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Entropy and lowest eigenvalue on evolving manifolds

by Hongxin Guo, Robert Philipowski and Anton Thalmaier

**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, 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.

We then derive explicit formula for the evolution equation of the lowest
eigenvalue of the Laplace-Beltrami operator with potential in the abstract
setting. The lowest eigenvalue is monotone under the same technical
assumption. In particular the lowest eigenvalue is nondecreasing in the above
mentioned geometric flows.

*Pacific J. Math. ***264** (2013) 61-82

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