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The classification of CR maps from hyperquadrics into tubes over null cones of symmetric forms

Nguyen Gia Hien, Michael Reiter, Duong Ngoc Son

TL;DR

We classify local CR maps from the Levi-degenerate hyperquadric $\mathbb{H}^{2n-1}_l$ to the tube model $\mathcal{X}^{2n+1}_l$ up to CR automorphisms, and deduce a parallel classification for local proper holomorphic maps from the generalized ball $\mathbb{B}^n_l$ to the generalized Lie ball $D^{\mathrm{IV}}_{m,l}$. The approach hinges on a robust normalization framework built from explicit automorphism groups, together with invariants such as the CR Ahlfors tensor and geometric rank to distinguish equivalence classes. For $n=3$, $l=1$ the authors establish six equivalence classes represented by explicit maps $H_j$ (plus an irrational $I$-type map); for $n\ge4$ only two isometry-type classes remain, represented by $\ell^{(n)}$ and $I^{(n)}$, each an isometry of canonical indefinite Kähler metrics. The paper also provides detailed local holomorphic map classifications to $D^{IV}_{n+1,l}$ and extends the results to higher codimension, yielding rigidity statements via a lifting construction and connections to known results in the literature.

Abstract

We classify CR maps from the hyperquadric of signature $l>0$ in $\mathbb{C}^n$, $n\geq 3$, to the local model for the tube over the null cone of a symmetric form in $\mathbb{C}^{n+1}$, up to CR automorphisms of the source and target. In contrast to the setting of the Heisenberg hypersurface in $\mathbb{C}^3$ (i.e., the case $l=0$), studied earlier in Reiter--Son [27], our analysis uncovers two new equivalence classes of CR maps of geometric rank one and one new class of geometric rank two in the case $n=3$. In the case $n\geq 4$, we establish that all maps extend to local isometries of certain indefinite Kähler metrics. We further derive a classification of (local) proper holomorphic maps from the generalized unit ball $\mathbb{B}^n_l$ into a generalized version of the Lie ball $D^{\mathrm{IV}}_{m,l}$ (the generalized classical domain of type~IV).

The classification of CR maps from hyperquadrics into tubes over null cones of symmetric forms

TL;DR

We classify local CR maps from the Levi-degenerate hyperquadric to the tube model up to CR automorphisms, and deduce a parallel classification for local proper holomorphic maps from the generalized ball to the generalized Lie ball . The approach hinges on a robust normalization framework built from explicit automorphism groups, together with invariants such as the CR Ahlfors tensor and geometric rank to distinguish equivalence classes. For , the authors establish six equivalence classes represented by explicit maps (plus an irrational -type map); for only two isometry-type classes remain, represented by and , each an isometry of canonical indefinite Kähler metrics. The paper also provides detailed local holomorphic map classifications to and extends the results to higher codimension, yielding rigidity statements via a lifting construction and connections to known results in the literature.

Abstract

We classify CR maps from the hyperquadric of signature in , , to the local model for the tube over the null cone of a symmetric form in , up to CR automorphisms of the source and target. In contrast to the setting of the Heisenberg hypersurface in (i.e., the case ), studied earlier in Reiter--Son [27], our analysis uncovers two new equivalence classes of CR maps of geometric rank one and one new class of geometric rank two in the case . In the case , we establish that all maps extend to local isometries of certain indefinite Kähler metrics. We further derive a classification of (local) proper holomorphic maps from the generalized unit ball into a generalized version of the Lie ball (the generalized classical domain of type~IV).
Paper Structure (11 sections, 25 theorems, 150 equations)

This paper contains 11 sections, 25 theorems, 150 equations.

Table of Contents

  1. Introduction
  2. Normalization for Theorem \ref{['thrmHn1Xn2']}
  3. The CR automorphism groups of $\mathbb{H}^{2n-1}_l$ and $\mathcal{X}^{2n+1}_l$
  4. A partial normalization
  5. Geometric rank and the CR Ahlfors tensor
  6. The CR Ahlfors tensor
  7. Geometric rank
  8. Isometries
  9. The classification of CR maps from $\mathbb{H}^{2n-1}_l$ to $\mathcal{X}^{2n+1}_l$
  10. On proper holomorphic maps from $\mathbb{B}^{n}_l$ to $D^{\mathrm{IV}}_{n+1,l}$
  11. Higher codimensional case

Key Result

Theorem 1.1

Let $n\geq 3, 1\leq l<n-1$ and $U$ be an open subset of $\mathbb{H}^{2n-1}_l\subset\mathbb{C}^{n}$. Let $H\colon U\to\mathcal{X}^{2n+1}_l\subset\mathbb{C}^{n+1}$ be a transversal CR map. Then

Theorems & Definitions (55)

  • Theorem 1.1
  • Remark 1
  • Corollary 1.2
  • Remark 2
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Remark 3
  • Proposition 2.3
  • ...and 45 more