On the Hawking mass for CMC surfaces in positive curved 3-manifolds

Melo, Luiz Ricardo

doi:10.1090/proc/17014

On the Hawking mass for CMC surfaces in positive curved 3-manifolds
HTML articles powered by AMS MathViewer

by Luiz Ricardo Abreu Melo;

Proc. Amer. Math. Soc. 152 (2024), 5373-5380

DOI: https://doi.org/10.1090/proc/17014

Published electronically: October 10, 2024

HTML | PDF | Request permission

Abstract:

In this paper, we present two rigidity results for stable constant mean curvature surfaces immersed in $3$-manifolds with positive scalar curvature, assuming that the Hawking mass is zero. In the first result, we assume the surface to be approximately round, while in the second result, we consider surfaces invariant under an even symmetry.

References

Richard Arnowitt, Stanley Deser, and Charles W. Misner, Republication of: The dynamics of general relativity, Gen. Relativity Gravitation 40 (2008), 1997–2027, DOI 10.1007/ s10714-008-0661-1.
Robert Bartnik, Mass and 3-metrics of non-negative scalar curvature, Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002) Higher Ed. Press, Beijing, 2002, pp. 231–240. MR 1957036
Simon Brendle, Fernando C. Marques, and Andre Neves, Deformations of the hemisphere that increase scalar curvature, Invent. Math. 185 (2011), no. 1, 175–197. MR 2810799, DOI 10.1007/s00222-010-0305-4
Jingyi Chen, Ailana Fraser, and Chao Pang, Minimal immersions of compact bordered Riemann surfaces with free boundary, Trans. Amer. Math. Soc. 367 (2015), no. 4, 2487–2507. MR 3301871, DOI 10.1090/S0002-9947-2014-05990-4
Otis Chodosh, Large isoperimetric regions in asymptotically hyperbolic manifolds, Comm. Math. Phys. 343 (2016), no. 2, 393–443. MR 3477343, DOI 10.1007/s00220-015-2457-y
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
A. El Soufi and S. Ilias, Majoration de la seconde valeur propre d’un opérateur de Schrödinger sur une variété compacte et applications, J. Funct. Anal. 103 (1992), no. 2, 294–316 (French, with English summary). MR 1151550, DOI 10.1016/0022-1236(92)90123-Z
Fengbo Hang and Xiaodong Wang, Rigidity theorems for compact manifolds with boundary and positive Ricci curvature, J. Geom. Anal. 19 (2009), no. 3, 628–642. MR 2496569, DOI 10.1007/s12220-009-9074-y
S. W. Hawking, Gravitational radiation in an expanding universe, J. Mathematical Phys. 9 (1968), no. 4, 598–604. MR 3960907, DOI 10.1063/1.1664615
H. Blaine Lawson Jr., The unknottedness of minimal embeddings, Invent. Math. 11 (1970), 183–187. MR 287447, DOI 10.1007/BF01404649
Davi Máximo and Ivaldo Nunes, Hawking mass and local rigidity of minimal two-spheres in three-manifolds, Comm. Anal. Geom. 21 (2013), no. 2, 409–432. MR 3043752, DOI 10.4310/CAG.2013.v21.n2.a6
Richard Schoen and Shing Tung Yau, On the proof of the positive mass conjecture in general relativity, Comm. Math. Phys. 65 (1979), no. 1, 45–76. MR 526976, DOI 10.1007/BF01940959
Yuguang Shi, Jiacheng Sun, Gang Tian, and Dongyi Wei, Uniqueness of the mean field equation and rigidity of Hawking mass, Calc. Var. Partial Differential Equations 58 (2019), no. 2, Paper No. 41, 16. MR 3910093, DOI 10.1007/s00526-019-1496-1
Jiacheng Sun, Rigidity of Hawking mass for surfaces in three manifolds, Pacific J. Math. 292 (2018), no. 2, 479–504. MR 3733981, DOI 10.2140/pjm.2018.292.479
Edward Witten, A new proof of the positive energy theorem, Comm. Math. Phys. 80 (1981), no. 3, 381–402. MR 626707, DOI 10.1007/BF01208277