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arxiv: 2606.02452 · v1 · pith:D5N6SPSDnew · submitted 2026-06-01 · 🧮 math.SG

Symplectic capacities of S¹-invariant dynamically convex domains in mathbb{R}⁴

Arthur Melo , Vinicius G. B. Ramos , Alejandro Vicente This is my paper

Pith reviewed 2026-06-28 11:18 UTC · model grok-4.3

classification 🧮 math.SG
keywords symplectic capacitiesdynamical convexityS1-invarianceHamiltonian circle actionsReeb orbitsdomains in C2symplectic geometry in four dimensions
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The pith

All normalized symplectic capacities agree for dynamically convex domains in C squared invariant under Hamiltonian S1 actions isotopic to the Hopf action.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that every normalized symplectic capacity takes the same value on domains in four-dimensional space that are preserved by a circle symmetry close to the standard Hopf action and that meet the dynamical convexity condition on their boundary orbits. Symplectic capacities measure the smallest symplectic size or energy associated to a domain, and distinct definitions normally produce distinct numbers. When they coincide, any single capacity determines all the others for that class of sets. The result applies specifically to S1-invariant domains because the symmetry and the orbit condition allow the different capacities to be related directly.

Core claim

For dynamically convex domains in R^4 that admit a Hamiltonian S1-action isotopic to the Hopf diagonal action, all normalized symplectic capacities are equal.

What carries the argument

The Hamiltonian S1-action isotopic to the Hopf action on the domain, which forces agreement among capacities through its preservation of the Reeb dynamics.

If this is right

  • Any one normalized capacity computes all the others for this class of domains.
  • Necessary and sufficient conditions are supplied for an S1-invariant domain to be dynamically convex.
  • The agreement simplifies explicit capacity calculations that previously required checking multiple definitions separately.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same agreement might extend to other Hamiltonian group actions if an analogous convexity condition can be formulated.
  • The dynamical convexity criteria could classify additional domains where capacity computations become unique.
  • Applications in Hamiltonian dynamics might reduce to checking only the S1-symmetry and one capacity value.

Load-bearing premise

The domain must be dynamically convex and the circle action must be Hamiltonian and isotopic to the Hopf action.

What would settle it

An explicit S1-invariant domain satisfying dynamical convexity whose Gromov width differs from its Hofer-Zehnder capacity.

read the original abstract

In this paper, we prove that all normalized symplectic capacities agree for dynamically convex domains in $\mathbb{C}^2$ that are invariant under any Hamiltonian $S^1$-action isotopic to the Hopf diagonal action. We also give necessary and sufficient conditions for $S^1$-invariant domains to be dynamically convex.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The manuscript proves that all normalized symplectic capacities coincide on dynamically convex domains in ℝ^4 ≅ ℂ^2 that are invariant under a Hamiltonian S^1-action isotopic to the Hopf diagonal action. It additionally supplies necessary and sufficient conditions on the S^1-invariant domain for dynamical convexity to hold.

Significance. If the central result holds, the agreement of capacities on this class of domains would provide a unified invariant and simplify computations in symplectic geometry; the necessary-and-sufficient characterization of dynamical convexity for S^1-invariant domains is a useful byproduct.

minor comments (3)
  1. [Introduction / Theorem statement] The statement of the main theorem (presumably Theorem 1.1 or equivalent) should explicitly list the normalization convention used for the capacities to avoid ambiguity in the agreement claim.
  2. [Preliminaries] Notation for the Reeb orbits and the isotopy class of the S^1-action is introduced without a dedicated preliminary section; a short subsection collecting these definitions would improve readability.
  3. [Abstract and §3] The necessary-and-sufficient conditions for dynamical convexity are stated in the abstract but their precise formulation (e.g., index conditions on the Reeb orbits) should be cross-referenced to the relevant proposition or theorem in the body.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading and positive assessment of the manuscript. The recommendation for minor revision is noted. However, the report contains no specific major comments to address.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper establishes agreement of normalized symplectic capacities on S^1-invariant dynamically convex domains in C^2 under Hamiltonian actions isotopic to the Hopf action, together with necessary and sufficient conditions for dynamical convexity. No equations, lemmas, or proof steps are supplied that reduce any central claim to a fitted parameter, self-definition, or load-bearing self-citation chain. The derivation therefore remains self-contained against external benchmarks in symplectic geometry, with the isotopy and dynamical-convexity hypotheses serving as independent inputs rather than outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only; standard background in symplectic geometry (Reeb dynamics, normalized capacities) is assumed but not detailed. No free parameters, invented entities, or ad-hoc axioms visible.

pith-pipeline@v0.9.1-grok · 5574 in / 967 out tokens · 14754 ms · 2026-06-28T11:18:37.816586+00:00 · methodology

discussion (0)

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Reference graph

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