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On generalized Gauss maps of minimal surfaces sharing hypersurfaces in a projective variety

Si Duc Quang, Do Thi Thuy Hang

Abstract

In this article, we study the uniqueness problem for the generalized gauss maps of minimal surfaces (with the same base) immersed in $\mathbb R^{n+1}$ which have the same inverse image of some hypersurfaces in a projective subvariety $V\subset\mathbb P^n(\mathbb C)$. As we know, this is the first time the unicity of generalized gauss maps on minimal surfaces sharing hypersurfaces in a projective varieties is studied. Our results generalize and improve the previous results in this field.

On generalized Gauss maps of minimal surfaces sharing hypersurfaces in a projective variety

Abstract

In this article, we study the uniqueness problem for the generalized gauss maps of minimal surfaces (with the same base) immersed in which have the same inverse image of some hypersurfaces in a projective subvariety . As we know, this is the first time the unicity of generalized gauss maps on minimal surfaces sharing hypersurfaces in a projective varieties is studied. Our results generalize and improve the previous results in this field.
Paper Structure (4 sections, 12 theorems, 91 equations)

This paper contains 4 sections, 12 theorems, 91 equations.

Table of Contents

  1. Introduction and Main results
  2. Main lemmas
  3. Holomorphic curves from complex discs into projective varieties
  4. Uniqueness theorems for Gauss maps

Key Result

Theorem 1.1

Let $V$ be an $\ell$-dimension projective subvariety of ${\mathbb{P}}^n({\mathbb{C}})$. Let $S_1,S_2$ be non-flat minimal surfaces immersed in ${\mathbb{R}}^{n+1}$ with the Gauss maps $G_1,G_2$ into $V$, respectively. Assume that there are conformal diffeomorphisms $\Phi_i$ of $S_1$ onto $S_2$. Let Suppose that $f^1$ is linear nondegenerate over $I_d(V)$. If $S^1$ is complete and where $M=H_d(V)

Theorems & Definitions (19)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1: cf. QA
  • Theorem 2.2: cf. Q21,Q23,Q24
  • Theorem 2.3: cf. Fu93
  • Theorem 2.4
  • proof
  • Lemma 2.5: Generalized Schwarz's Lemma A38
  • Lemma 3.1
  • proof
  • ...and 9 more