Mean curvature 1 surfaces in hyperbolic 3-space with low total curvature. I

Abstract

A complete surface of constant mean curvature 1 (CMC-1) in hyperbolic 3- space with constant curvature $-1$ has two natural notions of ‘‘total curvature’’—one is the total absolute curvature which is the integral over the surface of the absolute value of the Gaussian curvature, and the other is the dual total absolute curvature which is the total absolute curvature of the dual CMC-1 surface. In this paper, we completely classify CMC-1 surfaces with dual total absolute curvature at most $4\pi$. Moreover, we give new examples and partially classify CMC-1 surfaces with dual total absolute curvature at most $8\pi$.