  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.
