Contraction of convex hypersurfaces in Euclidean space

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Abstract

We consider a class of fully nonlinear parabolic evolution equations for hypersurfaces in Euclidean space. A new geometrical lemma is used to prove that any strictly convex compact initial hypersurface contracts to a point in finite time, becoming spherical in shape as the limit is approached. In the particular case of the mean curvature flow this provides a simple new proof of a theorem of Huisken.

This work was carried out while the author was supported by an Australian Postgraduate Research Award and an ANUTECH scholarship.