Vol. 264, No. 1, 2013

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ISSN: 0030-8730
Entropy and lowest eigenvalue on evolving manifolds

Hongxin Guo, Robert Philipowski and Anton Thalmaier

Vol. 264 (2013), No. 1, 61–81
Abstract

We determine the first two derivatives of the classical Boltzmann–Shannon entropy of the conjugate heat equation on general evolving manifolds. Based on the second derivative of the Boltzmann–Shannon entropy, we construct Perelman’s and 𝒲 entropy in abstract geometric flows. Monotonicity of the entropies holds when a technical condition is satisfied.

This condition is satisfied on static Riemannian manifolds with nonnegative Ricci curvature, for Hamilton’s Ricci flow, List’s extended Ricci flow, Müller’s Ricci flow coupled with harmonic map flow and Lorentzian mean curvature flow when the ambient space has nonnegative sectional curvature.

Under the extra assumption that the lowest eigenvalue is differentiable along time, we derive an explicit formula for the evolution of the lowest eigenvalue of the Laplace–Beltrami operator with potential in the abstract setting.

Keywords
Ricci flow, conjugate heat equation, entropy, eigenvalue
Mathematical Subject Classification 2010
Primary: 53C44
Milestones
Received: 30 January 2012
Revised: 23 October 2012
Accepted: 26 December 2012
Published: 5 July 2013
Authors
Hongxin Guo
School of Mathematics and Information Science
Wenzhou University
Chashan University Town
Wenzhou, Zhejiang 325035
China
Robert Philipowski
Mathematics Research Unit
University of Luxembourg
6, rue Richard Coudenhove-Kalergi
L-1359 Luxembourg
Luxembourg
Anton Thalmaier
Mathematics Research Unit
University of Luxembourg
6, rue Richard Coudenhove-Kalergi
L-1359 Luxembourg
Luxembourg
http://math.uni.lu/thalmaier/