Open Access
September 2013 The round sphere minimizes entropy among closed self-shrinkers
Tobias Holck Colding, Tom Ilmanen, William P. Minicozzi II, Brian White
J. Differential Geom. 95(1): 53-69 (September 2013). DOI: 10.4310/jdg/1375124609


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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Tobias Holck Colding. Tom Ilmanen. William P. Minicozzi II. Brian White. "The round sphere minimizes entropy among closed self-shrinkers." J. Differential Geom. 95 (1) 53 - 69, September 2013.


Published: September 2013
First available in Project Euclid: 29 July 2013

zbMATH: 1278.53069
MathSciNet: MR3128979
Digital Object Identifier: 10.4310/jdg/1375124609

Rights: Copyright © 2013 Lehigh University

Vol.95 • No. 1 • September 2013
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