A family of parameter-dependent diffeomorphisms acting on function spaces over a Riemannian manifold and applications to geometric flows

TL;DR

This paper develops a technical framework for analyzing regularity of solutions to parabolic equations on manifolds, demonstrating joint analyticity in time and space for various geometric flows.

Contribution

It introduces a new method to study regularity and analyticity of solutions to geometric flows on Riemannian manifolds.

Findings

Solutions to Ricci-DeTurck, surface diffusion, and mean curvature flows are jointly analytic in time and space.

Solutions to Ricci flow are temporally analytic.

The technique advances understanding of regularity in geometric PDEs.

Abstract

It is the purpose of this article to establish a technical tool to study regularity of solutions to parabolic equations on manifolds. As applications of this technique, we prove that solutions to the Ricci-DeTurck flow, the surface diffusion flow and the mean curvature flow enjoy joint analyticity in time and space, and solutions to the Ricci flow admit temporal analyticity.