  We study minimal immersions of closed surfaces (of genus $g \ge 2$) in
hyperbolic 3-manifolds, with prescribed data $(\sigma, t\alpha)$, where
$\sigma$ is a conformal structure on a topological surface $S$, and $\alpha
dz^2$ is a holomorphic quadratic differential on the surface $(S,\sigma)$. We
show that, for each $t \in (0,\tau_0)$ for some $\tau_0 > 0$, depending only on
$(\sigma, \alpha)$, there are at least two minimal immersions of closed surface
of prescribed second fundamental form $Re(t\alpha)$ in the conformal structure
$\sigma$. Moreover, for $t$ sufficiently large, there exists no such minimal
immersion. Asymptotically, as $t \to 0$, the principal curvatures of one
minimal immersion tend to zero, while the intrinsic curvatures of the other
blow up in magnitude.
