Modified defect relation for Gauss maps of minimal surfaces with hypersurfaces of projective varieties in the subgeneral position

Abstract

In this paper, we establish some modified defect relations for the Gauss map $g$ of a complete minimal surface $S\subset R^m$ into a $k$-dimension projective subvariety $V\subset P^n\left(C\right)(n=m-1)$ with hypersurfaces $Q_1,\text{…},Q_q$ of $P^n\left(C\right)$ in $N$-subgeneral position with respect to $V(N\ge k)$. In particular, we give the upper bound for the number $q$ if the image $g\left(S\right)$ intersects each hypersurface $Q_1,\text{…},Q_q$ a finite number of times and $g$ is nondegenerate over $I_d\left(V\right)$, where $d=\text{lcm}(deg{Q}_1,\text{…},deg{Q}_q)$, that is, the image of $g$ is not contained in any hypersurface $Q$ of degree $d$ with $V\not\subset Q$. Our results extend and generalize the previous results for the case of the Gauss map and hyperplanes in a projective space. The results and the method of this paper have been applied by some authors to study the unicity problem of the Gauss maps sharing families of hypersurfaces.