Minimal Surfaces with Planar Boundary Curves

In 1956, Shiffman [Sh] proved that any compact minimal annulus with two convex boundary curves (resp. circles) in parallel planes is foliated by convex planar curves (resp. circles) in the intermediate planes. In 1978, Meeks conjectured that the assumption the minimal surface is an annulus is unnecessary [M]; that is, he conjectured that any compact connected minimal surface with two planar convex boundary curves in parallel planes must be an annulus. Partial results have been proven in the direction of this conjecture. Schoen [Sc1] proved the Meeks conjecture in the case where the two boundary curves share a pair of reflectional symmetries in planes perpendicular to the planes containing the boundary curves. Another interesting result related to the Meeks conjecture has been proven by Meeks and White (Theorem 1.2, [MW2]). Recall that a minimal surface M is called stable if, with respect to any normal variation that vanishes on ∂M , the second derivative of the area functional is positive. The minimal surface is unstable if there exists such a variation with negative second derivative for the area functional, and it is almost-stable if the second derivative is nonnegative for all such variations and is zero for some nontrivial variation. Recall also that a subset of R is called extremal if it is contained in the boundary of its convex hull. The result of Meeks and White is that if Γ is an extremal pair of smooth disjoint convex curves in distinct planes, then exactly one of the following holds: