CHEEGER ISOPERIMETRIC CONSTANTS OF GROMOV-HYPERBOLIC SPACES WITH QUASI-POLES

2000 ◽  
Vol 02 (04) ◽  
pp. 511-533 ◽  
Author(s):  
JIANGUO CAO
Keyword(s):  
Martin Boundary ◽  
Hyperbolic Manifolds ◽  
Hyperbolic Spaces ◽  
Connected Components ◽  
Local Geometry ◽  
Complete Manifold ◽  
Cheeger Constant ◽  
Gromov Hyperbolic ◽  
Geometric Boundary ◽  
Dirichlet Problem At Infinity

Let X be a non-compact complete manifold (or a graph) which admits a quasi-pole and has bounded local geometry. Suppose that X is Gromov-hyperbolic and the diameters (for a fixed Gromov metric) of the connected components of X(∞) have a positive lower bound. Under these assumptions we show that X has positive Cheeger isoperimetric constant. Examples are also constructed to show that the Cheeger constant h(X) may be zero if any of the above assumption on X is removed. Applications of this isoperimetric estimate include the solvability of the Dirichlet problem at infinity for non-compact Gromov-hyperbolic manifolds X above. In addition, we show that the Martin boundary ∂ΔX of such a space X is homeomorphic to the geometric boundary X(∞) of X at infinity.

scholarly journals Cheeger isoperimetric constant of Gromov hyperbolic manifolds and graphs

2018 ◽  
Vol 20 (05) ◽  
pp. 1750050 ◽  
Author(s):  
Alvaro Martínez-Pérez ◽  
José M. Rodríguez
Keyword(s):  
Isoperimetric Inequality ◽  
Riemannian Manifolds ◽  
Martin Boundary ◽  
Hyperbolic Manifolds ◽  
Local Geometry ◽  
Gromov Hyperbolic ◽  
Relationship Of ◽  
The Relationship ◽  
Dirichlet Problem At Infinity ◽  
Gromov Boundary

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. Furthermore, we characterize the trees with isoperimetric inequality (without any hypothesis). As an application of our results, we obtain the solvability of the Dirichlet problem at infinity for these Riemannian manifolds and graphs, and that the Martin boundary is homeomorphic to the Gromov boundary.

scholarly journals A note on isoperimetric inequalities of Gromov hyperbolic manifolds and graphs

2021 ◽  
Vol 115 (3) ◽  
Author(s):  
Álvaro Martínez-Pérez ◽  
José M. Rodríguez
Keyword(s):  
Isoperimetric Inequality ◽  
Riemannian Manifolds ◽  
Hyperbolic Manifolds ◽  
Necessary Condition ◽  
Local Geometry ◽  
Gromov Hyperbolic ◽  
Relationship Of ◽  
The Relationship ◽  
Gromov Boundary

AbstractWe study in this paper the relationship of isoperimetric inequality and hyperbolicity for graphs and Riemannian manifolds. We obtain a characterization of graphs and Riemannian manifolds (with bounded local geometry) satisfying the (Cheeger) 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 to have isoperimetric inequality and, therefore, it can be removed as hypothesis.

scholarly journals The asymptotic Dirichlet problems on manifolds with unbounded negative curvature

2018 ◽  
Vol 167 (01) ◽  
pp. 133-157
Author(s):  
RAN JI
Keyword(s):  
Dirichlet Problem ◽  
Negative Curvature ◽  
Probabilistic Method ◽  
Martin Boundary ◽  
Dirichlet Problems ◽  
Curvature Condition ◽  
New Approach ◽  
Harnack Inequalities ◽  
Geometric Boundary ◽  
Analytical Proof

