This paper investigates the question of stability for a class of Ricci flows which start at possibly non-smooth metric spaces. We show that if the initial metric space is Reifenberg and locally bi-Lipschitz to Euclidean space, then two solutions to the Ricci flow whose Ricci curvature is uniformly bounded from below and whose curvature is bounded by c⋅t−1 converge to one another at an exponential rate once they have been appropriately gauged. As an application, we show that smooth three dimensional, complete, uniformly Ricci-pinched Riemannian manifolds with bounded curvature are either compact or flat, thus confirming a conjecture of Hamilton and Lott.

Related project(s):
75Solutions to Ricci flow whose scalar curvature is bounded in L^p II

 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 \(W^{2,2}\)(M) and satisfy  \(\frac{1}{a}h \leq g \leq ah\)  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  \(W^{2,2}\)(M)  metrics  \(g\)  on closed four manifolds which are bounded in the \(L^{\infty}\) sense by \(\frac{1}{a}h \leq g \leq ah\)  for some \(1<a<\infty\)  and some smooth Riemannian metric \(h\) on  M.

Related project(s):
75Solutions to Ricci flow whose scalar curvature is bounded in L^p II

In this paper, we establish the existence and uniqueness of Ricci flow that admits an embedded closed convex surface in $\mathbb{R}^3$ as metric initial condition. The main point is a family of smooth Ricci flows starting from smooth convex surfaces whose metrics converge uniformly to the metric of the initial surface in intrinsic sense.

Related project(s):
75Solutions to Ricci flow whose scalar curvature is bounded in L^p II