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

Si Duc Quang

Abstract

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 $Q_1,\ldots,Q_q$ 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 $Q_1,\ldots,Q_q$ a finite number of times and $g$ is nondegenerate over $I_d(V)$, where $d=lcm(°Q_1,\ldots,°Q_q)$, i.e., 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.

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

Abstract

In this paper, we establish some modified defect relations for the Gauss map of a complete minimal surface into a -dimension projective subvariety with hypersurfaces of in -subgeneral position with respect to . In particular, we give the upper bound for the number if the image intersects each hypersurfaces a finite number of times and is nondegenerate over , where , i.e., the image of is not contained in any hypersurface of degree with . 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.

Paper Structure

This paper contains 4 sections, 22 theorems, 167 equations.

Table of Contents

  1. Introduction and Main results
  2. Defect relation of holomorphic curves from a punctured disk into a projective variety with a family of hypersurfaces
  3. Sum to Product inequality for holomorphic curves and families of hypersurfaces
  4. Modified defect relation for Gauss map with hypersurfaces

Key Result

Theorem 1.2

Let $S$ be a complete minimal surface in ${\mathbb{R}}^m$. Let $V$ be a projective subvariety of dimension $k$ of ${\mathbb{P}}^{n}({\mathbb{C}})\ (n=m-1)$. Let $Q_1,\ldots,Q_q$ be hypersurfaces of ${\mathbb{P}}^n({\mathbb{C}})$ in $N$-subgeneral position with respect to $V$. Let $d$ be the least co where $M=H_V(d)-1$. Then $S$ has finite total curvature.

Theorems & Definitions (32)

  • Theorem 1.2
  • Corollary 1.3
  • Theorem 1.4
  • Corollary 1.5
  • proof
  • Corollary 1.6
  • proof
  • Proposition 2.1: cf. Fu85
  • Lemma 2.2: cf. No81
  • Lemma 2.3: cf. QA
  • ...and 22 more