A CMC existence result for expanding cosmological spacetimes
HTML articles powered by AMS MathViewer
- by Gregory J. Galloway and Eric Ling;
- Trans. Amer. Math. Soc.
- DOI: https://doi.org/10.1090/tran/9471
- Published electronically: June 10, 2025
- PDF | Request permission
Abstract:
We establish a new CMC (constant mean curvature) existence result for cosmological spacetimes, by which we mean globally hyperbolic spacetimes with compact Cauchy surfaces. If a cosmological spacetime satisfying the strong energy condition contains an expanding Cauchy surface and is future timelike geodesically complete, then the spacetime contains a CMC Cauchy surface. This result settles, under certain circumstances, a conjecture of the authors and a conjecture of Dilts and Holst. Our proof relies on the construction of barriers in the support sense, and the CMC Cauchy surface is found as the asymptotic limit of mean curvature flow. Analogous results are also obtained in the case of a positive cosmological constant $\Lambda > 0$. Lastly, we include some comments concerning the future causal boundary for cosmological spacetimes which pertain to the CMC conjecture of the authors.References
- Lars Andersson, Thierry Barbot, François Béguin, and Abdelghani Zeghib, Cosmological time versus CMC time in spacetimes of constant curvature, Asian J. Math. 16 (2012), no. 1, 37–87. MR 2904912, DOI 10.4310/AJM.2012.v16.n1.a2
- Lars Andersson and Vincent Moncrief, Future complete vacuum spacetimes, The Einstein equations and the large scale behavior of gravitational fields, Birkhäuser, Basel, 2004, pp. 299–330. MR 2098919
- Lars Andersson, Gregory J. Galloway, and Ralph Howard, A strong maximum principle for weak solutions of quasi-linear elliptic equations with applications to Lorentzian and Riemannian geometry, Comm. Pure Appl. Math. 51 (1998), no. 6, 581–624. MR 1611140, DOI 10.1002/(SICI)1097-0312(199806)51:6<581::AID-CPA2>3.3.CO;2-E
- Robert Bartnik, Existence of maximal surfaces in asymptotically flat spacetimes, Comm. Math. Phys. 94 (1984), no. 2, 155–175. MR 761792
- Robert Bartnik, Remarks on cosmological spacetimes and constant mean curvature surfaces, Comm. Math. Phys. 117 (1988), no. 4, 615–624. MR 953823
- John K. Beem, Paul E. Ehrlich, and Kevin L. Easley, Global Lorentzian geometry, 2nd ed., Monographs and Textbooks in Pure and Applied Mathematics, vol. 202, Marcel Dekker, Inc., New York, 1996. MR 1384756
- Antonio N. Bernal and Miguel Sánchez, Smoothness of time functions and the metric splitting of globally hyperbolic spacetimes, Comm. Math. Phys. 257 (2005), no. 1, 43–50. MR 2163568, DOI 10.1007/s00220-005-1346-1
- Piotr T. Chruściel, James Isenberg, and Daniel Pollack, Initial data engineering, Comm. Math. Phys. 257 (2005), no. 1, 29–42. MR 2163567, DOI 10.1007/s00220-005-1345-2
- J. Dilts and M. Holst, When do spacetimes have constant mean curvature slices?, 2021, arXiv:2104.02136.
- Fernando Dobarro and Bülent Ünal, Curvature of multiply warped products, J. Geom. Phys. 55 (2005), no. 1, 75–106. MR 2157416, DOI 10.1016/j.geomphys.2004.12.001
- Klaus Ecker, On mean curvature flow of spacelike hypersurfaces in asymptotically flat spacetimes, J. Austral. Math. Soc. Ser. A 55 (1993), no. 1, 41–59. MR 1231693
- Klaus Ecker and Gerhard Huisken, Parabolic methods for the construction of spacelike slices of prescribed mean curvature in cosmological spacetimes, Comm. Math. Phys. 135 (1991), no. 3, 595–613. MR 1091580
- J.-H. Eschenburg, Comparison theorems and hypersurfaces, Manuscripta Math. 59 (1987), no. 3, 295–323. MR 909847, DOI 10.1007/BF01174796
- Gregory J. Galloway, Some rigidity results for spatially closed spacetimes, Mathematics of gravitation, Part I (Warsaw, 1996) Banach Center Publ., vol. 41, Polish Acad. Sci. Inst. Math., Warsaw, 1997, pp. 21–34. MR 1466507
- Gregory J. Galloway, Existence of CMC Cauchy surfaces and spacetime splitting, Pure Appl. Math. Q. 15 (2019), no. 2, 667–682. MR 4047388, DOI 10.4310/PAMQ.2019.v15.n2.a2
- Gregory J. Galloway and Eric Ling, Existence of CMC Cauchy surfaces from a spacetime curvature condition, Gen. Relativity Gravitation 50 (2018), no. 9, Paper No. 108, 7. MR 3840968, DOI 10.1007/s10714-018-2428-7
- Gregory J. Galloway and Eric Ling, Remarks on the existence of CMC Cauchy surfaces, Developments in Lorentzian geometry, Springer Proc. Math. Stat., vol. 389, Springer, Cham, [2022] ©2022, pp. 93–104. MR 4539753, DOI 10.1007/978-3-031-05379-5_{6}
- Gregory J. Galloway and Carlos Vega, Achronal limits, Lorentzian spheres, and splitting, Ann. Henri Poincaré 15 (2014), no. 11, 2241–2279. MR 3268829, DOI 10.1007/s00023-013-0305-1
- Gregory J. Galloway and Carlos Vega, Hausdorff closed limits and rigidity in Lorentzian geometry, Ann. Henri Poincaré 18 (2017), no. 10, 3399–3426. MR 3697198, DOI 10.1007/s00023-017-0594-x
- Claus Gerhardt, $H$-surfaces in Lorentzian manifolds, Comm. Math. Phys. 89 (1983), no. 4, 523–553. MR 713684
- Steven G. Harris, Discrete group actions on spacetimes: causality conditions and the causal boundary, Classical Quantum Gravity 21 (2004), no. 4, 1209–1236. MR 2036151, DOI 10.1088/0264-9381/21/4/032
- S. W. Hawking and G. F. R. Ellis, The large scale structure of space-time, Cambridge Monographs on Mathematical Physics, No. 1, Cambridge University Press, London-New York, 1973. MR 424186
- James Isenberg, Constant mean curvature solutions of the Einstein constraint equations on closed manifolds, Classical Quantum Gravity 12 (1995), no. 9, 2249–2274. MR 1353772
- Hermann Karcher, Riemannian comparison constructions, Global differential geometry, MAA Stud. Math., vol. 27, Math. Assoc. America, Washington, DC, 1989, pp. 170–222. MR 1013810
- Eric Ling and Argam Ohanyan, Examples of cosmological spacetimes without CMC Cauchy surfaces, Lett. Math. Phys. 114 (2024), no. 4, Paper No. 96, 27. MR 4771098, DOI 10.1007/s11005-024-01843-7
- E. Ling and A. Ohanyan, Vacuum cosmological spacetimes without CMC cauchy surfaces, arXiv:2407.21524 (2024).
- Barrett O’Neill, Semi-Riemannian geometry, Pure and Applied Mathematics, vol. 103, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1983. With applications to relativity. MR 719023
- F. J. Tipler, A new condition implying the existence of a constant mean curvature foliation, Directions in General Relativity: Proceedings of the 1993 International Symposium, Maryland: Papers in Honor of Dieter Brill 2 (1993), no. 10, 306–315.
- Bülent Ünal, Multiply warped products, J. Geom. Phys. 34 (2000), no. 3-4, 287–301. MR 1762779, DOI 10.1016/S0393-0440(99)00072-8
Bibliographic Information
- Gregory J. Galloway
- Affiliation: University of Miami, Coral Gables, Florida
- MR Author ID: 189210
- Email: galloway@math.miami.edu
- Eric Ling
- Affiliation: Copenhagen Centre for Geometry and Topology (GeoTop), Department of Mathematical Sciences, University of Copenhagen, Denmark
- MR Author ID: 1231921
- ORCID: 0000-0002-9989-132X
- Email: el@math.ku.dk
- Received by editor(s): November 13, 2024
- Received by editor(s) in revised form: February 13, 2025, February 18, 2025, February 19, 2025, and February 22, 2025
- Published electronically: June 10, 2025
- Additional Notes: The first author was supported by the Simons Foundation, Award No. 850541. The second author was supported by Carlsberg Foundation CF21-0680 and Danmarks Grundforskningsfond CPH-GEOTOP-DNRF151. Part of the research on this paper was supported by the National Science Foundation under Grant No. DMS-1928930 while the authors were in residence at the Simons Laufer Mathematical Sciences Institute (formerly MSRI) in Berkeley, California, during the Fall 2024 semester.
- © Copyright 2025 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
- MSC (2020): Primary 53-XX; Secondary 83-XX
- DOI: https://doi.org/10.1090/tran/9471