Nonexistence of Edge-Cone Yamabe Metrics at Small Cone Angles

Definitions and mathematical statement

Let $\Sigma=S^{n-2}\subset S^n$ be the singular locus and let $M=S^n\setminus\Sigma$. The source denotes by $h_\beta$ the standard edge-cone metric on $(S^n,\Sigma)$ with transverse cone angle $2\pi\beta$. A transverse cone of this angle has metric \[ {\,\mathrm{d}} r^2+\beta^2r^2{\,\mathrm{d}}\theta^2, \qquad \theta\in{\mathbb{R}}/(2\pi{\mathbb{Z}}). \] This specifies the cone-angle convention; the supplied record does not give the full metric $h_\beta$, its precise asymptotic regularity, or the admissible conformal factors at $\Sigma$.

Under the variational interpretation suggested by the supplied literature assessment, an edge-cone Yamabe metric means an admissible metric attaining the Yamabe infimum in the prescribed conformal class. More explicitly, for a fixed dimension $n\geq3$, let $\mathcal G_\beta$ denote the intended class of admissible edge-cone metrics \[ g=u^{4/(n-2)}h_\beta,\qquad u>0\text{ on }M, \] with the prescribed cone angle and with finite positive volume and an integrable scalar curvature. Write \[ \mathcal Y(g)= \frac{\displaystyle\int_M\operatorname{Scal}_g\,{\,\mathrm{d}} V_g} {\displaystyle\operatorname{Vol}_g(M)^{(n-2)/n}}, \qquad Y_\beta=\inf_{g\in\mathcal G_\beta}\mathcal Y(g). \] The proposed nonexistence statement is \[ \forall\beta\in(1,2),\qquad \nexists\,g\in\mathcal G_\beta \quad\text{such that}\quad \mathcal Y(g)=Y_\beta. \] Thus the conjectured nonexistence concerns cone angles strictly between $2\pi$ and $4\pi$; neither endpoint belongs to the requested range.

Unresolved qualifications. This is a conditional variational formalization: the source does not specify the dimension range, define the full standard metric, or determine $\mathcal G_\beta$, including regularity and boundary behavior at $\Sigma$. The convention $n\geq3$ is required for the displayed usual Yamabe formulation and is not an explicitly supplied dimension hypothesis. The record uses both "edge-cone Yamabe metric" and "Yamabe minimizer"; the displayed statement follows the minimizer interpretation. Nonexistence of all constant-scalar-curvature edge-cone metrics would be a different, stronger assertion and is not established by this formulation.

Short English statement

For every $1<\beta<2$, the conjecture asks whether the standard conformal class on $(S^n,S^{n-2})$, with cone angle $2\pi\beta$, has no edge-cone metric attaining its Yamabe infimum. The dimension range, precise standard metric, and admissible behavior at the singular locus require clarification. The minimizer interpretation follows the supplied assessment.

Sources

• Local UnsolvedMath registry, data/unsolvedmath/problems.json, permanent ID $30003545$, exact source problem code catalogue entry; supplied title: "Nonexistence of Edge-Cone Yamabe Metrics at Small Cone Angles." No individual proposer is identified in the supplied record.
• Supplied report citation: Analysis, Geometry and Topology of Positive Scalar Curvature Metrics (2018), Oberwolfach Reports, pp. 2223--2298, DOI 10.4171/owr/2017/36. Original OWR report: https://doi.org/10.4171/owr/2017/36. The record separately gives proposed year 2017.
• Supplied literature reference: Akutagawa--Mondello, "singular Yamabe minimizers," https://arxiv.org/abs/1909.09367. This descriptive reference label is reproduced from the record; no fuller bibliographic citation is supplied.
• Literature assessment dated 2026-08-21: the supplied curation labels the problem partially solved and asserts nonexistence of an edge-cone Yamabe minimizer for cone angle at least $4\pi$, equivalently $\beta\geq2$, while reporting the range $2\pi<2\pi\beta<4\pi$ as unresolved. This is supplied dated curation, not an independently verified literature assessment.
• The supplied statement-verification field reports that the official report was checked and that it proves the $\beta\geq2$ result and states the remaining range as a conjecture. Its label corrected_verified records supplied provenance; no source visit or independent verification is claimed here.