Entropy and lowest eigenvalue on evolving manifolds

TL;DR

This paper analyzes the derivatives of entropy functionals on evolving manifolds, constructs related entropies, and derives eigenvalue evolution formulas, with applications to various geometric flows and curvature conditions.

Contribution

It introduces a framework for the derivatives of entropy functionals on evolving manifolds and connects these to eigenvalue evolution, extending Perelman's entropy concepts.

Findings

Monotonicity of entropies under certain geometric flows.

Explicit formula for the evolution of the lowest eigenvalue.

Applicability to static and dynamic curvature conditions.

Abstract

In this note 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 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 nonnegative Ricci curvature, for Hamilton's Ricci flow, List's extended Ricci flow, M\"uller'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.