Constant Mean Curvature 1 Surfaces with Low Total Curvature in Hyperbolic 3-Space

Abstract

Surfaces of constant mean curvature one in hyperbolic 3-space have quite similar properties to minimal surfaces in Euclidean 3-space. We shall list the possibilities of constant mean curvature one surfaces in hyperbolic 3-space with low total absolute curvature, or low dual total absolute curvature, and compare them with the known classification of minimal surfaces with low total curvature. Complete proofs of the new results will be published in two forthcoming papers (listed in the bibliography).