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.[1] 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.[2]
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.[3]
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.
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. ISBN0-8247-9324-2
Yvonne Choquet-Bruhat. General relativity and the Einstein equations. Oxford Mathematical Monographs. Oxford University Press, Oxford, 2009. xxvi+785 pp. ISBN978-0-19-923072-3
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. ISBN0-691-08777-6
Éric Gourgoulhon. 3 + 1 formalism in general relativity. Bases of numerical relativity. Lecture Notes in Physics, 846. Springer, Heidelberg, 2012. xviii+294 pp. ISBN978-3-642-24524-4
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 L3. Tokyo J. Math. 6 (1983), no. 2, 297–309. doi:10.3836/tjm/1270213872
April 18, 2023
maximal, surface, mathematical, field, differential, geometry, maximal, surface, certain, kind, submanifold, lorentzian, manifold, precisely, given, lorentzian, manifold, maximal, surface, spacelike, submanifold, whose, mean, curvature, zero, such, maximal, su. 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 1 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 2 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 3 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 Alias Alfonso Romero and Miguel Sanchez 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 4 References EditFootnotes 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 Eric 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 4Articles 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 Alias Alfonso Romero and Miguel Sanchez 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 L3 Tokyo J Math 6 1983 no 2 297 309 doi 10 3836 tjm 1270213872 Retrieved from https en wikipedia org w index php title Maximal surface amp oldid 986305308, wikipedia, wiki, book, books, library,
