no. 3
Maximizing properties of extremal surfaces in general relativity
Annales de l'institut Henri Poincaré. Section A, Physique Théorique, Tome 28 (1978) no. 3, pp. 335-347. 
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     author = {Brill, Dieter and Flaherty, Frank},
     title = {Maximizing properties of extremal surfaces in general relativity},
     journal = {Annales de l'institut Henri Poincar\'e. Section A, Physique Th\'eorique},
     pages = {335--347},
     publisher = {Gauthier-Villars},
     volume = {28},
     number = {3},
     year = {1978},
     mrnumber = {479299},
     zbl = {0375.53002},
     language = {en},
     url = {http://www.numdam.org/item/AIHPA_1978__28_3_335_0/}
}
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Brill, Dieter; Flaherty, Frank. Maximizing properties of extremal surfaces in general relativity. Annales de l'institut Henri Poincaré. Section A, Physique Théorique, Tome 28 (1978) no. 3, pp. 335-347. http://www.numdam.org/item/AIHPA_1978__28_3_335_0/

[1] R. Arnowitt, S. Deser, C.W. Misner, in : Gravitation (ed. L. Witten), New York, Wiley, 1962 ; D. Brill, S. Deser, Ann. Phys. New York, t. 50, 1968, p. 548 ; J. York, N. O'Murchadha, J. Math. Phys., t. 14, 1973, p. 1551 ; E. Schücking, talk given at I. C. T. P., Trieste, July 1975 ; Y. Choquet-Bruhat, A.E. Fischer, J.E. Marsden, Proceedings of 1976 « E. Fermi » school of Physics. | MR

[2] D. Brill, F. Flaherty, Comm. Math. Phys., t. 50, 1976, p. 157. | MR | Zbl

[3] E. Heinz, Math. Ann., t. 127, 1954, p. 258 ; M. Miranda, Proc. Symp. Pure Math., XXIII, 1973, p. 1 ; D. Brill, J. Isenberg, to be published. | MR | Zbl

[4] A.J. Goddard, Ph. D. Thesis, Oxford, 1975. G. R. G. Journal, t. 8, 1977, p. 525.

[5] The operators which we define on the normal bundle would correspond to operators acting on scalars in the usual [1] « 3 + 1 decomposition ». See appendix of [2] for more detail. Among the advantages of using the normal bundle are that e. g. the mean curvature vector is independent of the choice of normal direction, and that the approach can more easily be generalized to hypersurfaces of higher codimension.

[6] See, for example, R. Courant, D. Hilbert, Methods of Mathematical Physics, Vo. II, New York, Wiley, 1962.

[7] M. Morse, The Calculus of Variations in the Large, New York. Amer. Math. Soc., 1934. | JFM | Zbl

[8] J. Simons, Ann. Math. (USA), t. 88, 1968, p. 62. | MR | Zbl

[9] S. Hawking, G. Ellis, The large scale structure of spacetime, Cambridge, University Press, 1973. | MR | Zbl

[10] F. Tipler, J. Math. Phys., t. 18, 1977, p. 1568. | Zbl

[11] A.H. Taub, Ann. Math. (USA), t. 53, 1951, p. 472; C.W. Misner, A.H. Taub, J. E. T. P., t. 28, 1968, p. 122. | Zbl

[12] D.R. Brill, Phys. Rev. B, t. 133, 1964, p. 845. | MR | Zbl

[13] We use the convention of earlier publications [11, 12], without a factor 1/2. The « unit » 3-sphere then has radius 2 rather than 1.

[14] A. Lichnerowicz, Problèmes globaux en Mécanique Relativiste, Paris, Herman, 1939 ; Y. Choquet-Bruhat, J. Rat., Mech. Anal., t. 5, 1956, p. 951. | Zbl