Problem 1: A jordan curve on the unit sphere in ${\bf R}^3$ is known to admit a spanning minimal embedded disc. At the same time, the spherical domains complementary to the curve are constant mean curvature (cmc) discs with $H =\pm 1$. It is natural to ask whether the ball can be foliated by cmc discs spanning the curve.
Problem 2: After deleting the north and south poles from the unit sphere in ${\bf R}^3$, consider a foliation of the now twice punctured sphere by jordan curves. Each admits a minimal spanning disc. Do these foliate the ball?
A unified approach to these and similar questions will be presented.