Nonlinear evolution by mean curvature and isoperimetric inequalities

2006/06/27 by Felix Schulze, Schulze, Felix · 2 citations
Mathematics · #28A75 #49Q20 #53C44 #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #math.AP #math.DG #msc:28A75 #msc:49Q20 #msc:53C44

paper · pdf · doi:10.48550/arxiv.math/0606675

42 pages

arxiv created 2006/06/27 · arxiv updated 2009/12/01

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.

Cited by

• Mean curvature flow through singularities
• Variational Nonlinear and Nonlocal Curvature Flows

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