Foliations of hyperbolic space by constant mean curvature surfaces sharing ideal boundary

Let γ be a Jordan curve in 8 2, considered as the ideal boundary of H³. Under certain circumstances, it is known that for any c ϵ (-1, 1), there is a disc of constant mean curvature c embedded in H³ with γ as its ideal boundary. Using analysis and numerical experiments, we examine whether or not these surfaces in fact foliate H³, and to what extent the known conditions on the curve can be relaxed.