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A general construction of conformally covariant tridifferential operators

Jeffrey S. Case, Opal Cieslak

TL;DR

This work addresses the problem of constructing nonlinear conformally covariant operators by extending the Graham–Jenne–Mason–Sparling framework to tridifferential operators using Fefferman–Graham ambient space methods. The authors define ambient operators of leading order $2k$ acting on three inputs, and, by solving recurrence relations derived from commutator identities with the ambient dilation generator, prove tangential descent to conformally invariant tridifferential operators on the base manifold; a chain-complex analysis yields dimension counts for generic weights. They show the existence of a $(k+1)$-parameter family of ambient tridifferential operators that descend to conformally invariant operators, with weight-dependent multiplicities persisting even on the sphere, and extend the construction to differential and bidifferential variants; under the weight condition $w_1=w_2=w_3=-\frac{n-2k}{4}$, symmetrized ambient operators are formally self-adjoint. The results provide a structured source of nonunique higher-order conformal invariants with potential applications to variational curvature prescription and broader conformal geometry problems.

Abstract

We construct a large family of conformally covariant tridifferential operators as tangential operators in the Fefferman--Graham ambient space. Our construction is analogous to the linear and bilinear constructions of Graham--Jenne--Mason--Sparling and Case--Lin--Yuan, respectively. We also show that the symmetrization of our ambient operators are formally self-adjoint when acting on densities of the correct weight.

A general construction of conformally covariant tridifferential operators

TL;DR

This work addresses the problem of constructing nonlinear conformally covariant operators by extending the Graham–Jenne–Mason–Sparling framework to tridifferential operators using Fefferman–Graham ambient space methods. The authors define ambient operators of leading order acting on three inputs, and, by solving recurrence relations derived from commutator identities with the ambient dilation generator, prove tangential descent to conformally invariant tridifferential operators on the base manifold; a chain-complex analysis yields dimension counts for generic weights. They show the existence of a -parameter family of ambient tridifferential operators that descend to conformally invariant operators, with weight-dependent multiplicities persisting even on the sphere, and extend the construction to differential and bidifferential variants; under the weight condition , symmetrized ambient operators are formally self-adjoint. The results provide a structured source of nonunique higher-order conformal invariants with potential applications to variational curvature prescription and broader conformal geometry problems.

Abstract

We construct a large family of conformally covariant tridifferential operators as tangential operators in the Fefferman--Graham ambient space. Our construction is analogous to the linear and bilinear constructions of Graham--Jenne--Mason--Sparling and Case--Lin--Yuan, respectively. We also show that the symmetrization of our ambient operators are formally self-adjoint when acting on densities of the correct weight.

Paper Structure

This paper contains 4 sections, 13 theorems, 109 equations.

Table of Contents

  1. Introduction
  2. Weyl operators
  3. Conformally covariant tridifferential operators
  4. Formal self-adjointness in the ambient space

Key Result

Theorem 1.1

Fix integers $k \geq 0$ and $n \geq 3$; if $n$ is even, then assume additionally that $n \geq 2k$. Let $w_1,w_2,w_3 \in \mathbb{R}$. There is a $(k+1)$-dimensional family of functions $A \in \mathscr{F}_k^5$ such that is tangential on $\widetilde{\mathcal{E}}[w_1] \otimes \widetilde{\mathcal{E}}[w_2] \otimes \widetilde{\mathcal{E}}[w_3]$, where $A_\alpha := A(\alpha)$ and

Theorems & Definitions (29)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 2.1
  • proof
  • proof : Proof of \ref{['tridifferential']}
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • Proposition 3.3
  • ...and 19 more