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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

We consider solutions to Ricci flow defined on manifolds M for a time interval (0,T) whose Ricci curvature is bounded uniformly in time

from below, and for which the norm of the full curvature tensor at time t  is bounded by c/t for some fixed constant c>1 for all t in (0,T).

From previous works, it is known that if the solution is complete for all times t>0, then there is a limit metric space (M,d_0), as time t approaches zero. We show : if there is a region V on which (V,d_0) is smooth, then the solution can be extended smoothly to time zero on V.

Related project(s):
31Solutions to Ricci flow whose scalar curvature is bounded in Lp.

Given a three-dimensional Riemannian manifold containing a ball with an explicit lower bound on its Ricci curvature and positive lower bound on its volume, we use Ricci flow to perturb the Riemannian metric on the interior to a nearby Riemannian metric still with such lower bounds on its Ricci curvature and volume, but additionally with uniform bounds on its full curvature tensor and all its derivatives. The new manifold is near to the old one not just in the Gromov-Hausdorff sense, but also in the sense that the distance function is uniformly close to what it was before, and additionally  we have Hölder/Lipschitz equivalence of the old and new manifolds.

One consequence is that we obtain a local bi-Hölder correspondence between Ricci limit spaces in three dimensions and smooth manifolds.

This is more than a complete resolution of the three-dimensional case of the conjecture of Anderson-Cheeger-Colding-Tian, describing how Ricci limit spaces in three dimensions must be homeomorphic to manifolds, and we obtain this in the most general, locally non-collapsed case.

The proofs build on results and ideas  from recent papers of Hochard and the current authors.

Related project(s):
31Solutions to Ricci flow whose scalar curvature is bounded in Lp.