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.