Total curvature of a complete minimal surface and the modified defect relation of a Fermat hypersurface for the Gauss map

Abstract

In this paper, we establish some modified defect relations for the Gauss map $g$ of a complete minimal surface $S\subset {\mathbb{R}}^m$ into ${\mathbb{P}}^n(\mathbb{C})(n=m-1)$ with only a single Fermat hypersurface $Q$ of ${\mathbb{P}}^n(\mathbb{C})$. In particular, we show that $S$ must have finite total curvature if the image $g(S)$ intersects $Q$ with only a finite number of times and the degree of $Q$ is sufficiently large.

arXiv comment19 pages