Some scalar curvature warped product splitting theorems

Galloway, Gregory; Jang, Hyun

doi:10.1090/proc/14922

Some scalar curvature warped product splitting theorems
HTML articles powered by AMS MathViewer

by Gregory J. Galloway and Hyun Chul Jang

Proc. Amer. Math. Soc. 148 (2020), 2617-2629

DOI: https://doi.org/10.1090/proc/14922

Published electronically: February 18, 2020

PDF | Request permission

Abstract:

We present several rigidity results for Riemannian manifolds $(M,g)$ with scalar curvature $S \ge -n(n-1)$ (or $S\ge 0$), and having compact boundary $N$ satisfying a related mean curvature inequality. The proofs make use of results on marginally outer trapped surfaces applied to appropriate initial data sets. One of the results involves an analysis of Obata’s equation on manifolds with boundary. This result is relevant to recent work of Lan-Hsuan Huang and the second author concerning the rigidity of asymptotically locally hyperbolic manifolds with zero mass.

References

Lucas C. Ambrozio, Rigidity of area-minimizing free boundary surfaces in mean convex three-manifolds, J. Geom. Anal. 25 (2015), no. 2, 1001–1017. MR 3319958, DOI 10.1007/s12220-013-9453-2
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
Lars Andersson, Michael Eichmair, and Jan Metzger, Jang’s equation and its applications to marginally trapped surfaces, Complex analysis and dynamical systems IV. Part 2, Contemp. Math., vol. 554, Amer. Math. Soc., Providence, RI, 2011, pp. 13–45. MR 2884392, DOI 10.1090/conm/554/10958
Lars Andersson, Marc Mars, and Walter Simon, Stability of marginally outer trapped surfaces and existence of marginally outer trapped tubes, Adv. Theor. Math. Phys. 12 (2008), no. 4, 853–888. MR 2420905
Lars Andersson and Jan Metzger, The area of horizons and the trapped region, Comm. Math. Phys. 290 (2009), no. 3, 941–972. MR 2525646, DOI 10.1007/s00220-008-0723-y
Lars Andersson and Jan Metzger, Curvature estimates for stable marginally trapped surfaces, J. Differential Geom. 84 (2010), no. 2, 231–265. MR 2652461
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
Alessandro Carlotto, Otis Chodosh, and Michael Eichmair, Effective versions of the positive mass theorem, Invent. Math. 206 (2016), no. 3, 975–1016. MR 3573977, DOI 10.1007/s00222-016-0667-3
Otis Chodosh, Michael Eichmair, and Vlad Moraru, A splitting theorem for scalar curvature, Comm. Pure Appl. Math. 72 (2019), no. 6, 1231–1242. MR 3948556, DOI 10.1002/cpa.21803
P. T. Chruściel and E. Delay, The hyperbolic positive energy theorem, 2019, arXiv.1901.05263v2.
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
Christopher B. Croke and Bruce Kleiner, A warped product splitting theorem, Duke Math. J. 67 (1992), no. 3, 571–574. MR 1181314, DOI 10.1215/S0012-7094-92-06723-8
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
Gregory J. Galloway, Rigidity of marginally trapped surfaces and the topology of black holes, Comm. Anal. Geom. 16 (2008), no. 1, 217–229. MR 2411473
Gregory J. Galloway, Stability and rigidity of extremal surfaces in Riemannian geometry and general relativity, Surveys in geometric analysis and relativity, Adv. Lect. Math. (ALM), vol. 20, Int. Press, Somerville, MA, 2011, pp. 221–239. MR 2906927
Gregory J. Galloway, Rigidity of outermost MOTS: the initial data version, Gen. Relativity Gravitation 50 (2018), no. 3, Paper No. 32, 7. MR 3768955, DOI 10.1007/s10714-018-2353-9
Gregory J. Galloway and Richard Schoen, A generalization of Hawking’s black hole topology theorem to higher dimensions, Comm. Math. Phys. 266 (2006), no. 2, 571–576. MR 2238889, DOI 10.1007/s00220-006-0019-z
Mikhael Gromov and H. Blaine Lawson Jr., Positive scalar curvature and the Dirac operator on complete Riemannian manifolds, Inst. Hautes Études Sci. Publ. Math. 58 (1983), 83–196 (1984). MR 720933
L.-H. Huang and H. C. Jang, Mass rigidity for asymptotically locally hyperbolic manifolds with boundary, in preparation.
L.-H. Huang, H. C. Jang, and D. Martin, Mass rigidity for hyperbolic manifolds, arXiv (2019).
Masahiko Kanai, On a differential equation characterizing a Riemannian structure of a manifold, Tokyo J. Math. 6 (1983), no. 1, 143–151. MR 707845, DOI 10.3836/tjm/1270214332
J. Lohkamp, The higher dimensional positive mass theorem II, 2016, arXiv:1612.0750.
Mario Micallef and Vlad Moraru, Splitting of 3-manifolds and rigidity of area-minimising surfaces, Proc. Amer. Math. Soc. 143 (2015), no. 7, 2865–2872. MR 3336611, DOI 10.1090/S0002-9939-2015-12137-5
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
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
R. Schoen and S. T. Yau, On the structure of manifolds with positive scalar curvature, Manuscripta Math. 28 (1979), no. 1-3, 159–183. MR 535700, DOI 10.1007/BF01647970
Richard Schoen and Shing Tung Yau, Proof of the positive mass theorem. II, Comm. Math. Phys. 79 (1981), no. 2, 231–260. MR 612249
Richard Schoen and Shing Tung Yau, Positive scalar curvature and minimal hypersurface singularities, 2017, arXiv:1704.05490.
Yoshihiro Tashiro, Complete Riemannian manifolds and some vector fields, Trans. Amer. Math. Soc. 117 (1965), 251–275. MR 174022, DOI 10.1090/S0002-9947-1965-0174022-6
Xiaodong Wang, The mass of asymptotically hyperbolic manifolds, J. Differential Geom. 57 (2001), no. 2, 273–299. MR 1879228