Bolza Surface Maximization of the First Eigenvalue

Definitions and mathematical statement

The supplied registry asks whether the first eigenvalue for genus-$2$ surfaces is maximized by the Bolza surface. It does not specify the metric normalization, the differential operator, or the eigenvalue indexing convention. Moreover, its dated literature assessment says that the underlying extracted conjecture is truncated, whereas its statement-verification field describes the displayed formulation as complete and verified. Consequently, the following is a conditional interpretation requiring source verification, not an unqualified reconstruction of the original conjecture.

Assume that the intended parameter space is the Teichmüller space $\mathcal T_2$ of marked, connected, closed, oriented genus-$2$ surfaces equipped with hyperbolic metrics of constant Gaussian curvature $-1$, with the usual equivalence by marking-preserving isometry up to isotopy. These surfaces have no boundary. For a representative $(S,g)$, let $\Delta_g=-\operatorname{div}_g\nabla_g$ be the nonnegative Laplace--Beltrami operator on scalar functions, with eigenvalues, counted with multiplicity, \[ 0=\lambda_0(S,g)<\lambda_1(S,g)\leq\lambda_2(S,g)\leq\cdots. \] Thus $\lambda_1(S,g)$ denotes the first strictly positive eigenvalue. Let $(B,g_B)$ denote the surface called the Bolza surface in the supplied record, equipped, under this interpretation, with its curvature-$-1$ hyperbolic metric. The record provides no intrinsic characterization of $B$.

Under these explicit additional conventions, the proposed statement is \[ \forall [(S,g)]\in\mathcal T_2,\qquad \lambda_1(S,g)\leq\lambda_1(B,g_B). \] No uniqueness of the maximizer is asserted. The hyperbolic metric convention, curvature normalization, scalar operator, and positive-eigenvalue indexing must be confirmed against a complete source before this can be treated as the intended mathematical statement.

Short English statement

Under an explicitly conditional interpretation using closed genus-$2$ hyperbolic surfaces of curvature $-1$, the conjecture asks whether the Bolza surface has the largest first positive scalar Laplace eigenvalue. The supplied record leaves these conventions unspecified and contains conflicting claims about the completeness of the original extraction.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30001955$, source problem code catalogue entry, title: Bolza Surface Maximization of the First Eigenvalue. The supplied record labels the problem open and its statement corrected_verified; these are registry labels, not independently verified mathematical facts.
• Mini-Workshop: Boundary Value Problems and Spectral Geometry (2012), Oberwolfach Reports, pp. 43--76, DOI 10.4171/owr/2012/02. Supplied source URL and literature-source link, labeled "original Oberwolfach report": https://doi.org/10.4171/owr/2012/02. No individual proposer is identified in the supplied record.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open but states that the extraction ends after "we conjecture that the first" and does not contain a complete mathematical problem, so no resolution is verifiable. This is supplied dated curation, not an independently verified assessment. The background explicitly requests independent verification of time-sensitive status and cited claims.