Minimal maps between the hyperbolic discs and generalized Gauss maps of maximal surfaces in the anti-de-Sitter 3-space (Q1594911)

A smooth map \(u:M\to N\) between Riemannian manifolds is called a minimal map if its graph is a minimal submanifold of \(M\times N\). For general dimensions, the authors prove a Ruh-Vilms type theorem for minimal Gauss maps. A corollary is that the Gauss map of an immersed surface in Euclidean \(3\)-space is minimal if and only if \(H/(1-K)\) is constant or \(K\equiv 1\). The main part of the paper is concerned with the study of minimal maps between hyperbolic discs. For the Dirichlet problem at infinity for minimal diffeomorphisms, an existence result is proved: Any quasi-symmetric function with sufficiently small dilatation has a quasi-conformal minimal diffeomorphic extension to the hyperbolic disc. Finally, a representation formula for minimal diffeomorphisms between hyperbolic discs is given by means of the generalized Gauss map of a complete maximal surface in anti-de-Sitter \(3\)-space.