A conformal-factor cocycle proof of fact 2

Definitions and mathematical statement

Problem $18$ asks to deduce "fact $2$" from "fact $1$," or conversely, or to give a Lie-algebraic proof of fact $2$. The facts themselves are absent from the canonical excerpt. The aid interprets fact $1$ as critical conformal covariance of the flat operator $P=\Delta^{n/2}$ in even dimension, and fact $2$ as \[ \Delta^{n/2}\log\Omega=0 \] for a positive flat-to-flat conformal scale. This reconstruction and its conventions must be checked against the preceding source text.

For a local Euclidean conformal map $\phi$, write $\phi^*g_0=\Omega_\phi^2g_0$ and $b_\phi=\log\Omega_\phi$. The intended critical covariance is \[ P(f\circ\phi)=\Omega_\phi^n(Pf)\circ\phi, \] so conformal pullback preserves $\ker P$. Logarithmic scales satisfy \[ b_{\phi\circ\psi}=b_\psi+b_\phi\circ\psi. \] For a conformal Killing field $X$, the infinitesimal scale is $\sigma_X=(\operatorname{div}X)/n$; in Euclidean dimension at least three it is affine and hence belongs to $\ker\Delta^{n/2}$.

The aid's abstract integration lemma says: if a connected Lie group acts continuously and smoothly on a function space preserving the closed kernel of a linear differential operator $P$, and a smooth function-valued group $1$-cocycle has every infinitesimal value in that kernel, then all its finite cocycle values lie in the kernel. Applied to logarithmic scales on a compatible conformal transformation domain, it yields the claimed implication from covariance to fact $2$. Passing from a connected transformation-group argument to all local flat-to-flat changes requires the usual local classification/generator and component checks; these details are not in the summary. The aid also reports an independent Schouten recurrence. In dimension two the separate holomorphic/antiholomorphic description gives harmonic logarithmic scale wherever the derivative is nonzero. No inference is made across singularities of a conformal map.

The appended Eastwood--Slovák discussion records a formal identity $(38)$, whose original scale/weight conventions are missing. In its notation $\widehat P_{ab}$ is the hatted Schouten tensor, $\widehat P$ its trace, and $\Upsilon$ a logarithmic scale derivative; it prints \[ \begin{split} \Delta^2\log\Omega={}&-\widehat\Delta\widehat P -(n-2)\widehat P_{ab}\widehat P^{ab}+2\widehat P^2\\ &+2(n-4)\Upsilon^a\widehat\nabla_a\widehat P +2(n-4)(\Upsilon^a\Upsilon_a)\widehat P\\ &-(n-2)(n-4)\Upsilon^a\Upsilon^b\widehat P_{ab} +\tfrac14(n-2)(n-4)(\Upsilon^a\Upsilon_a)^2. \end{split} \] This is retained as the source formula, not used as a tensorially normalized identity without the original metric/weight identifications. The source guarantees its intended use only in the conformally flat case, observes agreement with the general dimension-four expression \[ Q=2P^2-2P_{ab}P^{ab}-\Delta P, \] and notes that repeated differentiation produces complete contractions of Schouten tensors, hatted derivatives and $\Upsilon$, becoming rapidly complicated. It explicitly says this naive method fails for a general metric in dimension six; flat/conformally-flat calculations therefore do not yield the general higher-dimensional $Q$-curvature. The source's recompilation footnote and page number $26$ provide no missing definitions or theorem hypotheses.

Short English statement

The aid reconstructs the missing facts as critical flat covariance and polyharmonicity of a flat-to-flat logarithmic conformal scale. A cocycle lemma supplies the Lie-algebraic bridge, with local and component qualifications. The appended Schouten formulas remain convention-dependent and explicitly do not give general six-dimensional $Q$-curvature by naive differentiation.

Sources

• Local UnsolvedMath registry, permanent ID $20002038$, source code catalogue entry; canonical collection aim-geometry-notes.json, notes[375], zero-based index $375$.
• AIM workshop "Conformal structure in geometry, analysis, and physics," Problem $18$, followed by formulas $(38),(39)$: https://aimath.org/WWN/confstruct/confstruct.pdf. The cited facts lie outside the supplied excerpt. No PDF was opened in this drafting session.
• Eastwood and Slovák are credited for the displayed formula. The source calls the section a recompilation of conundra and asks readers to consult an original document, without a complete reference here. Original remarks and literature are empty; no proposer or year is supplied.
• Supplied research attempt $1$, result full_solution, run status "solved," rigor label "proved," novelty confidence "low." Registry status "solved," classification SOLVED-IN-LITERATURE, updated 14 August 2026. The aid's asserted solution is for the reconstructed flat-factor facts and cocycle argument.