Radial graphs with constant mean curvature in the hyperbolic space
Let \(S\) be a geodesic sphere in hyperbolic space form \(H^{n+1}\), \(\Omega_a\) a domain whose closure contained in an open hemisphere of \(S\), and \(\Gamma=\partial \Omega\). Using the Poincaré model of \(H^{n+1}\), the radial graph \(\Sigma\) \[ \Sigma=\{X=\exp(| u| x)x \mid x\in\Omega\} \] has constant mean curvature \(H\) if and only if \(e^{u}<\frac1a\) and \[ Q_H(u)= \text{div} (\frac{\nabla u}{w(u)})-\frac{n}{a^2w(u)} +\frac{2ne^u}{a(1-d^2e^{2u})}\Big(H-\frac{ae^u}{w(u)} \Big)=0, \] where \(w(u)=\sqrt{1+u^2|\nabla u|^2}\). The author proves if \(-\inf H_{\Gamma}<H\leq 0\), then the Dirichlet problem \[ Q_H(u)=0,\quad \text{in } \Omega,\quad u=0 \quad \text{in} \quad\Gamma \] has unique solution. It is equivalent to the existence and uniqueness of the radial \(H\)-graphs with boundary \(\Gamma\).
scientific article; zbMATH DE number 1845443