In this paper we study n-dimensional Ricci flows  (M,g(t)), t in [0,T),   where  T is finite,  and   potentially a singular time, and for which the spatial L^p norm,  p>n/2,  of the scalar curvature is uniformly  bounded on [0,T). 

 In the case that M is closed, we show that non-collapsing  and non-inflating estimates hold. If we further assume   that  n=4 or that  M^n is Kähler, we explain how  these non-inflating/non-collapsing estimates can be combined  with    integral bounds on the Ricci and full curvature tensor of  the  prequel paper   to   obtain  an improved space time integral bound of the Ricci curvature.  

  As an application of these estimates,  we show  that if we further restrict to n=4, then  the solution convergences to an orbifold as t approaches T and  that the flow can be extended    using the Orbifold Ricci flow to the time interval    [0,T+a)$ for some a>0.

  We also prove local versions of many of the  results mentioned above. 

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

 In this   paper we  prove   localised weighted curvature   integral estimates for solutions to the Ricci flow 
in the setting of a  smooth four dimensional Ricci flow or a closed n-dimensional Kähler Ricci flow. 
These integral   estimates improve and extend  the integral curvature estimates shown by the second author  in an earlier paper. If  
the scalar curvature is uniformly bounded in the spatial L^p sense for some p>2, then the estimates imply a uniform bound on the spatial L^2 norm of the Riemannian curvature  tensor. Stronger integral estimates are shown to hold if one further assumes a weak non-inflating condition.    
In a sequel paper, we show that in many natural settings,  a   non-inflating condition holds.

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

In this paper, we consider the stability of the generalized Lagrangian mean curvature flow of graph case in the cotangent bundle, which is first defined by Smoczyk-Tsui-Wang [14]. By new estimates of derivatives along the flow, we weaken the initial condition and remove the positive curvature condition in [14]. More precisely, we prove that if the graph induced by a closed $1$-form is a special Lagrangian submanifold in the cotangent bundle of a Riemannian manifold, then the generalized Lagrangian mean curvature flow is stable near it.

Related project(s):
31Solutions to Ricci flow whose scalar curvature is bounded in Lp.75Solutions to Ricci flow whose scalar curvature is bounded in L^p II

We consider a general class of non-homogeneous contracting flows of convex hypersurfaces in \(\mathbb R^{n+1}\), and prove the existence and regularity of the flow before extincting to a point in finite time.

Related project(s):
31Solutions to Ricci flow whose scalar curvature is bounded in Lp.75Solutions to Ricci flow whose scalar curvature is bounded in L^p II

In this paper, we show the relation between the existence of twisted conical Kähler-Ricci solitons and the greatest log Bakry-Emery-Ricci lower bound on Fano manifolds. This is based on our proofs of some openness theorems on the existence of twisted conical Kähler-Ricci solitons, which generalize Donaldson's existence conjecture and openness theorem of the conical Kähler-Einstein metrics to the conical soliton case.

Related project(s):
31Solutions to Ricci flow whose scalar curvature is bounded in Lp.75Solutions to Ricci flow whose scalar curvature is bounded in L^p II

In this paper, by using smooth approximation, we give a new proof of Donaldson's existence conjecture that there exist conical Kähler-Einstein metrics with positive Ricci curvatures on Fano manifolds. 

Related project(s):
31Solutions to Ricci flow whose scalar curvature is bounded in Lp.75Solutions to Ricci flow whose scalar curvature is bounded in L^p II

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 study the stability of the conical Kähler-Ricci flows on Fano manifolds. That is, if there exists a conical Kähler-Einstein metric with cone angle $2\pi\beta$ along the divisor, then for any $\beta'$ sufficiently close to $\beta$, the corresponding conical Kähler-Ricci flow converges to a conical Kähler-Einstein metric with cone angle $2\pi\beta'$ along the divisor. Here, we only use the condition that the Log Mabuchi energy is bounded from below. This is a weaker condition than the properness that we have adopted to study the convergence before. As corollaries, we give parabolic proofs of Donaldson's openness theorem and his existence conjecture for the conical Kähler-Einstein metrics with positive Ricci curvatures.

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