Compact surfaces with boundary with prescribed mean curvature depending on the Gauss map

Abstract

Given a C 1 function H defined in the unit sphere S 2 , an H-surface M is a surface in the Euclidean space R 3 whose mean curvature H M satisfies H M (p) = H(N p ), p ∈ M, where N is the Gauss map of M. Given a closed simple curve Γ ⊂ R 3 and a function H, in this paper we investigate the geometry of compact H-surfaces spanning Γ in terms of Γ. Under mild assumptions on H, we prove non-existence of closed H-surfaces, in contrast with the classical case of constant mean curvature. We give conditions on H that ensure that if Γ is a circle, then M is a rotational surface. We also establish the existence of estimates of the area of H-surfaces in terms of the height of the surface.