One-sided minimal surfaces with a given boundary

Abstract

One-sided surfaces present themselves quite naturally in the theory of minimal surfaces; they were first studied systematically by Sophus Liet under the name of Minimaldoppelflachen. Lie had interpreted the formulas of Monge for a minimal surface in the now well known geometric way: that every minimal surface can be generated by translating one minimal curvet along another. An equivalent construction is to take the mid-points of all line segments whose ends lie respectively on any two fixed minimal curves gi, g2; their locus is a minimal surface, which is real whenever ,ut, A2 are conjugate complex. If li and A2 coincide in a minimal curve A, this construction becomes the taking of the locus of the midpoints of all chords of ,u. The resulting minimal surface, real whenever / is its own conjugate complex curve, is of the type designated as Minimaldoppelflache by Lie. If this surface is not periodic, i.e. does not go over into itself by a certain translation and its repetitions, then it is a one-sided surface in the sense of topology: a,material point moving continuously on the surface can pass from any position to that directly beneath without crossing over any boundary of the surface (M6bius strip); or, if we ascribe a certain arrow to the normal at any fixed point and follow the continuous variation of this sensed normal as the point moves on the surface, it is possible to describe a closed path which will reverse this arrow. In particular, the surface cannot be periodic if it is algebraic; therefore every algebraic double minimal surface is one-sided. Algebraic character can be secured for the surface by taking the minimal curve pu to be algebraic; this in turn can be done by taking as algebraic the arbitrary function f(t) in the Weierstrass formulas for a minimal curve:?