Given a C1 function H defined in the unit sphere S2, an H-surface M is a surface in the Euclidean space R3 whose mean curvature HM satisfies HM(p) = H(Np), p ∈ M, where N is the Gauss map of M. Given a closed simple curve Γ ⊂ R3 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.
Mathematics Subject Classification: 53A10, 53C42, 35J93, 35B06, 35B50.