The short time asymptotics of Nash entropy

TL;DR

This paper investigates the short-time asymptotic behavior of Nash entropy and its derivative on complete Riemannian manifolds with Ricci curvature bounds, providing precise formulas as time approaches zero.

Contribution

It derives the asymptotic formulas for Nash entropy and its derivative at zero time and establishes a Hamilton-type Laplacian bound for heat equation solutions.

Findings

Asymptotic formulas for Nash entropy as t approaches 0

Asymptotic behavior of the derivative of Nash entropy at t=0

Hamilton-type Laplacian upper bound for heat solutions

Abstract

Let $(M^n, g)$ be a complete Riemannian manifold with $Rc\geq -Kg$, $H(x, y, t)$ is the heat kernel on $M^n$, and $H= (4\pi t)^{-\frac{n}{2}}e^{-f}$. Nash entropy is defined as $N(H, t)= \int_{M^n} (fH) d\mu(x)- \frac{n}{2}$. We studied the asymptotic behavior of $N(H, t)$ and $\frac{\partial}{\partial t}\Big[N(H, t)\Big]$ as $t\rightarrow 0^{+}$, and got the asymptotic formulas at $t= 0$. In the Appendix, we got Hamilton-type upper bound for Laplacian of positive solution of the heat equation on such manifolds, which has its own independent interest.