The aim of this project is to further investigate the types of finite time singularities that occur for the Ricci flow in four dimensions in the real case, and higher dimensions in the Kaehler case, when the scalar curvature is bounded in the L^p norm.

## Publications

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):****75**Solutions 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):****75**Solutions 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):****75**Solutions to Ricci flow whose scalar curvature is bounded in L^p II

## Team Members

**Dr. Florian Litzinger**

Researcher

Otto-von-Guericke-Universität Magdeburg

florian.litzinger(at)ovgu.de

**Prof. Dr. Miles Simon**

Project leader

Otto-von-Guericke-Universität Magdeburg

miles.simon(at)ovgu.de

**M.Sc. Priyamvada Vishwamitra**

Doctoral student

Otto-von-Guericke-Universität Magdeburg

priyamvada.vishwamitra(at)ovgu.de

## Former Members

**Dr. Jiawei Liu**

Researcher

Otto-von-Guericke-Universität Magdeburg