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The equivariant Minkowski problem in Minkowski space

The classical Minkowski problem in Minkowski space asks, for a positive function $ϕ$ on $\mathbb{H}^d$, 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 $\mathbb{H}^d$ 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 $\mathbb{H}^d$, given a $Γ$ invariant Radon measure $μ$, given a 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 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 surface of constant Gauss-Kronecker curvature equal to $1$.