Curves Minimizing $\lambda_{1}(g,C)$

Definitions and mathematical statement

Let \[ G=(1/4,1]\ \cup\ \left(( -\infty,0)\setminus\{-1\}\right). \] The supplied problem asks, for each fixed real $g\in G$, to determine the curve $C$ minimizing the quantity denoted $\lambda_1(g,C)$. The admissible class of curves and the definition of this spectral quantity are not supplied. Denoting that intended class by $\mathcal C$, the variational target is therefore only formally expressible as \[ m(g)=\inf_{C\in\mathcal C}\lambda_1(g,C),\qquad \operatorname{Argmin}_{C\in\mathcal C}\lambda_1(g,C) =\{C\in\mathcal C:\lambda_1(g,C)=m(g)\}. \] Determine the minimizing curves for every $g\in G$, including the existence of a minimizer where this is part of the intended question. If the infimum is not finite or not attained for some admissible $g$, that possibility cannot be excluded merely by the source's use of "minimizing." The minimizer may depend on $g$; the statement does not assert a single curve minimizes for every parameter, or that a minimizer is unique.

The parameter restrictions are literal: $g=1$ is included, $g=1/4$, $g=0$ and $g=-1$ are excluded, and all other negative $g$ are included. No claim about the excluded values or other positive values is added.

For a fully specified spectral problem one needs the operator whose eigenvalue is $\lambda_1$, its Hilbert space and domain, its boundary or periodic conditions, and the eigenvalue numbering convention. One also needs the ambient space, regularity, closedness, simplicity or embeddedness of $C$, any length or other geometric normalization, and the equivalences under which curves are to be classified. In particular, a geometric scale constraint cannot be guessed from the notation, and no curvature-dependent operator is inserted without source evidence.

The dated literature assessment checked on 2026-08-21 reports that identifying minimizing curves remains unresolved in exactly these parameter ranges and cites a later isoperimetric eigenvalue survey. This is retained as the supplied open-status assessment, not independently verified. The source-verification note reports only removal of a redundant problem label and balancing of mathematical notation; it does not supply the missing operator or admissibility conditions.

Short English statement

For each $g\in(1/4,1]$ or negative $g\ne-1$, identify the curve minimizing $\lambda_1(g,C)$. The operator and class of curves are absent, so a precise variational problem still requires those definitions and its geometric normalization. The parameter ranges are preserved exactly.

Sources

• Local UnsolvedMath registry, permanent ID 30000150, source code catalogue entry, "Curves Minimizing $\lambda_1(g,C)$," including original statement, cleaned statement, verification note and dated literature assessment.
• Spectral Analysis of Partial Differential Equations (2005), Oberwolfach Reports, pp. 2839--2912, DOI 10.4171/owr/2004/53: https://doi.org/10.4171/owr/2004/53. This is the supplied "original OWR report." The report citation uses 2005, while the DOI and proposed-year field use 2004.
• Supplied reference "isoperimetric eigenvalue survey": https://doi.org/10.1007/s13373-011-0017-0. No fuller citation is supplied.
• Literature triage checked 2026-08-21; local record updated 2026-09-16. No external reference has been visited or independently checked here.