The short time asymptotics of Nash entropy

Guoyi Xu

Vol. 266 (2013), No. 2, 423–447

DOI: 10.2140/pjm.2013.266.423

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∕2e^−f. Nash entropy is defined as N(H,t) = ∫ _Mⁿ(fH)dμ(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