1. Modified defect relation for Gauss maps of minimal surfaces with hypersurfaces of projective varieties in the subgeneral position.
- Author
-
Quang, Si Duc
- Subjects
- *
GAUSS maps , *HYPERSURFACES , *MINIMAL surfaces , *PROJECTIVE spaces , *HYPERPLANES - Abstract
In this paper, we establish some modified defect relations for the Gauss map g$g$ of a complete minimal surface S⊂Rm$S\subset \mathbb {R}^m$ into a k$k$‐dimension projective subvariety V⊂Pn(C)(n=m−1)$V\subset \mathbb {P}^n(\mathbb {C})\ (n=m-1)$ with hypersurfaces Q1,…,Qq$Q_1,\ldots,Q_q$ of Pn(C)$\mathbb {P}^n(\mathbb {C})$ in N$N$‐subgeneral position with respect to V(N≥k)$V\ (N\ge k)$. In particular, we give the upper bound for the number q$q$ if the image g(S)$g(S)$ intersects each hypersurface Q1,…,Qq$Q_1,\ldots,Q_q$ a finite number of times and g$g$ is nondegenerate over Id(V)$I_d(V)$, where d=lcm(degQ1,…,degQq)$d=\text{lcm}(\deg Q_1,\ldots,\deg Q_q)$, that is, the image of g$g$ is not contained in any hypersurface Q$Q$ of degree d$d$ with V⊄Q$V\not\subset Q$. 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. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF