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arxiv: 2606.28316 · v1 · pith:FP3AGGNLnew · submitted 2026-06-26 · 🧮 math.SG · math.DS· math.GT

Infinite ECH Capacities and Anosov Flows

Gabriel Beiner This is my paper

Pith reviewed 2026-06-29 01:18 UTC · model grok-4.3

classification 🧮 math.SG math.DSmath.GT
keywords embedded contact homologyECH capacitiesAnosov flowssymplectic four-manifoldscotangent bundlesReeb dynamicsLagrangian submanifoldsFloer theory
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The pith

Embedded contact homology capacities are infinite for cotangent disk bundles over surfaces of genus at least two and obstruct Anosov flows.

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

This paper shows that the ECH capacities of many symplectic four-manifolds are infinite. Examples include the cotangent disk bundles over closed oriented surfaces of genus two or higher. Infinite capacities imply that these manifolds do not admit Reeb Anosov flows or Hamiltonian Anosov flows. The work therefore settles the four-dimensional version of a question raised by Herman in 1998. It also produces obstructions for three-manifolds to carry Anosov flows and limits the possible high-genus Lagrangians in symplectic four-manifolds.

Core claim

We show that in many cases the ECH capacities of a symplectic 4-manifold are infinite, including cotangent disk bundles over closed oriented surfaces of genus at least two. We prove that ECH obstructs Reeb Anosov and Hamiltonian Anosov flows, addressing the four-dimensional case of a question posed by Herman in 1998. Further, we obtain Floer-theoretic obstructions to a 3-manifold admitting any Anosov flow. As an application, we give new constraints on the existence of embedded Lagrangians of genus at least two in symplectic 4-manifolds. In an appendix, some related results in all dimensions are proved for capacities constructed from rational symplectic field theory.

What carries the argument

ECH capacities, the sequence of numerical invariants from embedded contact homology whose values become infinite and thereby rule out Anosov Reeb flows and Hamiltonian Anosov flows.

If this is right

  • Cotangent disk bundles over closed oriented surfaces of genus at least two have infinite ECH capacities.
  • Reeb Anosov flows are obstructed on the boundaries of these bundles.
  • Hamiltonian Anosov flows are obstructed on these symplectic four-manifolds.
  • Three-manifolds admitting Anosov flows cannot bound such manifolds with infinite capacities.
  • Embedded Lagrangians of genus at least two face new existence constraints inside these symplectic four-manifolds.

Where Pith is reading between the lines

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

  • The rational SFT capacity results in the appendix may produce analogous obstructions to Anosov flows in dimensions greater than four.
  • Other symplectic fillings of contact three-manifolds known to support Anosov flows could be checked for infinite ECH capacities with similar methods.
  • Combining these Floer obstructions with existing classifications of Anosov flows on three-manifolds might narrow the list of admissible manifolds further.
  • The infiniteness property might extend to cotangent bundles over surfaces with boundary or non-orientable surfaces.

Load-bearing premise

The standard definitions and computation rules for ECH capacities apply directly to the listed manifolds without additional restrictions that would render the capacities finite.

What would settle it

An explicit computation showing finite ECH capacities for the cotangent disk bundle over a genus-two surface, or the construction of a Reeb Anosov flow on the boundary of such a bundle.

Figures

Figures reproduced from arXiv: 2606.28316 by Gabriel Beiner.

Figure 1
Figure 1. Figure 1: On the left, an illustration of the stable and unstable direc￾tions of the Anosov splitting and how vectors evolve under the Ansov flow X. On the right, an illustration of the bi-contact structure ξ± = ker α± determined by the Anosov flow within a tangent space. Everything is oriented with respect to the right hand rule. Remark 2.16. The Anosov Liouville domains are of interest partly because they were the… view at source ↗
Figure 2
Figure 2. Figure 2: An illustration of the proof of Theorem 3.1. If c1(Y1, λ1) < ∞, compatibility of the U maps and cobordism map implies the existence of a holomorphic curve which does not respect the action filtration. Let UE denote the U map on ∂E. By compatibility of the U maps with cobordisms, one has U L E ◦ Φ L (X \ E, ω, 0)(γ ⊗ c(Y2, ξ2) ⊗ · · · ⊗ c(Yn, ξn)) = ΦL (X \ E, ω, 0) ◦ U L 1 (γ ⊗ c(Y2, ξ2) ⊗ · · · ⊗ c(Yn, ξn… view at source ↗
Figure 3
Figure 3. Figure 3: A point-constrained curve that might plausibly exist in an Anosov Liouville domain I × Y and be seen by c Alt 1 (I × Y, ω). Note that the content of the proof of Theorem 1.2 is not sufficient to resolve these questions in the affirmative. And in fact, the author’s best guess is that the answers to Questions 4.15 and 4.17 are no, at least for some of the Anosov Liouville domains. Roughly speaking, in provin… view at source ↗
read the original abstract

