The ECH and alternative ECH capacities of closed symplectic 4-manifolds

Abstract: We show that the ECH capacities of every closed symplectic 4-manifold with a rational symplectic form are infinite. We also give the first examples of symplectic 4-manifolds whose alternative ECH capacities are infinite. We verify the ECH Weyl law holds with bounded subleading asymptotics for the alternative ECH capacities of any closed symplectic 4-manifold with $b_2<sup>+=1$ or any smooth domain therewithin.

• The paper proves that every closed rational symplectic 4-manifold has infinite ECH capacities, using Donaldson divisors, Biran decomposition, and prequantization-boundary estimates.
• The paper constructs the first examples with infinite alternative ECH capacities through Zehnder torus hypersurfaces and a neck-stretching argument that exploits the absence of periodic characteristic orbits.
• The paper proves that alternative capacities are finite for closed manifolds with b₂⁺=1 and satisfy the Weyl law cₖᵃˡᵗ=√(4 vol(X,ω)k)+O(1), with extensions to smoothly embedded domains and conjectures for subleading terms.

This paper by Gabriel Beiner establishes several foundational results concerning the finiteness and asymptotics of embedded contact homology (ECH) capacities and their "alternative" counterparts for closed symplectic 4-manifolds. The three principal contributions are: (1) the ECH capacities of every closed symplectic 4-manifold with a rational symplectic form are infinite; (2) the first examples of closed symplectic 4-manifolds with infinite alternative ECH capacities are produced, via Zehnder torus hypersurfaces; and (3) for any closed symplectic 4-manifold with $b_2^+=1$ (or any smooth domain therein), the alternative ECH capacities are finite and satisfy a Weyl law with a bounded error term, $$c_k^{Alt}(X,\omega)=\sqrt{4\,\mathrm{vol}(X,\omega)\,k}+O(1)$$.

Background on the two capacity systems

ECH capacities, introduced by Hutchings, are defined via the action filtration on ECH of the contact boundary of Liouville domains. For an arbitrary symplectic 4-manifold they are defined indirectly as a supremum over embedded weak Liouville domains, which is precisely why their values on closed manifolds have been so difficult to determine — even for $\mathbb{C}P^2$ and $S^2\times S^2$. The alternative ECH capacities, also due to Hutchings, are defined instead by max-min problems over point-constrained holomorphic curves: $c_k^{Alt}$ is a supremum over admissible almost-complex structures and point constraints of the minimal energy of a holomorphic curve through those points. They require no Seiberg–Witten theory, only Hutchings' ECH index inequality, Gromov–Taubes compactness, and elementary curve analysis. They are bounded above by the ECH capacities and agree with them in many cases, but lower bounds are hard to obtain since one must rule out all holomorphic curves.

Infinite ECH capacities of rational closed 4-manifolds

The main theorem states that if $(X,\omega)$ is a closed rational symplectic 4-manifold — meaning some multiple of $[\omega]$ is integral — then all ECH capacities of $(X,\omega)$ are infinite. An independent proof appears in contemporaneous work of Chen. The argument combines Donaldson's existence of symplectic divisors Poincaré dual to large multiples of $[\omega]$ with Biran's decomposition of $(X,\omega)$ into a Liouville domain glued to a symplectic disk bundle over the divisor. The maximal Liouville compactification $$c_k^{Alt}(X,\omega)=\sqrt{4\,\mathrm{vol}(X,\omega)\,k}+O(1)$$0 has boundary the prequantization circle bundle over the divisor $$c_k^{Alt}(X,\omega)=\sqrt{4\,\mathrm{vol}(X,\omega)\,k}+O(1)$$1, with Euler number $$c_k^{Alt}(X,\omega)=\sqrt{4\,\mathrm{vol}(X,\omega)\,k}+O(1)$$2 and Liouville form scaled by $$c_k^{Alt}(X,\omega)=\sqrt{4\,\mathrm{vol}(X,\omega)\,k}+O(1)$$3. A key lemma shows that the first element of the ECH spectrum of a prequantization bundle satisfies $$c_k^{Alt}(X,\omega)=\sqrt{4\,\mathrm{vol}(X,\omega)\,k}+O(1)$$4, and equals infinity when $$c_k^{Alt}(X,\omega)=\sqrt{4\,\mathrm{vol}(X,\omega)\,k}+O(1)$$5 (via the Milnor–Wood inequality, hypertaut foliations in the sense of Eliashberg–Thurston, and Beiner's earlier result that such spectra are infinite). Since conformality gives $$c_k^{Alt}(X,\omega)=\sqrt{4\,\mathrm{vol}(X,\omega)\,k}+O(1)$$6, monotonicity forces $$c_k^{Alt}(X,\omega)=\sqrt{4\,\mathrm{vol}(X,\omega)\,k}+O(1)$$7 to dominate an arbitrarily large quantity.

