Let S be a compact, orientable surface of hyperbolic type. Let $$(k_+,k_-)$$ ( k + , k - ) be a pair of negative numbers and let $$(g_+, g_-)$$ ( g + , g - ) be a pair of marked metrics over S of constant curvature equal to $$k_+$$ k + and $$k_-$$ k - respectively. Using a functional introduced by Bonsante, Mondello and Schlenker, we show that there exists a unique affine deformation $$\Gamma :=(\rho ,\tau )$$ Γ : = ( ρ , τ ) of a Fuchsian group such that $$(S,g_+)$$ ( S , g + ) and $$(S, g_-)$$ ( S , g - ) embed isometrically as locally strictly convex Cauchy surfaces in the future and past complete components respectively of the quotient by $$\Gamma $$ Γ of an open subset $$\Omega $$ Ω 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 $$(S, g_+)$$ ( S , g + ) and $$(S, g_-)$$ ( S , g - ) are realised as the third fundamental forms of future- and past-oriented, locally strictly convex graphs.

中文翻译：

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闵可夫斯基空间中不变常曲率浸入的注记

设 S 是一个紧凑的、可定向的双曲型曲面。让 $$(k_+,k_-)$$ ( k + , k - ) 是一对负数，让 $$(g_+, g_-)$$ ( g + , g - ) 是一对标记恒定曲率 S 上的度量分别等于 $$k_+$$ k + 和 $$k_-$$ k -。使用 Bonsante、Mondello 和 Schlenker 引入的泛函，我们证明存在一个 Fuchsian 群的独特仿射变形 $$\Gamma :=(\rho ,\tau )$$ Γ : = ( ρ , τ ) 使得 $ $(S,g_+)$$ ( S , g + ) 和 $$(S, g_-)$$ ( S , g - ) 分别作为局部严格凸柯西曲面嵌入由 Minkowski 空间的开子集 $$\Omega $$Ω 的 $$\Gamma $$Γ 商。这样的商被称为全局双曲线、最大、柯西紧凑 Minkowski 时空，自然是由 Danciger 引入的半管空间的对偶。当翻译成后一个框架时，我们的结果表明存在一个独特的、标记的、准 Fuchsian 半管道空间，其中 $$(S, g_+)$$ ( S , g + ) 和 $$(S, g_ -)$$ ( S , g - ) 被实现为面向未来和过去的局部严格凸图的第三种基本形式。