On the Moduli Space of (CMC) 1-immersions of a Closed Surface Into Hyperbolic 3-Manifolds

Constant Mean Curvature (CMC) immersions of a closed surface S into hyperbolic 3-manifolds emerged by the work of Uhlenbeck in connection with irreducible rep- resentations of the fundamental group into the Mobious group. Moreover, Bryant revealed a bi-holomorphic (cousin) relation between (CMC) 1-immersions of sur- faces into the hyperbolic 3-space (Bryant surfaces) and minimal immersions into the Euclidian 3-space. In this note, we survey recent results concerning the existence and uniqueness of (CMC) 1-immersions of a closed surface into hyperbolic 3-manifolds, labelled by Dolbeault co-homology classes. While, (CMC) c-immersions of a surface S (closed, orientable, with genus g ≥ 2) into hyperbolic 3- manifolds are always available when |c| < 1 (and described in terms of the tangent bundle of the Teich- mueller space of S) we find that (CMC) 1-immersions can be attained only as limits of (CMC) c-immersions for |c| → 1. However, the passage to the limit can be prevented by possible blow-up phenomena, so that (after scaling) we end up with a (CMC) 1- immersion with (finitely many) conical singularities, consistently with the presence of smooth ends in Bryant surfaces. We see how to encompass the blow-up situation in terms of suitable orthogonality conditions, involving the image Z of the Kodaira map for genus g = 2, and the (g − 1)-secant variety of Z , for genus g = 3. Consequently, we can ensure the passage to the limit under an appropriate generic condition (sharp for genus g = 2), yielding to (CMC) 1-immersions into suitable (germs) of hyperbolic 3-manifolds.