Einstein factorization and the trace-free-Ricci obstruction to Paneitz positivity

Definitions and mathematical statement

Let $(M^4,g)$ be a smooth closed connected Riemannian four-manifold. Use the nonnegative Laplacian $\Delta=d^*d=-\operatorname{div}\operatorname{grad}$, put $E=\operatorname{Ric}-(R/4)g$, and write $\delta=d^*$ on one-forms. With these conventions the Paneitz operator is \[ P_4u=\Delta^2u+\delta\left((\tfrac23Rg-2\operatorname{Ric})\,{\,\mathrm{d}} u\right). \] Here a symmetric tensor acts on ${\,\mathrm{d}} u$ using the metric. "Positive with kernel consisting of constants" means nonnegative quadratic form and strict positivity on nonconstant functions modulo constants, rather than positivity on every nonzero function.

Problem 7. The source reports Gursky's theorem: if the scalar curvature is positive and $\int_MQ_4\,{\,\mathrm{d}} V_g>0$, then $P_4$ is nonnegative and $\ker P_4={\mathbb{R}}$. Can this known conclusion also be derived directly from the construction of the general GJMS operators, without the nonlinear-PDE estimates used in the cited proof? The request concerns a proof method, not whether the positivity conclusion itself is conjectural. Closedness and connectedness are made explicit; otherwise boundary conditions or componentwise constants are needed.

Supplied direct special cases. If $g$ is Einstein with constant $R>0$, the aid gives the GJMS factorization \[ P_4=\Delta(\Delta+R/6),\qquad \lambda_1(P_4)=\lambda_1(\Delta)\left(\lambda_1(\Delta)+R/6\right), \qquad\ker P_4={\mathbb{R}}, \] where $\lambda_1$ denotes the first positive eigenvalue in this representative. Critical conformal covariance transfers nonnegativity and the kernel to other metrics in a conformal class containing a positive Einstein metric; the numerical eigenvalue formula is not asserted to be conformally invariant.

For any constant-scalar-curvature representative, the exact defect identity supplied by the aid is \[ P_4=\Delta(\Delta+R/6)-2\delta(E\,{\,\mathrm{d}}), \] \[ \int_MuP_4u\,{\,\mathrm{d}} V_g =\int_M\left((\Delta u)^2+\tfrac R6|{\,\mathrm{d}} u|^2 -2E(\nabla u,\nabla u)\right)\,{\,\mathrm{d}} V_g. \] Let $\Lambda=\max_{x\in M}\lambda_{\max}(E_x)$, with $E_x$ viewed as a self-adjoint endomorphism. The aid's sufficient condition is \[ 2\Lambda<\lambda_1(\Delta)+R/6. \] Under this strict condition it asserts $P_4\ge0$ and $\ker P_4={\mathbb{R}}$. The pinching statement itself does not require an additional positive-$R$ premise beyond the displayed inequality.

With the compatible standard $Q_4$ normalization, a constant-scalar-curvature metric satisfies \[ \int_MQ_4\,{\,\mathrm{d}} V_g =\frac{R^2}{24}\operatorname{Vol}_g(M)-\frac12\int_M|E|^2\,{\,\mathrm{d}} V_g. \] Thus positive total $Q$ gives an integral bound on $|E|^2$, whereas the sufficient condition uses the largest pointwise directional eigenvalue. The supplied algebraic factorization does not establish that pointwise condition from the original hypotheses, and this gap is not a counterexample to Gursky's theorem. Obtaining the full construction-based proof remains separate from these Einstein and pinched special cases. The trailing name "Claude LeBrun" is preserved as a source line whose attachment to the problem or next contribution is not specified here.

Short English statement

The problem asks for a construction-based proof of Gursky's known Paneitz-positivity theorem. The supplied aid gives an Einstein factorization and a constant-scalar-curvature pinching criterion. Positive total $Q$ supplies an integral curvature bound but does not by itself provide the pointwise estimate used in that criterion, so the requested full proof method remains unestablished.

Sources

• Local UnsolvedMath registry, permanent ID 20002026, source code catalogue entry; canonical aim-geometry-notes.json, notes[363].
• AIM Conformal structure in geometry, analysis, and physics, Problem 7: https://aimath.org/WWN/confstruct/confstruct.pdf. The extract ends with the line "Claude LeBrun".
• Gursky, "The principal eigenvalue of a conformally invariant differential operator, with an application to semilinear elliptic PDE", Comm. Math. Phys. 207(1) (1999), pp. 131--143, as supplied.
• Registry research attempt 1, dated 2026-08-14: partial result, status at run open, rigor "proved", novelty confidence low. The explicitly machine-generated, non-peer-reviewed aid requires independent verification before citation.