Generalized Q-operators on closed forms

Definitions and mathematical statement

This record is contextual prose between Problems 34 and 35, not a standalone open problem. The supplied verification restores its opening as "Solutions to Problem 34 have a role to play" in characterizing $Q$-curvature and GJMS operators.

Let $M^n$ be a compact oriented even-dimensional manifold, not necessarily connected; assume no boundary for the stated integration identities. Write $\Omega^k(M)$ for smooth forms and $\mathcal C^k=\ker({\,\mathrm{d}}:\Omega^k\to\Omega^{k+1})$. The source describes generalized $Q$-operators $T_g^k:\Omega^k\to\Omega^{n-k}$, for $0\leq k\leq n/2-1$, with their intended weighted/Hodge identifications. A precise version of the supplied abstract descent lemma assumes \[ T_{e^{2\omega}g}^k(c)=T_g^k(c)+L_k(\omega c)\qquad(c\in\mathcal C^k), \quad L_k={\,\mathrm{d}} A_k{\,\mathrm{d}}, \] where $A_k:\Omega^{k+1}\to\Omega^{n-k-1}$, and $L_k$ is conformally invariant and formally self-adjoint for the integrated form pairing. Multiplication by $\omega$ occurs before applying $L_k$; it is not the scalar expression $(L_k\omega)c$.

The aid reports the resulting invariant space and map \[ \mathcal H_g^k=\{c\in\mathcal C^k:{\,\mathrm{d}} T_g^k(c)=0\},\qquad c\longmapsto[T_g^k(c)]\in H_{\mathrm{dR}}^{n-k}(M). \] The space $\mathcal H_g^k$ and this map are independent of representative, since ${\,\mathrm{d}} L_k=0$ and the change $L_k(\omega c)$ is exact. For $u\in\ker L_k$ and $c\in\mathcal C^k$, the pairing \[ \int_Mu\wedge T_g^k(c) \] is also conformally invariant by formal self-adjointness. The source additionally reports formal self-adjointness of the generalized $Q$-operators themselves and nontriviality of the induced map on some conformal manifolds; those properties are not inferred from factorization alone.

For $k=0$, let $q_g=Q_g\,{\,\mathrm{d}} V_g$ and $T_g^0(c)=c q_g$, with $c$ locally constant. Its affine law is $q_{e^{2\omega}g}=q_g+P_n\omega$, where the critical operator has the indicated exact-form factorization and kills locally constant functions. The lemma implies that $\int_Mu c q_g$ is invariant for $u\in\ker P_n$. Taking $c$ to be each connected-component indicator shows that every component's total $Q$-curvature is separately invariant, not merely their sum.

The source attributes the geometric operator construction to Branson--Gover and writes $L_k$ as a constant multiple of ${\,\mathrm{d}} M_{k+1}^g{\,\mathrm{d}}$, with conventions to be supplied from their paper. At degree $n/2-1$ it gives \[ M^g={\,\mathrm{d}}\delta+2J-4P^{\sharp}: \Omega^{n/2-1}\longrightarrow\Omega^{n/2-1}[-2], \] where $\delta$ is the metric codifferential, $J$ the Schouten trace, and $P^{\sharp}$ its induced action on forms; conformal Hodge duality gives the complementary-degree version. It reports that these operators cannot generally be made conformally invariant merely by subtracting a divergence, invoking the leading Maxwell operator. This is retained as construction-specific background, not a consequence of the minimal lemma or a new unresolved question.

Short English statement

The record explains generalized $Q$-operators rather than posing an independent question. The supplied aid isolates an affine-law/factorization lemma yielding conformal-harmonic classes and nullspace pairings. In degree zero it gives separate conformal invariance of total $Q$-curvature on every connected component.

Sources

• Context between Problems 34 and 35, 2003 AIM workshop Conformal Structure in Geometry, Analysis, and Physics: https://aimath.org/WWN/confstruct/confstruct.pdf.
• T. Branson and R. Gover, supplied arXiv identifier math.DG/0309085, for the ambient/tractor generalized $Q$-operator construction. No full title or publication entry is included in the record.
• Local registry data/unsolvedmath/problems.json, permanent ID 20002059, code catalogue entry; canonical aim-geometry-notes.json, notes[396]; updated 14 August 2026. The supplied verification identifies lost opening words and the contextual nature of the extract.
• Attempt 1: context only, problem status at run not a problem, rigor label proved, novelty confidence low; current registry nevertheless says open/open triage. The machine-generated aid is not peer reviewed and requires independent verification before citation.