The equivariant Minkowski problem in Minkowski space
Annales de l'Institut Fourier, Volume 67 (2017) no. 3, p. 1035-1113
The classical Minkowski problem in Minkowski space asks,given a positive function φ on d , for a convex set K in Minkowski space with C 2 space-like boundary S, such that φ(η) -1 is the Gauss–Kronecker curvature at the point with normal η. Analogously to the Euclidean case, it is possible to formulate a weak version of this problem: given a Radon measure μ on d the generalized Minkowski problem in Minkowski space asks for a convex subset K such that the area measure of K is μ.In the present paper we look at an equivariant version of the problem: given a uniform lattice Γ of isometries of d , a Γ invariant Radon measure μ and an isometry group Γ τ of Minkowski space with Γ as linear part, there exists a unique convex set with area measure μ, invariant under the action of Γ τ . The proof uses a functional which is the covolume associated to every invariant convex set.This result translates as a solution of the Minkowski problem in flat space times with compact hyperbolic Cauchy surface. The uniqueness part, as well as the regularity results, follow from properties of the Monge–Ampère equation. The existence part can be translated as an existence result for Monge–Ampère equation.The regular version was proved by T. Barbot, F. Béguin and A. Zeghib for d=2 and by V. Oliker and U. Simon for Γ τ =Γ. Our method is totally different. Moreover, we show that those cases are very specific: in general, there is no smooth Γ τ -invariant hypersurface of constant Gauss–Kronecker curvature equal to 1.
La forme classique du problème de Minkowski dans l’espace de Minkowski est la recherche, pour une fonction positive φ sur d , d’un ensemble convexe K dans l’espace de Minkowski, avec un bord S C 2 de type espace, tel que φ(η) -1 soit la courbure de Gauss–Kronecker au point dont la normale est η. De façon analogue au cas euclidien, il est possible de formuler une version faible de ce problème  : étant donné un mesure de Radon μ sur d , le problème de Minkowski généralisé dans l’espace de Minkowski est la recherche d’un ensemble convexe K tel que la mesure d’aire de K est μ.Dans ce travail nous regardons une version équivariante de ce problème  : étant donné un réseau uniforme Γ d’isométries de d , étant donné une mesure de Radon μ invariante pour l’action de Γ, étant donné un groupe d’isométries Γ τ de l’espace de Minkowski, dont Γ est la partie linéaire, il existe un unique ensemble convexe de mesure d’aire μ, et qui est invariant sous l’action de Γ τ . La preuve utilise une fonctionnelle, le covolume associé à tout ensemble convexe invariant.Ce résultat donne une solution du problème de Minkowski dans des espace-temps plats avec des surfaces de Cauchy compactes. L’unicité de la solution, ainsi que les résultats à propos de la régularité de la solution, sont des conséquences de propriétés de l’équation de Monge–Ampère. L’existence de la solution se traduit en un résultat d’existence pour l’équation de Monge–Ampère.La version régulière de notre résultat a été montrée par T. Barbot, F. Béguin et A. Zeghib dans le cas d=2 et par V. Oliker et U. Simon quand Γ τ =Γ. Notre méthode est totalement différente. De plus, nous montrons que ces deux cas sont vraiment spéciaux  : en général, il n’existe pas d’hypersurface lisse invariante sous l’action de Γ τ ayant une courbure de Gauss–Kronecker constante égale à 1.
Received : 2016-02-04
Revised : 2016-09-08
Accepted : 2016-09-20
Published online : 2017-05-31
DOI : https://doi.org/10.5802/aif.3105
Classification:  53C50,  52A20,  53C42,  35J60
Keywords: Minkowski problem, Lorentzian geometry, covolume, Monge–Ampère equation
@article{AIF_2017__67_3_1035_0,
     author = {Bonsante, Francesco and Fillastre, Fran\c cois},
     title = {The equivariant Minkowski problem in Minkowski space},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {67},
     number = {3},
     year = {2017},
     pages = {1035-1113},
     doi = {10.5802/aif.3105},
     language = {en},
     url = {https://aif.centre-mersenne.org/item/AIF_2017__67_3_1035_0}
}
The equivariant Minkowski problem in Minkowski space. Annales de l'Institut Fourier, Volume 67 (2017) no. 3, pp. 1035-1113. doi : 10.5802/aif.3105. https://aif.centre-mersenne.org/item/AIF_2017__67_3_1035_0/

