Graduate Seminar
Nonlinear Evolution by Mean Curvature and the Isoperimetric Inequality
Felix Schulze
13 Nov 2005, 17:00 – 19:00
Consider a family of smooth compact hypersurfaces in R^(n+1) evolving with normal speed equal to a positive power k of the mean curvature. For k > n-1, smooth solutions to such flows improve a certian 'isoperimetric difference'. If a smooth flow exists until the volume decreases to zero, this proves the isoperimetric inequality for the initial configuration. In general, singularities will develop before the volume goes to zero. To deal with this problem, we develop a weak level-set formulation for such flows and show that the above monotonicity is still valid. This proves the isoperimetric inequality for n < 7. Extending this to complete, simply connected 3- dimensional manifolds with nonpositive sectional curvature, we give a new proof for the Euclidian isoperimetric inequality on such manifolds.
http://geometricanalysis.mi.fu-berlin.de/os/os-ws0506.htm
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