The Entropy Formula for Linear Heat Equation
نویسندگان
چکیده
§0 Introduction. In a recent paper of Perelman[P1], an entropy formula for Ricci flow was derived. The formula turns out being of fundamental importance in the study of Ricci flow (cf. [P1, Sections 3, 4, 10]) as well as the Kähler-Ricci flow [P2]. The derivation of the entropy formula in [P1, Section 9] resembles the gradient estimate for the linear heat equation proved by Li-Yau in another fundamental paper [L-Y] on the linear parabolic equation. This suggests that one might has similar entropy formula for the linear parabolic equation. This note is to confirm this and show some applications of the entropy formula for the linear case. Let M be a complete Riemannian manifold. We study the heat equation
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The Entropy Formula for Linear Heat Equation 87 Corollary 1
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