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


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

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

First available in Project Euclid: 29 July 2013

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


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.

Export citation