On \(p\)-parabolicity of Riemannian manifolds and graphs
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 properties invariant under quasi-isometries, including polynomial volume growth of metric $r$-balls as \(r \rightarrow \infty\) and positivity of the isoperimetric and Sobolev constants \(I_{m}(X), m \geq max\{\dim~X, \dim~Y \}\) and \(S_{\ell , m}(X), \ell \geq m / m-1,1 \leq m \leq \infty\). The Cheeger and Buser constants arise when \(m = \infty\) and \(\ell = m = 2\) respectively. Kanai's strategy was to reduce the original geometric problem to a simpler discrete version of it. A discrete set $P$ in a Riemannian manifold $X$ is said to be an \(\epsilon\)-net if it is a maximal \(\epsilon\)-separated set, equipped with a set \(N_{(p)} = \{q \in P : d(p,q) < 3 \epsilon \}\) of neighbors of $p$ for every \(p \in P\). By the maximality of $P$ it follows that $P$ is \(\epsilon\)-dense in $X$, and it follows from the definition that the inclusion map \(i : P \rightarrow X\) is a quasi-isometry. In particular, if $P$ and $Q$ are \(\epsilon\)-nets in quasi-isometric Riemannian manifolds $X$ and $Y$, then $P$ and $Q$ are quasi-isometric. Given a quasi-isometry \(f : X \rightarrow Y\) and a geometric property of $X$ and $Y$ Kanai considered discrete analogues of the property on \(\epsilon\)-nets $P$ of $X$ and $Q$ of $Y$. Using combinatorial arguments he showed in several cases that a geometric property is a quasi-isometric invariant of $X$ and $Y$ if and only if the discrete analogue of this property is a quasi-isometric invariant of $P$ and $Q$. The discrete property is often easier to deal with than the original property. Let \(\Delta : C^{\infty}(X) \rightarrow C^{\infty}(X)\) denote the Laplace operator on a Riemannian manifold $X$. The manifold $X$ is said to be \(\mathit{parabolic}\) if the only smooth positive functions $u$ on $X$ with \(\Delta(u) \leq 0\) are constant functions. Let $X$ and $Y$ be quasi-isometric complete Riemannian manifolds with Ricci curvature bounded below and with positive injectivity radius. Using the discretizing strategy described above Kanai proved that $X$ is parabolic \(\Leftrightarrow\) $Y$ is parabolic. In the present article the authors use a similar discretizing strategy to show that under certain conditions on the local geometry $p$-parabolicity is a quasi-isometric invariant of complete Riemannian manifolds. A Riemannian manifold is $p$-parabolic for \(1 < p < \infty\) if there are no nonconstant smooth positive functions u on M such that \(\Delta_{p}(u) \leq 0\), where \(\Delta_{p}(u) = - \operatorname{div}(|\nabla(u)|^{p-2}~\nabla(u))\). The authors replace the Kanai assumptions of Ricci curvature bounded below and positive injectivity radius with three weaker conditions that help to regulate the local geometry. (1) A metric space $X$ is said to be \textit{doubling} if there exists a positive integer $M$ such that the metric ball \(B_{2 \epsilon}(x)\) can be covered by at most $M$ metric \(\epsilon\)-balls for every \(\epsilon > 0\) and every \(x \in X\). (2) For \(1 < p < \infty\) and \(\epsilon > 0\) a metric space $X$ is said to be \((\epsilon, p)\) \textit{nice} if there exist positive constants \(k_{1}, k_{2}\) such that for every \(x_{0} \in X\) there exists a positive number \(V(x_{0})\) with the property that \(k_{1} V(x_{0})^{\frac{p-1}{p}} \leq vol~B_{\epsilon}(x) \leq \mathrm{vol}~B_{4 \epsilon}(x) \leq k_{2}~V(x_{0})\) for every \(x \in \overline{B_{3 \epsilon}(x_{0})}\). (3) A Riemannian manifold $X$ is said to satisfy the local Poincare inequality if for every \(\epsilon > 0\) and every smooth function $u$ on $M$ with compact support there exists a positive number \(\beta\) such that \(\int_{B_{\epsilon}(x)} ~|\nabla(u)| \geq \beta \int_{B_{\epsilon}(x)}~|u - u^{*}|\), where \(u^{*}\) is the average value of u on \(B_{\epsilon}(x)\). The main result of this paper is the following: Theorem. Let $X$ and $Y$ be complete, quasi-isometric Riemannian manifolds. Let $X$ be doubling, satisfy the local Poincaré inequality and be \((\epsilon, p)\)-nice for some \(1 < p < \infty\) and some \(\epsilon > 0\). Let $Y$ have Ricci curvature bounded below. If $X$ is $p$-parabolic, then $Y$ is $p$-parabolic. The authors also define $p$-parabolicity on graphs and trees and obtain several results, including a characterization of $p$-parabolicity for a large class of trees.
scientific article; zbMATH DE number 7464331
scientific article