In the mathematical field of differential geometry, a maximal surface is a certain kind of submanifold of a Lorentzian manifold. Precisely, given a Lorentzian manifold (M, g), a maximal surface is a spacelike submanifold of M whose mean curvature is zero. As such, maximal surfaces in Lorentzian geometry are directly analogous to minimal surfaces in Riemannian geometry. The difference in terminology between the two settings has to do with the fact that small regions in maximal surfaces are local maximizers of the area functional, while small regions in minimal surfaces are local minimizers of the area functional.
In 1976, Shiu-Yuen Cheng and Shing-Tung Yau resolved the "Bernstein problem" for maximal hypersurfaces of Minkowski space which are properly embedded, showing that any such hypersurface is a plane. This was part of the body of work for which Yau was awarded the Fields medal in 1982. The Bernstein problem was originally posed by Eugenio Calabi in 1970, who proved some special cases of the result. Simple examples show that there are a number of hypersurfaces of Minkowski space of zero mean curvature which fail to be spacelike.
By an extension of Cheng and Yau's methods, Kazuo Akutagawa considered the case of spacelike hypersurfaces of constant mean curvature in Lorentzian manifolds of positive constant curvature, such as de Sitter space. Luis Alías, Alfonso Romero, and Miguel Sánchez proved a version of Cheng and Yau's result, replacing Minkowski space by the warped product of a closed Riemannian manifold with an interval.
As a problem of partial differential equations, Robert Bartnik and Leon Simon studied the boundary-value problem for maximal surfaces in Minkowski space. The general existence of maximal hypersurfaces in asymptotically flat Lorentzian manifolds, due to Bartnik, is significant in Demetrios Christodoulou and Sergiu Klainerman's renowned proof of the nonlinear stability of Minkowski space under the Einstein field equations. They use a maximal slicing of a general spacetime; the same approach is common in numerical relativity.
References
Footnotes
- Beem, Ehrlich, and Easley, section 6.3
- Choquet-Bruhat, pg. 745
- Kobayashi (1983), section 5
- Gourgoulhon, chapter 10.2
Books
- John K. Beem, Paul E. Ehrlich, and Kevin L. Easley. Global Lorentzian geometry. Second edition. Monographs and Textbooks in Pure and Applied Mathematics, 202. Marcel Dekker, Inc., New York, 1996. xiv+635 pp. ISBN 0-8247-9324-2
- Yvonne Choquet-Bruhat. General relativity and the Einstein equations. Oxford Mathematical Monographs. Oxford University Press, Oxford, 2009. xxvi+785 pp. ISBN 978-0-19-923072-3
- Demetrios Christodoulou and Sergiu Klainerman. The global nonlinear stability of the Minkowski space. Princeton Mathematical Series, 41. Princeton University Press, Princeton, NJ, 1993. x+514 pp. ISBN 0-691-08777-6
- Éric Gourgoulhon. 3 + 1 formalism in general relativity. Bases of numerical relativity. Lecture Notes in Physics, 846. Springer, Heidelberg, 2012. xviii+294 pp. ISBN 978-3-642-24524-4
Articles
- Kazuo Akutagawa. On spacelike hypersurfaces with constant mean curvature in the de Sitter space. Math. Z. 196 (1987), no. 1, 13–19. doi:10.1007/BF01179263
- Luis J. Alías, Alfonso Romero, and Miguel Sánchez. Uniqueness of complete spacelike hypersurfaces of constant mean curvature in generalized Robertson–Walker spacetimes. Gen. Relativity Gravitation 27 (1995), no. 1, 71–84. doi:10.1007/BF02105675
- Robert Bartnik and Leon Simon. Spacelike hypersurfaces with prescribed boundary values and mean curvature.
Comm. Math. Phys. 87 (1982), no. 1, 131–152. doi:10.1007/bf01211061 - Eugenio Calabi. Examples of Bernstein problems for some nonlinear equations. Proc. Sympos. Pure Math., Vol. XV (1970), pp. 223–230. Global Analysis. Amer. Math. Soc., Providence, R.I. doi:10.1090/pspum/015
- Shiu Yuen Cheng and Shing Tung Yau. Maximal space-like hypersurfaces in the Lorentz–Minkowski spaces. Ann. of Math. (2) 104 (1976), no. 3, 407–419. doi:10.2307/1970963
- Osamu Kobayashi. Maximal surfaces in the 3-dimensional Minkowski space L. Tokyo J. Math. 6 (1983), no. 2, 297–309. doi:10.3836/tjm/1270213872