Connectivity and Geometry of Hofer-Metric Annuli

Definitions and mathematical statement

Let $\operatorname{Ham}$ be the Hamiltonian diffeomorphism group of the symplectic manifold intended by the report, with Hofer distance $\rho$. In the standard setting the Hofer length of a Hamiltonian path generated by $H_s$ is \[ \ell_H=\int_0^1(\max H_s-\min H_s)\,{\,\mathrm{d}} s, \] and the distance is the infimum of such lengths for paths joining the prescribed endpoints. Compact-support or normalization conventions depend on the unspecified ambient manifold.

For $t>0$, retain the closed annulus \[ A_t=\{f\in\operatorname{Ham}:t\le\rho(\operatorname{id},f)\le2t\}. \] Its intrinsic length distance is conventionally \[ \rho_t(f,g)=\inf\{\operatorname{Length}_\rho(\gamma): \gamma\text{ joins }f\text{ to }g\text{ within }A_t\}, \] where path length is the supremum of sums of $\rho$-distances over finite parameter partitions, and an empty path class gives distance $+\infty$.

The source asks:

1. Is $A_t$ path connected for all sufficiently large $t$?
2. What is the large-$t$ geometry of $(A_t,\rho_t)$? In particular, is \[ \operatorname{diam}(A_t,\rho_t) =\sup_{f,g\in A_t}\rho_t(f,g)<\infty \] for the intended large values of $t$?

Finiteness for each $t$ is not a uniform bound as $t\to\infty$, and no linear growth estimate is supplied. The intrinsic metric must not be replaced by the restriction of the ambient distance: paths are constrained to remain in the annulus. Ordinary path connectedness also does not by itself specify finite-length joining paths.

The symplectic manifold, support conditions, topology for path connectedness, and admissible path class in the length definition are omitted. The record does not guarantee nonempty annuli for arbitrarily large $t$ or state which dimension/manifold class is intended. These are necessary context, rather than an authorization to assert the question for every Hamiltonian group. The large-scale geometry request beyond finite diameter has no additional precise invariant specified.

Short English statement

The question asks whether sufficiently large Hofer annuli are path connected and whether traveling within such an annulus has finite intrinsic diameter. It concerns paths constrained to the annulus, with no uniform diameter bound supplied and no ambient symplectic manifold identified.

Sources

• Local UnsolvedMath registry, permanent ID 30000544, code catalogue entry, "Connectivity and Geometry of Hofer-Metric Annuli." Proposed year 2006; proposer unspecified.
• Geometric Group Theory, Hyperbolic Dynamics and Symplectic Geometry (2007), Oberwolfach Reports, pp. 1991--2058, DOI 10.4171/owr/2006/33; https://doi.org/10.4171/owr/2006/33. The displayed report year 2007 differs from the DOI and proposed year 2006.
• Supplied "Spaeth, length-minimizing Hamiltonian paths": https://arxiv.org/abs/math/0703738.
• Statement status corrected_verified; the supplied note says editorial and LaTeX cleanup preserved meaning and added no source-dependent content.
• Literature triage checked 2026-08-21: open. It reports local path-connectedness and length-minimizing neighborhoods, while saying these do not settle the global annulus questions. Exact hypotheses and current status require independent verification. Created 2026-08-25; updated 2026-09-16.
• Registry curation by Ulam AI / UnsolvedMath Contributors, CC BY 4.0; underlying report and literature provenance remain as supplied. Unreviewed statement-only draft.