$\gamma$-Topology Limits of Hamiltonians with Singular Sets

Definitions and mathematical statement

Let $(M,\omega)$ denote the symplectic manifold implicit in the source, and let $V\subseteq M$ be the singular set. The group $\operatorname{Ham}(M,\omega)$ consists of Hamiltonian diffeomorphisms, with the support and normalization conventions still to be specified. Write $\widehat{\operatorname{Ham}}^{\mathrm{Hum}}(M,\omega)$ as a placeholder for the Humilière completion intended by the source; its precise construction and its relation to the stated $\gamma$-topology are not specified in the record.

The supplied question concerns a continuous Hamiltonian $H$ on $M\setminus V$ and a sequence of smooth Hamiltonians $(H_k)_{k\geq 1}$ with time-one maps $\phi_k\in\operatorname{Ham}(M,\omega)$, such that $H_k\to H$ uniformly on every compact subset of $M\setminus V$. For Hamiltonians with the spatial domains displayed in the source, this approximation condition is \[ \forall K\subseteq M\setminus V\text{ compact},\qquad \lim_{k\to\infty}\sup_{x\in K}|H_k(x)-H(x)|=0. \] The source does not specify whether time-dependent Hamiltonians are intended; if they are, their time domain and the required uniformity in time must also be supplied.

Question. For every Hamiltonian and every approximating sequence satisfying the intended hypotheses above, must there exist \[ \Phi\in\widehat{\operatorname{Ham}}^{\mathrm{Hum}}(M,\omega) \quad\text{such that}\quad \phi_k\xrightarrow[k\to\infty]{\gamma}\Phi? \] The question asks for convergence of the given sequence. It does not explicitly ask whether different approximating sequences have the same limit.

Unresolved hypotheses. The accompanying literature assessment describes the unrestricted target as concerning arbitrary compact singular sets $V$; compactness is absent from the displayed source statement. No restriction on $M$, definition or normalization of the $\gamma$-topology, or precise choice of completion is supplied. The domain and support conditions on the approximating Hamiltonians, and the hypotheses ensuring their flows exist through time one, are also unspecified. Thus the display records the intended convergence question, but these omissions prevent a fully determined mathematical formulation.

Supplied qualification. The dated curation labels the problem partially solved and reports convergence when the singular set has dimension less than $n$, and more generally when it is nowhere $\gamma$-coisotropic, while reporting the arbitrary compact singular-set case as open. The record does not define the dimension used, the parameter $n$, or the condition nowhere $\gamma$-coisotropic. In particular, an interpretation such as $\dim M=2n$ would require confirmation. These reported results and their applicability are supplied literature claims, not independently verified assertions in this draft.

Short English statement

The question asks whether every smooth approximation of a continuous Hamiltonian away from a singular set has time-one maps converging in the $\gamma$-topology to an element of the intended Humilière completion. The supplied curation reports partial results and leaves arbitrary compact singular sets open. The ambient hypotheses, time dependence, topology, completion, and singular-set conditions need clarification before the question is fully precise.

Sources

• Local UnsolvedMath registry, permanent ID $30004746$, source problem code catalogue entry, title: $\gamma$-Topology Limits of Hamiltonians with Singular Sets. The supplied record belongs to the Oberwolfach Reports Open Problems collection. No individual proposer is identified; the proposed year is recorded as 2021.
• Supplied report citation: Dynamische Systeme (2022), Oberwolfach Reports, pp. 1735--1803, DOI 10.4171/owr/2021/33. https://doi.org/10.4171/owr/2021/33. The background describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions. The supplied 2022 citation year is retained alongside the record's 2021 proposed year and DOI year component; their relationship has not been checked.
• Supplied literature source, labeled "Humiliere completion criterion": https://doi.org/10.24033/bsmf.2560. No authors, article title, or theorem locator are supplied for this reference.
• Supplied literature source, labeled "gamma-coisotropic extension": https://doi.org/10.4171/owr/2021/33. No precise contribution, page, or theorem locator for this extension is supplied.
• Literature assessment checked at 2026-08-22: the supplied dated curation labels the problem partially solved, reports convergence for singular sets of dimension less than $n$ and for nowhere $\gamma$-coisotropic singular sets, and describes the arbitrary compact singular-set case as open. This is supplied dated curation derived from the OWR record, not an independently verified literature assessment; the background explicitly requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and reports a manual semantic, typographic, and LaTeX review claiming self-containment and preservation of the target. These are supplied record annotations and do not resolve the formulation gaps identified here.