Nonlinear evolution by mean curvature and isoperimetric inequalities

Felix Schulze FU Berlin, Arnimallee 6

Differential Geometry mathscidoc:1609.10069

Journal of Differential Geometry, 79, (2), 197-241, 2008
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 6 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.
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@inproceedings{felix2008nonlinear,
  title={NONLINEAR EVOLUTION BY MEAN CURVATURE AND ISOPERIMETRIC INEQUALITIES},
  author={Felix Schulze},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160909132138126861726},
  booktitle={Journal of Differential Geometry},
  volume={79},
  number={2},
  pages={197-241},
  year={2008},
}
Felix Schulze. NONLINEAR EVOLUTION BY MEAN CURVATURE AND ISOPERIMETRIC INEQUALITIES. 2008. Vol. 79. In Journal of Differential Geometry. pp.197-241. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160909132138126861726.
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