Totally umbilic surfaces in hyperbolic 3-manifolds of finite volume

Let S be a connected surface with finite negative Euler characteristic and let H be a real number with absolute value less than one.

In this talk we show that S appears as a properly embedded, totally umbilic surface with mean curvature H in a hyperbolic 3-manifold of finite volume.
Conversely, a complete, totally umbilic surface with mean curvature H, embedded in a hyperbolic 3-manifold of finite volume
must be proper and have finite, negative Euler characteristic.
Joint work with Colin Adams and William Meeks.