Topology of Closed Hypersurfaces of Small Entropy

Abstract

We use a weak mean curvature flow together with a surgery procedure to show that all closed hypersurfaces in ${\mathbb{R}}^4$ with entropy less than or equal to that of ${\mathbb{S}}^2\times \mathbb{R}$, the round cylinder in ${\mathbb{R}}^4$, are diffeomorphic to ${\mathbb{S}}^3$.

24 pages. Revised version

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• On self shrinkers of medium entropy in ${\mathbb{R}}^4$
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• Round spheres are Hausdorff stable under small perturbation of entropy
• Superconvexity of the Heat Kernel on Hyperbolic Space
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