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Solutions to Ricci flow whose scalar curvature is bounded in Lp (II)

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

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

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## Team Members

Dr. Jiawei Liu
Researcher
Otto-von-Guericke-Universität Magdeburg
jiawei.liu(at)ovgu.de

Prof. Dr. Miles Simon