The Gauss Map of Algebraic Complete Minimal Surfaces Omits Hypersurfaces in Subgeneral Position

Abstract

The first aim of this paper is to show a second main theorem for algebraic curves into the n-dimensional projective space sharing hypersurfaces in subgeneral position. We then use it to study the value distribution of the generalized Gauss map of the complete (regular) minimal surfaces in \(\mathbb {R}^{m}\) with finite total curvature, as well as the unicity problem, sharing hypersurfaces in subgeneral position. Our results generalize and complete previous results in this area, especially the works of Chern and Osserman (J. Anal. Math. 19, 15–34, 1967), Jin and Ru (Differ. Geom. Appl. 25: 701–712, 2007).