Closed Orbits of Dynamically Convex Reeb Flows: Towards the HZ- and Multiplicity Conjectures

Erman Çineli, Viktor L. Ginzburg, Başak Z. Gürel

correctmedium confidence

Category: math.DS
Journal tier: Strong Field
Processed: Sep 28, 2025, 12:56 AM
arXiv Links: Abstract ↗PDF ↗

Audit review

The paper proves (M1) ≥ n prime closed orbits under dynamical convexity, and (M2) exactly n non‑alternating primes with uniform parity in the nondegenerate case, plus the centrally symmetric n-or-∞ result yielding exactly n symmetric, non‑alternating primes under finiteness; these are explicitly stated and supported by filtered symplectic homology barcode arguments and index recurrence. By contrast, the model overreaches in (i): from the CP^∞ parity pattern it incorrectly infers that the S^1‑equivariant differential vanishes and hence that there are exactly n simple orbits in general (i.e., no additional alternating primes), contradicting the paper’s allowance for more alternating primes in the non‑symmetric case. The model also replaces the paper’s core filtered barcode machinery with a parity-only argument that is not logically sufficient.

Referee report (LaTeX)

\textbf{Recommendation:} no revision

\textbf{Journal Tier:} strong field

\textbf{Justification:}

This manuscript makes a substantial advance in the multiplicity problem for dynamically convex Reeb flows on star-shaped hypersurfaces in R\^{2n}. It proves the sharp lower bound of n prime closed orbits under dynamical convexity with finiteness and refines the nondegenerate case by identifying exactly n non‑alternating primes and uniform parity. In the centrally symmetric setting, it establishes an n-or-∞ dichotomy and, under finiteness, exactly n symmetric, non‑alternating primes. The proofs leverage filtered symplectic homology as a persistence module, delivering new rigidity (one-dimensionality for every action threshold) and fine structural consequences (degree/action ordering and constant A/μ̂ ratio). The results are significant, well-situated in the literature, and technically sound based on the presented arguments.