Minimal immersions of closed surfaces in hyperbolic three-manifolds

We study minimal immersions of closed incompressible surfaces in hyperbolic 3-manifolds, with
prescribed data $(\sigma, t\alpha)$, where $\sigma$ is a conformal structure on a closed surface $S$, and
$\alpha dz^2$ is a holomorphic quadratic differential on the marked 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 genus $g \ge 2$, of prescribed second fundamental form
$Re(t\alpha)$ in the conformal structure $\sigma$. Moreover, for $t$ sufficiently large, there exists no such
Minimal immersions. Asymptotic geometry of the solutions are also investigated, as well as the case of CMC
immersions. This is joint work with Marcllo Lucia.