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Hermitian Rank and Rigidity of Holomorphic Mappings

Yun Gao

Abstract

Huang's Lemma is an important tool in CR geometry to study rigidity problems. This paper introduces a generalization of Huang's Lemma based on the rigidity properties of holomorphic mappings preserving certain orthogonality on projective spaces, which is optimal for the case of partial linearity. By exploring the intricate relationship between rigidity and Huang's Lemma, we establish that the rigidity properties of Segre maps or proper holomorphic mappings between generalized balls with Levi-degenerate boundaries can be inferred from those between generalized balls with Levi-non-degenerate boundaries by in a coordinate-free and more geometric manner.

Hermitian Rank and Rigidity of Holomorphic Mappings

Abstract

Huang's Lemma is an important tool in CR geometry to study rigidity problems. This paper introduces a generalization of Huang's Lemma based on the rigidity properties of holomorphic mappings preserving certain orthogonality on projective spaces, which is optimal for the case of partial linearity. By exploring the intricate relationship between rigidity and Huang's Lemma, we establish that the rigidity properties of Segre maps or proper holomorphic mappings between generalized balls with Levi-degenerate boundaries can be inferred from those between generalized balls with Levi-non-degenerate boundaries by in a coordinate-free and more geometric manner.
Paper Structure (4 sections, 12 theorems, 32 equations)

This paper contains 4 sections, 12 theorems, 32 equations.

Table of Contents

  1. Introduction
  2. Orthogonal pair and orthogonal map
  3. Rigidity of orthogonal maps and pairs
  4. Proof of Theorem \ref{['pair']}

Key Result

Theorem 1.1

For $r, s\in{\mathbb Z_{\ge 0}}, n'\in \mathbb N$, let $\{\phi_j\}_{j=1}^{n'},$$\{\psi_j\}_{j=1}^{n'}$ be holomorphic functions in $z\in \mathbb C^{r+s}$ near the origin and let $H(z,\bar{z})$ be a complex-valued real-analytic function such that If $r+s\leq n'\le 2(r+s)-3$ and $H(z,\bar{z}) \not\equiv 0$, then there exist holomorphic functions $h_1(z)$ and $h_2(z)$ such that $H(z,\bar{z})=h_1(z)\

Theorems & Definitions (23)

  • Theorem 1.1
  • Example 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Theorem 1.5
  • Definition 2.1
  • Definition 2.2: Ga
  • Proposition 3.1
  • proof
  • Definition 3.2
  • ...and 13 more