Entropy and lowest eigenvalue on evolving manifolds
Entropy and lowest eigenvalue on evolving manifolds
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 Ᏺ and ᐃ entropy in abstract geometric flows.Monotonicity of the entropies holds when a technical condition is satisfied.This condition …