On the Weyl Problem in Minkowski Space

Let $S$ be a closed surface of hyperbolic type. We show that, for every pair $(g_+,g_-)$ of negatively curved metrics over $S$, there exists a unique globally hyperbolic, maximal, and Cauchy compact Minkowski spacetime $X$ into which $(S,g_+)$ and $(S,g_-)$ isometrically embed as Cauchy surfaces in the future and past components, respectively.

[1]  G. Mess.  Lorentz spacetimes of constant curvature , 2007, 0706.1570.

[2]  Kevin P. Scannell,et al.  Notes on a paper of Mess , 2007, 0706.0640.

[3]  Shing-Tung Yau,et al.  On univalent harmonic maps between surfaces , 1978 .

[4]  P. Bassanini,et al.  Elliptic Partial Differential Equations of Second Order , 1997 .

[5]  D. Mumford.  A remark on Mahler’s compactness theorem , 1971 .

[6]  L. Bers.  SIMULTANEOUS UNIFORMIZATION , 2022 .

[7]  A. Tromba.  Teichmüller Theory in Riemannian Geometry , 2004 .

[8]  S. Trapani,et al.  One-harmonic maps on Riemann surfaces , 1995 .

[9]  Jean-Marc Schlenker,et al.  A cyclic extension of the earthquake flow II , 2012, 1208.1738.

[10]  Ronald Calinger,et al.  Classics of Mathematics , 1994 .

[11]  P. Buser,et al.  Geometry and Spectra of Compact Riemann Surfaces , 1992 .

[12]  Michael Wolf.  The Teichmüller theory of harmonic maps , 1989 .

[13]  F. Bonsante,et al.  On Codazzi tensors on a hyperbolic surface and flat Lorentzian geometry , 2015, 1501.04922.

[14]  F. Labourie.  Métriques prescrites sur le bord des variétés hyperboliques de dimension 3 , 1992 .

[15]  Franccois Fillastre,et al.  Quasi-Fuchsian Co-Minkowski Manifolds , 2018, In the Tradition of Thurston.

[16]  Hyperbolic manifolds with convex boundary , 2002, math/0205305.

[17]  L. Ahlfors,et al.  Lectures on quasiconformal mappings , 1966 .

[18]  F. Bonsante,et al.  On the volume of Anti-de Sitter maximal globally hyperbolic three-manifolds , 2017, Geometric and Functional Analysis.

[19]  Louis Nirenberg,et al.  Complex Analytic Coordinates in Almost Complex Manifolds , 1957 .

[20]  F. Labourie.  Problème de Minkowski et surfaces à courbure constante dans les variétés hyperboliques , 1991 .

[21]  Graham Smith,et al.  Group actions and scattering problems in Teichmüller theory , 2016, 1605.04563.

[22]  François Fillastre,et al.  A note on invariant constant curvature immersions in Minkowski space , 2018, Geometriae Dedicata.

[23]  Andrea Tamburelli.  Prescribing Metrics on the Boundary of Anti-de Sitter 3-Manifolds , 2016 .

[24]  Flat spacetimes with compact hyperbolic Cauchy surfaces , 2003, math/0311019.