Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The equivariant Minkowski problem in Minkowski space   The classical Minkowski problem in Minkowski space asks, for a positive
function $\phi$ on $\mathbb{H}^d$, for a convex set $K$ in Minkowski space with
$C^2$ space-like boundary $S$, such that $\phi(\eta)^{-1}$ 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 $\mathbb{H}^d$ the generalized Minkowski problem
in Minkowski space asks for a convex subset $K$ such that the area measure of
$K$ is $\mu$.
  In the present paper we look at an equivariant version of the problem: given
a uniform lattice $\Gamma$ of isometries of $\mathbb{H}^d$, given a $\Gamma$
invariant Radon measure $\mu$, given a 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
regularity results, follow from properties of the Monge--Amp\`ere equation. The
existence part can be translated as an existence result for Monge--Amp\`ere
equation.
  The regular version was proved by T.~Barbot, F.~B\'eguin 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 surface of constant
Gauss-Kronecker curvature equal to $1$.

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The categories of this paper are: supr-con
5
The categories of this paper are: gr-qc
31
The categories of this paper are: ao-sci
7
The categories of this paper are: hep-th
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The categories of this paper are: adap-org
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The categories of this paper are: cmp-lg
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The categories of this paper are: eess
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The categories of this paper are: alg-geom
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The categories of this paper are: dg-ga
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The categories of this paper are: acc-phys
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The categories of this paper are: econ
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The categories of this paper are: astro-ph
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The categories of this paper are: stat
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The categories of this paper are: solv-int
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The categories of this paper are: cs
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The categories of this paper are: plasm-ph