[Paper Review] Conformally invariant operators, differential forms, cohomology and a generalisation of Q-curvature

Thomas Branson, A. Rod Gover|ArXiv.org|Sep 5, 2003

Geometric Analysis and Curvature Flows35 references4 citations

TL;DR

This paper constructs new conformally invariant differential complexes on even-dimensional conformal manifolds by introducing higher-order conformally invariant operators on differential forms, generalizing Q-curvature and extending the critical order conformal Laplacian. The key contribution is a geometric construction via the ambient metric that yields formally self-adjoint, factorizable operators $L_k$ and associated gauge companions $G_k$, forming elliptically coercive systems that define finite-dimensional conformally stable subspaces of form harmonics.

ABSTRACT

On conformal manifolds of even dimension $n\geq 4$ we construct a family of new conformally invariant differential complexes. Each bundle in each of these complexes appears either in the de Rham complex or in its dual. Each of the new complexes is elliptic if the signature is Riemannian. We also construct gauge companion operators which complete the exterior derivative to a conformally invariant and (in the case of Riemannian signature) elliptically coercive system. These (operator,gauge) pairs are used to define finite dimensional conformally stable form subspaces which are are candidates for spaces of conformal harmonics. These constructions are based on a family of operators on closed forms which generalise in a natural way the Q-curvature. We give a universal construction of these new operators and show that they yield new conformally invariant global pairings between differential form bundles. Finally we give a geometric construction of a family of conformally invariant differential operators between density-valued differential form bundles and develop their properties (including their ellipticity type in the case of definite conformal signature). The construction is based on the ambient metric of Fefferman and Graham, and its relationship to tractor bundles. For each form order, our derivation yields an operator of every even order in odd dimensions, and even order operators up to order $n$ in even dimension $n$. In the case of unweighted forms as domain, these operators are the natural form analogues of the critical order conformal Laplacian of Graham et al., and are key ingredients in the new differential complexes mentioned above.

Motivation & Objective

• To construct new conformally invariant differential complexes on even-dimensional conformal manifolds with coboundary operators of order greater than one.
• To generalize Branson’s Q-curvature to differential forms using a universal geometric construction based on the ambient metric.
• To define finite-dimensional, conformally stable subspaces of differential forms—candidates for conformal harmonics—via conformally invariant and elliptically coercive operator-gauge pairs.
• To extend the critical order conformal Laplacian to form-valued operators of all even orders in odd dimensions and up to order $n$ in even dimensions $n$.
• To establish a geometric link between tractor bundles, ambient metrics, and conformally invariant global pairings between form bundles.

Proposed method

• Utilizes the ambient metric of Fefferman and Graham to construct conformally invariant differential operators between density-valued differential forms.
• Derives a family of operators $L_k$ on true $k$-forms with principal part $(\delta d)^{n/2-k}$, factorizable as $\delta M d$, ensuring formal self-adjointness and conformal invariance.
• Introduces gauge companion operators $G_k$ of order $n-2k+1$ with principal part $\delta(d\delta)^{n/2-k}$, forming conformally invariant and elliptically coercive systems $(L_k, G_k)$ in Riemannian signature.
• Establishes a correspondence between tractor bundles and homogeneous ambient forms, enabling a geometric realization of the operators and their conformal transformation laws.
• Uses exterior calculus identities and the homogeneity of $\log\tilde{\sigma}$ to derive ambient expressions for the generalized Q-curvature operators $Q_k^\sigma$, linking them to the ambient Laplacian and contraction operators.
• Demonstrates that the ambient expression for $Q_k^\sigma$ reduces to the known Q-curvature in the scalar case and generalizes it to form-valued operators via $\tilde{\mathbb{Q}}^{\ell,\sigma}_k$.

Experimental results

Research questions

• RQ1 How can conformally invariant differential complexes be constructed on even-dimensional conformal manifolds that include higher-order coboundary operators?
• RQ2 What is the geometric construction of conformally invariant operators on differential forms that generalize the critical order conformal Laplacian and Q-curvature?
• RQ3 How can gauge companion operators be defined to render the system $(L_k, G_k)$ conformally invariant and elliptically coercive?
• RQ4 What is the relationship between the ambient metric, tractor bundles, and the construction of conformally invariant form operators?
• RQ5 How do the generalized Q-curvature operators $Q_k^\sigma$ extend the classical Q-curvature and relate to ambient differential operators?

Key findings

• The paper constructs a new family of conformally invariant differential complexes on even-dimensional conformal manifolds of dimension $n \geq 4$, each containing a coboundary operator of order greater than one.
• The operators $L_k$ on $k$-forms are formally self-adjoint, factorizable as $\delta M d$, and have principal part $(\delta d)^{n/2-k}$, generalizing the Maxwell operator.
• In Riemannian signature, the complexes are elliptic, and the operators $L_k$ are non-elliptic but positive semidefinite, necessitating gauge companions for coercivity.
• The gauge companion operators $G_k$ of order $n-2k+1$ are constructed to form conformally invariant and elliptically coercive systems $(L_k, G_k)$, enabling the definition of finite-dimensional conformally stable subspaces of form harmonics.
• The ambient metric construction yields a universal family of conformally invariant operators on density-valued forms, with $L_k$ being the natural form analogues of the critical order conformal Laplacian.
• The generalized Q-curvature operators $Q_k^\sigma$ are derived via ambient geometry and shown to reduce to the classical Q-curvature in the scalar case, with their ambient expression involving $\mbox{\boldmath{$\Delta$}}/^{\ell}$ and $\log\tilde{\sigma}$.