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

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