The Flux Formula for Constant Mean Curvature Surfaces

Abstract

The problem that we treat in this chapter concerns the question of whether or not there are limitations on the possible values H of mean curvatures for the existence of an H-surface spanning a given boundary curve Γ. In general, the geometry of Γ imposes restrictions to the values of mean curvatures H, indeed, we will see that not all values of H are admissible. In this chapter, we shall obtain a flux formula for a compact cmc surface M immersed in $\mathbb{R}^{3}$ that relates integrals of M and of its boundary ∂M, linking information between the geometry of M and ∂M. We will combine the flux formula and the tangency principle to derive some geometric configurations when the surface is contained in an Euclidean ball or a cylinder whose size is related with the value of mean curvature of the surface. In the last section, we will prove that if an embedded surface with convex boundary is transverse to the boundary plane then it lies on one side of this plane.