Behavior of some CMC capillary surfaces at convex corners (Q883838)

The paper concerns the conjecture of Concus and Finn which says that a capillary surface over a domain with a corner with opening angle \(2\alpha\), \(0<2\alpha<\pi\), is discontinuous near the corner if the contact angles on the adjacent curves of the corner differ by more then \(\pi-2\alpha\). The results of the paper support this conjecture. The authors construct examples of nonparametric surfaces of zero mean curvature (minimal surfaces) over domains with a corner which make contact angles \(\gamma_1, \gamma_2\) such that \(\gamma_2-\gamma_1>\pi-2\alpha\) and which are discontinuous at the corner. The construction is based on Weierstrass representation formulae for minimal surfaces and on a carefully discussion of an associated Riemann-Hilbert problem.