Values of Gauss maps of complete minimal surfaces in ℝ^{𝕞} on annular ends

Abstract

Let M M be a complete minimal surface in R m {\mathbb {R}}^{m} and let A A be an annular end of M M which is conformal to { z | 0 > 1 / r > | z | > r } \{z~|~ 0 > 1/r>|z|>r\} , where z z is the conformal coordinate. Let G G be the generalized Gauss map of M M . We show that G ( A ) G(A) must intersect every hyperplane in P m − 1 ( C ) {\mathbb {P}}^{m-1}({\mathbb {C}}) , with the possible exception of m ( m + 1 ) / 2 m(m+1)/2 hyperplanes in general position.