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Infinitesimal Conformal Rigidity on Damek-Ricci Spaces

Hiroyasu Satoh, Hemangi Madhusudan Shah

TL;DR

The paper proves infinitesimal conformal rigidity for Damek-Ricci spaces by recasting the conformal Killing equation as a local PDE on the solvable Lie group model and performing a detailed holomorphic-harmonic analysis of the coefficients. It shows that any conformal vector field must have vanishing conformal factor $\\rho$, hence is Killing, extending a prior rigidity result for complex hyperbolic spaces to the full class of Damek-Ricci spaces. The approach avoids global symmetry arguments and relies on a constructive, analytic investigation of the overdetermined system, exploiting a holomorphic structure in adapted coordinates. This highlights a strong rigidity phenomenon in noncompact harmonic Einstein spaces, contrasting with the richer conformal structure seen in real hyperbolic spaces and aligning with Lichnerowicz-type phenomena in related settings.

Abstract

We show that every conformal vector field on a Damek-Ricci space is necessarily Killing, establishing a strong form of infinitesimal conformal rigidity. Although this rigidity phenomenon is classically known in the Einstein setting, our proof follows a completely different approach. We formulate the conformal Killing condition as an explicit system of partial differential equations on the solvable Lie group model and analyze it directly. This local and analytic method yields a constructive proof of rigidity without relying on global arguments or transformation groups.

Infinitesimal Conformal Rigidity on Damek-Ricci Spaces

TL;DR

The paper proves infinitesimal conformal rigidity for Damek-Ricci spaces by recasting the conformal Killing equation as a local PDE on the solvable Lie group model and performing a detailed holomorphic-harmonic analysis of the coefficients. It shows that any conformal vector field must have vanishing conformal factor , hence is Killing, extending a prior rigidity result for complex hyperbolic spaces to the full class of Damek-Ricci spaces. The approach avoids global symmetry arguments and relies on a constructive, analytic investigation of the overdetermined system, exploiting a holomorphic structure in adapted coordinates. This highlights a strong rigidity phenomenon in noncompact harmonic Einstein spaces, contrasting with the richer conformal structure seen in real hyperbolic spaces and aligning with Lichnerowicz-type phenomena in related settings.

Abstract

We show that every conformal vector field on a Damek-Ricci space is necessarily Killing, establishing a strong form of infinitesimal conformal rigidity. Although this rigidity phenomenon is classically known in the Einstein setting, our proof follows a completely different approach. We formulate the conformal Killing condition as an explicit system of partial differential equations on the solvable Lie group model and analyze it directly. This local and analytic method yields a constructive proof of rigidity without relying on global arguments or transformation groups.

Paper Structure

This paper contains 7 sections, 3 theorems, 60 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Conformal vector fields
  4. Damek--Ricci spaces
  5. Conformal Killing equations on Damek--Ricci spaces
  6. Holomorphic and harmonic structure of coefficients of $Z$ and $A$
  7. Main result and its proof

Key Result

Theorem 1

Let $(M, g)$ be a Damek--Ricci space and $\xi$ a conformal vector field. Then the potential function $\rho$ of $\xi$ must vanish identically. That is, $\xi$ is a Killing vector field.

Theorems & Definitions (20)

  • Theorem 1
  • Definition 2
  • Lemma 3
  • proof
  • Example 4
  • Example 5
  • Example 6
  • Remark 7
  • Definition 8
  • Definition 9
  • ...and 10 more