A note on invariant constant curvature immersions in Minkowski space

Abstract

Let S be a compact, orientable surface of hyperbolic type. Let $$(k_+,k_-)$$ be a pair of negative numbers and let $$(g_+, g_-)$$ be a pair of marked metrics over S of constant curvature equal to $$k_+$$ and $$k_-$$ respectively. Using a functional introduced by Bonsante, Mondello and Schlenker, we show that there exists a unique affine deformation $$\Gamma :=(\rho ,\tau )$$ of a Fuchsian group such that $$(S,g_+)$$ and $$(S, g_-)$$ embed isometrically as locally strictly convex Cauchy surfaces in the future and past complete components respectively of the quotient by $$\Gamma $$ of an open subset $$\Omega $$ of Minkowski space. Such quotients are known as Globally Hyperbolic, Maximal, Cauchy compact Minkowski spacetimes and are naturally dual to the half-pipe spaces introduced by Danciger. When translated into this latter framework, our result states that there exists a unique, marked, quasi-Fuchsian half-pipe space in which $$(S, g_+)$$ and $$(S, g_-)$$ are realised as the third fundamental forms of future- and past-oriented, locally strictly convex graphs.