Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

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

Request Permissions   Purchase Content 
 
 

 

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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • [1] Michael T. Anderson, On area-minimizing hypersurfaces in manifolds of nonnegative curvature, Indiana Univ. Math. J. 32 (1983), no. 5, 745-760. MR 711865 (85e:53077), https://doi.org/10.1512/iumj.1983.32.32049
  • [2] Lars Andersson, Mingliang Cai, and Gregory J. Galloway, Rigidity and positivity of mass for asymptotically hyperbolic manifolds, Ann. Henri Poincaré 9 (2008), no. 1, 1-33. MR 2389888 (2009e:53054), https://doi.org/10.1007/s00023-007-0348-2
  • [3] Hubert Bray, Simon Brendle, and Andre Neves, Rigidity of area-minimizing two-spheres in three-manifolds, Comm. Anal. Geom. 18 (2010), no. 4, 821-830. MR 2765731 (2012a:53067)
  • [4] Mingliang Cai, Volume minimizing hypersurfaces in manifolds of nonnegative scalar curvature (Baltimore, MD, 1999), Adv. Stud. Pure Math., vol. 34, Math. Soc. Japan, Tokyo, 2002, pp. 1-7. MR 1925731 (2003f:53104)
  • [5] Mingliang Cai and Gregory J. Galloway, Rigidity of area minimizing tori in 3-manifolds of nonnegative scalar curvature, Comm. Anal. Geom. 8 (2000), no. 3, 565-573. MR 1775139 (2001j:53051)
  • [6] Jeff Cheeger and Detlef Gromoll, The splitting theorem for manifolds of nonnegative Ricci curvature, J. Differential Geometry 6 (1971/72), 119-128. MR 0303460 (46 #2597)
  • [7] Doris Fischer-Colbrie and Richard Schoen, The structure of complete stable minimal surfaces in $ 3$-manifolds of nonnegative scalar curvature, Comm. Pure Appl. Math. 33 (1980), no. 2, 199-211. MR 562550 (81i:53044), https://doi.org/10.1002/cpa.3160330206
  • [8] G. W. Gibbons, Some comments on gravitational entropy and the inverse mean curvature flow, Classical Quantum Gravity 16 (1999), no. 6, 1677-1687. MR 1697098 (2000j:53089), https://doi.org/10.1088/0264-9381/16/6/302
  • [9] Gerhard Huisken and Tom Ilmanen, The inverse mean curvature flow and the Riemannian Penrose inequality, J. Differential Geom. 59 (2001), no. 3, 353-437. MR 1916951 (2003h:53091)
  • [10] Ernst Heintze and Hermann Karcher, A general comparison theorem with applications to volume estimates for submanifolds, Ann. Sci. École Norm. Sup. (4) 11 (1978), no. 4, 451-470. MR 533065 (80i:53026)
  • [11] I. Nunes, Rigidity of area-minimizing hyperbolic surfaces in three-manifolds, J. Geom. Anal. 23 (2013), no. 3, 1290-1302. MR 3078354
  • [12] R. Schoen and Shing Tung Yau, Existence of incompressible minimal surfaces and the topology of three-dimensional manifolds with nonnegative scalar curvature, Ann. of Math. (2) 110 (1979), no. 1, 127-142. MR 541332 (81k:58029), https://doi.org/10.2307/1971247
  • [13] Ying Shen and Shunhui Zhu, Rigidity of stable minimal hypersurfaces, Math. Ann. 309 (1997), no. 1, 107-116. MR 1467649 (98g:53113), https://doi.org/10.1007/s002080050105
  • [14] James Simons, Minimal varieties in riemannian manifolds, Ann. of Math. (2) 88 (1968), 62-105. MR 0233295 (38 #1617)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 49Q05, 53C24, 26D10

Retrieve articles in all journals with MSC (2010): 49Q05, 53C24, 26D10


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.

American Mathematical Society