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We review recent results on classifying complete constant mean curvature 1(CMC 1) surfaces in hyperbolic 3-space with low total curvature. There are twonatural notions of "total curvature" -- one is the total absolute curvature,which is the integral over the surface of the absolute value of the Gaussiancurvature, and the other is the dual total absolute curvature, which is thetotal absolute curvature of the dual CMC 1 surface. Here we discuss results onboth notions (proven in two other papers by the authors), and we introduce newresults as well.

Period problems for mean curvature one surfaces in $H^3$ (with application to surfaces of low total curvature)