2408.10687
Structure of optimal gradient flows bifurcations on closed surfaces
Illia Ovtsynov, Alexandr Prishlyak
correctmedium confidence
- Category
- Not specified
- Journal tier
- Specialist/Solid
- Processed
- Sep 28, 2025, 12:56 AM
- arXiv Links
- Abstract ↗PDF ↗
Audit review
The paper proves that C-diagrams (marked-arc chord diagrams) classify and realize optimal SN-flows, and T-diagrams classify and realize optimal SC-flows, with explicit statements and proofs (via separatrix diagrams and Peixoto’s classification). The candidate solution reaches the same main conclusions with a different, more constructive gluing/extension approach. Minor caveats: the model implicitly assumes p=q=1 for optimal SC-flows on all surfaces, while the paper notes a sphere exception; and the model’s flow-box extension lemma is used but not cited from the paper.
Referee report (LaTeX)
\textbf{Recommendation:} minor revisions
\textbf{Journal Tier:} specialist/solid
\textbf{Justification:}
The paper introduces concise, workable invariants (C- and T-diagrams) for codimension-one gradient flows on closed surfaces, proves completeness and realization, and supplies enumerations by genus. Proofs rely on standard separatrix-diagram methods and Peixoto classification, and the handling of special sphere cases is careful. Exposition could be streamlined (definitions, color rules, language). With minor editorial and explanatory improvements, the paper is suitable for publication.