  Initiated by the work of Uhlenbeck in late 1970s, we study questions about
the existence, multiplicity and asymptotic behavior for minimal immersions of
closed surface in some hyperbolic three-manifold, with prescribed conformal
structure on the surface and second fundamental form of the immersion. We prove
several results in these directions. In particular, we determine when exactly
the solution is unique and when multiple solutions appear. Moreover, we analyze
in detail the asymptotic behavior of the solutions when (and how) blowing up
might occur. Interestingly the blow-up analysis exhibit different behaviors
when the surface is of genus two or greater. Furthermore, we extend this
program to consider similar problems where the total extrinsic curvature is
prescribed and we prove an existence result.

Differential Geometry;Analysis of PDEs;Functional Analysis