  We review recent results on classifying complete constant mean curvature 1
(CMC 1) surfaces in hyperbolic 3-space with low total curvature. There are 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. Here we discuss results on
both notions (proven in two other papers by the authors), and we introduce new
results as well.
