Christoffel and Minkowski problems in Minkowski space
Séminaire de théorie spectrale et géométrie, Volume 32  (2014-2015), p. 97-114

For convex sets in the Lorentzian Minkowski space bounded by space-like hyperplanes, it is possible to define area measures, similarly to the classical definition for convex bodies in the Euclidean space. Here the measures are defined on the hyperbolic space rather than on the round sphere. We are particularly interested by convex sets invariant under the action of isometries groups of the Minkowski space, so that the measures can be defined on compact hyperbolic manifolds. We can then look at the Christoffel and the Minkowski problems (i.e. particular measures are prescribed) in a general setting. In dimension $\left(2+1\right)$, the Christoffel problem include a famous construction by G. Mess. In this dimension, the smooth version of the Minkowski problem already had a positive answer, and we show that this is a specificity of dimension $\left(2+1\right)$, while the general problem has a solution in all dimensions.

DOI : https://doi.org/10.5802/tsg.305
Keywords: Area measures, convex sets, Lorentzian geometry, Globally hyperbolic spacetime
@article{TSG_2014-2015__32__97_0,
author = {Fillastre, Fran\c cois},
title = {Christoffel and Minkowski problems in Minkowski space},
journal = {S\'eminaire de th\'eorie spectrale et g\'eom\'etrie},
publisher = {Institut Fourier},
volume = {32},
year = {2014-2015},
pages = {97-114},
doi = {10.5802/tsg.305},
language = {en},
url = {http://www.numdam.org/item/TSG_2014-2015__32__97_0}
}

Fillastre, François. Christoffel and Minkowski problems in Minkowski space. Séminaire de théorie spectrale et géométrie, Volume 32 (2014-2015) , pp. 97-114. doi : 10.5802/tsg.305. http://www.numdam.org/item/TSG_2014-2015__32__97_0/

[1] Barbot, T. Globally hyperbolic flat space-times, J. Geom. Phys., Tome 53 (2005) no. 2, pp. 123-165 | Article | MR 2110829 | Zbl 1087.53065

[2] Barbot, T.; Béguin, F.; Zeghib, A. Prescribing Gauss curvature of surfaces in 3-dimensional spacetimes: application to the Minkowski problem in the Minkowski space, Ann. Inst. Fourier (Grenoble), Tome 61 (2011) no. 2, pp. 511-591 | Article | Numdam | MR 2895066 | Zbl 1234.53019

[3] Barbot, T.; Fillastre, F. An assymmetric norm on ${H}^{1}\left(\Gamma ,{ℝ}^{d+1}\right)$ (In preparation)

[4] Berg, C. Corps convexes et potentiels sphériques, Mat.-Fys. Medd. Danske Vid. Selsk., Tome 37 (1969) no. 6, pp. 64 pp. (1969) | MR 254789 | Zbl 0181.38303

[5] Bonnesen, T.; Fenchel, W. Theory of convex bodies, BCS Associates, Moscow, ID (1987), pp. x+172 (Translated from the German and edited by L. Boron, C. Christenson and B. Smith) | MR 920366 | Zbl 0628.52001

[6] Bonsante, F. Flat spacetimes with compact hyperbolic Cauchy surfaces, J. Differential Geom., Tome 69 (2005) no. 3, pp. 441-521 http://projecteuclid.org/getRecord?id=euclid.jdg/1122493997 | MR 2170277 | Zbl 1094.53063

[7] Bonsante, F.; Fillastre, F. The equivariant Minkowski problem in Minkowski space (2014) (https://arxiv.org/abs/1405.4376v1)

[8] Bonsante, F.; Seppi, A. Spacelike convex surfaces with prescribed curvature in (2+1)-Minkowski space (2015) (https://arxiv.org/abs/1505.06748)

[9] 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 | MR 3245885 | Zbl 1298.83125

[10] Carlier, G. On a theorem of Alexandrov, J. Nonlinear Convex Anal., Tome 5 (2004) no. 1, pp. 49-58 | MR 2049056 | Zbl 1077.49029

[11] Fenchel, W.; Jessen, B. Mengenfunktionen und konvexe Körper., Danske Videnks. Selsk. Math.-fys. Medd., Tome 16 (1938) no. 3, pp. 1-31

[12] Fillastre, F. Fuchsian convex bodies: basics of Brunn-Minkowski theory, Geom. Funct. Anal., Tome 23 (2013) no. 1, pp. 295-333 | Article | MR 3037901 | Zbl 1271.52009

[13] Fillastre, F.; Veronelli, G. Lorentzian area measures and the Christoffel problem (2013) (to appear Ann. Scuola Norm. Sup. Pisa Cl. Sci., https://arxiv.org/abs/1302.6169v1)

[14] Gutiérrez, C. The Monge-Ampère equation, Birkhäuser Boston Inc., Boston, MA, Progress in Nonlinear Differential Equations and their Applications, 44 (2001), pp. xii+127 | Article | Zbl 0989.35052

[15] Khovanskiĭ, A.; Timorin, V. On the theory of coconvex bodies, Discrete Comput. Geom., Tome 52 (2014) no. 4, pp. 806-823 | Article | MR 3279551

[16] Mess, G. Lorentz spacetimes of constant curvature, Geom. Dedicata, Tome 126 (2007), pp. 3-45 | Article | MR 2328921 | Zbl 1206.83117

[17] Oliker, V. I.; Simon, U. 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 | MR 703485 | Zbl 0502.53038

[18] Trudinger, N.; Wang, X.-J. The Monge-Ampère equation and its geometric applications, Handbook of geometric analysis. No. 1, Int. Press, Somerville, MA (Adv. Lect. Math. (ALM)) Tome 7 (2008), pp. 467-524 | MR 2483373