ENTROPY AND LOWEST EIGENVALUE ON EVOLVING MANIFOLDS

By Hongxin Guo, Robert Philipowski, Anton Thalmaier. 2013

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HONGXIN GUO, ROBERT PHILIPOWSKI, AND ANTON THALMAIERAbstract. In this note we determine the first two derivatives of the clas-sical Boltzmann-Shannon entropy of the conjugate heat equation on generalevolving manifolds. Based on the second derivative of the Boltzmann-Shannonentropy, we construct Perelman’s F 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 nonnegativeRicci curvature, for Hamilton’s Ricci flow, List’s extended Ricci flow, Mu¨ller’sRicci flow coupled with harmonic map flow and Lorentzian mean curvatureflow when the ambient space has nonnegative sectional curvature.Under the extra assumption that the lowest eigenvalue is differentiablealong time, we derive an explicit formula for the evolution of the lowest eigen-value of the Laplace-Beltrami operator with potential in the abstract setting.

Subfields: Applied Mathematics, Astronomy and Astrophysics

View Paper: https://www.semanticscholar.org/paper/888921ae8c43c9babe7c2ae3d17c7ae107f20a0b