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

TL;DR

This paper extends the value distribution theory of Gauss maps on annular ends of minimal surfaces, establishing a modified defect relation for hypersurfaces in subgeneral position, with implications for the uniqueness of Gauss maps.

Contribution

It introduces the first study of Gauss map value distribution on annular ends with hypersurface targets, providing a new inequality and unicity results.

Findings

The image of the Gauss map cannot omit all hypersurfaces under certain conditions.

Established a new defect relation for Gauss maps on annular ends.

Applied results to prove unicity of Gauss maps in minimal surface theory.

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 $Q_1,\ldots,Q_q$ 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 $\deg Q_1,\ldots,\deg Q_q$. 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].