On \(p\)-parabolicity of Riemannian manifolds and graphs
type: Item
instance of: scholarly article
title: On \(p\)-parabolicity of Riemannian manifolds and graphs
zbMATH Open document ID: 1492.53061
author: Álvaro Martínez-Pérez
author: José M. Rodríguez
published in: Revista Matemática Complutense
publication date: 2022-01-26T00:00:00Z
review text: Let \(f : X \rightarrow Y\) be a quasi-isometry between complete Riemannian manifolds. Let the Ricci curvature of $Y$ be bounded below. Let \(\epsilon > 0\) and $p$ with \(1 < p < \infty\) be given, and let $X$ satisfy certain conditions, depending on \(\epsilon\) and $p$, on the volume of small metric balls of $X$. The authors show that $Y$ is $p$-parabolic if $X$ is $p$-parabolic. This work and the strategy of the proof are related to results of \textit{M. Kanai} [Lect. Notes Math. 1201, 122--137 (1986; Zbl 0593.53026)]. Let $X$ and $Y$ be metric spaces. A function \(f : X \rightarrow Y\) is a \textit{quasi-isometry} if (1) there exist constants \(a \geq 1\) and \(b \geq 0\) such that \(\frac{1}{a}~ d(x_{1}, x_{2}) - b \leq d(f(x_{1}) , f(x_{2})) \leq a~d(x_{1}, x_{2}) + b\) for all \(x_{1},x_{2} \in X\), (2) there exists a positive constant r such that every point of $Y$ is within distance r of some point of \(f(X)\). Being quasi-isometric is an equivalence relation on metric spaces. It is of interest to determine geometric quantities of metric spaces that are preserved by quasi-isometries, particularly when the metric spaces are complete Riemannian manifolds. Note that local geometry is not preserved by quasi-isometries; for example, if $Y$ is a compact metric space and $X$ is any subset of $Y$, then the inclusion map \(i : X \rightarrow Y\) is a quasi-isometry, where \(a = 1\) and \(b = 0\). Hence quasi-isometric invariants must involve large scale geometry. One expects that some additional restrictions on the local geometry of $X$ and $Y$ will be needed. In the 1980s Kanai considered complete, noncompact Riemannian manifolds $X$ and $Y$, where the Ricci curvature of $X$ is bounded below and the injectivity radius is positive. The latter requirement implies that there exists \(\epsilon > 0\) such that every metric ball of radius \(r < \epsilon\) in $X$ is diffeomorphic to an $r$-ball in \(\mathbb{R}^{n}, n = \dim~X\). Kanai determined a number of geometric prop…
Mathematics Subject Classification ID: 53C23
Mathematics Subject Classification ID: 53C21
Mathematics Subject Classification ID: 58C40
Mathematics Subject Classification ID: 30F45
Mathematics Subject Classification ID: 53C20
Mathematics Subject Classification ID: 30F20
Mathematics Subject Classification ID: 31C12
zbMATH DE Number: 7464331
zbMATH Keywords: parabolicity
zbMATH Keywords: \(p\)-parabolicity
zbMATH Keywords: bounded local geometry
zbMATH Keywords: quasi-isometries
reviewed by: Patrick Eberlein
Wikidata QID: Q115376162
MaRDI profile type: Publication
full work available at URL: s13163-021-00387-x
OpenAlex ID: W3135746189
cites work: Estimates for nonlinear harmonic ''measures'' on trees
cites work: On harmonic functions on trees
cites work: Quasi-isometries and isoperimetric inequalities in planar domains
cites work: Differential equations on riemannian manifolds and their geometric applications
cites work: Isometric Riemannian manifolds at infinity
cites work: Parabolic Harnack inequality and estimates of Markov chains on graphs
cites work: Q3706275
cites work: On the Existence of Green's Function in Riemannian Manifolds
cites work: The exponent of convergence of Riemann surfaces. Bass Riemann surfaces
cites work: Area growth and Green's function of Riemann surfaces
cites work: Stability of the volume growth rate under quasi-isometries
cites work: Stability of the injectivity radius under quasi-isometries and applications to isoperimetric inequalities
cites work: Q3772406
cites work: Sur les groupes hyperboliques d'après Mikhael Gromov. (On the hyperbolic groups à la M. Gromov)
cites work: Q3996137
cites work: Rough isometries and \(p\)-harmonic functions with finite Dirichlet integral
cites work: \(p\)-harmonic functions on graphs and manifolds
cites work: Trees and ultrametric spaces: A categorical equivalence
cites work: A framework for non-homogeneous analysis on metric spaces, and the RBMO space of Tolsa
cites work: Rough isometries, and combinatorial approximations of geometries of non- compact Riemannian manifolds
cites work: Rough isometries and the parabolicity of Riemannian manifolds
cites work: Q3723243
cites work: Existence et approximation des solutions des équations aux dérivées partielles et des équations de convolution
cites work: Uniformly continuous maps between ends of \(\mathbb{R}\)-trees
cites work: Cheeger isoperimetric constant of Gromov hyperbolic manifolds and graphs
cites work: Isoperimetric inequalities in Riemann surfaces and graphs
cites work: Gromov hyperbolicity through decomposition of metrics spaces. II
cites work: A characterization of Gromov hyperbolicity of surfaces with variable negative curvature
cites work: Isoperimetric inequalities and Dirichlet functions of Riemann surfaces
cites work: Two remarks on Riemann surfaces
cites work: Classification theory of Riemannian manifolds. Harmonic, quasiharmonic and biharmonic functions
cites work: Potential theory on infinite networks
cites work: Graphs and Gromov hyperbolicity of non-constant negatively curved surfaces
DOI: 10.1007/S13163-021-00387-X
description: scientific article; zbMATH DE number 7464331