Skip to Main Content

Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles 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 1.48 .

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.


Self-shrinkers with a rotational symmetry
HTML articles powered by AMS MathViewer

by Stephen Kleene and Niels Martin Møller PDF
Trans. Amer. Math. Soc. 366 (2014), 3943-3963 Request permission


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.

Similar Articles
  • Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 53C44
  • Retrieve articles in all journals with MSC (2010): 53C44
Additional Information
  • Stephen Kleene
  • Affiliation: Department of Mathematics, Johns Hopkins University, Baltimore, Maryland 21218
  • MR Author ID: 915857
  • Email:
  • Niels Martin Møller
  • Affiliation: Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
  • Email:
  • 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:
  • MathSciNet review: 3206448