On finiteness of the number of boundary slopes of immersed surfaces in   3-manifolds
  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. 
Joel Hass, Shicheng Wang, Qing Zhou