The short time asymptotics of Nash entropy

Abstract

Let $(M^n,g)$ be a complete Riemannian manifold with $Rc\ge -Kg$, $H(x,y,t)$ 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 $\frac{\partial }{\partial t}[N(H,t)]$ as $t\to 0^{+}$, 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.

arXiv comment22pp, reference added, to appear on Pacific Journal of Math