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Gerhard Huisken : Parabolic Evolution Equations for the Deformation of Hypersurfaces

A smooth one-parameter family F0 : Mnx [0,T) ---> (Nn+1,g) of hypersurfaces in a Riemannian manifold (N(n+1),g) is said to move by its curvature if it satisfies an evolution equation of the form
(d/dt) F(p,t) = fNu(p,t) p in Mn, t in [0,T),
such that at each point of the surface its speed in normal direction Nu is a function $f$ of the extrinsic curvature of the hypersurface. Examples such as the flow by mean curvature, flow by Gauss curvature or flow by inverse mean curvature arise naturally both in Differential Geometry, where they exhibit fascinating interactions between the extrinsic curvature of the surfaces and intrinsic geometric properties of the ambient manifold, and in Mathematical Physics, where they serve as models for the evolution of interfaces in phase transitions. The first lecture gives a general introduction to the main examples and phenomena and highlights some recent results. The second lecture shows how parabolic rescaling techniques can be combined with a priori estimates to study and in some cases classify possible singularities of the mean curvature flow. The series concludes with applications of hypersurface families in General relativity, including a recent proof of an optimal lower bound for the total energy of an isolated gravitating system by Huisken and Ilmanen.

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