A Constructive Approach to Infinitesimal Conformal Rigidity on Complex Hyperbolic Space

TL;DR

The paper proves a local, constructive rigidity: on complex hyperbolic space $\mathbb{C}H^n$ with $n\ge2$, every conformal vector field is Killing. It achieves this by modeling $\mathbb{C}H^n$ as a Damek–Ricci solvable space, formulating the conformal Killing condition as an explicit PDE system in left-invariant coordinates, and solving it completely in the case $n=2$ before extending the method to all $n$. The core idea is to express conformal fields via four scalar functions, analyze their PDEs through harmonic expansions of $F_3$ and $F_4$, and show that nontrivial conformal deformations cannot occur. The result highlights a strong infinitesimal conformal rigidity for $\mathbb{C}H^n$, with potential extensions to the broader family of Damek–Ricci spaces and local rigidity phenomena in noncompact symmetric spaces.

Abstract

We prove that every conformal vector field on the complex hyperbolic space $\mathbb{C}H^n$ is Killing for all $n\ge 2$. Although this rigidity is classically known, our proof is entirely different in nature: it is local, analytic, and fully constructive. Our approach is local, analytic, and constructive: we view $\mathbb{C}H^2$ through its solvable Lie group model and express the conformal Killing equation as an explicit system of partial differential equations. By solving this system completely, we show that any conformal vector field must be determined by a Killing field. The analysis in complex dimension $2$ naturally extends to arbitrary $n$, yielding a unified and fully explicit proof of this rigidity phenomenon.

Theorems & Definitions (25)

• Theorem 1
• Remark 2
• Definition 3
• Lemma 4
• proof
• Definition 5
• Proposition 6
• proof
• Corollary 7
• Remark 8