A stronger statement holds under the topological hypothesis $$c_k^{Alt}(X,\omega)=\sqrt{4\,\mathrm{vol}(X,\omega)\,k}+O(1)$$8: then $$c_k^{Alt}(X,\omega)=\sqrt{4\,\mathrm{vol}(X,\omega)\,k}+O(1)$$9 contains an embedded Liouville subdomain whose ECH capacities are infinite. This follows from the adjunction formula, which under the hypothesis yields $\mathbb{C}P^2$0 for a Donaldson divisor of genus $\mathbb{C}P^2$1, placing its prequantization boundary in the infinite-spectrum regime. The hypothesis covers most symplectic 4-manifolds: everything with $\mathbb{C}P^2$2, everything non-Kähler, and Kähler surfaces with $\mathbb{C}P^2$3 and non-negative Kodaira dimension (Enriques, Dolgachev, Barlow, hyperelliptic surfaces), as well as ruled surfaces and blow-ups with suitably chosen forms. The paper explicitly notes that this should not be read as ECH being blind to embeddings into closed manifolds: Hutchings' completed ECH capacities remain finite for $\mathbb{C}P^2$4 and $\mathbb{C}P^2$5 and obstruct embeddings there.

Infinite alternative capacities via Zehnder tori

The paper gives the first examples of symplectic 4-manifolds with infinite alternative ECH capacities. A Zehnder torus hypersurface is a 3-torus $\mathbb{C}P^2$6 on which $\mathbb{C}P^2$7 takes the constant-coefficient form determined by a triple $\mathbb{C}P^2$8 rationally independent, so that the characteristic flow is a minimal irrational rotation with no periodic orbits. Such hypersurfaces exist for almost every constant-coefficient form on $\mathbb{C}P^2$9, and, following Usher's construction, on simply connected elliptic surfaces with $S^2\times S^2$0 after perturbation near a Gompf fibre sum region.

The proof of infiniteness is a neck-stretching argument. A Fredholm/transversality lemma arranges that any somewhere-injective curve through a generic point cannot lie entirely in the neighbourhood $S^2\times S^2$1: such a curve would represent a class in the image of $S^2\times S^2$2, which is isotropic and pairs trivially with $S^2\times S^2$3, forcing the Gromov–Taubes index $S^2\times S^2$4, contradicting $S^2\times S^2$5. Stretching the neck along $S^2\times S^2$6 and assuming $S^2\times S^2$7, one obtains curves of energy below $S^2\times S^2$8 exiting the stretched region; exhaustive Gromov–Taubes compactness produces a limiting proper holomorphic curve in the semi-infinite symplectization, which must be $S^2\times S^2$9-asymptotic to trivial cylinders over closed orbits of the stable Hamiltonian structure. Since the Zehnder flow has none, contradiction.

Two caveats are stated plainly. First, Usher's non-stable Hamiltonian hypersurfaces (torus bundles failing nearby existence without stability) are expected to yield the same conclusion, but the argument would require feral-curve adiabatic techniques, which the author does not address. Second, the analogous statement in dimension six and above is much easier: negative curvature or Calabi–Yau/aspherical conditions force Fredholm indices at most zero, making the elementary capacities $c_k^{Alt}$0 infinite immediately.

