A Constructive Approach to Infinitesimal Conformal Rigidity on Complex Hyperbolic Space

TL;DR

This paper provides a new local, analytic, and constructive proof that all conformal vector fields on complex hyperbolic spaces are Killing fields, extending known rigidity results to all dimensions using explicit PDE analysis.

Contribution

It introduces a novel local and constructive method to prove conformal rigidity on complex hyperbolic space, applicable in all dimensions.

Findings

All conformal vector fields on $ ext{CH}^n$ are Killing fields for $n  2$.

The proof uses explicit PDE systems derived from the solvable Lie group model.

The approach extends naturally from complex dimension 2 to higher dimensions.

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.