Making the curvature constant does not fix the density – what Yamabe’s theorem can and cannot do to the component light leaves behind.
Paper 111 wrote that light fixes only nine of the ten components of a metric, and Paper 150 identified the remaining one as a density. Can geometry fix that one? Yamabe’s theorem says that every conformal class contains a metric of constant scalar curvature. This paper makes that theorem say what it can and cannot settle. NO NEW MATHEMATICAL THEOREM IS CLAIMED. The conformal covariance of the conformal Laplacian, the Yamabe problem (Yamabe 1960, Trudinger 1968, Aubin 1976, Schoen 1984), the Aubin-Talenti family of solutions, and the dependence of uniqueness on the sign of the invariant are all standard.
First, the conformal Laplacian really is conformally covariant. That the scalar curvature of the flat metric, rescaled by a power of u, equals the conformal Laplacian applied to u was checked symbolically until the difference vanished identically, and the difference was computed separately in dimensions three through seven, giving zero each time (computed here). The agreement of the exponents – that minus one minus four over n minus two equals minus n plus two over n minus two – is what the covariance consists of.
Second, the exponent of the nonlinearity is the critical one, and two dimensions lie outside the problem. The exponent is n plus two over n minus two, which sits exactly where the Sobolev embedding loses compactness: five in three dimensions, three in four, two in six. In two dimensions the denominator vanishes and the question becomes a different one, that of Gauss and Bonnet. The difficulty of the problem is therefore not a technical accident but a consequence of where the exponent is, and the special behaviour of the sphere has the same origin.
Third, two of the corpus’s numbers sit inside the Yamabe invariant of the sphere. The volume of the unit sphere is 4π in two dimensions and 2π² in three (computed here). The latter is exactly the number Paper 46 identified when it wrote that of 16π², only Ω₃ = 2π² is geometry. What the corpus found inside a measure in quantum field theory sits here inside a Yamabe invariant. The address must be written, though: a shared number is not a shared root. One is a measure on four-dimensional momentum space and the other the volume of a three-sphere, and what they share is the single quantity the volume of the unit three-sphere – not an accident, and the same geometric object, but put to different uses.
Fourth, and this is the core. Imposing constant curvature does not fix the density. On flat space take the family of functions carrying a parameter lambda – the Aubin-Talenti bubbles – and rescale the metric by them. Computing the scalar curvature of the result gives twenty-four in three dimensions, forty-eight in four, eighty in five and one hundred and twenty in six, each the constant belonging to a sphere of radius one half (computed here). What matters is that the answer depends on neither the radial coordinate nor lambda. Moving lambda moves the metric – volumes and distances change – and the curvature does not move. So the demand make the scalar curvature constant does not select one density: on the component light left behind, curvature places a constraint but does not supply a value. The size of the family can be counted too: the conformal group of the sphere modulo isometries leaves n plus one parameters in n dimensions, so five on the four-sphere.
Fifth, the fence is the sign of the Yamabe invariant. If the invariant is negative or zero the solution is unique up to scale, so the density really is fixed. Only when it is positive need uniqueness fail, the sphere being the example. The separator is therefore not whether the manifold is a sphere but the sign of a single number. This is also a half-correction to Paper 150, which wrote that a standard of volume must be carried in from outside: in a conformal class whose invariant is not positive, geometry carries it in; when it is positive, it must still come from outside.
Finally, one roll-call of names. Yamabe holds two theorems: the Gleason-Yamabe theorem treated in Paper 90 (Hilbert’s fifth problem, that a topological group is a Lie group) and the Yamabe problem here. They are unrelated, and this is the same shape Paper 125 found for Hopf, one person with two theorems; writing Yamabe’s theorem is not yet an address. And one more: the remaining case of the Yamabe problem was closed by Schoen in 1984 using the positive mass theorem of general relativity. A problem in geometry was settled by leaning on a theorem from physics.
IMPORTANT (scope): a census of established standard material (Layer 2). What is not claimed: any new mathematical theorem (there is none). What is added: asking what geometry can do to the component Paper 150 said must be carried in from outside, exhibiting the failure to fix it as the disappearance of lambda from the answer, writing that the fence is the sign of the Yamabe invariant, and calling the roll on Yamabe as one person with two theorems. What remains: light teaches only the conformal class and leaves one number behind. Asking curvature to pick that number up, it sometimes can and sometimes cannot – and what separates the two cases is not the name of the manifold but the sign of a single number. What remains when it cannot is a parameter called lambda, which vanishes from the answer.
On the making of this work: The ideas and content of this work stem from the author’s own considerations. Assistance from an AI (a large language model) was used for structuring, English translation, and checking the algebra. Any remaining errors or misinterpretations are solely the author’s. Feedback and corrections are sincerely appreciated.
Reference
Yamagishi, Y. (2026). Making the Curvature Constant Does Not Fix the Density [Paper 23]. Zenodo.
DOI: 10.5281/zenodo.22090231
ORCID: 0009-0006-8587-3450