[1] Alexandrov, Aleksandr D. On the theory of mixed volumes II, Mat. Sbornik, Tome 44 (1937), pp. 1205-1238 (in Russian, translated in [2])

[2] Alexandrov, Aleksandr D. Selected works. Part I, Gordon and Breach Publishers, Classics of Soviet Mathematics, Tome 4 (1996), x+322 pages (Selected scientific papers, Translated from the Russian by P. S. V. Naidu, Edited and with a preface by Yu. G. Reshetnyak and S. S. Kutateladze)

[3] Barbot, Thierry Globally hyperbolic flat space-times, J. Geom. Phys., Tome 53 (2005) no. 2, pp. 123-165 | Article

[4] Barbot, Thierry Domaines globalement hyperboliques de l’espace de Minkowski et de l’espace anti-de Sitter, Algèbre, dynamique et analyse pour la géométrie : aspects récents, Ellipses (2010), pp. 101-138 (Proceedings des Écoles de Géométrie et Systèmes dynamiques, Algérie, 2004-2007)

[5] Barbot, Thierry; Béguin, François; Zeghib, Abdelghani Prescribing Gauss curvature of surfaces in 3-dimensional spacetimes: application to the Minkowski problem in the Minkowski space, Ann. Inst. Fourier, Tome 61 (2011) no. 2, pp. 511-591 | Article

[6] Belraouti, Mehdi Asymptomatic convergence of level sets of quasi-concave times in a space-time of constant curvature, Université d’Avignon (France) (2013) (Ph. D. Thesis)

[7] Belraouti, Mehdi Asymptotic behavior of Cauchy hypersurfaces in constant curvature space-times (2015) (https://arxiv.org/abs/1503.06343 )

[8] Bertrand, Jérôme Prescription of Gauss curvature on compact hyperbolic orbifolds, Discrete Contin. Dyn. Syst., Tome 34 (2014) no. 4, pp. 1269-1284 | Article

[9] Bishop, Robert L.; O’Neill, Barrett Manifolds of negative curvature, Trans. Am. Math. Soc., Tome 145 (1969), pp. 1-49 | Article

[10] Bonnesen, T.; Fenchel, Werner Theory of convex bodies, BCS Associates, Moscow, ID (1987), x+172 pages (Translated from the German and edited by L. Boron, C. Christenson and B. Smith)

[11] Bonsante, Francesco Flat spacetimes with compact hyperbolic Cauchy surfaces, J. Differ. Geom., Tome 69 (2005) no. 3, pp. 441-521 http://projecteuclid.org/getRecord?id=euclid.jdg/1122493997 | Article

[12] Bonsante, Francesco; Meusburger, Catherine; Schlenker, Jean-Marc Recovering the geometry of a flat spacetime from background radiation, Ann. Henri Poincaré, Tome 15 (2014) no. 9, pp. 1733-1799 | Article

[13] Carlier, Guillaume On a theorem of Alexandrov, J. Nonlinear Convex Anal., Tome 5 (2004) no. 1, pp. 49-58

[14] Cheng, Shiu-Yuen; Yau, Shing-Tung On the regularity of the Monge-Ampère equation det ( 2 u/x i sx j )=F(x,u), Commun. Pure Appl. Math., Tome 30 (1977) no. 1, pp. 41-68 | Article

[15] Choquet-Bruhat, Yvonne; Geroch, Robert Global aspects of the Cauchy problem in general relativity, Commun. Math. Phys., Tome 14 (1969), pp. 329-335 | Article

[16] Fillastre, François Fuchsian convex bodies: basics of Brunn-Minkowski theory, Geom. Funct. Anal., Tome 23 (2013) no. 1, pp. 295-333 | Article

[17] Fillastre, François; Veronelli, Giona Lorentzian area measures and the Christoffel problem, Ann. Sc. Norm. Super. Pisa Cl. Sci., Tome 16 (2016) no. 2, pp. 383-467

