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Some remarks on the existence of spacelike hypersurfaces of constant mean curvature

Published online by Cambridge University Press:  24 October 2008

A. J. Goddard
A. J. Goddard
Affiliation:
Mathematical Institute, Oxford
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Abstract

Bernstein's theorem states that the only complete minimal graphs in R3 are the hyperplanes. We shall produce evidence in favour of some conjectural generalizations of this theorem for the cases of spacelike hypersurfaces of constant mean curvature in Minkowski space and in de Sitter space. The results suggest that the class of asymptotically simple space-times admitting a complete spacelike hypersurface of constant mean curvature may well be considerably smaller than the general class of asymptotically simple space-times.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 82 , Issue 3 , November 1977 , pp. 489 - 495
Copyright
Copyright © Cambridge Philosophical Society 1977

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References

REFERENCES

(1)Almgren, F. J.Ann. of Math. 84 (1966), 277–292.CrossRefGoogle Scholar
(2)Bernstein, S. N.Math. Z. 26 (1927), 551–565.CrossRefGoogle Scholar
(3)Bombieri, E., de Giorgi, E. and Giusti, E.. Invent. Math. 7 (1969), 243–268.CrossRefGoogle Scholar
(4)Calabi, E. (Preprint.)Google Scholar
(5)Erdelyi, A., Magnus, W., Oberhettinger, F. and Tricomi, F. G.Higher Transcendental Functions (New York, McGraw-Hill, 1953).Google Scholar
(6)Fleming, W. H.Rend. Circ. Mat. Palermo 11 (1962), 69–89.CrossRefGoogle Scholar
(7)de Giorgi, E.Ann. Scuola Norm. Sup. Pisa (Scienze Fis. Mat. III), 19 (1965), 79–85.Google Scholar
(8)Hawking, S. W. and Ellis, G. F. R.The large-scale structure of space-time (Cambridge University Press, 1973).CrossRefGoogle Scholar
(9)Hoyle, F.Mon. Not. R.A.S. 108 (1948), 372–382.CrossRefGoogle Scholar
(10)Jorgens, K.Math. Ann. 127 (1954), 130–134.CrossRefGoogle Scholar
(11)Magnus, W., Oberhettinger, F. and Soni, R. P.Special functions of mathematics and physics 3rd ed. (Berlin, Springer-Verlag, 1966).Google Scholar
(12)Penrose, R. Relativistic symmetry groups. Group theory in nonlinear problems (ed. Barat, A. O.). N.A.T.O. Advanced Study Institute, no. 7 (Dordrecht, Reidel, 1974).Google Scholar
(13)Schucking, E. Talk given at the I.C.T.P., Trieste, 07 1975.Google Scholar
(14)Simons, J.Ann. of Math. 88 (1968), 62–105.CrossRefGoogle Scholar