Minimal immersions of closed surfaces in hyperbolic three-manifolds

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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.