Dimension four bounds total Q above and determines its exact range

Definitions and mathematical statement

Problem $9$ asks, for a compact manifold of even dimension greater than two, for smooth metrics $g_j$ with total $Q$-curvature $\int_M Q_{g_j}\,{\,\mathrm{d}} V_{g_j}\to+\infty$. In the usual closed-manifold interpretation, dimension four is included. The aid reports that this inclusion is genuinely in the source and refutes the universal positive-infinity assertion in that dimension.

For a closed smooth four-manifold, use the reported AIM normalization \[ Q_4=\frac{R^2}{6}-\frac{|\operatorname{Ric}|^2}{2} -\frac{\Delta R}{6}. \] The Laplacian sign affects its pointwise divergence term but not its integral. With $|W|^2$ the full tensor squared norm in the corresponding Chern--Gauss--Bonnet convention, \[ \mathcal Q(g):=\int_MQ_4(g)\,{\,\mathrm{d}} V_g =8\pi^2\chi(M)-\frac14\int_M|W_g|^2\,{\,\mathrm{d}} V_g \leq8\pi^2\chi(M). \] Thus no sequence on such a manifold has $\mathcal Q(g_j)\to+\infty$. Changing the Weyl norm to a curvature-operator convention would change the displayed coefficient, so its normalization is material.

The aid also supplies a stronger range statement. Define \[ \mathcal W_{\inf}(M)=\inf_g\int_M|W_g|^2\,{\,\mathrm{d}} V_g,\qquad S_Q(M)=8\pi^2\chi(M)-\frac14\mathcal W_{\inf}(M), \] where the infimum ranges over all smooth Riemannian metrics. It reports that for every background metric $g_0$ and prescribed coordinate ball $U\subset M$, there are metrics $g_j$ equal to $g_0$ outside $U$ with \[ \mathcal Q(g_j)\longrightarrow-\infty. \] The mechanism is compactly supported scale-invariant Weyl-energy bubbles. Together with path-connectedness of the smooth metric space and continuity of $\mathcal Q$, this gives the exact value set \[ \{\mathcal Q(g):g\text{ smooth Riemannian}\} =(-\infty,S_Q(M))\quad\text{or}\quad(-\infty,S_Q(M)]. \] The upper endpoint is present exactly when the infimum Weyl energy is attained. Neither its numerical value nor attainment is asserted for every manifold. This four-dimensional correction refutes the source's unrestricted claim but gives no conclusion here for all higher even dimensions or for boundary problems with additional terms.

Short English statement

The printed positive-infinity claim fails on closed four-manifolds: Gauss--Bonnet bounds total $Q_4$ above by topology minus Weyl energy. The aid reports local metric modifications making it tend to negative infinity and describes its exact interval of values, with the upper endpoint depending on Weyl-energy attainment.

Sources

• Local UnsolvedMath registry, permanent ID $20002028$, source code catalogue entry; canonical collection aim-geometry-notes.json, notes[365].
• AIM workshop "Conformal structure in geometry, analysis, and physics," Problem $9$: https://aimath.org/WWN/confstruct/confstruct.pdf. The record's trailing name "Robin Graham" is preserved without inferring an attribution not specified by the registry. The aid reports that the official source genuinely includes dimension four. No PDF was opened in this session.
• Chern--Gauss--Bonnet is the named curvature identity used by the aid; no full authoritative reference for the local Weyl-bubble or range statements is supplied. Original remarks and literature fields are empty, with proposer/year registry fields null.
• Supplied research attempt $1$, result "counterexample," run status "refuted," rigor label "proved," novelty confidence "low." Registry status "solved," classification SOLVED-IN-LITERATURE, updated 14 August 2026.