Foliations of Hyperbolic Space by Constant Mean Curvature Surfaces Sharing Ideal Boundary

Abstract

Let {\small $\gamma$} be a Jordan curve in {\small $\sph{2}$}, considered as the ideal boundary of {\small $\hyp{3}$}. Under certain circumstances, it is known that for any {\small $c \in (-1,1)$}, there is a disc of constant mean curvature c embedded in {\small $\hyp{3}$} with {\small $\gamma$} as its ideal boundary. Using analysis and numerical experiments, we examine whether or not these surfaces in fact foliate {\small $\hyp{3}$}, and to what extent the known conditions on the curve can be relaxed.