On the constructibility of a negatively curved complete surface of constant mean curvature in \(\mathbb{H}^3\) determined by a prescribed Hopf differential
It is first shown that given a constant \(c\in(-1,1)\), a holomorphic function \(A(z)\) on the unit disk \(B=\{z: | z| < 1\}\subset\mathbb C\), and a solution of the elliptic problem: \[ \begin{cases} 4 u_{z\overline z} = (1-c^2)| A| ^2 e^{-2w} + e^{2u} \\ u(z) \to\infty, \text{ as } z\to \partial B\end{cases} \tag{1} \] there exists an isometric immersion of \(B\) with the metric \(( \frac{1}{1-c^2}) c^{2u} dz^2\) into the hyperbolic space \(\mathbb H^3(-1)\) such that the image of the immersion is a negatively curved complete surface with mean cuvature \(c\). It is then shown, by a delicate analysis of (1), that growth conditions on \(K=(1-c^2)| A| ^2\) guarantee the existence and uniqueness of solutions of (1). By combining these facts with some unpublished results of C. H. Epstein, it is also shown that some conditions on \(| A| \) imply that the above immersion is an embedding.
scientific article; zbMATH DE number 1983439
scientific article