We study minimal immersions of closed surfaces (of genus g≥2) in
hyperbolic 3-manifolds, with prescribed data (σ,tα), where
σ is a conformal structure on a topological surface S, and αdz2 is a holomorphic quadratic differential on the surface (S,σ). We
show that, for each t∈(0,τ0​) for some τ0​>0, depending only on
(σ,α), there are at least two minimal immersions of closed surface
of prescribed second fundamental form Re(tα) in the conformal structure
σ. Moreover, for t sufficiently large, there exists no such minimal
immersion. Asymptotically, as t→0, the principal curvatures of one
minimal immersion tend to zero, while the intrinsic curvatures of the other
blow up in magnitude.Comment: 16 page