Linearized heat semigroups on Finsler measure spaces and some applications

It is known that the Finsler heat flow is a nonlinear flow.

This leads to the study of linearized heat semigroups for the Finsler heat flow.

In this paper, we give the properties of linearized heat semigroups and prove that the semigroup is conservative on complete Finsler measure spaces $(M, F, m)$ with weighted Ricci curvature Ric$_N$ bounded from below.

As applications, we give new proofs of Li-Yau's inequalities established in \cite{Xia2} and \cite{OS2} respectively in the compact case and extend them to the complete Finsler measure spaces with Ric$_N\geq K$ for $K\in \mathbb R$.

Finally we give several equivalent characterizations of Ric$_\infty\geq K (K\in \mathbb R)$ via the linearized heat semigroup approach and their applications.