Christoffel and Minkowski problems in Minkowski space
Séminaire de théorie spectrale et géométrie, Volume 32 (2014-2015), pp. 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 (2+1), 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 (2+1), while the general problem has a solution in all dimensions.

DOI: 10.5802/tsg.305
Keywords: Area measures, convex sets, Lorentzian geometry, Globally hyperbolic spacetime
Fillastre, François 1

1 Université de Cergy-Pontoise UMR CNRS 8088, 95000 Cergy-Pontoise (France)
@article{TSG_2014-2015__32__97_0,
     author = {Fillastre, Fran\c{c}ois},
     title = {Christoffel and {Minkowski} problems in {Minkowski} space},
     journal = {S\'eminaire de th\'eorie spectrale et g\'eom\'etrie},
     pages = {97--114},
     publisher = {Institut Fourier},
     address = {Grenoble},
     volume = {32},
     year = {2014-2015},
     doi = {10.5802/tsg.305},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/tsg.305/}
}
TY  - JOUR
AU  - Fillastre, François
TI  - Christoffel and Minkowski problems in Minkowski space
JO  - Séminaire de théorie spectrale et géométrie
PY  - 2014-2015
SP  - 97
EP  - 114
VL  - 32
PB  - Institut Fourier
PP  - Grenoble
UR  - http://www.numdam.org/articles/10.5802/tsg.305/
DO  - 10.5802/tsg.305
LA  - en
ID  - TSG_2014-2015__32__97_0
ER  - 
%0 Journal Article
%A Fillastre, François
%T Christoffel and Minkowski problems in Minkowski space
%J Séminaire de théorie spectrale et géométrie
%D 2014-2015
%P 97-114
%V 32
%I Institut Fourier
%C Grenoble
%U http://www.numdam.org/articles/10.5802/tsg.305/
%R 10.5802/tsg.305
%G en
%F 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/articles/10.5802/tsg.305/

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

[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), Volume 61 (2011) no. 2, pp. 511-591 | DOI | Numdam | MR | Zbl

[3] Barbot, T.; Fillastre, F. An assymmetric norm on H 1 (Γ, d+1 ) (In preparation)

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

[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 | Zbl

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

[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, Volume 15 (2014) no. 9, pp. 1733-1799 | DOI | MR | Zbl

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

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

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

[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, Progress in Nonlinear Differential Equations and their Applications, 44, Birkhäuser Boston Inc., Boston, MA, 2001, pp. xii+127 | DOI | Zbl

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

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

[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., Volume 342 (1983), pp. 35-65 | MR | Zbl

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

Cited by Sources: