The round sphere minimizes entropy among closed self-shrinkers

By T. Colding, T. Ilmanen, W. Minicozzi, B. White. 2012

109 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/73ee41f3347d274881899a8ada2919094de12cb4

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In Applied Mathematics

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Uniqueness of blowups and Łojasiewicz inequalities (2013, unknown): “f Athen follow from this and the interior estimates of Ecker and Huisken, [EH]. 5.3. A mean value inequality. In the next lemma, we will use that if Σt ﬂow by the rescaled MCF, then (see section 2 of [CIMW]) (∂t −L)φ= 0 where L= L+|A|2 + 1 2 (5.30) . Hence, (5.31) (∂t − L)φ2 = 2φ(∂t − L)φ− 2|∇φ|2 = ”
Embedded self-similar shrinkers of genus 0 (2014, background): “Furthermore, T. Colding, T. Ilmanen, W. Minicozzi, and B. White recently showed that the round sphere has smallest entropy among all compact self-similar shrinkers (see [8]).”
A sharp lower bound for the entropy of closed hypersurfaces up to dimension six (2014, methodology): “geometric quantity which measures the complexity of a hypersurface in Rn+1. It is non-increasing along the mean curvature ﬂow and so plays a signiﬁcant role in analyzing the dynamics of this ﬂow . In [5], Colding-IlmanenMinicozzi-White showed that within the class of closed smooth self-shrinking sol”