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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 {{math|(''M'', ''g'')}}, a maximal surface is a spacelike submanifold of {{mvar|M}} whose [[mean curvature]] is zero.<ref>Beem, Ehrlich, and Easley, section 6.3</ref> As such, maximal surfaces in Lorentzian geometry are directly analogous to [[minimal surface]]s 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.<ref>Choquet-Bruhat, pg. 745</ref>

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.<ref>Kobayashi (1983), section 5</ref>

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 manifold|closed]] [[Riemannian manifold]] with an interval.

As a problem of [[partial differential equation]]s, [[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]].<ref>Gourgoulhon, chapter 10.2</ref>

== References ==
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'''Footnotes'''
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'''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&nbsp;+&nbsp;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}} {{closed access}}
* 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}} {{closed access}}
* Robert Bartnik and Leon Simon. [https://projecteuclid.org/euclid.cmp/1103921909 ''Spacelike hypersurfaces with prescribed boundary values and mean curvature.''] {{free access}} Comm. Math. Phys. 87 (1982), no. 1, 131–152. {{doi|10.1007/bf01211061}} {{closed access}} 
* 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}} {{closed access}}
* 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}} {{closed access}}
* Osamu Kobayashi. ''Maximal surfaces in the 3-dimensional Minkowski space {{math|L<sup>3</sup>}}.'' Tokyo J. Math. 6 (1983), no. 2, 297–309. {{doi|10.3836/tjm/1270213872}} {{free access}}
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[[Category:Differential geometry]]