# Topics in Geometric Analysis

#### 13.06. bis 13.06.2019, 16.15  –  Free University Berlin, Arnimallee 3, HS 001 Seminar "Topics in Geometric Analysis"

Wolfgang Maurer, Melanie Graf

 16:15 Wolfgang Maurer Curvature Flows Without Singularities I'll give an overview of my work on curvature flows without singularities. The concept was introduced by Sáez and Schnürer in '14. They considered mean curvature flow of complete (but not necessarily entire) graphs over $/mathbb{R}^n$. They showed the existence of a smooth solution and were able to interpret the projection to $\mathbb{R}^n$ as a weak solution of mean curvature flow. I investigated regularity properties of the mean curvature flow without singularities. Moreover I applied the concept to solve mean curvature flow with Neumann boundary condition on a supporting hyperplane. I also studied the flow by powers of the mean curvature. 17:45 Melanie Graf Singularity theorems for C^1 -Lorentzian metricsThe classical singularity theorems of General Relativity show that a Lorentzian manifold with a smooth metric satisfying certain physically reasonable curvature and causality conditions cannot be causal geodesically complete. In my talk I will discuss current work concerning stability of causal geodesic completeness for metrics that are merely continuously differentiable - a regularity where one still has existence but not uniqueness for solutions of the geodesic equation. Together with careful estimates of the curvature of approximating smooth metrics this can be used to prove certain singularity theorems, such as Hawking’s theorem, for metrics of this regularity.
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