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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Self-shrinkers with a rotational symmetry
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by Stephen Kleene and Niels Martin Møller PDF
Trans. Amer. Math. Soc. 366 (2014), 3943-3963 Request permission

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
  • MR Author ID: 915857
  • Email: skleene@math.jhu.edu
  • Niels Martin Møller
  • Affiliation: Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
  • Email: moller@math.mit.edu
  • Received by editor(s): August 12, 2010
  • Received by editor(s) in revised form: September 9, 2011
  • Published electronically: March 26, 2014
  • © Copyright 2014 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Trans. Amer. Math. Soc. 366 (2014), 3943-3963
  • MSC (2010): Primary 53C44
  • DOI: https://doi.org/10.1090/S0002-9947-2014-05721-8
  • MathSciNet review: 3206448