition above is independent from the base point.
Therefore, the set of classes of geodesic rays is called geodesic boundary ofX,∂^gX.  Herein, we do not distinguish between the geodesic ray and its image.

A sequence of points{x_i}⊂ Xconverges to infinity
if 
    lim_i,j→∞ (x_i|x_j)_o=∞.
 This property is
independent of the choice ofosince

    |(x|x')_o-(x|x')_o'|≤ d(o,o')
 for anyx,x',o,o' ∈ X.



Two sequences{x_i},{x'_i}that converge to infinity are
equivalent if 
    lim_i→∞ (x_i|x'_i)_o=∞.

Using theδ-inequality, we easily see that this defines an
equivalence relation for sequences inXconverging to infinity.
The sequential boundary at infinity∂_∞ XofXis
defined to be the set of equivalence classes of sequences
converging to infinity.

Note that given a geodesic rayγ, the sequence{γ(n)}converges to infinity and two equivalent rays induce equivalent sequences.
Thus, in general,∂^g X⊆∂_∞ X.

We say that a metric space is proper if every closed ball is compact.
Every uniform graph and every complete Riemannian manifold are proper geodesic spaces.

<cit.> If X is a proper hyperbolic geodesic space, then the natural map from ∂^g X to ∂_∞ X is a bijection.



For everyξ, ξ' ∈∂_∞ X, its Gromov product
with respect to the base pointo∈ Xis defined as

    (ξ|ξ')_o =  inf lim inf_i→∞ (x_i|x'_i)_o,

where the infimum is taken over all sequences{x_i}∈ξ,{x'_i}∈ξ' .


A metricdon the sequential boundary at infinity∂_∞ XofXis said to be visual, if there areo∈ X,a > 1and positive
constantsc_1,c_2, such that

    c_1a^-(ξ|ξ')_o≤  d(ξ, ξ') ≤ c_2a^-(ξ|ξ')_o

for allξ, ξ' ∈∂_∞ X. In this case, we say thatdis a visual metric
with respect to the base pointoand the parametera.


<cit.> Let X be a hyperbolic space. Then for any o ∈ X,
there is a_0 > 1 such that for every a ∈ (1, a_0] there exists a metric d on
∂_∞ X, which is visual with respect to o and a.
 Notice that for any visual metric, ∂_∞ X is bounded and complete.

Given a hyperbolic uniform infinite graph Γ with a pole, then h(Γ)>0 if and only if ∂_∞Γ is uniformly perfect for some visual metric.


§ HYPERBOLIC MANIFOLDS



Let us recall the following definitions from <cit.>.

A subsetAin a metric space(X,d)is called r-separated,r>0, ifd(a,a')≥ rfor any distincta,a'∈ A. Note that
ifAis maximal with this property, then the union∪_a∈
A B_r(a)coversX. A maximalr-separated setAin a metric
spaceXis called anr-approximation ofX.

LetXbe a complete Riemannian manifold and denote bydthe induced metric.
Given anyε-approximationA_εofX,
the graphΓ_A_ε=(V,E)withV=A_εandE:={xy   |   x,y ∈ A_ε 0<d(x,y)≤ 2ε}is called anε-net.




<cit.> Suppose that X is a complete Riemannian manifold with bounded local geometry and let Γ be an ε-net in X.
Then, h(X)>0 if and only if h(Γ)>0.


Note that the results in <cit.> requireMto have positive injectivity radius and a lower bound on its Ricci curvature instead of bounded local geometry,
but the proofs in <cit.> just use that there are uniform lower and upper bounds for the volume of the ballsB(x,r)which do not depend onx∈ Mfor0<r<r_0(and we have these uniform bounds with bounded local geometry).
Hence, the results in <cit.> also hold with the weaker hypothesis of bounded local geometry.

<cit.>
Suppose that X is a complete Riemannian manifold with bounded local geometry and let Γ be an ε-net in X.
Then, X and Γ are quasi-isometric.
<cit.>
If f:X → Y is a quasi-isometry between geodesic spaces, then X is hyperbolic if and only if Y is hyperbolic.
<cit.>
Suppose X,Y are proper hyperbolic geodesic spaces and f:X → Y is a quasi-isometry.
If X has a pole in v, then Y has a pole in f(v).
<cit.> Every ε-net in a complete Riemannian manifold with bounded local geometry is uniform.
<cit.>
Let X be a non-compact complete Riemannian manifold with bounded local geometry.
Assume that X is hyperbolic and has a pole.
Then, h(X)>0 if and only if ∂_∞ X is uniformly perfect.




In <cit.> appear several sufficient conditions in order to guarantee that a Riemannian manifold is hyperbolic.
See also <cit.> for the case of Riemannian surfaces.


















§ MAIN RESULTS



In this section we prove that having a pole is a necessary condition for a hyperbolic uniform graph or a complete Riemannian manifold with bounded local geometry satisfying isoperimetric inequality. Therefore, it can be removed as hypothesis in the statements of Theorems <ref> and <ref>.

In fact, we are going to prove the following results.



Given a hyperbolic uniform graph Γ, then h(Γ)>0 if and only if ∂_∞Γ is uniformly perfect for some visual metric and Γ is an infinite graph with a pole.

Let X be a hyperbolic complete Riemannian manifold with bounded local geometry.
Then, h(X)>0 if and only if ∂_∞ X is uniformly perfect and X is non-compact and has a pole.


These results are a consequence of Theorems <ref>, <ref>, and the two following results.


Let Γ be a uniform graph. If Γ is hyperbolic and h(Γ)>0, then Γ is an infinite graph with a pole.

Let X be a complete Riemannian manifold with bounded local geometry.
If X is hyperbolic and h(X)>0, then X is non-compact and has a pole.



Let us now proceed with the proof of Theorem <ref>.


Since h(Γ)>0, we have that Γ is an infinite graph.


Seeking for a contradiction assume that Γ does not have a pole.
Let μ be a constant such that  is μ-uniform.

Fix v∈ V() and denote by K the union of the geodesic rays starting from v.
Since Γ does not have a pole, for each n there exists v_n∈ V() with d_(v_n,K) ≥ 4n.
Let η_n : [0,ℓ_n] → be a geodesic joining v_n with K and such that η_n(0)= v_n and
d_(v_n,K) = L(η_n) = ℓ_n  ≥ 4n.

Given s ∈, denote by ⌊ s ⌋ the lower integer part of s, i.e., the largest integer not greater than s.
Assume that the ball B_(η_n(t),⌊_̣th() ⌋+1 ) intersects every geodesic joining v_n with some point of K,
for some t ∈^+ with ⌊_̣th() ⌋  < t ≤ 4n ≤ℓ_n.
Thus, the connected component A_n of ∖ B_(η_n(t),⌊_̣th() ⌋+1 ) containing v_n satisfies
that its boundary A_n is contained in the sphere S_(η_n(t),⌊_̣th() ⌋).

Since  is μ-uniform,

|  A_n |
≤| S_(η_n(t),⌊_̣th() ⌋) |
≤μ^⌊_̣th() ⌋ .

Thus

t - ⌊_̣th() ⌋
= | V() ∩η_n ([ 0, t - ⌊_̣th() ⌋ - 1 ] ) |
≤ | A_n |
≤ h()^-1 |  A_n |
≤ h()^-1μ^⌊_̣th() ⌋ ,

and we conclude that

t ≤⌊_̣th() ⌋ + h()^-1μ^⌊_̣th() ⌋
=: M .


Hence, if 2n> M, there exists a geodesic _n from v_n to K such that
_n ∩ B_(η_n(2n),⌊_̣th() ⌋+1 ) = ∅.
Note that d_(η_n(2n),K) = ℓ_n - 2n ≥ 2n.
Let x_n and y_n be the endpoints of η_n and _n in K, respectively,
and consider the geodesic triangle T^n={v_n,x_n,y_n} in .
Since _̣th() < ⌊_̣th() ⌋+1,
we have

d_(η_n(2n),_n)
≥⌊_̣th() ⌋+1
> _̣th(),

and so, there exists z_n ∈ [x_ny_n] with d_(η_n(2n),z_n) ≤_̣th() (see Figure <ref>).


    < g r a p h i c s >
If there is a geodesic _n=[v_ny_n] from v_n to K which does not intersect the ball  B_(η_n(2n),'̣), where '̣:=⌊_̣th() ⌋+1, then there is a point z_n∈ [x_ny_n] which is far from K.


Consider now the geodesic triangle T_n={v,x_n,y_n} in , and z_n ∈ [x_ny_n].
Since [vx_n]∪ [vy_n] ⊂ K,
we have
_̣th()
≥ d_(z_n,[vx_n]∪ [vy_n])
≥ d_(z_n,K)
≥ d_(η_n(2n),K) - d_(η_n(2n),z_n)
≥ 2n - _̣th(),

and so, _̣th() ≥ n for every n.
This is the contradiction we were looking for, since  is hyperbolic, and we conclude that Γ has a pole.



Finally, let us now proceed with the proof of Theorem <ref>.


Since h(X)>0, we have that X is non-compact.

Let Γ be an ε-net in X.
Since h(X)>0, Proposition <ref> gives that h(Γ)>0.

Note that X is a proper geodesic space since it is a complete Riemannian manifold.
Also,  is a proper geodesic space since it is a uniform graph by Lemma <ref>.

By Proposition <ref>, X and Γ are quasi-isometric and so, Γ is hyperbolic by Theorem <ref>.
Theorem <ref> gives that Γ has a pole, and
by Proposition <ref>, X has a pole.
 Theorem <ref> also allows to improve Theorem 4.12 in <cit.> which is based in the same results. In the same way, having a pole is a necessary condition and can be removed as hypothesis.
 Theorem 5.11 in <cit.> states that for geodesic, visual, Gromov hyperbolic spaces, having uniformly perfect boundary at infinity is equivalent to being uniformly equilateral. Thus, theorems <ref> and <ref> can also be written using this property and without using the boundary at infinity.
 Acknowledgments:
We would like to thank to Paloma Martín for her support during this research.
This work was supported in part by two grants
from Ministerio de Economía y Competitividad, Agencia Estatal de
Investigación (AEI) and Fondo Europeo de Desarrollo Regional (FEDER) (MTM2015-63612P, MTM2016-78227-C2-1-P and MTM2017-90584-REDT), Spain.










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