Constant Mean Curvature Surfaces in a Half-Space of R With Boundary in the Boundary of the Half-Space

The structure of the set of compact constant mean curvature surfaces whose boundary is a given Jordan curve in R seems far from being understood. We shall consider a simple, but interesting, situation concerning this problem: Assume that M be an embedded compact H-surface in R+ = {x3 ≥ 0} with ∂M = Γ ⊂ P = {x3 = 0}. Then, there is little known about the geometry and topology of M in terms of Γ. For example, if Γ is convex, is M of genus zero? When Γ is a circle, it follows from Alexandrov [A] that M is neccesarily a spherical cap or the planar disk bounded by Γ. We first show that if Γn ⊂ P is a sequence of embedded (perhaps nonconnected) curves converging to a point p, and Mn ⊂ R+ is a sequence of 1-surfaces (H = 1), with ∂Mn = Γn, then some subsequence of Mn converges either to p or to the unit sphere tangent to P at p (the convergence being smooth in R − p). The same kind of result was obtained by Wente [W] when Γn is an arbitrary Jordan curve in R converging to a point p and Mn is an immersed topological disc bounded by Γn which minimizes area among disks bounding a fixed algebraic volume.