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The Ambient Obstruction Tensor and Q-Curvature

C. Robin Graham, Kengo Hirachi

TL;DR

This work identifies Branson's $Q$-curvature as a variational functional whose first variation is governed by the ambient obstruction tensor ${\mathcal{O}}_{ij}$, thereby linking $Q$-curvature to the obstruction to smooth ambient/Poincaré metric expansions in even dimensions. It provides a precise formula for the variation and uses conformal volume expansion to connect the log-term coefficient to ${\mathcal{O}}_{ij}$. The paper also delivers an invariant-theoretic classification showing that, modulo quadratic and higher curvature terms, the Weyl/Cotton family and the obstruction tensor (in the even case) are the only irreducible conformally invariant tensors, with the Boe–Collingwood classification underpinning the result. Together, these results illuminate the fundamental role of the ambient obstruction tensor in conformal geometry and its relation to $Q$-curvature.

Abstract

It is shown that the variational derivative of the integral of Branson's Q-curvature is the ambient obstruction tensor of Fefferman-Graham. A classification of irreducible conformally invariant tensors modulo quadratic and higher degree terms in curvature is established.

The Ambient Obstruction Tensor and Q-Curvature

TL;DR

This work identifies Branson's -curvature as a variational functional whose first variation is governed by the ambient obstruction tensor , thereby linking -curvature to the obstruction to smooth ambient/Poincaré metric expansions in even dimensions. It provides a precise formula for the variation and uses conformal volume expansion to connect the log-term coefficient to . The paper also delivers an invariant-theoretic classification showing that, modulo quadratic and higher curvature terms, the Weyl/Cotton family and the obstruction tensor (in the even case) are the only irreducible conformally invariant tensors, with the Boe–Collingwood classification underpinning the result. Together, these results illuminate the fundamental role of the ambient obstruction tensor in conformal geometry and its relation to -curvature.

Abstract

It is shown that the variational derivative of the integral of Branson's Q-curvature is the ambient obstruction tensor of Fefferman-Graham. A classification of irreducible conformally invariant tensors modulo quadratic and higher degree terms in curvature is established.

Paper Structure

This paper contains 4 sections, 5 theorems, 32 equations.

Table of Contents

  1. Introduction
  2. The Obstruction Tensor
  3. Proof of Theorem \ref{['variation']}
  4. Proof of Theorem \ref{['invtensors']}

Key Result

Theorem 1.1

If $g^t$ is a 1-parameter family of metrics on a compact manifold $M$ of even dimension $n\geq 4$, then where ${\dot {}} = \partial_t|_{t=0}$ and ${\mathcal{O}}_{ij}$ and $dv$ on the right hand side are with respect to $g^0$.

Theorems & Definitions (8)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 2.1
  • proof
  • Theorem 2.2
  • proof
  • Lemma 3.1
  • proof