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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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Smooth compactness of $f$-minimal hypersurfaces with bounded $f$-index
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by Ezequiel Barbosa, Ben Sharp and Yong Wei PDF
Proc. Amer. Math. Soc. 145 (2017), 4945-4961 Request permission

Abstract:

Let $(M^{n+1},g,e^{-f}d\mu )$ be a complete smooth metric measure space with $2\leq n\leq 6$ and Bakry-Émery Ricci curvature bounded below by a positive constant. We prove a smooth compactness theorem for the space of complete embedded $f$-minimal hypersurfaces in $M$ with uniform upper bounds on $f$-index and weighted volume. As a corollary, we obtain a smooth compactness theorem for the space of embedded self-shrinkers in $\mathbb {R}^{n+1}$ with $2\leq n\leq 6$. We also prove some estimates on the $f$-index of $f$-minimal hypersurfaces.
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Additional Information
  • Ezequiel Barbosa
  • Affiliation: Departamento de Matemática, Universidade Federal de Minas Gerais (UFMG), Caixa Postal 702, 30123-970, Belo Horizonte, MG, Brazil
  • Email: ezequiel@mat.ufmg.br
  • Ben Sharp
  • Affiliation: Department of Mathematics, University of Warwick, Coventry, CV4 7AL, United Kingdom
  • MR Author ID: 1008414
  • Email: b.sharp@warwick.ac.uk
  • Yong Wei
  • Affiliation: Mathematical Sciences Institute, Australian National University, John Dedman Building 27, Union Lane, Canberra ACT 2601, Australia
  • MR Author ID: 1036099
  • ORCID: 0000-0002-9460-9217
  • Email: yong.wei@anu.edu.au
  • Received by editor(s): July 11, 2016
  • Received by editor(s) in revised form: November 24, 2016
  • Published electronically: May 26, 2017
  • Additional Notes: The first author was supported by FAPEMIG and CNPq grants. The second author was supported by André Neves’ ERC and Leverhulme trust grants. The third author was supported by Jason D Lotay’s EPSRC grant
  • Communicated by: Guofang Wei
  • © Copyright 2017 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 145 (2017), 4945-4961
  • MSC (2010): Primary 53C42, 53C21
  • DOI: https://doi.org/10.1090/proc/13628
  • MathSciNet review: 3692008