The size of isoperimetric surfaces in $3$-manifolds and a rigidity result for the upper hemisphere
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Abstract:
We characterize the standard $\mathbb {S}^3$ as the closed Ricci-positive $3$-manifold with scalar curvature at least $6$ having isoperimetric surfaces of largest area: $4\pi$. As a corollary we answer in the affirmative an interesting special case of a conjecture of M. Min-Oo’s on the scalar curvature rigidity of the upper hemisphere.References
- 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
- Christophe Bavard and Pierre Pansu, Sur le volume minimal de $\textbf {R}^2$, Ann. Sci. École Norm. Sup. (4) 19 (1986), no. 4, 479–490 (French). MR 875084
- H. L. Bray: “The Penrose inequality in general relativity and volume comparison theorems involving scalar curvature,” thesis, Stanford University (1997). arXiv:0902.3241
- 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, DOI 10.1090/conm/071/954405
- Piotr T. Chruściel and Marc Herzlich, The mass of asymptotically hyperbolic Riemannian manifolds, Pacific J. Math. 212 (2003), no. 2, 231–264. MR 2038048, DOI 10.2140/pjm.2003.212.231
- Justin Corvino, Scalar curvature deformation and a gluing construction for the Einstein constraint equations, Comm. Math. Phys. 214 (2000), no. 1, 137–189. MR 1794269, DOI 10.1007/PL00005533
- A. Gray and L. Vanhecke, Riemannian geometry as determined by the volumes of small geodesic balls, Acta Math. 142 (1979), no. 3-4, 157–198. MR 521460, DOI 10.1007/BF02395060
- Fengbo Hang and Xiaodong Wang, Rigidity and non-rigidity results on the sphere, Comm. Anal. Geom. 14 (2006), no. 1, 91–106. MR 2230571
- F. Hang, X. Wang: “A rigidity theorem for the hemisphere,” preprint (2007). arXiv:0711.4595v2
- G. Huisken: “An isoperimetric concept for mass and quasilocal mass,” Oberwolfach Rep. 3, No. 1 (2006), 87–88.
- Pengzi Miao, Positive mass theorem on manifolds admitting corners along a hypersurface, Adv. Theor. Math. Phys. 6 (2002), no. 6, 1163–1182 (2003). MR 1982695, DOI 10.4310/ATMP.2002.v6.n6.a4
- Maung Min-Oo, Scalar curvature rigidity of asymptotically hyperbolic spin manifolds, Math. Ann. 285 (1989), no. 4, 527–539. MR 1027758, DOI 10.1007/BF01452046
- Maung Min-Oo, Scalar curvature rigidity of certain symmetric spaces, Geometry, topology, and dynamics (Montreal, PQ, 1995) CRM Proc. Lecture Notes, vol. 15, Amer. Math. Soc., Providence, RI, 1998, pp. 127–136. MR 1619128, DOI 10.1090/crmp/015/08
- Antonio Ros, The isoperimetric problem, Global theory of minimal surfaces, Clay Math. Proc., vol. 2, Amer. Math. Soc., Providence, RI, 2005, pp. 175–209. MR 2167260
- R. Schoen and S.-T. Yau, Lectures on differential geometry, Conference Proceedings and Lecture Notes in Geometry and Topology, I, International Press, Cambridge, MA, 1994. Lecture notes prepared by Wei Yue Ding, Kung Ching Chang [Gong Qing Zhang], Jia Qing Zhong and Yi Chao Xu; Translated from the Chinese by Ding and S. Y. Cheng; With a preface translated from the Chinese by Kaising Tso. MR 1333601
- Richard Schoen and Shing Tung Yau, Positivity of the total mass of a general space-time, Phys. Rev. Lett. 43 (1979), no. 20, 1457–1459. MR 547753, DOI 10.1103/PhysRevLett.43.1457
- Yuguang Shi and Luen-Fai Tam, Positive mass theorem and the boundary behaviors of compact manifolds with nonnegative scalar curvature, J. Differential Geom. 62 (2002), no. 1, 79–125. MR 1987378
- Xiaodong Wang, The mass of asymptotically hyperbolic manifolds, J. Differential Geom. 57 (2001), no. 2, 273–299. MR 1879228
- Edward Witten, A new proof of the positive energy theorem, Comm. Math. Phys. 80 (1981), no. 3, 381–402. MR 626707
Additional Information
- Michael Eichmair
- Affiliation: Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139-4307
- Email: eichmair@math.mit.edu
- Received by editor(s): December 3, 2007
- Received by editor(s) in revised form: September 17, 2008
- Published electronically: April 3, 2009
- Communicated by: Richard A. Wentworth
- © Copyright 2009
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 137 (2009), 2733-2740
- MSC (2000): Primary 53C20
- DOI: https://doi.org/10.1090/S0002-9939-09-09789-5
- MathSciNet review: 2497486