Entropy of closed surfaces and min-max theory
By Daniel Ketover, Xin Zhou. Journal of differential geometry, 2015
47 citations recorded by the source corpus.
Entropy is a natural geometric quantity measuring the complexity of a surface embedded in $\mathbb{R}^3$. For dynamical reasons relating to mean curvature flow, Colding-Ilmanen-Minicozzi-White conjectured that the entropy of any closed surface is at least that of the self-shrinking two-sphere. We prove this conjecture for all closed embedded $2$-spheres. Assuming a conjectural Morse index bound (announced recently by Marques-Neves), we can improve the result to apply to all closed embedded surfaces that are not tori. Our results can be thought of as the parabolic analog to the Willmore conjecture and our argument is analogous in many ways to that of Marques-Neves on the Willmore problem. The main tool is the min-max theory applied to the Gaussian area functional in $\mathbb{R}^3$ which we also establish. To any closed surface in $\mathbb{R}^3$ we associate a four parameter canonical family of surfaces and run a min-max procedure. The key step is ruling out the min-max sequence approaching a self-shrinking plane, and we accomplish this with a degree argument. To establish the min-max theory for $\mathbb{R}^3$ with Gaussian weight, the crucial ingredient is a tightening map that decreases the mass of non-stationary varifolds (with respect to the Gaussian metric of $\mathbb{R}^3$) in a continuous manner.
Subfields: Applied Mathematics, Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/1b2c464157e248a5a44e9b8f145e976fe145c83a
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Citations in the corpus
- Morse index of multiplicity one min-max minimal hypersurfaces (2015, methodology): “As an application, the main result of Ketover-Zhou [17] about the entropy of closed surfaces can be extended to surfaces with genus different than one.”
- Existence of hypersurfaces with prescribed mean curvature I – generic min-max (2018, background): “Consequently, the min-max program has seen a series of developments in various directions, including (but not limited to) [16, 28, 37, 44, 42, 15, 41, 36, 12, 46, 57].”
- Existence of infinitely many minimal hypersurfaces in closed manifolds (2018, methodology): “[41] X. Zhou.”
- On the entropy of closed hypersurfaces and singular self-shrinkers (2016, methodology): “Finally we thank Xin Zhou for several insightful conversations.”