Spectral Convergence Under Lower Ricci Bounds

Definitions and mathematical statement

The supplied record asks for an extension of convergence results for noncollapsed Ricci-bounded sequences. It does not specify the individual convergence theorems or all their hypotheses, so it determines an extension question rather than a unique fully quantified conjecture.

To explain the curvature notation, suppose that the spaces in question are smooth Riemannian manifolds $(X_i,g_i)$, and write $\operatorname{Ric}_{g_i}$ for their Ricci tensors. Under this interpretation, a uniform lower Ricci bound means that there is a common constant $K\geq 0$ such that \[ \operatorname{Ric}_{g_i}(v,v)\geq -K g_i(v,v) \qquad\text{for every }i,\ x\in X_i,\ v\in T_xX_i. \] The stronger hypothesis is written in the source as $\lvert\operatorname{Ric}_{X_i}\rvert\leq K$. The tensor norm or alternative convention intended by these absolute-value signs is not specified.

The requested extension can be expressed schematically as follows. For each intended spectral, tensor, or related convergence theorem, let $H$ denote its remaining hypotheses, including its precise noncollapse and convergence assumptions, and let $C$ denote its conclusion. Given the corresponding result under the source's two-sided curvature hypothesis, \[ H\ \text{ and }\ \lvert\operatorname{Ric}_{X_i}\rvert\leq K\ \text{ for all }i \quad\Longrightarrow\quad C, \] does the analogous assertion \[ H\ \text{ and }\ \operatorname{Ric}_{g_i}\geq -K g_i\ \text{ for all }i \quad\Longrightarrow\quad C \] hold? Here $H$ and $C$ are placeholders for the particular results intended in the cited report, not independently specified hypotheses and conclusions. The quantification ranges only over those intended results; the question is not an assertion that every theorem involving a two-sided Ricci bound admits this replacement.

The supplied record does not fix the dimension, compactness or boundary assumptions, volume normalization and scale defining noncollapse, convergence topology, limiting spaces, spectral operators, tensor classes, or precise convergence conclusions. These must be recovered from the cited source before the schematic question becomes a precise individual mathematical statement.

Short English statement

Can the spectral, tensor, and related convergence results intended in the source for noncollapsed sequences with uniformly bounded Ricci curvature be extended to a uniform lower Ricci bound, retaining their other hypotheses? The supplied record leaves the particular results and several essential hypotheses unspecified.

Sources

• Supplied local UnsolvedMath problem registry: permanent ID $30003153$, source problem code catalogue entry, title Spectral Convergence Under Lower Ricci Bounds; collection: Oberwolfach Reports Open Problems. No individual proposer is identified. The supplied proposed year is 2016.
• Supplied report citation: Geometrie (2017), Oberwolfach Reports, DOI 10.4171/owr/2016/28, pp. 1577--1634. https://doi.org/10.4171/owr/2016/28. The background describes these reports as records of mathematical workshops and their open-problem discussions. The citation's 2017 label is retained alongside the supplied 2016 proposed year and DOI identifier; their relationship has not been independently checked.
• Supplied literature source: Honda, Spectral convergence under bounded Ricci curvature. https://doi.org/10.1016/j.jfa.2017.05.009.
• Supplied literature assessment, checked on 2026-08-21: the record labels the problem open and states that the report identifies the one-sided lower Ricci bound version as an open extension, while the convergence theorems cited there do not settle all requested observations without a two-sided curvature bound. This is supplied dated curation derived from the OWR material, not independently verified. The supplied background explicitly requests independent verification of time-sensitive status and cited claims.