The equivariant Minkowski problem in Minkowski space

Abstract : The classical Minkowski problem in Minkowski space asks, given a positive function φ on Hd, for a convex set K in Minkowski space with C2 spacelike 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 Hd 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 Hd, a Γ invariant Radon measure µ and an isometry group Γτ of Minkowski space with Γ as linear part, there exists a unique convex set with area measure µ, invariant under the action of Γτ . 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 Γτ = Γ. Our method is totally different. Moreover, we show that those cases are very specific: in general, there is no smooth Γτ -invariant hypersurface of constant Gauss–Kronecker curvature equal to 1.