AbstractElton P. Hsu used probabilistic method to show that the asymptotic Dirichlet problem is uniquely solvable under the curvature condition −Ce(2−η)r(x) ≤ KM(x) ≤ −1 with η > 0. We give an analytical proof of the same statement. In addition, using this new approach we are able to establish two boundary Harnack inequalities under the curvature condition −Ce(2/3−η)r(x) ≤ KM(x) ≤ −1 with η > 0. This implies that there is a natural homeomorphism between the Martin boundary and the geometric boundary of M. As far as we know, this is the first result of this kind under unbounded curvature conditions. Our proof is a modification of an argument due to M. T. Anderson and R. Schoen.

scholarly journals FREELY INDECOMPOSABLE GROUPS ACTING ON HYPERBOLIC SPACES

2004 ◽  
Vol 14 (02) ◽  
pp. 115-171 ◽  
Author(s):  
ILYA KAPOVICH ◽  
RICHARD WEIDMANN
Keyword(s):  
Hyperbolic Group ◽  
Conjugacy Classes ◽  
Hyperbolic Spaces ◽  
Gromov Hyperbolic ◽  
Torsion Free

We obtain a number of finiteness results for groups acting on Gromov-hyperbolic spaces. In particular we show that a torsion-free locally quasiconvex hyperbolic group has only finitely many conjugacy classes of n-generated one-ended subgroups.

scholarly journals Hyperbolic Unfoldings of Minimal Hypersurfaces

2018 ◽  
Vol 6 (1) ◽  
pp. 96-128 ◽  
Author(s):  
Joachim Lohkamp
Keyword(s):  
Point Of View ◽  
Hyperbolic Spaces ◽  
Intrinsic Geometry ◽  
Minimal Hypersurfaces ◽  
Bounded Geometry ◽  
New Concepts ◽  
Gromov Hyperbolic ◽  
Area Minimizing ◽  
Conformal Deformations ◽  
Gromov Boundary

Abstract We study the intrinsic geometry of area minimizing hypersurfaces from a new point of view by relating this subject to quasiconformal geometry. Namely, for any such hypersurface H we define and construct a so-called S-structure. This new and natural concept reveals some unexpected geometric and analytic properties of H and its singularity set Ʃ. Moreover, it can be used to prove the existence of hyperbolic unfoldings of H\Ʃ. These are canonical conformal deformations of H\Ʃ into complete Gromov hyperbolic spaces of bounded geometry with Gromov boundary homeomorphic to Ʃ. These new concepts and results naturally extend to the larger class of almost minimizers.

scholarly journals Rough isometry between Gromov hyperbolic spaces and uniformization

2021 ◽  
Vol 46 (1) ◽  
pp. 449-464
Author(s):  
Jeff Lindquist ◽  
Nageswari Shanmugalingam
Keyword(s):  
Hyperbolic Spaces ◽  
Gromov Hyperbolic ◽  
Rough Isometry

scholarly journals On deformation spaces of nonuniform hyperbolic lattices

2016 ◽  
Vol 161 (2) ◽  
pp. 283-303 ◽  
Author(s):  
SUNGWOON KIM ◽  
INKANG KIM
Keyword(s):  
Hyperbolic Manifolds ◽  
Rigidity Theorem ◽  
Connected Components ◽  
Connected Component ◽  
Deformation Spaces ◽  
Representation Variety ◽  
The Real ◽  
Local Rigidity ◽  
Semialgebraic Subset ◽  
Hyperbolic Lattices

AbstractLet Γ be a nonuniform lattice acting on the real hyperbolic n-space. We show that in dimension greater than or equal to 4, the volume of a representation is constant on each connected component of the representation variety of Γ in SO(n, 1). Furthermore, in dimensions 2 and 3, there is a semialgebraic subset of the representation variety such that the volume of a representation is constant on connected components of the semialgebraic subset. Combining our approach with the main result of [2] gives a new proof of the local rigidity theorem for nonuniform hyperbolic lattices and the analogue of Soma's theorem, which shows that the number of orientable hyperbolic manifolds dominated by a closed, connected, orientable 3-manifold is finite, for noncompact 3-manifolds.

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