Limits of Foliations in Quasi-Fuchsian Manifolds

We study the asymptotic behavior of certain foliations of ends of quasi-Fuchsian manifolds. We introduce a class of such foliations, which we call asymptotically Poincaré families as they are asymptotic to a family of surfaces determined naturally by the Poincaré metric on the Riemann surface at infinity. We prove the limiting behavior of any asymptotically Poincaré family is completely determined by the geometry of the quasi-Fuchsian manifold M and the conformal structure on the ideal boundary of M. As an application of these results we establish a conjecture of Labourie regarding the asymptotic behavior of the k-surface foliation he constructs. We also show that the constant mean curvature foliation of Mazzeo and Pacard forms an asymptotically Poincaré family and so we can describe its asymptotics as well.