A New Entropy Formula and Gradient Estimates for the Linear Heat Equation on Static Manifold

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Abimbola Abolarinwa, A New Entropy Formula and Gradient Estimates for the Linear Heat Equation on Static Manifold, Int. J. Anal. Appl., 6 (1) (2014), 1-17.

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Abimbola Abolarinwa

Abstract

In this paper we prove a new monotonicity formula for the heat equation via a generalized family of entropy functionals. This family of entropy formulas generalizes both Perelman's entropy for evolving metric and Ni's entropy on static manifold. We show that this entropy satisfies a pointwise differential inequality for heat kernel. The consequences of which are various gradient and Harnack estimates for all positive solutions to the heat equation on compact manifold.

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