Harmonic Forms on Manifolds with Long Necks

Definitions and mathematical statement

The supplied record refers to a family of stretched manifolds $X(r)$, with stretching parameter $r\to+\infty$, but does not specify its construction, Riemannian metrics, or boundary conditions. Consequently, the following notation is conditional on those data being supplied.

For each admissible $r$ and differential-form degree $k$, let $\Delta_r^k$ denote the Hodge Laplacian associated with the intended metric and let $\mathcal D(\Delta_r^k)$ denote its intended domain, including any boundary, integrability, or decay conditions. Its differential expression is \[ \Delta_r^k={\,\mathrm{d}}\,{\,\mathrm{d}}_r^*+{\,\mathrm{d}}_r^*{\,\mathrm{d}}, \] where ${\,\mathrm{d}}$ is the exterior derivative and ${\,\mathrm{d}}_r^*$ its metric-dependent formal adjoint. Write \[ \mathcal H_r^k =\{\omega\in\mathcal D(\Delta_r^k):\Delta_r^k\omega=0\} \] for the resulting space of harmonic $k$-forms. The coefficient field, relevant degrees, operator realization, and any normalization of forms remain unspecified by the record.

Source question. Describe the asymptotic behavior of the spaces of harmonic differential forms on $X(r)$ as $r\to+\infty$.

A precise asymptotic assertion requires the construction of $X(r)$, the geometric and analytic hypotheses, and a specified means of comparing forms or spaces for different values of $r$. The source does not specify a limiting space, comparison maps, topology of convergence, or desired asymptotic estimates. Thus the question is retained as an unresolved formulation request; the notation above does not select the missing hypotheses or assert a convergence theorem.

Short English statement

Describe how harmonic differential forms behave when the stretching parameter of $X(r)$ tends to infinity. The supplied record leaves the stretching construction, metrics, analytic conditions, and meaning of asymptotic behavior unspecified, so a complete mathematical formulation requires those details. Its partially solved label is supplied dated curation, not an independently verified conclusion.

Sources

• Local registry: data/unsolvedmath/problems.json, permanent UnsolvedMath ID $30001501$, title Harmonic Forms on Manifolds with Long Necks, source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. The record assigns proposed year 2010 and supplies no individual proposer.
• Supplied original report citation: Analysis and Geometric Singularities (2010), Oberwolfach Reports, pp. 1625--1690, DOI 10.4171/owr/2010/28. https://doi.org/10.4171/owr/2010/28. The background describes these reports as documenting mathematical workshops and their open-problem discussions; no exact problem page or individual attribution is supplied.
• Supplied literature reference, labeled "Fritzsch adiabatic Hodge decomposition": https://arxiv.org/abs/1712.09832.
• Literature assessment date: 2026-08-21. This is supplied dated curation, not independently verified. The supplied status is partially_solved. The supplied assessment says that Fritzsch later developed an adiabatic decomposition of harmonic forms using a Čech--de Rham complex and generalized splicing maps, substantially answering the asymptotic question but not providing a complete description in every setting. The background characterizes this as dated literature triage derived from the supplied OWR curation and calls for independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified; its supplied verification explains only that the manifold and limiting parameter were placed in standard mathematical notation without changing the question. This label does not establish that the missing geometric hypotheses have been recovered.