The short time asymptotics of Nash entropy

Let $(M^n,g)$ be a complete Riemannian manifold with $Rc\ge -Kg$, $H(x, y, t)$istheheatkernelon$M^n$,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.