The critical Q-density fibre and the kernel of P

Definitions and mathematical statement

Problem $27$ asks when $Q$ determines a metric, within a fixed conformal class, up to constant rescaling. The displayed law takes values in top forms, so the source's convention is the density $q_g=Q_g\,{\,\mathrm{d}} V_g$, not the scalar $Q_g$. On a closed connected even-dimensional Riemannian manifold of dimension $n$, let $P_g:C^\infty(M)\to\Omega^n(M)$ be the critical formally self-adjoint elliptic operator, with $P_g1=0$ and principal part of type $\Delta^{n/2}$, satisfying \[ q_{e^{2u}g}=q_g+P_gu. \] The aid's exact fiber description is \[ \{\widehat g\in[g]:q_{\widehat g}=q_g\} =\{e^{2u}g:u\in\ker P_g\}. \] Consequently density rigidity up to one constant scale holds exactly when $\ker P_g={\mathbb{R}}\cdot1$. This recovers an equivalence suggested but cut off in the source, rather than classifying all conformal structures with that kernel property.

Let $V_g=\operatorname{Vol}_g(M)$, $K=\ker P_g$, and define \[ A_g(v)=\frac1n\log\left(V_g^{-1}\int_Me^{nv}\,{\,\mathrm{d}} V_g\right). \] Then $A_g(v+c)=A_g(v)+c$, and the aid's global parametrization of the same-volume fiber is \[ K/{\mathbb{R}}\ \longrightarrow\ \{\widehat g\in[g]:q_{\widehat g}=q_g,\ \operatorname{Vol}_{\widehat g}(M)=V_g\}, \qquad[v]\longmapsto e^{2(v-A_g(v))}g. \] It is a bijection with no arbitrary mean-zero representative choice. In this closed elliptic setting $K$ is finite-dimensional. Connectedness is material: on a disconnected manifold locally constant functions already allow independent component scales, distinct from one global homothety.

The aid separately treats scalar curvature data. Write $\mathscr P_g u=(P_gu)/{\,\mathrm{d}} V_g$, so \[ Q_{e^{2u}g}e^{nu}=Q_g+\mathscr P_gu. \] Assume $\mathscr P_g$ is nonnegative in the $L^2(g)$ quadratic form, has only constants in its kernel, and $Q_g\leq0$. If the scalar values agree pointwise, $Q_{e^{2u}g}=Q_g$, then $u$ is constant. When $Q_g\not\equiv0$, this constant is zero; when $Q_g\equiv0$, constant rescalings remain. The comparison uses $u(e^{nu}-1)\geq0$ and the sign of $Q_g$. This is a nonlinear scalar statement with extra sign/spectral hypotheses, distinct from the unconditional density-fiber identity. No general scalar-Q uniqueness is inferred.

Short English statement

For the source's density-valued $Q$, the entire conformal fiber is the critical operator's kernel, and rigidity up to scale is exactly a constant-only kernel. The aid gives an explicit same-volume parametrization by kernel modulo constants. Scalar $Q$ has a different nonlinear transformation and requires additional hypotheses for the reported rigidity.

Sources

• Local UnsolvedMath registry, permanent ID $20002049$, source code catalogue entry; canonical collection aim-geometry-notes.json, notes[386], source index $386$.
• AIM workshop "Conformal structure in geometry, analysis, and physics," Problem $27$: https://aimath.org/WWN/confstruct/confstruct.pdf. The original ends "this question is equivalent to" and does not include the next statement. Its operator is explicitly functions to $n$-forms. No PDF was opened in this session.
• Supplied research attempt $1$, partial_result, run status "open," rigor label "proved," novelty confidence "low." Registry status partially_solved, classification PARTIAL-PROGRESS, updated 14 August 2026. The candidate contribution is the normalized fixed-volume fiber formula.
• No full authoritative bibliography, original remarks, literature entry, proposer or proposed year is supplied.