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 >= n-1. As singularities may develop before the volume goes to zero, 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 Euclidean isoperimetric inequality on such manifolds.
- Publication:
-
arXiv Mathematics e-prints
- Pub Date:
- June 2006
- DOI:
- 10.48550/arXiv.math/0606675
- arXiv:
- arXiv:math/0606675
- Bibcode:
- 2006math......6675S
- Keywords:
-
- Mathematics - Differential Geometry;
- Mathematics - Analysis of PDEs;
- 28A75;
- 49Q20;
- 53C44
- E-Print:
- 42 pages