Convex surfaces with prescribed induced metrics in anti-de Sitter spacetimes (Q6663904)

The paper deals with \(3\)-dimensional maximal globally hyperbolic and spatially compact (MGHC) anti-de Sitter (AdS) spacetimes. If \(M\) is a MGHC AdS spacetime, it contains closed Cauchy surfaces, which are all homeomorphic to a fixed closed surface \(S\) which is assumed to be of genus at least 2. The identity component of the isometry group of the \(3\)-dimensional anti-de Sitter space identifies (up to finite index) with \(\mathrm{PSL}(2, \mathbb{R})\times \mathrm{PSL}(2, \mathbb{R})\), and, if \(M\) is a GHMC AdS spacetime, its holonomy representation \(\rho:\pi_{1}(M) \rightarrow\mathrm{PSL}(2, \mathbb{R})\times\mathrm{PSL}(2, \mathbb{R})\) is the product \(\rho=(\rho_{L},\rho_{R})\) of two Fuchsian representations, called its left and right representations. Those representations \(\rho_{L}\) and \(\rho_{R}\) are therefore the holonomy representations of two hyperbolic metrics \(h_{L}\) and \(h_{R}\) on \(S\) called the left and right metrics of \(M\). The main result of the paper is the following:\N\NTheorem. Let \(h\) be a complete Riemannian metric of curvature \(K<-1\) on \(S\), and let \(h_{0}\) be a hyperbolic metric on \(S\). Then there is a unique GHMC AdS spacetime \(M\) with left metric isotopic to \(h_{0}\) and containing a past-convex spacelike surface with induced metric isotopic to \(h\).