Removing Curvature Restrictions from Metric Measure Reconstruction

Definitions and mathematical statement

The source concerns metric-measure spaces, abbreviated mm-spaces, their isomorphism classes, and certain pseudometrics on those classes. The supplied excerpt does not define the admissible mm-spaces, their isomorphisms, or the pseudometrics. In particular, it does not specify the measure normalization, regularity assumptions, or underlying class of spaces.

For context, a pseudometric on a set $S$ is a function $\delta:S\times S\to[0,\infty)$ satisfying, for all $x,y,z\in S$, \[ \delta(x,x)=0,\qquad \delta(x,y)=\delta(y,x),\qquad \delta(x,z)\leq\delta(x,y)+\delta(y,z). \] It is a metric if it also satisfies $\delta(x,y)=0\Rightarrow x=y$. The excerpt asks whether its unspecified pseudometrics become metrics on suitable subclasses of isomorphism classes of mm-spaces. This general definition does not identify the particular pseudometrics used in the source.

Source question. Can the curvature restrictions in Theorem 2 of the cited contribution be removed from its metric-measure reconstruction result, while retaining its other hypotheses and its conclusion?

Only a schematic logical formulation is possible from the supplied record. Let $\mathcal X$ denote the domain of that theorem, let $H_{\mathrm{other}}(u)$ denote all its hypotheses other than its curvature restrictions, let $H_{\mathrm{curv}}(u)$ denote those restrictions, and let $C(u)$ denote its conclusion. Here $u$ denotes the entire collection of objects quantified over by the theorem. The requested removal would replace an implication of the form \[ \forall u\in\mathcal X,\qquad \left(H_{\mathrm{other}}(u)\land H_{\mathrm{curv}}(u)\right)\Longrightarrow C(u) \] by \[ \forall u\in\mathcal X,\qquad H_{\mathrm{other}}(u)\Longrightarrow C(u). \] These symbols are placeholders, not a reconstruction of Theorem 2. Its statement, curvature assumptions, remaining hypotheses, quantifiers, and reconstruction conclusion are absent from the supplied record. Consequently, a complete precise mathematical conjecture cannot be extracted without additional source material.

Short English statement

The question asks whether the curvature assumptions in a cited metric-measure reconstruction theorem can be dropped while keeping its other assumptions and conclusion. The surrounding discussion concerns when certain pseudometrics distinguish isomorphism classes of metric-measure spaces. The supplied excerpt omits Theorem 2 and the relevant definitions, so the formulation remains schematic and requires human mathematical review.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30005260$, source problem code catalogue entry, title: Removing Curvature Restrictions from Metric Measure Reconstruction. The record belongs to the Oberwolfach Reports Open Problems collection; no individual proposer is supplied. The proposed year is recorded as 2022.
• Supplied source citation: Heat Kernels, Stochastic Processes and Functional Inequalities (2023), Oberwolfach Reports, pp. 2811--2893, DOI 10.4171/owr/2022/49. https://doi.org/10.4171/owr/2022/49. The supplied excerpt identifies report page 2839 and refers to Theorem 2 and reference [6]; neither the theorem nor the bibliographic identity of [6] is supplied.
• Supplied literature source: Oberwolfach Report 49. https://ems.press/content/serial-article-files/46987.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem open and describes the curvature restrictions in Theorem 2 as possible proof artifacts, with no counterexample or removal theorem established. This is supplied dated curation, not an independently verified literature assessment. The background explicitly requests independent verification of time-sensitive status and cited claims.