Modified defect relation of Gauss maps on annular ends of minimal surfaces for hypersurfaces of projective varieties in subgeneral position

Abstract

Let $A$ be an annular end of a complete minimal surface $S$ in ${\mathbb{R}}^m$ and let $V$ be a $k$-dimension projective subvariety of ${\mathbb{P}}^n(\mathbb{C})(n=m-1)$. Let $g$ be the generalized Gauss map of $S$ into $V\subset {\mathbb{P}}^n(\mathbb{C})$. In this paper, we establish a modified defect relation of $g$ on the annular end $A$ for $q$ hypersurfaces $\{Q_i{\}}_{i=1}^q$ of ${\mathbb{P}}^n(\mathbb{C})$ in $N$-subgeneral position with respect to $V$. Our result implies that the image $g(A)$ cannot omit all $q$ hypersurfaces $Q_1,\dots ,Q_q$ if $g$ is nondegenerate over $I_d(V)$ and $q>\frac{(2N-k+1)(M+1)(M+2d)}{2d(k+1)}$, where $M=H_V(d)-1$ and $d$ is the least of common multiple of $°Q_1,\dots ,°Q_q$. As our best knowledge, it is the first time the value distribution of the Gauss map on an annular end of a minimal surfaces with hypersurface targets is studied, in particular the product into sum inequality for holomorphic curves on Riemann surfaces with hypersurfaces targets is presented. This our result has been used to study the unicity of the gauss maps in the recent work of C. Lu and X. Chen [14].

In this final version, some typos are corrected. This paper has been published in Journal of Mathematical Analysis and Applications