Journal of Differential Geometry

The round sphere minimizes entropy among closed self-shrinkers

Tobias Holck Colding, Tom Ilmanen, William P. Minicozzi, II, and Brian White

Full-text: Open access

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.

Article information

Source
J. Differential Geom., Volume 95, Number 1 (2013), 53-69.

Dates
First available in Project Euclid: 29 July 2013

Permanent link to this document
https://projecteuclid.org/euclid.jdg/1375124609

Digital Object Identifier
doi:10.4310/jdg/1375124609

Mathematical Reviews number (MathSciNet)
MR3128979

Zentralblatt MATH identifier
1278.53069

Citation

Colding, Tobias Holck; Ilmanen, Tom; Minicozzi, William P.; White, Brian. The round sphere minimizes entropy among closed self-shrinkers. J. Differential Geom. 95 (2013), no. 1, 53--69. doi:10.4310/jdg/1375124609. https://projecteuclid.org/euclid.jdg/1375124609


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