The Round Sphere Minimizes Entropy among Closed Self-Shrinkers
Author(s)Colding, Tobias; Minicozzi, William; White, Brian; Ilmanen, Tom
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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.
DepartmentMassachusetts Institute of Technology. Department of Mathematics
Journal of Differential Geometry
International Press of Boston, Inc.
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.