Surfaces of Constant Mean Curvature in Euclidean 3-space Orthogonal to a Plane along its Boundary

Abstract

We consider compact surfaces with constant nonzero mean curvature whose boundary is a convex planar Jordan curve. We prove that if such a surface is orthogonal to the plane of the boundary, then it is a hemisphere. and we show that, in the above conditions, if M is embedded and is convex, then M is a hemisphere. Explicitly we prove that:

Constant mean curvature surfaces with circular boundary in R³

“…The initial demonstration of this fact was made by H. Heinz (Heinz 1969) and it doesn't use the flux formula. The equality |H | = L 2A(R) is characterized by the author in (Hinojosa 2002) without the hypothesis that the curve of the boundary is a circle. Now we will consider spherical caps of mean curvature H, whose boundary is an unitary circle.…”