Hodge Spectral Gaps Under Ricci Bounded Convergence

Definitions and mathematical statement

Let $(X_i,g_i)$ be the sequence in the supplied record, with Gromov--Hausdorff limit $X$. The stated hypotheses are that the convergence is noncollapsed and that there exists a constant $K\geq 0$, independent of $i$, such that \[ \lvert\operatorname{Ric}_{g_i}\rvert\leq K\qquad\text{for every }i. \] The record does not specify the dimension, compactness, completeness, boundary conditions, meaning of noncollapse, or norm used for the Ricci bound. These omissions must be resolved before this becomes a fully specified mathematical problem.

For a space $Y$, the conventional real ordinary first Betti number is \[ b_1(Y;{\mathbb{R}}):=\dim_{{\mathbb{R}}}H^1_{\mathrm{sing}}(Y;{\mathbb{R}}). \] The source writes $b_1(Y)$ without specifying coefficients or finiteness; the real interpretation is explicit here and requires confirmation. On a smooth Riemannian manifold, the Hodge Laplacian on differential $1$-forms has differential expression \[ \Delta_{H,Y}^{(1)}=\mathrm d\delta+\delta\mathrm d, \] where $\delta$ is the formal adjoint of $\mathrm d$. Its operator domain and any boundary conditions must also be specified. On the possibly singular limit $X$, a definition of differential forms and a realization of the Hodge Laplacian must be chosen. If such a realization is supplied, define its harmonic first Betti number by \[ h_1(X):=\dim_{{\mathbb{R}}}\ker\Delta_{H,X}^{(1)}. \] This is an analytic definition and does not assume equality with ordinary cohomology.

Question recorded by the source. For every sequence satisfying the stated hypotheses, is there a uniform spectral gap above zero for the Hodge Laplacians on $1$-forms? Under the interpretation of a nonnegative self-adjoint realization on each $X_i$, a gap uniform along the sequence means \[ \exists c>0\ \forall i,\qquad \operatorname{Spec}\left(\Delta_{H,X_i}^{(1)}\right) \subseteq\{0\}\cup[c,\infty). \] The source does not specify whether $c$ may depend on the entire sequence or must depend only on fixed geometric parameters.

The source also asks whether the ordinary first Betti numbers converge to that of the limit and whether the harmonic first Betti number of the limit agrees with its ordinary first Betti number. With the explicit real-coefficient interpretation above, these targets are \[ \lim_{i\to\infty}b_1(X_i;{\mathbb{R}})=b_1(X;{\mathbb{R}}), \qquad h_1(X)=b_1(X;{\mathbb{R}}). \] The record describes the spectral-gap and Betti-number formulations as equivalent. That equivalence is retained as a source qualification, not asserted here: its precise meaning requires the missing geometric and analytic hypotheses, an appropriate spectral-convergence statement, and clarification of the relationship between harmonic forms and ordinary cohomology.

Short English statement

The supplied problem asks whether noncollapsed Gromov--Hausdorff convergence with a uniform two-sided Ricci bound prevents positive Hodge eigenvalues on $1$-forms from approaching zero, preserves ordinary first Betti numbers, and identifies harmonic and ordinary first Betti numbers on the limit. The geometric setting, limit operator, coefficient convention, and claimed equivalence require clarification.

Sources

• Local UnsolvedMath registry, supplied complete record: permanent ID $30003152$, exact source problem code catalogue entry, title Hodge Spectral Gaps Under Ricci Bounded Convergence, collection Oberwolfach Reports Open Problems. The proposed year is recorded as 2016; no individual proposer is supplied.
• Supplied report citation: Geometrie (2017), Oberwolfach Reports, pp. 1577--1634, DOI 10.4171/owr/2016/28. https://doi.org/10.4171/owr/2016/28. This is also the supplied literature reference labeled Oberwolfach Report 28. The background describes these reports as records of mathematical workshops and their open-problem discussions. The supplied title year 2017 and the recorded proposed year 2016 are retained without reconciliation.
• Supplied literature reference: Honda, Spectral convergence under bounded Ricci curvature. https://doi.org/10.1016/j.jfa.2017.05.009.
• The literature assessment dated exactly 2026-08-21 is supplied dated curation, not independently verified. It labels the problem open and states that Honda established spectral convergence under uniform two-sided Ricci bounds and related upper semicontinuity results, while the proposed uniform spectral gap and equality of ordinary and limit first Betti numbers remain unresolved under the stated hypotheses. The background explicitly requests independent verification of time-sensitive status and cited claims.
• The registry status is open. Its statement label corrected_verified is accompanied by the explanation that notation, typography, and sentence structure were repaired without changing mathematical content. These supplied labels do not constitute independent mathematical verification.