Large time behavior of the heat equation on complete manifolds with non- negative Ricci curvature
type: Item
instance of: scholarly article
title: Large time behavior of the heat equation on complete manifolds with non- negative Ricci curvature
zbMATH Open document ID: 0613.58032
DOI: 10.2307/1971385
author: Peter Li
published in: Annals of Mathematics. Second Series
publication date: 1986-01-01T00:00:00Z
review text: It is shown that if a complete Riemannian manifold M has nonnegative Ricci curvature and if the volume growth of M is of maximum order then \(\lim_{t\to \infty}V(B_ x(\sqrt{t}))H(x,y,t)\) exists, where \(V(B_ x(\sqrt{t}))\) denotes the volume of the geodesic ball centered at x of radius \(\sqrt{t}\) and H(x,y,t) is the heat kernel, i.e. H(x,y,t) solves the heat equation \[ (\Delta -\partial /\partial t)F(x,t)=0\quad on\quad M\times (0,\infty), \] with initial data \(F(x,0)=f(x)\), by setting \(F(x,t)=\int_{M}H(x,y,t)f(y)dy\). An interesting consequence of this result is derived: A nonnegatively Ricci-curved manifold with maximal volume growth must have finite fundamental group. Further, the asymptotic formula is applied to obtain a necessary and sufficient condition for the heat equation with bounded initial data to stabilize as t. Finally, the heat equation is used to study the behavior of bounded superharmonic functions on complete manifolds with nonnegative Ricci curvature.
Mathematics Subject Classification ID: 58J35
Mathematics Subject Classification ID: 35K05
Mathematics Subject Classification ID: 53C20
zbMATH DE Number: 3992498
zbMATH Keywords: heat kernel
zbMATH Keywords: heat equation
zbMATH Keywords: asymptotic formula
zbMATH Keywords: manifolds with nonnegative Ricci curvature
reviewed by: Marek Fila
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2307/1971385
OpenAlex ID: W2002133517
description: scientific article; zbMATH DE number 3992498