Modified isoperimetric constants, and large time heat diffusion in Riemannian manifolds
The paper is devoted to estimates on the minimal positive heat kernel \(p(x,y,t)\) on a noncompact Riemannian manifold \(M\) for \(x,y\in M\), \(t>0\). The authors introduce the notion of a modified isoperimetric constant \[ I_{v,\rho}(M)=\inf_ \Omega(A(\delta\Omega)/V(\Omega)^{(v- 1)/v)}\quad\text{ where }v>1,\;\rho>0, \] \(\Omega\) varies over open submanifolds in \(M\) with compact closure and smooth boundary \(\delta\Omega\) which contain a closed metric disk of radius \(\rho\), \(A\) and \(V\) are Riemannian measures on \(\delta\Omega\) and on \(\Omega\), respectively. The authors say that \(M\) has bounded geometry if its Ricci curvature is bounded uniformly from below and if its injectivity radius is bounded uniformly away from 0. It is shown in the main result of the paper that for a complete \(M\) with \(\dim \geq 2\) and with bounded geometry, if for any \(v>1\) and \(\rho>0\) we have \(I_{v,\rho}(M)>0\), then \[ p(x,y,t)\leq \text{const}_{v,s}t^{-v/2}e^{-d^ 2(x,y)/4(1+\delta)t} \] is valid for sufficiently large \(t\), where \(d\) is the distance on \(M\), and there exists a minimal positive Green's function satisfying a certain estimate with respect to \(d\). So the use of modified isoperimetric constants instead of usual ones avoids some difficulties, posed by strictly local phenomena. The estimates on \(p(x,y,t)\) with respect to time \(t\) are established, the corresponding results for finite connected graphs are obtained.
scientific article; zbMATH DE number 58537
scientific article