Splitting of 3-manifolds and rigidity of area-minimising surfaces
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- by Mario Micallef and Vlad Moraru PDF
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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.
References
- Michael T. Anderson, On area-minimizing hypersurfaces in manifolds of nonnegative curvature, Indiana Univ. Math. J. 32 (1983), no. 5, 745–760. MR 711865, DOI 10.1512/iumj.1983.32.32049
- 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, DOI 10.1007/s00023-007-0348-2
- 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, DOI 10.4310/CAG.2010.v18.n4.a6
- Mingliang Cai, Volume minimizing hypersurfaces in manifolds of nonnegative scalar curvature, Minimal surfaces, geometric analysis and symplectic geometry (Baltimore, MD, 1999) Adv. Stud. Pure Math., vol. 34, Math. Soc. Japan, Tokyo, 2002, pp. 1–7. MR 1925731, DOI 10.2969/aspm/03410001
- 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, DOI 10.4310/CAG.2000.v8.n3.a6
- Jeff Cheeger and Detlef Gromoll, The splitting theorem for manifolds of nonnegative Ricci curvature, J. Differential Geometry 6 (1971/72), 119–128. MR 303460
- 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, DOI 10.1002/cpa.3160330206
- 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, DOI 10.1088/0264-9381/16/6/302
- 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
- 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
- Ivaldo Nunes, Rigidity of area-minimizing hyperbolic surfaces in three-manifolds, J. Geom. Anal. 23 (2013), no. 3, 1290–1302. MR 3078354, DOI 10.1007/s12220-011-9287-8
- 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, DOI 10.2307/1971247
- Ying Shen and Shunhui Zhu, Rigidity of stable minimal hypersurfaces, Math. Ann. 309 (1997), no. 1, 107–116. MR 1467649, DOI 10.1007/s002080050105
- James Simons, Minimal varieties in riemannian manifolds, Ann. of Math. (2) 88 (1968), 62–105. MR 233295, DOI 10.2307/1970556
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