关于具有凸边界的$(2,1)$维反德西特时空
On $(2,1)$-dimensional anti-de Sitter spacetimes with convex boundary
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中文总结 AI 辅助
本文研究具有强凸边界的$(2,1)$维反德西特时空的逆问题，证明给定曲率$K<-1$的边界度量存在唯一解，并肯定解决Mess猜想，即给定填充$S$的测度叶状结构对，存在唯一GHMC反德西特时空，其凸核边界弯曲叶状结构恰为给定对。
中文摘要 AI 辅助
设$S$为一个亏格至少为$2$的闭定向曲面。设$X$为一个$(2,1)$维反德西特时空，其同胚于$S\times[-1,1]$，且具有光滑、类空、强凸边界。由高斯公式，$\partial X$的两个分支上的诱导度量具有曲率$K<-1$。本文研究逆问题。我们证明，对于$S$上任意两个具有曲率$K<-1$的光滑度量$h_-$和$h_+$，在$X:=S\times [-1,1]$上存在唯一的光滑反德西特度量，使得边界是局部强凸的，并且其过去分支上的诱导度量与$h_-$等距（isotopic），未来分支上的诱导度量与$h_+$等距（isotopic）。存在性已由Tamburelli证明。我们利用这一结果肯定地解决了Mess的一个猜想：给定任意两个共同填充$S$的测度叶状结构$l_-, l_+\in\mathcal{ML}_S$，存在唯一的$(2,1)$维、全局双曲、极大、柯西紧致（GHMC）反德西特时空，使得其凸核的过去和未来边界分支上的测度弯曲叶状结构分别为$l_-$和$l_+$。此处，存在性已由Bonsante--Schlenker证明。
英文摘要
Let $S$ be a closed oriented surface of genus at least $2$. Let $X$ be a $(2,1)$-dimensional anti-de Sitter spacetime homeomorphic to $S\times[-1,1]$ with smooth, spacelike, strongly convex boundary. By Gauss' formula, the induced metric on each of the two components of $\partial X$ has curvature $K<-1$. In this paper we address the inverse problem. We show that, for any two smooth metrics $h_-, h_+$ on $S$ with curvature $K<-1$, there exists a unique, smooth anti-de Sitter metric on $X:=S\times [-1,1]$ with respect to which the boundary is locally strongly convex and has induced metric isotopic to $h_-$ on its past component and isotopic to $h_+$ on its future component. Existence has already been proven by Tamburelli. We use this result to solve affirmatively a conjecture of Mess: given any two measured laminations $l_-, l_+\in\mathcal{ML}_S$ that together fill $S$, there exists a unique $(2,1)$-dimensional, globally hyperbolic, maximal, Cauchy compact (GHMC) anti-de Sitter spacetime such that the measured bending laminations of the past and future boundary components of its convex core are respectively $l_-$ and $l_+$. Here, existence has already been proven by Bonsante--Schlenker.
发表机构
- Universitat Autònoma de Barcelona（巴塞罗那自治大学）
- University of Luxembourg（卢森堡大学）
- Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio)（里约热内卢天主教大学）
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