  We consider compact connected minimal surfaces, with a pair of boundary
curves (not necessarily convex) in distinct planes, that have least-area
amongst all orientable surfaces with the same boundary. When the planes
containing these two boundary curves are either parallel or sufficiently close
to parallel, and when the boundary curves themselves are sufficiently close to
each other, we draw specific conclusions about the geometry and topology of the
surfaces. We also strength the following result: Let $M$ be any compact minimal
annulus with two planar boundary curves of diameters $d_1$ and $d_2$ in
parallel planes $P_1$ and $P_2$; if the distance between $P_1$ and $P_2$ is
$h$, then the inequality $h \leq {3/2}\max\{d_1,d_2\}$ is satisfied. We
strength it by removing the assumption that $M$ is an annulus and by showing
that the stronger conclusion $h \leq \max\{d_1,d_2\}$ holds. We also include a
similar result for nonminimal constant mean curvature surfaces.
