Minimal immersions of closed surfaces in hyperbolic three-manifolds

We study minimal immersions of closed surfaces (of genus a parts per thousand yen 2) in hyperbolic three-manifolds, with prescribed data (, ), where is a conformal structure on a topological surface , and (2) is a holomorphic quadratic differential on the surface (). We show that, for each for some (0) > 0, depending only on (, ), there are at least two minimal immersions of closed surface of prescribed second fundamental form ( ) in the conformal structure . Moreover, for sufficiently large, there exists no such minimal immersion. Asymptotically, as -> 0, the principal curvatures of one minimal immersion tend to zero, while the intrinsic curvatures of the other blow up in magnitude.