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

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

2008-05-24

arxiv.org/abs/0805.3761v1

Adv. Studies in Pure Math. 51 (2008), 335-387, Surveys on Geometry and Integrable Systems

Mathematics

Differential Geometry

Scientific paper

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.

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