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Archiv der Mathematik

, Volume 77, Issue 3, pp 278–288 | Cite as

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

  • G. Turowski
Article

Abstract.

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.

Keywords

Minimal Surface Support Surface Jordan Curve Generalize Graph Vertical Cylinder 

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Copyright information

© Birkhäuser Verlag, Basel 2001

Authors and Affiliations

  • G. Turowski
    • 1
  1. 1.Am Hahnsberg, D-53424 RemagenGermany

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