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The round sphere minimizes entropy among closed self-shrinkers

Tobias Holck Colding MIT Dept. of Mathematics Tom Ilmanen ETH Zentrum William P. Minicozzi II MIT Dept. of Mathematics Brian White Stanford University

Differential Geometry mathscidoc:1609.10326

Journal of Differential Geometry, 95, (1), 53-69, 2013
The entropy of a hypersurface is a geometric invariant that measures complexity and is invariant under rigid motions and dilations. It is given by the supremum over all Gaussian integrals with varying centers and scales. It is monotone under mean curvature flow, thus giving a Lyapunov functional. Therefore, the entropy of the initial hypersurface bounds the entropy at all future singularities. We show here that not only does the round sphere have the lowest entropy of any closed singularity, but there is a gap to the second lowest.
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@inproceedings{tobias2013the,
  title={THE ROUND SPHERE MINIMIZES ENTROPY AMONG CLOSED SELF-SHRINKERS},
  author={Tobias Holck Colding, Tom Ilmanen, William P. Minicozzi II, and Brian White},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160914081647635924988},
  booktitle={Journal of Differential Geometry},
  volume={95},
  number={1},
  pages={53-69},
  year={2013},
}
Tobias Holck Colding, Tom Ilmanen, William P. Minicozzi II, and Brian White. THE ROUND SPHERE MINIMIZES ENTROPY AMONG CLOSED SELF-SHRINKERS. 2013. Vol. 95. In Journal of Differential Geometry. pp.53-69. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160914081647635924988.
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