Magnetic-Laplacian Eigenvalue Infima on Closed Surfaces

Definitions and mathematical statement

Let $(M,g)$ be a fixed closed Riemannian surface, where closed means compact without boundary, and let $i>1$ be an integer. The supplied record does not specify connectedness, the admissible regularity of magnetic potentials, the spectral indexing convention, or the normalization denoted by the bar.

A conventional interpretation, requiring confirmation against the source, takes a real magnetic potential to be a smooth real differential one-form $A\in\Omega^1(M;{\mathbb{R}})$, and defines the magnetic Laplacian on complex-valued functions by \[ d_Au={\,\mathrm{d}} u+\mathrm{i}uA,\qquad \Delta_{g,A}=d_A^*d_A, \] where the adjoint uses the metric $g$ and its Riemannian volume measure. Under one-based indexing, its eigenvalues, repeated according to multiplicity, are written \[ 0\leq\lambda_1(M,g,A)\leq\lambda_2(M,g,A)\leq\cdots. \] A conventional surface normalization is \[ \overline\lambda_j(M,g,A)=\operatorname{Area}_g(M)\lambda_j(M,g,A). \] These conventions are explanatory candidates; they are not definitions verified from the supplied excerpt.

The principal requested statement is that, for a fixed $(M,g)$ and a fixed integer $i>1$, \[ \inf_A\overline\lambda_i(M,g,A)\geq C(M,g)>0, \] with the infimum taken over all admissible real magnetic potentials. Under the provisional conventions above, the infimum would be over $\Omega^1(M;{\mathbb{R}})$. The constant must be independent of $A$. Although the source writes $C(M,g)$, it does not explicitly resolve whether the constant must also be independent of $i$. Connectedness and the spectral conventions must be clarified before treating this as a fully specified conjecture.

The record additionally asks for a natural construction of critical metric-potential pairs $(g,A)$. It supplies neither the functional whose critical points are intended nor the allowed variations, constraints, or definition of criticality. Consequently this additional request cannot yet be formulated as a precise mathematical assertion.

Short English statement

For a fixed closed Riemannian surface and an eigenvalue index $i>1$, seek a positive lower bound for the normalized magnetic eigenvalue that holds for every admissible real magnetic potential. Also seek a natural construction of critical metric-potential pairs. The normalization, indexing, connectedness assumptions, admissible potentials, and meaning of criticality require confirmation.

Sources

• Local UnsolvedMath registry: supplied complete source record, permanent ID $30005619$, exact source problem code catalogue entry, title Magnetic-Laplacian Eigenvalue Infima on Closed Surfaces. The record belongs to the Oberwolfach Reports Open Problems collection; no individual proposer is supplied.
• Supplied source citation: Geometric Spectral Theory (2024), Oberwolfach Reports, DOI 10.4171/owr/2023/36, pp. 2017--2106. Supplied source URL and literature reference labeled "original OWR report": https://doi.org/10.4171/owr/2023/36. The record separately gives proposed year 2023 and the tag owr-year:2023; these are retained without reconciling them with the citation's 2024 label.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem open and states that available magnetic spectral results do not establish the fixed-surface lower bound in the stated generality. This is supplied dated curation, not an independently verified literature assessment. The background explicitly requests independent verification of time-sensitive status and cited claims.
• The supplied statement status is corrected_verified, with the supplied explanation that the statement was reconstructed and checked against the official Oberwolfach report. That verification is reported as source metadata; it has not been independently reproduced here.