The equivariant Minkowski problem in Minkowski space

Bonsante, Francesco; Fillastre, François

Annales de l'Institut Fourier, Tome 67 (2017) no. 3, pp. 1035-1113.

La forme classique du problème de Minkowski dans l’espace de Minkowski est la recherche, pour une fonction positive $\phi$ sur ${ℍ}^d$, d’un ensemble convexe $\mathbf{K}$ dans l’espace de Minkowski, avec un bord $S$ $C^2$ de type espace, tel que $\phi {\left(\eta \right)}^{-1}$ soit la courbure de Gauss–Kronecker au point dont la normale est $\eta$. De façon analogue au cas euclidien, il est possible de formuler une version faible de ce problème  : étant donné un mesure de Radon $\mu$ sur ${ℍ}^d$, le problème de Minkowski généralisé dans l’espace de Minkowski est la recherche d’un ensemble convexe $\mathbf{K}$ tel que la mesure d’aire de $\mathbf{K}$ est $\mu$.

Dans ce travail nous regardons une version équivariante de ce problème  : étant donné un réseau uniforme $\Gamma$ d’isométries de ${ℍ}^d$, étant donné une mesure de Radon $\mu$ invariante pour l’action de $\Gamma$, étant donné un groupe d’isométries ${\Gamma }_{\tau }$ de l’espace de Minkowski, dont $\Gamma$ est la partie linéaire, il existe un unique ensemble convexe de mesure d’aire $\mu$, et qui est invariant sous l’action de ${\Gamma }_{\tau }$. 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 ${\Gamma }_{\tau }=\Gamma$. 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 ${\Gamma }_{\tau }$ ayant une courbure de Gauss–Kronecker constante égale à $1$.

The classical Minkowski problem in Minkowski space asks,given a positive function $\phi$ on ${ℍ}^d$, for a convex set $\mathbf{K}$ in Minkowski space with $C^2$ space-like boundary $S$, such that $\phi {\left(\eta \right)}^{-1}$ is the Gauss–Kronecker curvature at the point with normal $\eta$. Analogously to the Euclidean case, it is possible to formulate a weak version of this problem: given a Radon measure $\mu$ on ${ℍ}^d$ the generalized Minkowski problem in Minkowski space asks for a convex subset $\mathbf{K}$ such that the area measure of $\mathbf{K}$ is $\mu$.

In the present paper we look at an equivariant version of the problem: given a uniform lattice $\Gamma$ of isometries of ${ℍ}^d$, a $\Gamma$ invariant Radon measure $\mu$ and an isometry group ${\Gamma }_{\tau }$ of Minkowski space with $\Gamma$ as linear part, there exists a unique convex set with area measure $\mu$, invariant under the action of ${\Gamma }_{\tau }$. 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 ${\Gamma }_{\tau }=\Gamma$. Our method is totally different. Moreover, we show that those cases are very specific: in general, there is no smooth ${\Gamma }_{\tau }$-invariant hypersurface of constant Gauss–Kronecker curvature equal to $1$.

Classification : 53C50, 52A20, 53C42, 35J60
Keywords: Minkowski problem, Lorentzian geometry, covolume, Monge–Ampère equation
Mot clés : Problème de Minkowski, géométrie lorentzienne, covolume, équation de Monge–Ampère

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