Abstract
Maximal surfaces and their implications for the ambient spacetime are studied. Our methods exploit the interplay between contact of the volume functional and energy conditions. Essentially, we find that in closed universes, maximal surfaces are unique; they maximize volume; and they yield future and past singularities.
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References
See, for example, Nitsche, J. C. C.: Vorlesungen über Minimalflächen. Berlin-Heidelberg-New York: Springer 1975
For an example, see, Reinhart, B. L.: J. Math. Phys.14, 719 (1973)
For existence proof see, Choquet-Bruhat, Y.: C. R. Acad. Sci. (Paris)280, 169 (1975);
Cantor, M., Fischer, A., Marsden, J., O'Murchadha, N., York, J.: Commun. math. Phys.49, 187 (1976)
Arnowitt, R., Deser, S., Misner, C. W.: In: Gravitation (ed. L. Witten). New York: Wiley 1962
Brill, D., Deser, S.: Ann. Phys. (NY)50, 548 (1968);
York, J., O'Murchadha, N.: J. Math. Phys.14, 1551 (1973);
Schücking, E.: Proceedings of Marcel Grossman Symposium, to appear
See also Frankel, T.: Bull. Amer. Math. Soc.81, 579 (1975)
See, for example, Hawking, S., Ellis, G.: The large scale structure of space-time, Cambridge: University Press 1973
See, for example, Misner, C. W., Thorne, K., Wheeler, J. A.: Gravitation. San Francisco: W. H. Freeman and Company 1973
Hsiang, W. Y.: Proc. Nat. Acad. Sci. (USA)56, 5 (1966)
Kasner, E.: Amer. J. Math.43, 217 (1921)
See, for example, Schrödinger, E.: Expanding Universes. London: Cambridge University Press 1956
See Gromoll, D., Klingenberg, W., Meyer, W.: Riemannsche Geometrie im Großen. Berlin-Heidelberg-New York: Springer 1968
Hicks, N.: Notes on Differential Geometry. Princeton: D. van Nostrand 1965
Frankel, T.: Pacific J. Math.11, 165 (1961)
Choquet-Bruhat, Y.: C.R. Acad. Sci. (Paris)281, 577 (1975)
Also see Choquet-Bruhat, Y.: C.R. Acad. Sci. (Paris)280, 169 (1975). For an existence proof in the case of a periodic closed space-time, see Avez, A.: Ann. Inst. Fourier13, 105 (1963)
See, for example, Corollary 3.6.1 of Simons, J.: Ann. Math. (USA)88, 62 (1968)
For example, see Spanier, E. H.: Algebraic topology. New York: McGraw-Hill Publishing Co. 1966
Simons, J.: Ann. Math. (USA)88, 62 (1968)
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Communicated by J. Ehlers
Supported in part by the National Science Foundation under grant No. PHY 70-02077A03 and by the Humboldt Foundation
Supported in part by the Sonderforschungsbereich (Theoretische Mathematik) of the University of Bonn
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Brill, D., Flaherty, F. Isolated maximal surfaces in spacetime. Commun.Math. Phys. 50, 157–165 (1976). https://doi.org/10.1007/BF01617993
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DOI: https://doi.org/10.1007/BF01617993