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

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 a $k$-dimension projective subvariety $V\subset {\mathbb{P}}^n(\mathbb{C})(n=m-1)$ with hypersurfaces $Q_1,\dots ,Q_q$ of ${\mathbb{P}}^n(\mathbb{C})$ 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 $g(S)$ intersects each hypersurfaces $Q_1,\dots ,Q_q$ a finite number of times and $g$ is nondegenerate over $I_d(V)$, where $d=lcm(\mathrm{deg}{Q}_1,\dots ,\mathrm{deg}{Q}_q)$, i.e., the image of $g$ is not contained in any hypersurface $Q$ of degree $d$ with $V\subset ̸Q$. 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.