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

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 $\frac{1}{a}h\le g\le ah$ for some $1<a<\infty$ and some smooth Riemann\-ian 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 $\ge k$ for $W^{2,2}$-metrics $g$ on closed four manifolds which are bounded in the $L^{\infty }$-sense by $\frac{1}{a}h\le g\le ah$ for some $1<a<\infty$ and some smooth Riemannian metric $h$ on $M$.