Minimal immersions of closed surfaces in hyperbolic three-manifolds

Abstract

We study minimal immersions of closed surfaces (of genus g ≥ 2) in hyperbolic three-manifolds, with prescribed data (σ, t α), where σ is a conformal structure on a topological surface S, and α dz 2 is a holomorphic quadratic differential on the surface (S, σ). We show that, for each $${t \in (0,\tau_0)}$$ for some τ 0 > 0, depending only on (σ, α), there are at least two minimal immersions of closed surface of prescribed second fundamental form Re(t α) 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.