The round sphere minimizes entropy among closed self-shrinkers

Tobias Holck Colding, Tom Ilmanen, William P. Minicozzi, Brian White
2013 Journal of differential geometry  
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,
more » ... sed singularity, but there is a gap to the second lowest.
doi:10.4310/jdg/1375124609 fatcat:rbpwjslu25gyti3no3v3rjjbzq