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Self-shrinkers with a rotational symmetry

Authors: Stephen Kleene and Niels Martin Møller
Journal: Trans. Amer. Math. Soc. 366 (2014), 3943-3963
MSC (2010): Primary 53C44
Published electronically: March 26, 2014
MathSciNet review: 3206448
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Abstract: In this paper we present a new family of non-compact properly embedded, self-shrinking, asymptotically conical, positive mean curvature ends $ \Sigma ^n\subseteq \mathbb{R}^{n+1}$ that are hypersurfaces of revolution with circular boundaries. These hypersurface families interpolate between the plane and half-cylinder in $ \mathbb{R}^{n+1}$, and any rotationally symmetric self-shrinking non-compact end belongs to our family. The proofs involve the global analysis of a cubic-derivative quasi-linear ODE.

We also prove the following classification result: a given complete, embedded, self-shrinking hypersurface of revolution $ \Sigma ^n$ is either a hyperplane $ \mathbb{R}^{n}$, the round cylinder $ \mathbb{R}\times S^{n-1}$ of radius $ \sqrt {2(n-1)}$, the round sphere $ S^n$ of radius $ \sqrt {2n}$, or is diffeomorphic to an $ S^1\times S^{n-1}$ (i.e. a ``doughnut'' as in the paper by Sigurd B. Angenent, 1992, which when $ n=2$ is a torus). In particular, for self-shrinkers there is no direct analogue of the Delaunay unduloid family. The proof of the classification uses translation and rotation of pieces, replacing the method of moving planes in the absence of isometries.

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Additional Information

Stephen Kleene
Affiliation: Department of Mathematics, Johns Hopkins University, Baltimore, Maryland 21218

Niels Martin Møller
Affiliation: Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139

Received by editor(s): August 12, 2010
Received by editor(s) in revised form: September 9, 2011
Published electronically: March 26, 2014
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.