Surfaces of constant mean curvature \(c\) in \(H^ 3(-c^ 2)\) with prescribed hyperbolic Gauss map
We denote by \(H^3\) the hyperbolic 3-space of constant sectional curvature \(-1\). Surfaces of constant mean curvature 1 in \(H^3\) have quite similar properties to minimal surfaces in \(\mathbb{R}^3\) and have been investigated by several authors. In particular, \textit{R. L. Bryant} [Astérisque 154-155, 321-437 (1988; Zbl 0635.53047)] established a representation formula for these surfaces in \(H^3\) as an analogy of the Weierstrass formula in minimal surface theory. The hyperbolic Gauss maps of such surfaces are also defined as holomorphic maps to \(P^1(\mathbb{C})\). We abbreviate the term ``constant mean curvature 1'' as ``CMC-1'' We show that the moduli space of conformal branched CMC-1 immersions of \(M^2\) into \(H^3\) of finite total curvature with a prescribed hyperbolic Gauss map can be identified with the moduli space of conformal pseudo-Riemannian metrics of constant curvature 1 with conical singularities on \(M^2\). Moreover, we give new examples of complete CMC-\(c\) surfaces of genus zero which have polyhedral or dihedral symmetries, as an application of the correspondence.
moduli space of conformal pseudo-Riemannian metrics of constant curvature 1
scientific article; zbMATH DE number 861910
scientific article