Ricci flow of $W^{2,2}$-metrics in four dimensions

TL;DR

This paper constructs solutions to Ricci DeTurck flow for non-smooth $W^{2,2}$ metrics in four dimensions, establishing regularity, estimates, and a framework for defining scalar curvature in this context.

Contribution

It introduces a method to evolve non-smooth $W^{2,2}$ metrics under Ricci flow, extending the flow's applicability to less regular initial data in four dimensions.

Findings

Constructed instantaneously smooth Ricci flow solutions from $W^{2,2}$ initial metrics.

Derived $L^p$ estimates essential for analyzing Ricci flow with low regularity.

Proposed a way to define scalar curvature $ ext{geq }k$ for $W^{2,2}$ metrics.

Abstract

In this paper we construct solutions to Ricci DeTurck flow in four dimensions on closed manifolds which are instantaneously smooth but whose initial values $g$ are (possibly) non-smooth Riemannian metrics whose components in smooth coordinates belong to $W^{2,2}$ and satisfy $ \frac{1}{a}h\leq g\leq a h$ for some $1<a<\infty$ and some smooth Riemannian metric $h$ on $M$. A Ricci flow related solution is constructed whose initial value is isometric in a weak sense to the initial value of the Ricci DeTurck solution. Results for a related non-compact setting are also presented. Various $L^p$ estimates for Ricci flow, which we require for some of the main results, are also derived. As an application we present a possible definition of scalar curvature $\geq k$ for $W^{2,2}$ metrics $g$ on closed four manifolds which are bounded in the $L^{\infty}$ sense by $ \frac{1}{a}h\leq g\leq a h$ for some $1<a<\infty$ and some smooth Riemannian metric $h$ on $M$.