A supplement to the condition of J. Douglas

Abstract

While a Jordan curve in Euclidean 3-space always spans a minimal surface of the type of the circular disc, the existence of a minimal surface of the type of the circular annulus, bounded by a system of two disjoint Jordan curves, very much depends on the configuration of these curves. A sufficient criterion for the existence has been given by J. Douglas in a famous paper [3] and, in different form, by R. Courant (see [2], esp. chapter IV). Roughly speaking, this condition guarantees the existence of a minimal surface of the type of the circular annulus, bounded by two Jordan curves Pi and P~ without common points, provided that these curves span a doubly-connected surface whose area is smaller than the sum of the areas ot the two simply-connected surfaces of least area bounded by P i and P~, respectively. Douglas' condition can be verified in many concrete cases. Nevertheless, even if Douglas' condition is violated, the problem of finding a minimal surface of the type of the annulus, bounded by two Jordan curves, may still be solvable. This fact is illustrated with the classical experiment where the curves Pi and P2 are coaxial unit circles in parallel planes, say the planes z = h and z = h (h ~ 0). (See for instance the presentation in (i. A. Bliss [1]). For h < ht : 0 . 5 2 7 7 . Douglas' condition is fulfilled. For ht ~< h ~ h~ = 0.6627... Douglas' condition is violated, but the circles Pi and r~ still span a doubly-connected minimal