Conformal Spinorial Eigenvalue Infimum on the Torus

Definitions and mathematical statement

Let $T$ be a smooth two-dimensional torus equipped with the spin structure called trivial in the supplied record. Write $g_{a,b}$ for the reference Riemannian metric appearing there. For a fixed such metric, define its smooth conformal class by \[ [g_{a,b}]=\{e^{2u}g_{a,b}:u\in C^\infty(T,{\mathbb{R}})\}. \] The record does not define the metric family $g_{a,b}$, its admissible parameter range, or the spectral quantity $\overline\mu_1(g)$, including its normalization and treatment of zero eigenvalues. Consequently, the following is a conditional transcription of the supplied conjecture; those definitions must be recovered before it specifies a complete mathematical problem.

For every admissible pair $(a,b)$ in the intended family, with $b>0$ as required by the displayed expression, the proposed identity is \[ \inf_{g\in[g_{a,b}]}\overline\mu_1(g) = \begin{cases} \dfrac{2\pi}{\sqrt b},&b>\pi,\\[4pt] 2\sqrt\pi,&0<b\leq\pi. \end{cases} \] Here the infimum is understood over smooth metrics under the displayed interpretation of the conformal class; the source record does not specify whether a larger class of metrics is intended. At $b=\pi$, the two displayed expressions agree.

The supplied original passage additionally conjectures that $b\geq\pi$ is the sharp condition for existence of minimizers, and that for $b<\pi$ there is no minimizing metric and a bubble forms. These attainment and bubbling assertions are additional source qualifications: the numerical infimum identity alone does not express them. The admissible regularity of a minimizer, the meaning of bubble formation, and the quantifiers and convergence conditions for any associated sequence are not supplied. The record identifies this discussion as Conjecture 3 and mentions Ammann's existence result without stating its hypotheses or conclusion.

Short English statement

For the spin structure designated trivial, the conjecture prescribes the infimum of a spinorial spectral quantity within each indicated conformal class of torus metrics, with a transition at $b=\pi$. The accompanying passage also predicts attainment exactly for $b\geq\pi$ and bubbling below that threshold. Missing definitions of the metric family, spectral normalization, and bubbling prevent a fully specified formulation.

Sources

• Local registry: data/unsolvedmath/problems.json, permanent UnsolvedMath ID $30005600$, source problem code catalogue entry, title "Conformal Spinorial Eigenvalue Infimum on the Torus." No individual proposer is identified; the supplied proposed year is 2023.
• Supplied report citation: Geometric Spectral Theory (2024), Oberwolfach Reports, DOI 10.4171/owr/2023/36, pp. 2017--2106. Supplied source URL and literature reference "Oberwolfach Report 36": https://doi.org/10.4171/owr/2023/36. The record attributes the conjectural discussion to Conjecture 3 in this report.
• The original passage refers to Ammann's existence result, but supplies no separate bibliographic reference or theorem statement.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem open and reports that the existence and bubbling dichotomy at the threshold remains unresolved. This is supplied dated curation, not an independently verified literature assessment.
• The registry labels the statement corrected_verified and says it was reconstructed and checked against the official report. These are supplied verification claims; no independent source inspection or mathematical verification is asserted here.