THE SHORT TIME ASYMPTOTICS OF NASH ENTROPY

Let (M-n, g) be a complete Riemannian manifold with Rc >= -Kg, H(x, y, t) be the heat kernel on M-n, and H = (4 pi t)(-n/2)e(-f). Nash entropy is defined as N(H, t) = integral(Mn) (f H) d mu(x) - n/2. We study the asymptotic behavior of N(H, t) and partial derivative N(H, t)/partial derivative t as t -> 0(+) and get the asymptotic formulas at t = 0. In the appendix, we get a Hamilton-type upper bound for the Laplacian of the positive solution to the heat equation on such manifolds, which is itself interesting.