The Dirichlet problem at infinity for harmonic map equations arising from constant mean curvature surfaces in the hyperbolic 3-space
Let \({\mathbb D}^m\) be the \(m\)-dimensional open unit ball with the Poincaré metric, and let \(\widehat{\mathbb D}^n_{\tau }\) denote the \(n\)-dimensional open unit ball \(\{ \mathbf{y} = (y_1,\cdots,y_n) \in {\mathbb R}^n; | \mathbf{y} | < 1 \}\) with the metric \[ \frac{4}{(1-| \mathbf{y}| ^2) ( 1+ \tau^2 | \mathbf{y}| ^2)} \sum_{i=1}^n (dy^i)^2. \] The authors prove existence, regularity and uniqueness results of the Dirichlet problem at infinity for proper harmonic maps from \({\mathbb D}^m\) to \(\widehat{{\mathbb D}}^n_{\tau }\). In the case of \(m = n = 2\), they construct a harmonic diffeomorphism, and as an application they give an existence result of complete surfaces with constant mean curvature in the hyperbolic 3-space.
scientific article; zbMATH DE number 1798867