Heat flow on time-dependent manifolds

TL;DR

This paper proves existence and uniqueness of heat flow on evolving manifolds, especially Ricci flows with scalar curvature bounds, using minimal assumptions related to volume measure derivatives.

Contribution

It introduces new estimates for heat flow on time-dependent manifolds that depend only on bounds of the volume measure's logarithmic derivative, applicable to Ricci flows with scalar curvature bounds.

Findings

Establishes existence and uniqueness of heat flow on time-dependent manifolds.

Provides estimates depending only on volume measure derivatives.

Applicable to Ricci flows with scalar curvature bounded below.

Abstract

We establish effective existence and uniqueness for the heat flow on time-dependent Riemannian manifolds, under minimal assumptions tailored towards the study of Ricci flow through singularities. The main point is that our estimates only depend on an upper bound for the logarithmic derivative of the volume measure. In particular, our estimates hold for any Ricci flow with scalar curvature bounded below, and such a lower bound of course depends only on the initial data.