Vol. 266, No. 2, 2013

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The short time asymptotics of Nash entropy

Guoyi Xu

Vol. 266 (2013), No. 2, 423–447
Abstract

Let (Mn,g) be a complete Riemannian manifold with Rc ≥−Kg, H(x,y,t) be the heat kernel on Mn, and H = (4πt)n∕2ef. Nash entropy is defined as N(H,t) = Mn(fH)(x) n∕2. We study the asymptotic behavior of N(H,t) and ∂N(H,t)∂t as t 0+ and get the asymptotic formulas at t = 0. In the appendix, we get a Hamilton-type upper bound for the Laplacian of the positive solution to the heat equation on such manifolds, which is itself interesting.

Keywords
Nash entropy, short time asymptotics
Mathematical Subject Classification 2010
Primary: 35K15, 53C44
Milestones
Received: 8 October 2012
Revised: 16 November 2012
Accepted: 3 December 2012
Published: 12 November 2013
Authors
Guoyi Xu
Mathematics Department
University of California, Irvine
340 Rowland Hall
Irvine, CA 92697
United States
Mathematical Sciences Center
Tsinghua University
100084 Beijing
China