Li-Yau gradient bounds on compact manifolds under nearly optimal curvature conditions

We prove Li-Yau type gradient bounds for the heat equation either on manifolds with fixed metric or under the Ricci flow. In the former case the curvature condition is $|Ric^-| \in L^p$ for some $p>n/2$, or $\sup_\M \int_\M |Ric^-|^2(y)d^{2-n}(x,y)dy<\infty$, where $n$ is the dimension of the manifold. In the later case, one only needs scalar curvature being bounded. We will explain why the conditions are nearly optimal and give an application. The Li-Yau bound for the heat equation on manifolds with fixed metric seems to be the first one allowing Ricci curvature not bounded from below.