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≤g≤aha1​h≤g≤ah for some 1<a<∞ 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 ≥k≥k for $W^{2,2}$-metrics g on closed four manifolds which are bounded in the L$^∞$-sense by $\frac{1}{a}$h≤g≤ah for some 1<a<∞ and some smooth Riemannian metric h on M