This article relates the theory of embedded contact homology (ECH) with the dynamics of Anosov flows. We show that in many cases the ECH capacities of a symplectic 4-manifold are infinite, including cotangent disk bundles over closed oriented surfaces of genus at least two. We prove that ECH obstructs Reeb Anosov and Hamiltonian Anosov flows, addressing the four-dimensional case of a question posed by Herman in 1998. Further, we obtain Floer-theoretic obstructions to a 3-manifold admitting any Anosov flow. As an application, we give new constraints on the existence of embedded Lagrangians of genus at least two in symplectic 4-manifolds. In an appendix, some related results in all dimensions are proved for capacities constructed from rational symplectic field theory.

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 paper relates embedded contact homology (ECH) with the dynamics of Anosov flows on 3-manifolds. It proves that ECH capacities are infinite for many symplectic 4-manifolds, including cotangent disk bundles over closed oriented surfaces of genus at least two. This infiniteness is used to show that ECH obstructs the existence of Reeb Anosov flows and Hamiltonian Anosov flows, addressing the four-dimensional case of a question posed by Herman in 1998. Additional results include Floer-theoretic obstructions to the existence of any Anosov flow on a 3-manifold and new constraints on embedded Lagrangians of genus at least two in symplectic 4-manifolds. An appendix establishes related results in all dimensions for capacities arising from rational symplectic field theory.

Significance. If the central claims hold, the work provides a novel bridge between ECH theory and Anosov dynamics, yielding concrete obstructions that resolve the 4D case of Herman's question and supply new applications to Lagrangian embeddings. The appendix extends the approach via rational SFT capacities to higher dimensions. Strengths include the use of established ECH definitions to derive infiniteness without ad-hoc parameters and the production of falsifiable dynamical obstructions.

minor comments (3)
  1. Clarify the precise statement of the main theorem on infiniteness of ECH capacities (likely Theorem 1.1 or equivalent) to specify the class of contact forms or fillings for which the result applies.
  2. In the appendix, ensure that the rational SFT capacities are defined with explicit reference to the relevant literature on rational SFT to avoid notation ambiguity.
  3. Figure captions or diagrams illustrating the cotangent disk bundles (if present) should include explicit genus labels for the base surfaces to aid readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive evaluation of our manuscript and for recommending minor revision. The report accurately summarizes our results on ECH capacities for cotangent disk bundles, obstructions to Anosov flows in 4D (resolving Herman's question), Floer obstructions on 3-manifolds, constraints on high-genus Lagrangians, and the appendix on rational SFT capacities. No major comments are provided in the report, so we have no specific points requiring response or revision at this stage.

Circularity Check

0 steps flagged

No significant circularity; derivation relies on established external ECH theory

full rationale

The paper claims to show infinite ECH capacities for cotangent disk bundles over genus >=2 surfaces and other manifolds using standard ECH definitions and computation rules, then derives obstructions to Reeb/Hamiltonian Anosov flows. This chain depends on pre-existing ECH capacity theory (independent of the present work) rather than any self-definitional loop, fitted parameter renamed as prediction, or load-bearing self-citation. No equations or steps in the provided abstract reduce the result to its inputs by construction. The argument is self-contained against external benchmarks in symplectic geometry.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on the established theory of embedded contact homology and properties of Anosov flows; no free parameters, new entities, or ad-hoc axioms are indicated in the abstract.

axioms (2)
  • standard math Standard axioms and properties of embedded contact homology (ECH) as developed in symplectic geometry.
    The infiniteness claims and obstructions depend on the pre-existing definition and functoriality of ECH capacities.
  • domain assumption Standard definitions and dynamical properties of Anosov flows on 3-manifolds.
    Used to formulate the obstruction statements.

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