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On the uniqueness of the foliation of spheres of constant mean curvature in asymptotically flat 3-manifolds


Authors: Jie Qing and Gang Tian
Journal: J. Amer. Math. Soc. 20 (2007), 1091-1110
MSC (2000): Primary 53C20; Secondary 58E20, 83C99
Published electronically: March 9, 2007
MathSciNet review: 2328717
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Abstract | References | Similar Articles | Additional Information

Abstract: In this note we study constant mean curvature surfaces in asymptotically flat 3-manifolds. We prove that, outside a given compact subset in an asymptotically flat 3-manifold with positive mass, stable spheres of given constant mean curvature are unique. Therefore we are able to conclude that the foliation of stable spheres of constant mean curvature in an asymptotically flat 3-manifold with positive mass outside a given compact subset is unique.


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  • [ADM] R. Amowitt, S. Deser and C. Misner, Coordinate invariant and energy expression in general relativity, Phys. Rev. 122 (1966) 997-1006.
  • [B] Robert Bartnik, The mass of an asymptotically flat manifold, Comm. Pure Appl. Math. 39 (1986), no. 5, 661–693. MR 849427, 10.1002/cpa.3160390505
  • [Br] H. Bray, The Penrose inequality in general relativity and volume comparison theorem involving scalar curvature, Stanford Thesis 1997.
  • [CY] D. Christodoulou and S.-T. Yau, Some remarks on the quasi-local mass, Mathematics and general relativity (Santa Cruz, CA, 1986) Contemp. Math., vol. 71, Amer. Math. Soc., Providence, RI, 1988, pp. 9–14. MR 954405, 10.1090/conm/071/954405
  • [MS] J. H. Michael and L. M. Simon, Sobolev and mean-value inequalities on generalized submanifolds of 𝑅ⁿ, Comm. Pure Appl. Math. 26 (1973), 361–379. MR 0344978
  • [HY] Gerhard Huisken and Shing-Tung Yau, Definition of center of mass for isolated physical systems and unique foliations by stable spheres with constant mean curvature, Invent. Math. 124 (1996), no. 1-3, 281–311. MR 1369419, 10.1007/s002220050054
  • [Ka] Katsuei Kenmotsu, Surfaces with constant mean curvature, Translations of Mathematical Monographs, vol. 221, American Mathematical Society, Providence, RI, 2003. Translated from the 2000 Japanese original by Katsuhiro Moriya and revised by the author. MR 2013507
  • [QT] Jie Qing and Gang Tian, Bubbling of the heat flows for harmonic maps from surfaces, Comm. Pure Appl. Math. 50 (1997), no. 4, 295–310. MR 1438148, 10.1002/(SICI)1097-0312(199704)50:4<295::AID-CPA1>3.0.CO;2-5
  • [SSY] R. Schoen, L. Simon, and S. T. Yau, Curvature estimates for minimal hypersurfaces, Acta Math. 134 (1975), no. 3-4, 275–288. MR 0423263
  • [Si] Leon Simon, Existence of Willmore surfaces, Miniconference on geometry and partial differential equations (Canberra, 1985) Proc. Centre Math. Anal. Austral. Nat. Univ., vol. 10, Austral. Nat. Univ., Canberra, 1986, pp. 187–216. MR 857667
  • [Sj] James Simons, Minimal varieties in riemannian manifolds, Ann. of Math. (2) 88 (1968), 62–105. MR 0233295
  • [Ye] Rugang Ye, Foliation by constant mean curvature spheres on asymptotically flat manifolds, Geometric analysis and the calculus of variations, Int. Press, Cambridge, MA, 1996, pp. 369–383. MR 1449417

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Additional Information

Jie Qing
Affiliation: Department of Mathematics, University of California, Santa Cruz, Santa Cruz, California 95064
Email: qing@ucsc.edu

Gang Tian
Affiliation: Department of Mathematics, Princeton University, Princeton, New Jersey 08544
Email: tian@math.princeton.edu

DOI: http://dx.doi.org/10.1090/S0894-0347-07-00560-7
Keywords: Asymptotically flat 3-manifold, stable sphere of constant mean curvature, uniqueness, center of mass, asymptotic analysis
Received by editor(s): September 24, 2005
Published electronically: March 9, 2007
Additional Notes: The first author was partially supported by DMS 0402294
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.