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

The aim of this project is to understand singularities of the Ricci flow in four dimensions if we assume restrictions on the topology and / or geometry of the solutions we are considering.

In particular we will consider the cases:

  1. \(R\le \frac{c}{(T-t)^\alpha}\) for some small \(\alpha >0\),
  2.  \(\int_M\mid R\mid^p<c\) for all \(t\in\left[0,T\right)\) for some fixed large \(p>0\),

where \(R\) denotes the scalar curvature.

Using estimates / results / ideas from previous works we aim to show that in certain cases this restricts the type of singularities that may occur as \(t\to T\).


Publications

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.

 

JournalAnnali 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→∞.

 

Related project(s):
21Stability and instability of Einstein manifolds with prescribed asymptotic geometry29Curvature flows without singularities30Nonlinear evolution equations on singular manifolds31Solutions to Ricci flow whose scalar curvature is bounded in Lp.

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.

 

JournalThe 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.

 

JournalCalculus of Variations and Partial Differential Equations
Link to preprint version

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
wolfgang.meiser(at)ovgu.de

Prof. Dr. Miles Simon
Project leader
Otto-von-Guericke-Universität Magdeburg
miles.simon(at)ovgu.de

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