differential geometry

Linearized heat semigroups on Finsler measure spaces and some applications

arXiv:2307.00272

summary

The paper investigates linearized heat semigroups on Finsler measure spaces, proving their conservativeness under lower bounds on weighted Ricci curvature and using this framework to re‑prove and extend Li‑Yau type inequalities and characterize Ricci curvature lower bounds.

Abstract

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

The title of paper has been changed since we reorganized the paper in which some proofs for theorems have been modified. All suggestions and comments are welcome

Topics & keywords

#finsler geometry #heat semigroup #ricci curvature #li-yau inequality #geometric analysislinearized heat semigroupFinsler measure spaceweighted Ricci curvature Ric_NconservativenessRicci curvature lower boundLi-Yau inequality