[18] Guan, Bo; Jian, Huai-Yu; Schoen, Richard M. Entire spacelike hypersurfaces of prescribed Gauss curvature in Minkowski space, J. Reine Angew. Math., Tome 595 (2006), pp. 167-188 | Article

[19] Gutiérrez, Cristian The Monge-Ampère equation, Birkhäuser, Boston, MA, Progress in Nonlinear Differential Equations and their Applications, Tome 44 (2001), xii+127 pages | Article

[20] Hiriart-Urruty, Jean-Baptiste; Lemaréchal, Claude Convex analysis and minimization algorithms. I, Springer-Verlag, Berlin, Grundlehren der Mathematischen Wissenschaften, Tome 305 (1993), xvii+417 pages (Fundamentals)

[21] Horn, Roger A.; Johnson, Charles R. Matrix analysis, Cambridge University Press (1990), xiii+561 pages (Corrected reprint of the 1985 original)

[22] Khovanskiĭ, Askold; Timorin, Vladlen On the theory of coconvex bodies, Discrete Comput. Geom., Tome 52 (2014) no. 4, pp. 806-823 | Article

[23] Li, An-Min Spacelike hypersurfaces with constant Gauss–Kronecker curvature in the Minkowski space, Arch. Math., Tome 64 (1995) no. 6, pp. 534-551 | Article

[24] Mess, Geoffrey Lorentz spacetimes of constant curvature, Geom. Dedicata, Tome 126 (2007), pp. 3-45 | Article

[25] Nikolaev, Igor G.; Shefelʼ, S. Z. Convex surfaces with positive bounded specific curvature, and a priori estimates for Monge-Ampère equations, Sibirsk. Mat. Zh., Tome 26 (1985) no. 4, p. 120-136, 205

[26] Oliker, Vladimir I.; Simon, Udo Codazzi tensors and equations of Monge-Ampère type on compact manifolds of constant sectional curvature, J. Reine Angew. Math., Tome 342 (1983), pp. 35-65

[27] Pogorelov, Aleksey V. Extrinsic geometry of convex surfaces, American Mathematical Society, Translations of Mathematical Monographs, Tome 35 (1973), vi+669 pages (Translated from the Russian by Israel Program for Scientific Translations)

[28] Pogorelov, Aleksey V. The Minkowski multidimensional problem, V. H. Winston & Sons, Washington, D.C. (1978), 106 pages (Translated from the Russian by Vladimir Oliker, Introduction by Louis Nirenberg, Scripta Series in Mathematics)

[29] Ratcliffe, John G. Foundations of hyperbolic manifolds, Springer-Verlag, New York, Graduate Texts in Mathematics, Tome 149 (2006), xii+779 pages

[30] Rauch, Jeffrey; Taylor, Bert A. The Dirichlet problem for the multidimensional Monge-Ampère equation, Rocky Mt. J. Math., Tome 7 (1977) no. 2, pp. 345-364 | Article

[31] Rockafellar, R. Tyrrell Convex analysis, Princeton University Press, Princeton Landmarks in Mathematics (1997), xviii+451 pages (Reprint of the 1970 original, Princeton Paperbacks)

[32] Rockafellar, R. Tyrrell; Wets, Roger J.-B. Variational analysis, Springer-Verlag, Berlin, Grundlehren der Mathematischen Wissenschaften, Tome 317 (1998), xiv+733 pages | Article

[33] Scannell, Kevin P. Infinitesimal deformations of some SO (3,1) lattices, Pac. J. Math., Tome 194 (2000) no. 2, pp. 455-464 | Article

[34] Schneider, Rolf Convex bodies: the Brunn–Minkowski theory, Cambridge University Press, Encyclopedia of Mathematics and its Applications, Tome 44 (1993), xiii+490 pages | Article

[35] Trudinger, Neil S.; Wang, Xu-Jia The Monge-Ampère equation and its applications, Handbook of geometric analysis. No. 1, Int. Press, Somerville, MA (Adv. Lect. Math. (ALM)) Tome 7 (2008), pp. 467-524