Structure Theorems for Constant Mean Curvature Surfaces Bounded by a Planar Curve

3 is the boundary of two spherical caps of constant mean curvature H for any positive number H, which is at most the radius of C. It is natural to ask whether spherical caps are the only possible examples. Some examples of constant mean curvature immersed tori by Wente [7] indicate that there are compact genus-one immersed constant mean curvature surfaces with boundary C that are approximated by compact domains in Wente tori; however, this has not been proved. Still one has the conjecture: Conjecture 1 A compact constant mean curvature surface bounded by a circle is a spherical cap if either of the following conditions hold: 1. The surface has genus 0 and is immersed; 2. The surface is embedded. If M is a compact embedded constant mean curvature surface in 3 with boundary C and M is contained in one of the two halfspaces determined by the plane containing C, the Alexander reflection method [1] immediately proves M has the planar reflectional symmetries of C; hence, M is a surface of revolution. Since the only compact constant mean curvature surfaces of revolution are spherical caps (by Delaunay's classification [3] of constant mean curvature surfaces of revolution), Conjecture 1 holds for the subclass of surfaces that are embedded and contained in a halfspace. It is therefore of interest to obtain natural geometric conditions that force a compact embedded constant mean curvature surface to be contained in a halfspace. One result of our paper is to give the following sufficient condition for a compact constant mean curvature surface to be contained in a halfspace. Theorem 1 Let C be a convex curve in a plane P and let M be a compact connected surface with boundary C. Assume M is embedded, of constant mean curvature, and transverse to P along C. Then M is contained in one of the halfspaces of 3 determined by P. 2 Theorem 1 can be generalized to the case where the planar curve C is not necessarily convex.