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

DOI:
https://doi.org/10.1090/S0894-0347-07-00560-7

Published electronically:
March 9, 2007

MathSciNet review:
2328717

Full-text PDF

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.

**[ADM]**R. Amowitt, S. Deser and C. Misner, Coordinate invariant and energy expression in general relativity, Phys. Rev. 122 (1966) 997-1006.**[B]**R. Barnik, The mass of an asymptotically flat manifold, Comm. Pure. Appl. Math. 39 (1986) 661-693. MR**849427 (88b:58144)****[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), 9-14, Contemp. Math., 71, Amer. Math. Soc., Providence, RI, 1988. MR**954405 (89k:83050)****[MS]**J. Michael and L. Simon, Sobolev and mean-value inequalities on general submanifolds of , Comm. Pure. Appl. Math. 26 (1973) 361-379. MR**0344978 (49:9717)****[HY]**G. Huisken and S. T. Yau, Definition of center of mass for isolated physical systems and unique foliations by stable spheres of constant mean curvature, Invent. Math. 124 (1996) 281-311. MR**1369419 (96m:53037)****[Ka]**K. Kenmotsu, ``Surfaces with constant mean curvature'', Translations of Math. Monographs, v. 221, AMS, 2003. MR**2013507 (2004m:53014)****[QT]**Jie Qing and Gang Tian, Bubbling of the heat flow for harmonic maps from surfaces, Comm. Pure and Appl. Math., Vol. L (1997), 295-310. MR**1438148 (98k:58070)****[SSY]**R. Schoen, L. Simon and S.T. Yau, Curvature estimates for minimal hypersurfaces, Acta Math. 134 (1975) 275-288. MR**0423263 (54:11243)****[Si]**L. Simon, Existence of Willmore surface, Proc. Centre for Math. Anal. 10 (1985) 187-216. MR**857667 (87m:53006)****[Sj]**J. Simons, Minimal Varieties in Riemannian manifolds, Ann. of Math. 88 (1968) 62-105. MR**0233295 (38:1617)****[Ye]**R. Ye, Foliation by constant mean curvature spheres on asymptotically flat manifolds, Geom. Anal. and the Calculus of Variations, 369-383, Intern. Press, Cambridge, MA, 1996. MR**1449417 (98e:53040)**

Retrieve articles in *Journal of the American Mathematical Society*
with MSC (2000):
53C20,
58E20,
83C99

Retrieve articles in all journals with MSC (2000): 53C20, 58E20, 83C99

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:
https://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.