Heat flow in Riemannian manifolds with non-negative Ricci curvature

2015/06/23 by M. van den Berg, Berg, Michiel van den
Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.1506.07063

openalex publication_date 2015/06/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let Ω be an open set in a geodesically complete, non-compact, m-dimen-sional Riemannian manifold M with non-negative Ricci curvature, and without boundary. We study the heat flow from Ω into M-Ω if the initial temperature distribution is the characteristic function of Ω. We obtain a necessary and sufficient condition which ensures that an open set Ω with infinite measure has finite heat content for all t>0. We also obtain upper and lower bounds for the heat content of Ω in M. Two-sided bounds are obtained for the heat loss of Ω in M if the measure of Ω is finite.

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