A note on the Gauss map of complete nonorientable minimal surfaces

Abstract

has achieved many important advances and also has given rise to manyproblems in recent decades. The most interesting question is to determinethe size of the spherical image of such a surface under its Gauss map.R. Osserman was the person who started the systematic development ofthis theory, and so, in 1961 he proved that the set omitted by the imageof a complete non flat orientable minimal surface by the Gauss map haslogarithmic capacity zero. In 1981 F. Xavier [12] proved that this set coversthe sphere except six values at the most, and nally in 1988 H. Fujimoto[3, 4] obtained the best possible theorem, and proved that the number ofexceptional values of the Gauss map is four at the most. An interestingextension of Fujimoto’s theorem was proved in 1990 by X. Mo and R. Os-serman [7]. They showed that if the Gauss map of a complete orientableminimal surface takes on ve distinct values only a nite number of times,then the surface has nite total curvature.There are many kinds of complete orientable minimal surfaces whoseGauss map omits four points of the sphere. Among these examples we em-phasize the classical Scherk’s doubly periodic surface and those described byK. Voss in [10] (see also [8]). The rst author of this paper in [5] constructsorientable examples with non trivial topology.Under the additional hypothesis of nite total curvature, R. Osserman [9]proved that the number of exceptional values is three at the most.In the nonorientable case, the Gauss map of the two sheeted orientablecovering surface induces, in a natural way, a