Finiteness and Weyl law for $c_k^{Alt}$1

In contrast, when $c_k^{Alt}$2 the alternative capacities are always finite and obey $c_k^{Alt}$3. The upper bound uses the Seiberg–Witten/Gromov–Taubes machinery: wall-crossing computations (following Usher) give $c_k^{Alt}$4, hence non-vanishing Gromov–Taubes invariants, which bound $c_k^{Alt}$5 above by pairings $c_k^{Alt}$6 for classes $c_k^{Alt}$7 with controlled index; optimizing over $c_k^{Alt}$8 bounds the error term uniformly. For irrational forms, the argument approximates $c_k^{Alt}$9 by rational classes $(X,\omega)$0 represented by integral forms $(X,\omega)$1, using Li–Liu's extension of Taubes' $(X,\omega)$2 equivalence to handle exceptional-class issues, and controls the error through explicit norm estimates. The lower bound comes from Buse–Hind–Opshtein packing stability plus Edtmair's recent packing stability theorem, together with the known bounded error terms for unions of balls.

An immediate corollary extends the Weyl law to any compact symplectic 4-manifold with smooth boundary embedding into a closed $(X,\omega)$3 manifold. Notably, the corresponding statement for ECH capacities is false: by the second main theorem, any closed rational 4-manifold with $(X,\omega)$4 and $(X,\omega)$5 (e.g. projective Kähler surfaces of non-negative Kodaira dimension with $(X,\omega)$6) contains embedded Liouville subdomains with infinite ECH capacities. The paper also highlights concrete applications: the ECH Weyl law was previously unknown even for the blown-up ball $(X,\omega)$7, where only the crude bounds $(X,\omega)$8 were available; the new results show the left inequality is an equality for $(X,\omega)$9. Similarly, precise bounded-error Weyl laws now hold for cotangent disk bundles of all non-orientable surfaces, not just those with Euler characteristic $[\omega]$0 or $[\omega]$1.

Subleading asymptotics

The paper analyzes the error term $[\omega]$2 and conjectures, for closed $[\omega]$3 manifolds,

$[\omega]$4

where $[\omega]$5 is a pseudonorm detecting primitivity in the integral lattice; in particular, $[\omega]$6 would converge exactly when the form is irrational. This parallels Hutchings' Ruelle invariant conjecture for star-shaped domains. Partial results confirm the conjecture when $[\omega]$7 (monotone/Fano cases, Enriques and hyperelliptic surfaces, Godeaux-type surfaces of general type): then $[\omega]$8 exactly, proved via a Fredholm-index lower bound combined with the Gromov–Taubes upper bounds. Combined with Hutchings' Ruelle conjecture, this would imply a constraint $[\omega]$9 on generic full fillings.

The proof of the Weyl law also motivates a four-dimensional analogue of Irie's equidistribution question: do the holomorphic currents realizing $(X,\omega)$0 become equidistributed after rescaling by $(X,\omega)$1? Donaldson's equidistribution theorem for his divisors provides positive evidence, and the claim holds on $(X,\omega)$2 for divisors realizing the capacities.

Limitations and open questions

Several restrictions are acknowledged. The infiniteness results for ECH capacities require rationality of the symplectic form; the simplest unresolved case is $(X,\omega)$3 with an irrational area ratio, though many irrational examples are covered by earlier work. The sharpness conjecture — that when $(X,\omega)$4 every embedded Liouville domain has finite ECH capacities — is established only for Weinstein domains, relying on Mark–Tosun's divisor-avoidance theorem; extending it to general Liouville domains remains open. The Zehnder torus argument does not cover Usher's non-stable Hamiltonian hypersurfaces. The tame elementary capacities are shown infinite for Kähler surfaces with $(X,\omega)$5 using Lee–Parker structure theory, but it is unknown whether $(X,\omega)$6 holds, since ECH has not been developed for merely tame almost-complex structures. Finally, the general subleading asymptotics conjecture is verified only in special cases, and the equidistribution question is entirely open beyond $(X,\omega)$7.

Conclusion

The paper resolves the basic finiteness question for ECH capacities of closed rational symplectic 4-manifolds in the affirmative-negative direction: they are always infinite, while identifying precisely the topological regime ($(X,\omega)$8) where embedded Liouville subdomains may still have finite capacities. Simultaneously it delineates the reach of the alternative capacities, exhibiting both their first infinite examples and a complete Weyl law with bounded error on $(X,\omega)$9 manifolds and their subdomains — a law unavailable for ECH capacities in comparable generality. The subleading analysis and the equidistribution question frame a concrete program connecting these capacities to Reeb dynamics and to the geometry of holomorphic currents on symplectic 4-manifolds.