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Solutions to Ricci flow whose scalar curvature is bounded in L^p.

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

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.

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.

In this paper, we study the limiting flow of conical Kähler-Ricci flows when the cone angles tend to $0$. We prove the existence and uniqueness of this limiting flow with cusp singularity on compact Kähler manifold $M$ which carries a smooth hypersurface $D$ such that the twisted canonical bundle $K_M+D$ is ample. Furthermore, we prove that this limiting flow converge to a unique cusp Kähler-Einstein metric.

 Journal Annali di Matematica Pura ed Applicata (1923 -) Link to preprint version

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

We consider the long-time behaviour of the mean curvature flow of spacelike hypersurfaces in the Lorentzian product manifold M×R, where M is asymptotically flat. If the initial hypersurface F0⊂M×R is uniformly spacelike and asymptotic to M×{s} for some s∈R at infinity, we show that the mean curvature flow starting at F0 exists for all times and converges uniformly to M×{s} as t→∞.

In this paper, we study the long-time behavior of modified Calabi flow to study the existence of generalized Kähler-Ricci soliton. We first give a new expression of the modified $K$-energy and prove its convexity along weak geodesics. Then we extend this functional to some finite energy spaces. After that, we study the long-time behavior of modified Calabi flow.

 Journal The Journal of Geometric Analysis Link to preprint version

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

In this paper, by providing the uniform gradient estimates for approximating equations, we prove the existence, uniqueness and regularity of conical parabolic complex Monge-Ampère equation with weak initial data. As an application, we obtain a regularity estimate, that is, any $L^{\infty}$-solution of the conical complex Monge-Ampère equation admits the $C^{2,\alpha,\beta}$-regularity.

 Journal Calculus of Variations and Partial Differential Equations Link to preprint version

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.

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## Team Members

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

Wolfgang Meiser
Doctoral student
Otto-von-Guericke-Universität Magdeburg

Prof. Dr. Miles Simon