Behaviour of doubly connected minimal surfaces at the edges of the support surface

摘要

This paper is the second part of our investigations on doubly connected minimal surfaces which are stationary in a boundary configuration $ (Gamma, S) $ in $ Bbb R ^3 $ . The support surface S is a vertical cylinder above a simple closed polygon $ P(S) $ in the x,y-plane. The surrounding Jordan curve $ Gamma $ is chosen as a generalized graph above its convex projection curve $ P(Gamma) $ . In [23] we have proved the existence of nonparametric minimal surfaces X of annulus type spanning such boundary configurations. We study the behaviour of these minimal surfaces at the edges of the support surface S. In particular we discuss the phenomenon of edge-creeping, i. e. the fact that the free trace of X may attach to an edge of S in a full interval. We prove that a solution X cuts any intruding edge of S perpendicularly. On the other hand, we derive a condition which forces X to exhibit the edge-creeping behaviour. Depending on the symmetries of $ (Gamma, S) $ we give bounds on the number of edges where edge-creeping occurs. Let $ (x,y,hbox {Z} (x,y)) $ for $ (x,y)in G $ be the nonparametric representation of X. Then at every vertex Q of $ P(S) $ the radial limits of Z from all directions in G exist.