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Proceedings of the American Mathematical Society

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

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Volume estimate about shrinkers
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by Xu Cheng and Detang Zhou PDF
Proc. Amer. Math. Soc. 141 (2013), 687-696 Request permission

Abstract:

We derive a precise estimate on the volume growth of the level set of a potential function on a complete noncompact Riemannian manifold. As applications, we obtain the volume growth rate of a complete noncompact self-shrinker and a gradient shrinking Ricci soliton. We also prove the equivalence of weighted volume finiteness, polynomial volume growth and properness of an immersed self-shrinker in Euclidean space.
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Additional Information
  • Xu Cheng
  • Affiliation: Instituto de Matematica, Universidade Federal Fluminense, Niterói, RJ 24020, Brazil
  • Email: xcheng@impa.br
  • Detang Zhou
  • Affiliation: Instituto de Matematica, Universidade Federal Fluminense, Niterói, RJ 24020, Brazil
  • Email: zhou@impa.br
  • Received by editor(s): July 4, 2011
  • Published electronically: September 27, 2012
  • Additional Notes: Both authors were partially supported by CNPq and FAPERJ, Brazil.
  • Communicated by: Chuu-Lian Terng
  • © Copyright 2012 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 141 (2013), 687-696
  • MSC (2010): Primary 53C20
  • DOI: https://doi.org/10.1090/S0002-9939-2012-11922-7
  • MathSciNet review: 2996973