Convex Fuchs surfaces in Lorentz spaces of constant curvature (Q1974129)

\textit{M. Gromov} shows in [``Partial differential relations'', Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge Bd. 9, Springer-Verlag (1986; Zbl 0651.53001)] the existence (but not unique) of an isometric Fuchs equivariant immersion in \({\mathbb{H}}^3\) of any surface \((\Sigma,\sigma)\) of genus \(g\geq 1\) and curvature \(K>1\). The Lorentz case is considered in the paper under review, where an analogous result is proved, but when the uniqueness of the immersion follows under the hypothesis of convexity and other geometric conditions. More precisely, the main result of the paper is the following Theorem: Let \(K_0\in{\mathbb{R}}\), \(S\) be a compact surface and \(\sigma \) be a metric on \(S\) of curvature \(K<K_0\), such that the length of every contractible closed geodesic is less than \(2\pi\). Then there is a unique convex Fuchs end of curvature \(K_0\) which has \((\Sigma,\sigma)\) as a bord. The proof uses interesting technics, including some from [\textit{J.-M. Schlenker}, Commun. Anal. Geom. 4, No. 2, 285-331 (1996; Zbl 0864.53016)].