Title: Evolution of Hypersurfaces by Their Curvature in Riemannian Manifolds

We study hypersurfaces in Riemannian manifolds moving in normal direction with a speed depending on their curvature. The deformation laws considered are motivated by concrete geometrical and physical phenomena and lead to second order nonlinear parabolic systems for the evolving surfaces. For selected examples of such flows the article investigates local and global geometric properties of solutions. In particular, it discusses recent results on the singularity formation in mean curvature flow of meanconvex surfaces (joint with C. Sinestrari) and applications of inverse mean curvature flow to asymptotically flat manifolds used for the modelling of isolated systems in General Relativity (joint with T. Ilmanen).

1991 Mathematics Subject Classification: 53A10, 53A35, 58G11

Keywords and Phrases: Geometric evolution equations, Mean curvature flow, Inverse mean curvature flow, Penrose inequality

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