Flow Limits of Local Mass Invariants

Definitions and mathematical statement

Retain the source notation $M_C^0(g,\Phi,\varphi_r,r)$. The supplied record does not define this quantity, the admissible geometric data $g$, the role of $\Phi$, or the family $(\varphi_r)_r$. Write $m_{\mathrm{iso}}$ for Huisken's isoperimetric mass and $m_{\mathrm{cap}}$ for Jauregui's isocapacitary mass, referring respectively to the source passage's references [7, Definition 11] and [8, Definition 4]. These are labels for the cited invariants; their definitions and domains are not supplied.

Source question. Determine how the limit \[ \lim_{r\to\infty}M_C^0(g,\Phi,\varphi_r,r) \] relates to Huisken's isoperimetric mass and Jauregui's isocapacitary mass for the geometric data intended in the source report.

Unresolved formulation qualifications. The record does not specify the dimension, manifold or end, metric regularity, asymptotic conditions, admissibility or normalization of $\Phi$ and $\varphi_r$, or any boundary conditions. It also does not specify hypotheses guaranteeing existence or finiteness of the limit or of the comparison masses. The intended relationship could concern equality, inequalities, or a comparison under additional hypotheses; no particular assertion or universal quantifier over a geometric class can be extracted reliably. Consequently, this is a statement of the supplied comparison question, pending recovery of the missing definitions and hypotheses.

Short English statement

The question asks how the limiting local mass expression compares with Huisken's isoperimetric mass and Jauregui's isocapacitary mass. The supplied record omits the definitions and geometric hypotheses needed for a precise quantified conjecture. Its supplied status is partially solved; this draft does not independently verify that assessment.

Sources

• Supplied local registry record, permanent UnsolvedMath ID $30005580$, source problem code catalogue entry, title Flow Limits of Local Mass Invariants. Project registry locations: data/unsolvedmath/problems.json and catalog/problems.json. No individual proposer is identified in the supplied record.
• Supplied report citation: Differentialgeometrie im Grossen (2024), Oberwolfach Reports, pp. 1617--1670, DOI 10.4171/owr/2023/29. https://doi.org/10.4171/owr/2023/29. The background describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions. The record gives proposed year 2023; the relationship between that year and the citation's 2024 designation is not explained.
• The original passage attributes isoperimetric mass to Huisken, citing [7, Definition 11], and isocapacitary mass to Jauregui, citing [8, Definition 4]. Bibliographic entries for these numbered references are absent from the supplied record.
• Supplied literature source: Burkhardt--Guim, https://arxiv.org/abs/2208.14550.
• Supplied literature source: Benatti, https://arxiv.org/abs/2511.11155.
• Supplied literature source labelled "original OWR report": https://doi.org/10.4171/owr/2023/29.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem partially solved, reports identification of the $C^0$ local mass with ADM mass in the smooth asymptotically flat setting, reports recent work relating isoperimetric and isocapacitary masses, and states that a general direct identification with the $C^0$ flow limit remains open. This is supplied dated curation, not independently verified here; its hypotheses and time-sensitive claims require independent checking.
• The record carries the supplied statement-status label corrected_verified and asserts that the official report was checked directly and that the clean statement restores definitions while removing extraction artifacts or source-dependent wording. These are provenance claims supplied by the record; the definitions are absent from the material provided here, and no source visit or independent verification is claimed by this draft.