Asymptotic flexibility of globally hyperbolic manifolds (Q424762)

Two globally hyperbolic manifolds \((M,g)\) and \((N,h)\) are called future-isometric (resp. past-isometric) iff there exists Cauchy hypersurfaces \(S\) of \((M,g)\) and \(T\) of \((N,h)\) such that \(I^+(S)\) is isometric to \(I^+(T)\) (resp. \(I^-(S)\) is isometric to \(I^-(T)\)). If \(J(g,h)\) is the set of globally hyperbolic manifolds past-isometric to \(g\) and future-isometric to \(h\), any metric in \(J(g,h)\) is an asymptotic join of \(g\) and \(h\).NEWLINENEWLINETheorem 3. Let \((M,g)\) and \((M,h)\) be globally hyperbolic, let the Cauchy hypersurfaces of \(g\) be diffeomorphic to those of \(h\). Then \(J(g,h)\) is nonempty. In particular, for any globally hyperbolic \((M,g)\), there is a globally hyperbolic ultrastatic metric \(u\) on \(M\) such that \(J(g,u)\) is nonempty.