Infinite Values of the $kappa$-Invariant

Definitions and mathematical statement

A closed manifold is a compact manifold without boundary. The supplied record asks about an invariant of a closed manifold $M$, using Riemannian metrics on $M$ and numbers $\mu_i(g)$ described as Dirac eigenvalues. It does not specify the Dirac operator, the geometric structures needed to define it, or the convention for indexing its eigenvalues.

Conditional on fixing the intended admissible class of manifolds and the intended spectral data $(\mu_i(g))_{i\geq 0}$, the displayed definition is \[ K(M)=\left\{k\in{\mathbb{Z}}_{\geq 0}:\ \forall\varepsilon>0\ \exists\text{ a Riemannian metric }g_\varepsilon\text{ on }M,\quad \mu_k(g_\varepsilon)=1\ \text{ and }\ \forall i\in{\mathbb{Z}},\ 0\leq i<k\Longrightarrow |\mu_i(g_\varepsilon)|<\varepsilon\right\}, \] \[ \kappa(M)=\begin{cases}\min K(M),&K(M)\neq\varnothing,\\ \infty,&K(M)=\varnothing.\end{cases} \] Here nonnegative indices follow the zero-based indexing displayed in the supplied statement; for $k=0$, the inequalities for earlier eigenvalues are vacuous.

The question is whether there exists a closed manifold $M$ in the intended admissible class for which $\kappa(M)=\infty$. Equivalently, does there exist such an $M$ satisfying \[ \forall k\in{\mathbb{Z}}_{\geq 0}\ \exists\varepsilon>0\ \forall\text{ Riemannian metrics }g\text{ on }M,\quad \mu_k(g)\neq 1\ \text{ or }\ \exists i\in{\mathbb{Z}}\ \left(0\leq i<k\ \text{ and }\ |\mu_i(g)|\geq\varepsilon\right)? \] This transcription remains conditional: the missing operator, admissibility hypotheses, and spectral indexing must be resolved before it defines a unique mathematical invariant. In particular, the cited conformal Laplacian paper raises an unresolved issue about the supplied description of the eigenvalues as Dirac eigenvalues.

Short English statement

Can the supplied $\kappa$-invariant be infinite on some closed manifold? Finiteness means that one fixed nonnegative index $k$ works for every positive tolerance: a metric makes the indexed eigenvalue exactly $1$ and all earlier indexed eigenvalues arbitrarily small in absolute value. The intended operator, manifold hypotheses, and eigenvalue ordering require clarification.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30002632$, source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. No individual proposer is supplied; the recorded proposed year is 2014.
• Supplied report citation: Analysis, Geometry and Topology of Positive Scalar Curvature Metrics (2015), Oberwolfach Reports, pp. 1991--2046, DOI 10.4171/owr/2014/36. https://doi.org/10.4171/owr/2014/36. The record attributes the problem to the workshop report and its open-problem discussions.
• Supplied literature reference, labeled "Bär--Dahl conformal Laplacian paper": https://doi.org/10.1007/s00039-003-0419-6. No full bibliographic details are supplied.
• Supplied literature reference, labeled "original OWR report": https://doi.org/10.4171/owr/2014/36.
• Literature assessment checked at 2026-08-21: the supplied curation labels the problem open and states that $\kappa$ is finite for simply connected manifolds of dimension at least five, has surgery and genus bounds, and has neither a known infinite example nor a known proof excluding all such examples. This is supplied dated curation derived from the OWR record, not an independently verified literature assessment. The supplied background explicitly requests independent verification of time-sensitive status and cited claims.