Some gradient estimates for the heat equation on domains and for an equation by Perelman

Details Some gradient estimates for the heat equation on domains and for an equation by Perelman Some gradient estimates for the heat equation on domains and for an equation by Perelman

2006-05-18

arxiv.org/abs/math/0605518v3

International Math. Research Notices 2006

Mathematics

Differential Geometry

Scientific paper

In the first part, we derive a sharp gradient estimate for the log of Dirichlet heat kernel and Poisson heat kernel on domains, and a sharpened local Li-Yau gradient estimate that matches the global one. In the second part, without explicit curvature assumptions, we prove a global upper bound for the fundamental solution of an equation introduced by G. Perelman, i.e. the heat equation of the conformal Laplacian under backward Ricci flow. Further, under nonnegative Ricci curvature assumption, we prove a qualitatively sharp, global Gaussian upper bound.

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