Complete Surfaces of Constant Mean Curvature-1 in the Hyperbolic 3-space

In the study of minimal surfaces in the euclidean 3-space, the Weierstrass representation plays an important role. Bryant Br showed that an analogue of the Weierstrass-representation formula holds for surfaces of mean in the hyperbolic 3-space X3. In this article we abbreviate the term constant mean curvature-i as CMC-1. Like minimal surfaces in the euclidean space, the hyperbolic Gauss map of CMC-1 surfaces is defined as a holomorphic map to C U oo. However, in contrast to the euclidean case, the hyperbolic Gauss map of a CMC-1 surface may not be extended across the ends even if the total Gaussian curvature is finite. We call a complete CMC-1 surface, whose Gauss map can be extended across all of its ends, a CMC-1 surface of regular ends. In this article we produce an explicit tool to construct CMC-1 surfaces of regular ends. In Section 2 we show that such surfaces are constructed by solving some ordinary differential equations with regular singularity. In our CMC-1 category, Ossermann's inequality is not expected and the Cohn-Vossen inequality is the best possible one. We show in Section 4 that the equality of the Cohn-Vossen inequality never holds for complete CMC-1 surfaces in XH3. In Section 5 we give a necessary and sufficient condition that a regular end of a CMC-1 surface be embedded. In Section 6 we classify complete CMC-1 surfaces of genus 0 with two regular ends. Our classification contains new examples. Furthermore, in Section 7, we construct several new CMC-1 surfaces with regular embedded ends. Each of these examples has a nontrivial deformation, which is mentioned in Section 3. It should be remarked that our construction does not work for surfaces with irregular ends. But there is another construction: By perturbing minimal surfaces in the euclidean 3-space, the authors constructed CMC-1 surfaces, all of whose ends are irregular (see UY1).