Abstract
We derive the entropy formula for the linear heat equation on general Riemannian manifolds and prove that it is monotone non-increasing on manifolds with nonnegative Ricci curvature. As applications, we study the relation between the value of entropy and the volume of balls of various scales. The results are simpler version, without Ricci flow, of Perelman ’s recent results on volume non-collapsing for Ricci flow on compact manifolds. We also prove that if the entropy for the heat kernel achieves its maximum value zero at some positive time, on any complete Riamannian manifold with nonnegative Ricci curvature, if and only if the manifold is isometric to the Euclidean space.
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