The Round Sphere Minimizes Entropy among Closed Self-Shrinkers
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Minicozzi_The round.pdf
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208.75 KB
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Author(s)
Colding, Tobias
Minicozzi, William
White, Brian
Ilmanen, Tom
Date Issued
July 2013
Journal
Journal of Differential Geometry
Publisher
International Press of Boston, Inc.
Citation
Colding, Tobias Holck, Tom Ilmanen, William P. Minicozzi, and Brian White. "The Round Sphere Minimizes Entropy Among Closed Self-Shrinkers." J. Differential Geom. 95.1 (2013): 53-69.
Version
Original manuscript
Abstract
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.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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Creative Commons Attribution-Noncommercial-Share Alike
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DOI of Published Version
http://projecteuclid.org/euclid.jdg/1375124609
