Nonlinear evolution by mean curvature and isoperimetric inequalities
Abstract
Evolving smooth, compact hypersurfaces in R^{n+1} with normal speed equal to a positive power k of the mean curvature improves a certain 'isoperimetric difference' for k >= n1. As singularities may develop before the volume goes to zero, we develop a weak levelset 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 3dimensional manifolds with nonpositive sectional curvature, we give a new proof for the Euclidean isoperimetric inequality on such manifolds.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 June 2006
 arXiv:
 arXiv:math/0606675
 Bibcode:
 2006math......6675S
 Keywords:

 Mathematics  Differential Geometry;
 Mathematics  Analysis of PDEs;
 28A75;
 49Q20;
 53C44
 EPrint:
 42 pages