  For any hyperbolic 3-manifold $M$ with totally geodesic boundary, there are
finitely many boundary slopes for essential immersed surfaces of a given genus.
There is a uniform bound for the number of such boundary slopes if the genus of
$\partial M$ or the volume of $M$ is bounded above. When the volume is bounded
above, then area of $\partial M$ is bounded above and the length of closed
geodesic on $\partial M$ is bounded below.
