Nonpositively Curved Cusped Manifolds from Tropical Curves

Definitions and mathematical statement

For an integer $m\geq 0$, write $T^m=(S^1)^m$ for the real $m$-torus. The supplied record denotes a tropical curve by $\Sigma^{(1)}\subset\Sigma_{\mathbb{P}^n}$, but does not define this notation, the admissible tropical curves, or the permitted values of $n$. The requested torus dimension requires $n\geq 1$.

The intended existence question is: for every tropical curve $\Sigma^{(1)}\subset\Sigma_{\mathbb{P}^n}$ satisfying the source's intended, unspecified hypotheses, can one construct a nonpositively curved cusped manifold $L_{\Sigma^{(1)}}$ with precisely $n+1$ boundary components, each a real $(n-1)$-torus? The stated boundary requirement is \[ \partial L_{\Sigma^{(1)}}\cong\bigsqcup_{j=1}^{n+1}T^{n-1}. \] Here this equation records the source's boundary terminology; it does not resolve whether the boundary is an actual manifold boundary or the boundary of a core obtained by truncating cusp ends. Under the latter interpretation, the equation would instead apply to the truncated core. The record also does not specify whether nonpositive curvature means sectional curvature of a Riemannian metric or a metric curvature condition such as local $\operatorname{CAT}(0)$, nor does it specify completeness, volume, or cusp geometry. Consequently, this is a qualified statement of the requested construction, not a fully determined geometric existence assertion.

The original extract additionally mentions an undefined object $P$, which it suggests may require choosing an augmentation for the Legendrian link of the cusp locus of $L_{\Sigma^{(1)}}$. This contextual expectation refers to an unidentified reference [2]. It supplies neither a definition of $P$ nor an explicit additional requirement on the construction above.

Short English statement

For each tropical curve of the intended class in $\Sigma_{\mathbb{P}^n}$, seek a nonpositively curved cusped manifold with $n+1$ torus boundary components of dimension $n-1$. The admissible curves, curvature convention, and meaning of cusp boundary require clarification. This statement-only draft requires human mathematical review.

Sources

• Local UnsolvedMath registry, permanent ID $30006298$, source problem code catalogue entry, title Nonpositively Curved Cusped Manifolds from Tropical Curves; supplied complete source record in the Oberwolfach Reports Open Problems collection. The record gives proposed year 2025 and identifies no individual proposer.
• Toric Geometry (2025), Oberwolfach Reports, DOI 10.4171/owr/2025/19, pp. 883--940. Supplied report provenance and literature-source link: https://doi.org/10.4171/owr/2025/19. No precise problem page or speaker attribution is supplied.
• Haney, Cusped hyperbolic Lagrangians as mirrors to lines in three-space. Supplied literature-source link: https://arxiv.org/abs/2402.03296.
• Supplied literature assessment dated 2026-08-22: the general construction is described as not established by the located literature, while Haney's special one-vertex example is described as evidence rather than a general solution. The record labels the problem open. These are supplied dated curation claims, not independently verified mathematical or literature conclusions.
• The original extract cites [2] and discusses an augmentation for a Legendrian link, but does not supply the bibliographic identity of [2]. Its relationship to the listed Haney paper cannot be established from the supplied record alone.