Heat flow on Finsler manifolds

TL;DR

This paper explores two methods to analyze heat flow on Finsler manifolds, establishing regularity and comparison results for solutions, and connecting gradient flow approaches in both $L^2$ and Wasserstein spaces.

Contribution

It introduces two gradient flow frameworks for heat flow on Finsler manifolds and proves regularity and comparison results for the solutions.

Findings

Solutions are $ ext{C}^{1,eta}$-regular, but generally not $ ext{C}^2$.

Established pointwise comparison results similar to Cheeger-Yau.

Derived integrated upper Gaussian estimates akin to Davies.

Abstract

We present two approaches to the heat flow on a Finsler manifold $(M,F)$: either as gradient flow on $L^2(M,m)$ for the energy; or as gradient flow on the reverse $L^2$-Wasserstein space $\mathcal{P}_2(M)$ of probability measures on $M$ for the relative entropy. Both approaches depend on the choice of a measure $m$ on $M$ and then lead to the same nonlinear evolution semigroup. We prove $\mathcal{C}^{1,\alpha}$-regularity for solutions to the (nonlinear) heat equation on the Finsler space $(M,F,m)$. Typically, solutions to the heat equation will not be $\mathcal{C}^2$. Moreover, we derive pointwise comparison results a la Cheeger-Yau and integrated upper Gaussian estimates a la Davies.