Topology of Yamabe Asymptotic Sets

Definitions and mathematical statement

The source specifies the subset \[ S=\{u\in{\mathbb{R}}^p\setminus\{0\}:(Au,u)\leq0\} \] and asks for its topology in the setting of sign-changing Yamabe asymptotes. The origin is excluded, the zero level is included, and no positivity restriction on the coordinates of $u$ is stated. The set carries the subspace topology from ${\mathbb{R}}^p$.

If $A$ denotes a fixed real linear operator on ${\mathbb{R}}^p$ and the parentheses denote the Euclidean inner product, the literal object is \[ S_A=\{u\ne0:u^{\mathsf T}Au\leq0\}, \qquad u^{\mathsf T}Au=u^{\mathsf T}\frac{A+A^{\mathsf T}}2u. \] In this interpretation only the symmetric part contributes, and the set is invariant under positive scaling. Its radial description is \[ S_A\cong(0,\infty)\times\{v\in S^{p-1}:v^{\mathsf T}Av\leq0\}, \qquad u\longmapsto(\|u\|,u/\|u\|). \] These observations specify the literal quadratic-set reading; they do not recover which operator, parameter dependence, or extra asymptotic information the source intends. In particular, a question about one fixed quadratic cone must not be identified without explanation with a possibly richer family of Yamabe asymptotic sets.

The supplied record does not define $A$, $p$, the inner product, or the relation to the Yamabe problem. It also does not say whether "determine the topology" requests a homeomorphism type, homotopy type, homology, connectedness, or behavior in a parameter family. The precise source data and target invariant are therefore unresolved. The supplied assessment checked 2026-08-21 reports that the report calls the topology problem open and that no resolution in that formulation was located; this dated label is not independently certified for the underspecified quadratic-set interpretation.

Short English statement

Determine the topology of the punctured nonpositive set defined by $(Au,u)$, in the intended Yamabe asymptotic setting. The operator and its geometric role are missing; the literal fixed-matrix reading is kept separate from an unspecified richer asymptotic problem.

Sources

• Local UnsolvedMath registry, permanent ID 30000264, catalogue entry, Topology of Yamabe Asymptotic Sets.
• Topological and Variational Methods for Differential Equations (2006), Oberwolfach Reports, pp. 1601--1678, DOI https://doi.org/10.4171/owr/2005/29.
• Supplied report-text link identifying the topology problem: https://oa.tib.eu/renate/bitstreams/91dd0822-361f-44ee-a77f-fa862bb21ab6/download.
• Supplied literature triage checked 2026-08-21; no definition of the operator or separate theorem citation is reproduced.