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Higher symmetries of the conformal powers of the Laplacian on conformally flat manifolds

A. Rod Gover, Josef Silhan

TL;DR

This work provides a universal, tractor-calculus–based framework to characterize all symmetries of the conformally invariant powers of the Laplacian, the GJMS operators $P_k$, on conformally flat manifolds. By constructing a canonical map from generalized conformal Killing tensors to symmetry pairs $(S_\varphi,S'_\varphi)$, the authors show that every symmetry of $Δ^k$ (and hence of $P_k$ in the flat setting) arises from this universal construction, modulo trivial symmetries. They then give an explicit, computable algebraic description of the symmetry algebra as a quotient of the tensor algebra on $\frak{so}_{s+1,s'+1}$ by a two-sided ideal generated by concrete relations, including the highest Cartan component. The combination of universal symmetry construction, leading-term classification, and precise algebraic structure provides a comprehensive, calculable picture of higher conformal symmetries with potential extensions to broader parabolic geometries and to curved settings via the ambient/tractor calculus.

Abstract

On locally conformally flat manifolds we describe a construction which maps generalised conformal Killing tensors to differential operators which may act on any conformally weighted tensor bundle; the operators in the range have the property that they are symmetries of any natural conformally invariant differential operator between such bundles. These are used to construct all symmetries of the conformally invariant powers of the Laplacian (often called the GJMS operators) on manifolds of dimension at least 3. In particular this yields all symmetries of the powers of the Laplacian $Δ^k$, $k\in \mathbb{Z}>0$, on Euclidean space $\mathbb{E}^n$. The algebra formed by the symmetry operators is described explicitly.

Higher symmetries of the conformal powers of the Laplacian on conformally flat manifolds

TL;DR

This work provides a universal, tractor-calculus–based framework to characterize all symmetries of the conformally invariant powers of the Laplacian, the GJMS operators , on conformally flat manifolds. By constructing a canonical map from generalized conformal Killing tensors to symmetry pairs , the authors show that every symmetry of (and hence of in the flat setting) arises from this universal construction, modulo trivial symmetries. They then give an explicit, computable algebraic description of the symmetry algebra as a quotient of the tensor algebra on by a two-sided ideal generated by concrete relations, including the highest Cartan component. The combination of universal symmetry construction, leading-term classification, and precise algebraic structure provides a comprehensive, calculable picture of higher conformal symmetries with potential extensions to broader parabolic geometries and to curved settings via the ambient/tractor calculus.

Abstract

On locally conformally flat manifolds we describe a construction which maps generalised conformal Killing tensors to differential operators which may act on any conformally weighted tensor bundle; the operators in the range have the property that they are symmetries of any natural conformally invariant differential operator between such bundles. These are used to construct all symmetries of the conformally invariant powers of the Laplacian (often called the GJMS operators) on manifolds of dimension at least 3. In particular this yields all symmetries of the powers of the Laplacian , , on Euclidean space . The algebra formed by the symmetry operators is described explicitly.

Paper Structure

This paper contains 13 sections, 17 theorems, 100 equations.

Table of Contents

  1. Introduction
  2. The main theorems
  3. Symmetries and triviality
  4. Symmetries of $\Delta^k$ on $\mathbb{E}^{s,s'}$
  5. Conformal geometry and the GJMS operators
  6. Algebraic structure
  7. Conformal tractor calculus
  8. Tractor bundles
  9. Key differential operators
  10. The double-D and conformally invariant operators
  11. A Construction of symmetries
  12. Classification of leading terms of symmetries
  13. Algebra of symmetries

Key Result

Theorem 2.1

Let us fix $k\in \mathbb{Z}_+$. For the Laplacian power $\Delta^k$ on $\mathbb{E}^{s,s'}$ we have the following. For each $\varphi$, a solution of (ck), there is canonically associated a symmetry $(S_\varphi,S'_\varphi)$ for $\Delta^k$ with $S_\varphi$ and $S'_\varphi$ each having leading term $p\in \mathbb{Z}_{\geq 0}$, $r\in\{ 0,1,\cdots ,k-1\}$. Modulo trivial symmetries, any symmetry of $\Del

Theorems & Definitions (30)

  • Theorem 2.1
  • Proposition 2.2
  • Theorem 2.4
  • Theorem 2.5
  • Proposition 4.1
  • proof
  • Theorem 4.2
  • proof
  • Corollary 4.3
  • proof
  • ...and 20 more