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Splitting of 3-manifolds and rigidity of area-minimising surfaces


Authors: Mario Micallef and Vlad Moraru
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
Published electronically: March 17, 2015
MathSciNet review: 3336611
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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.


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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

DOI: https://doi.org/10.1090/S0002-9939-2015-12137-5
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
Article copyright: © Copyright 2015 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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