A GAP IN THE HOFER METRIC BETWEEN INTEGRABLE AND AUTONOMOUS HAMILTONIAN DIFFEOMORPHISMS ON SURFACES

Michael Khanevsky

incompletemedium confidence

Category: Not specified
Journal tier: Specialist/Solid
Processed: Sep 28, 2025, 12:56 AM
arXiv Links: Abstract ↗PDF ↗

Audit review

The paper’s disk and higher‑genus cases are carried out with adequate detail (using the non‑autonomous braid Bna, explicit integrable construction g = hΔ ∘ h, and an adaptation of Alves–Meiwes braid stability), but the S2 case is presented only as a brief scheme (16‑strand braid and key claims left without proof), so the argument for all surfaces is incomplete. The model mirrors the paper’s approach and correctly identifies the key steps (non‑autonomous braid obstruction, integrable disk construction, Hofer‑small braid stability, extension to general surfaces and S2), but it also omits crucial admissibility/regularity checks and treats the S2 modifications at a purely schematic level; it even misstates one set of winding numbers and the moon/planet assignment. Hence, both are incomplete for the stated “every compact surface” scope.

Referee report (LaTeX)

\textbf{Recommendation:} major revisions

\textbf{Journal Tier:} specialist/solid

\textbf{Justification:}

The contribution is well-motivated and leverages modern Floer-theoretic stability to produce integrable maps that are Hofer-far from autonomous ones. The disk and higher-genus arguments are clear and likely correct. However, the S\^2 case is only sketched (16-strand construction; key claims after configuration-space reduction are stated without proof). For a complete “every compact surface” theorem, these steps must be supplied in full. With those details added and minor clarifications on admissibility and degeneracy handled explicitly, the paper would be suitable for publication.