A Picard theorem with an application to minimal surfaces. II

Abstract

We prove a Picard theorem for holomorphic maps from C to a quadric hypersurface. This implies a theorem on the number of directions in general position omitted by the normals to a minimal surface of the conformal type of C. The distribution of the normals to a two-dimensional minimal surface in Rn has been studied by Chern and Osserman [ 1]. This paper is concerned only with a special case of their theorem. For a minimal surface of the type of the entire plane C, Chern and Osserman prove that, if the normals to the surface omit n + 1 directions in general position, where n is the dimension of the ambient space, the image of the Gauss map lies in a proper linear subspace of CPn I . Theorem 1 of this paper improves on their result in two ways. First, it is only assumed that the normals omit n directions in general position. Secondly, we prove that the image of the Gauss map lies in a linear subspace of codimension two; in consequence, the minimal surface decomposes into a holomorphic function and a minimal surface in R -2 , in a sense that will be made precise below. The method, which derives from a paper of M. L. Green [4], is to apply value-distribution theory to maps into a quadric hypersurface instead of maps into projective space. In the definitions that follow we adopt the notation used by Hoffman and Osserman in their memoir [5], to which we refer for details. Let M be a Riemann surface and f: M Rn, where n > 2, be a nonconstant smooth map, with components (f1, ... , fn) . If z = x + iy is a local coordinate on M, let afk _ *fk k = X a The map f is called a minimal surface if the (Pk are holomorphic and satisfy the equation of conformality (1) + ..+ + 2 ? If the vector (P 1 ..P. ,k) is nonzero then it gives the homogeneous coordinates of some point in the complex projective space CPn-I . Since the (k are Received by the editors June 19, 1987. Presented to the Special Session on Differential Geometry, April 25, 1987, at the Society's 834th meeting in Newark, New Jersey (Abstract 834-53-28). 1980 Mathematics Subject Classification (1985 Revision). Primary 53A10; Secondary 30D35. (?) 1989 American Mathematical Society 0002-9947/89 $1.00 + $.25 per page