Nonlinear evolution by mean curvature and isoperimetric inequalities

Abstract

Evolving smooth, compact hypersurfaces in ℝn+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. © 2008 Applied Probability Trust.