The value distribution of the hyperbolic Gauss map

Abstract

In this paper, we investigate the hyperbolic Gauss map of a complete CMC-1 surface in H3(−1), and prove that it cannot omit more than four points unless the surface is a horosphere. 0. Introduction In minimal surface theory, the value distribution of the Gauss map has been studied for a long time. An early result is the classical Bernstein Theorem: Any complete minimal graph in R is a plane. Replacing the graph by some geometric conditions, R. Osserman [3] showed that if the complete minimal surface is nonflat, then the Gauss map cannot omit a set of positive logarithmic capacity, which answered a conjecture of Nirenberg. Afterwards, it was generalized by F. Xavier [8] that the Gauss map of such a surface can omit at most six points. In 1989, H. Fujimoto [2] proved the best result finally that the Gauss map of the nonflat complete minimal surface in R can omit at most four points. It is natural to consider the similar properties of minimal surfaces in H. An interesting feature is that there exists a family of absolutely area-minimizing hypersurfaces in H and only one of them is totally geodesic (see [7]). The question seems to be how to raise an adequate Bernstein problem in hyperbolic space. Professor Y. L. Xin pointed out to me that the striking work done by R. Bryant [1] supplies a framework to solve the Bernstein problem in hyperbolic space. In this paper, we shall be concerned with the surfaces in hyperbolic space of constant mean curvature one. We abbreviate “constant mean curvature one” by CMC-1. These surfaces share many properties with minimal surfaces in R. They possess the “Weierstrass representation” in terms of holomorphic data. This formula was discovered by R. Bryant [1]. Many other properties may be found in papers by M. Umhara and K. Yamada ([4], [5]). Here we try to investigate the hyperbolic analogue of the Gauss map. It is a natural question how the values of the hyperbolic Gauss map distribute. Using Bryant’s representation formula we are able to answer this question as follows. Theorem. The hyperbolic Gauss map of nonflat complete CMC-1 surfaces in H3(−1) can omit at most four points. Received by the editors November 1, 1995 and, in revised form, April 2, 1996. 1991 Mathematics Subject Classification. Primary 53A10; Secondary 53C42.