Minimal surfaces in a wedge II.
Stefan Hildebrandt, Friedrich Sauvigny · 1997
In this paper we continue our investigations of the boundary behaviour of minimal surfaces X which are stationary in a configuration hG;si. Here s is the boundary of a wedge wa with the edge g and the opening angle pa ap, and G is a Jordan arc contained in wa whose endpoints P1 and P2 lie on distinct faces of s. First we construct a minimal surface of this type whose free boundary on s attaches to the edge g in a full interval. Secondly we derive a condition which forces minimal surfaces stationary in hG;si to exhibit this edge-creeping behaviour. Apparently this phenomenon was discovered by Y. W. Chen when treating an exterior free boundary value problem for minimal surfaces. Later S. Hildebrandt and J. C. C. Nitsche proved existence and optimal regularity of solutions to a partially free boundary value problem where edge creeping occurs. They derived a uniqueness theorem proving that, under suitable assumptions on G and for aa 1, the solutions can be viewed as graphs over two- dimensional slit domains. In this paper we continue our investigations of the boundary behaviour of minimal surfaces X which are stationary in a configuration hG;si. Here s is the boundary of a wedge wa with the edge g and the opening angle pa ap, and G is a Jordan arc contained in wa whose endpoints P1 and P2 lie on distinct faces of s. First we construct a minimal surface of this type whose free boundary on s attaches to the edge g in a full interval. Secondly we derive a condition which forces minimal surfaces stationary inhG;si to exhibit this edge-creeping behaviour.