On CMC-immersions of surfaces into Hyperbolic 3-manifolds

I shall discuss the so called “moduli space” of Constant Mean Curvature (CMC) c-immersions of a closed surface S (orientable and of genus at least 2) into hyperbolic 3-manifolds. Interestingly when |c|<1, such space admits a nice parametrization described by elements of  the tangent bundle of the Teichmueller space of S. Indeed, for any such element we shall see how  to determine uniquely the pullback metric and the second fundamental form of the immersion by solving the “constrained” Gauss - Codazzi equations. This is attained by showing that the associated action functional ( known as the  “Donaldson -functional” in Gonsalves-Uhlenbeck (2007)) admits a global minimum as its unique critical point. In addition I shall discuss the asymptotic behavior of those minimizers and obtain  “convergence” to a (CMC) 1-immersion  in terms of the Kodaira map. Please note that (CMC) 1-immersion into the hyperbolic space are particularly relevant  in hyperbolic geometry  in view of their analogies  with minimal immersions into the Euclidean space. For example, we show that for  genus 2,  it is possible to catch at the limit a “regular “ CMC 1-immersions into an hyperbolic 3-manifold, except in very rare  situations which relate to the image, under the Kodaira map, of the six Weierstrass points of S. If time permits, I shall mention further  progress for higher genus obtained in collaboration with S. Trapani.