A sharp lower bound for the entropy of closed hypersurfaces up to dimension six

By J. Bernstein, Lu Wang. 2014

62 citations recorded by the source corpus.

The entropy is a natural geometric quantity which measures the complexity of a hypersurface in $$\mathbb R^{n+1}$$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 (Colding et al., J Differ Geom 95(1):53–69, 2013), Colding–Ilmanen–Minicozzi-White showed that within the class of closed smooth self-shrinking solutions of the mean curvature flow in $$\mathbb R^{n+1}$$Rn+1, the entropy is uniquely minimized at the round sphere. They conjectured that, for $$2\le n\le 6$$2≤n≤6, the round sphere minimizes the entropy among all closed hypersurfaces. Using an appropriate weak mean curvature flow, we prove their conjecture. For these dimensions, our approach also gives a new proof of the main result of Colding et al. (J Differ Geom 95(1):53–69, 2013) and extends its conclusions to compact singular self-shrinking solutions.

Subfields: Applied Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/250eae818921fa9986e21e4c46108f2b755f641a

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Citations in the corpus

Ancient asymptotically cylindrical flows and applications (2019, background): “12Our notion of separating is very closely related to the notion of strongly noncollapsed from [BW16].”
Entropy of closed surfaces and min-max theory (2015, background): “ural index bound for min-max limits that has recently been announcd by Marques-Neves, Theorem 1.2 can be extended to all closed surfaces with genus not equal to 1. Remark 1.4. Recently Bernstein-Wang [BW],[BW2] haveprovedConjecture 1.1usinga completely different method. They have further shown that ”
On the entropy of closed hypersurfaces and singular self-shrinkers (2016, methodology): “Similarly we extend the gap theorem of Bernstein-Wang [3] for the entropy of compact singular self-shrinkers to all dimensions n ≥ 2, which itself generalised the main theorem of Colding-Ilmanen-Minicozzi-White [5] to the singular setting for 2 ≤ n ≤ 6.”