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Explicit formulas for GJMS-operators and $Q$-curvatures

Andreas Juhl

Abstract

We describe GJMS-operators as linear combinations of compositions of natural second-order differential operators. These are defined in terms of Poincaré-Einstein metrics and renormalized volume coefficients. As special cases, we find explicit formulas for conformally covariant third and fourth powers of the Laplacian. Moreover, we prove related formulas for all Branson's $Q$-curvatures. The results settle and refine conjectural statements in earlier works. The proofs rest on the theory of residue families.

Explicit formulas for GJMS-operators and $Q$-curvatures

Abstract

We describe GJMS-operators as linear combinations of compositions of natural second-order differential operators. These are defined in terms of Poincaré-Einstein metrics and renormalized volume coefficients. As special cases, we find explicit formulas for conformally covariant third and fourth powers of the Laplacian. Moreover, we prove related formulas for all Branson's -curvatures. The results settle and refine conjectural statements in earlier works. The proofs rest on the theory of residue families.

Paper Structure

This paper contains 37 sections, 42 theorems, 652 equations.

Table of Contents

  1. Introduction and formulation of the main results
  2. The inversion formula
  3. Residue families
  4. Residue families in terms of GJMS-operators
  5. The pure $P$-terms
  6. The pure $\bar{P}$-terms
  7. The mixed terms
  8. Factorization relations for $D_{2N}(\lambda)$
  9. The restriction property and the commutator relations
  10. The principal part of ${\mathcal{M}}_{2N}$
  11. Description of the principal part
  12. Proof of Theorem \ref{['base-c']}
  13. The zeroth-order term of ${\mathcal{M}}_{2N}$
  14. General metrics
  15. Conformally flat metrics
  16. ...and 22 more sections

Key Result

Theorem 1.1

All GJMS-operators on Riemannian manifolds $(M,g)$ of dimension $n \ge 3$ can be written in the form Moreover, all operators ${\mathcal{M}}_{2N}(g)$ are (at most) second-order and are given by the formula Here $\delta$ is defined with respect to $g$, and we regard $g_r$ as an endomorphism on one-forms using $g$. Finally, $w$ is regarded as a function on $X$.

Theorems & Definitions (93)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 2.1
  • proof
  • Lemma 2.1
  • proof
  • Example 2.1
  • Example 2.2
  • Example 2.3
  • Example 2.4
  • ...and 83 more