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 | References | Similar Articles | Additional Information

Abstract: In this paper we present a new family of non-compact properly embedded, self-shrinking, asymptotically conical, positive mean curvature ends that are hypersurfaces of revolution with circular boundaries. These hypersurface families interpolate between the plane and half-cylinder in , 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 is either a hyperplane , the round cylinder of radius , the round sphere of radius , or is diffeomorphic to an (i.e. a ``doughnut'' as in the paper by Sigurd B. Angenent, 1992, which when 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

Email:
skleene@math.jhu.edu

**Niels Martin Møller**

Affiliation:
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139

Email:
moller@math.mit.edu

DOI:
https://doi.org/10.1090/S0002-9947-2014-05721-8

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.