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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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Splitting of 3-manifolds and rigidity of area-minimising surfaces
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by Mario Micallef and Vlad Moraru PDF
Proc. Amer. Math. Soc. 143 (2015), 2865-2872 Request permission

Abstract:

In this paper we modify an argument of Bray, Brendle and Neves to prove an area comparison result (Theorem 2) for certain totally geodesic surfaces in 3-manifolds with a lower bound on the scalar curvature.

This theorem is a variant of a comparison theorem (Theorem 3.2 (d) in the 1978 paper) of Heintze and Karcher for minimal hypersurfaces in manifolds of nonnegative Ricci curvature. Our assumptions on the ambient manifold are weaker, but the assumptions on the surface are considerably more restrictive.

We then use our comparison theorem to provide a unified proof of various splitting theorems for 3-manifolds with lower bounds on the scalar curvature that were first proved by Cai and Galloway; Bray, Brendle and Neves; and Nunes.

References
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Additional Information
  • Mario Micallef
  • Affiliation: Mathematics Institute, University of Warwick, Coventry, CV4 7AL, United Kingdom
  • Email: M.J.Micallef@warwick.ac.uk
  • Vlad Moraru
  • Affiliation: Mathematics Institute, University of Warwick, Coventry, CV4 7AL, United Kingdom
  • Email: V.Moraru@warwick.ac.uk
  • Received by editor(s): March 10, 2012
  • Received by editor(s) in revised form: December 29, 2012
  • Published electronically: March 17, 2015
  • Communicated by: Michael Wolf
  • © Copyright 2015 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Proc. Amer. Math. Soc. 143 (2015), 2865-2872
  • MSC (2010): Primary 49Q05, 53C24; Secondary 26D10
  • DOI: https://doi.org/10.1090/S0002-9939-2015-12137-5
  • MathSciNet review: 3336611