Orateur
Description
Constant Mean Curvature (CMC) c-immersions of a closed orientable surface S (with genus g ≥ 2) into hyperbolic 3-manifolds emerged by the work of Uhlenbeck in connection with irreducible representations of the fundamental group of S into the Mobious group.
In view of Bryant surfaces, the value c =1 of the mean curvature enters as a “critical” (yet significant) parameter in this context. It is responsible for natural “blow-up” phenomena, and for this reason the moduli space of such (CMC) 1-immersions remained elusive for long time.
In recent work, we showed how to encompass the blow-up situation in terms of sharp orthogonality conditions, involving the image Z of the Kodaira map, for genus g = 2, and the (g −1)-secant variety of Z, for larger genus.
In this way, under a generic condition, we can ensure existence and uniqueness of (CMC) 1-immersions of S into (germs) of hyperbolic 3-manifolds, and obtain a parametrization of the corresponding moduli space in terms of the tangent bundle of the Teichmueller space of S.