Continuous maximal regularity on uniformly regular Riemannian manifolds

TL;DR

This paper proves continuous maximal regularity for parabolic operators on Riemannian manifolds and demonstrates that solutions to the Yamabe flow instantly become real analytic in space and time.

Contribution

It introduces a new approach using parameter-dependent diffeomorphisms to establish regularity results for geometric flows on manifolds.

Findings

Solutions to Yamabe flow become real analytic instantly.

Established maximal regularity for parabolic operators on tensor bundles.

Developed a novel method combining diffeomorphisms and maximal regularity.

Abstract

We establish continuous maximal regularity results for parabolic differential operators acting on sections of tensor bundles on Riemannian manifolds. As an application, we show that solutions to the Yamabe flow instantaneously regularize and become real analytic in space and time. The regularity result is obtained by introducing a family of parameter-dependent diffeomorphims acting on functions over Riemannian manifolds in conjunction with maximal regularity and the implicit function theorem.