ECH Spectral Invariants

• ECH spectral invariants are quantitative measures derived from embedded contact homology, defined via minimax procedures on chain complexes generated by Reeb orbits and J-holomorphic curves.
• They provide rigorous tools to study dynamics, with max-min energy approaches yielding asymptotic volume recovery and closing lemmas that ensure the density of periodic Reeb orbits.
• These invariants lead to sharp symplectic embedding obstructions, with ECH capacities linking analytical, algebraic, and combinatorial methods to optimize geometric constraints.

ECH spectral invariants are quantitative symplectic-topological measurements arising from embedded contact homology (ECH) and related theories, capturing fine invariants of contact and symplectic manifolds, and offering sharp information about dynamics, embedding obstructions, and geometric structures in low dimensions. The formalism is grounded in minimax procedures applied to chain complexes generated by Reeb orbits or holomorphic curves, with spectral invariants encoding the minimal symplectic action or energy required to realize specified homology classes under various filtrations. This notion serves as a bridge between contact geometry, symplectic topology, Hamiltonian dynamics, and Floer-theoretic invariants, and has inspired a range of analogues and elementary alternatives utilizing max-min energy of pseudoholomorphic curves.

1. Formal Definition and Chain Complex Framework

ECH spectral invariants are defined for closed three-dimensional contact manifolds $(Y, \lambda)$, where $\lambda$ is a nondegenerate contact form with associated Reeb vector field. The ECH chain complex is generated by admissible orbit sets, each comprising embedded Reeb orbits with multiplicities subject to adjacency and index constraints. The differential counts embedded $J$-holomorphic curves in the symplectization $(\mathbb{R} \times Y, d(e^s \lambda))$.

Given a nonzero class $\sigma$ in $\mathrm{ECH}_*(Y, \xi)$, the ECH spectral invariant is

$$c^{ECH}_\sigma(Y, \lambda) = \inf \{ L \mid \sigma \text{ lies in the image of the inclusion } \mathrm{ECH}^L(Y, \lambda) \rightarrow \mathrm{ECH}(Y, \xi) \},$$

where $\mathrm{ECH}^L(Y, \lambda)$ is the subcomplex generated by orbit sets with total action $< L$.

Properties include monotonicity ($c_\sigma(f \lambda) \geq c_\sigma(\lambda)$ for $f > 0$), conformality ($c_\sigma(a\lambda) = a \cdot c_\sigma(\lambda)$), and continuity under smooth perturbations. The operator $U$ acting on ECH further enables relations between spectral invariants of graded classes.

2. Max-Min Energy and Elementary Alternatives

Recent developments define alternative "elementary" spectral invariants via max-min energy of $J$-holomorphic curves in symplectizations or symplectic cobordisms, avoiding deep aspects of Seiberg-Witten theory. For a closed contact three-manifold $(Y, \lambda)$, the $k$th max-min invariant is

$$c_k(Y, \lambda) = \sup_{R > 0} \sup_{J, x_1, \dots, x_k} \inf_{u \in \mathcal{M}^J([-R,0] \times Y; x_1, \dots, x_k)} \mathcal{A}(\alpha_+),$$

where $\mathcal{M}^J([-R,0] \times Y; x_1, \dots, x_k)$ is the moduli space of $J$-holomorphic curves meeting $k$ prescribed points, and $\mathcal{A}(\alpha_+)$ is the total action of the positive asymptotic orbit set. These invariants obey conformality, subadditivity, and a spectral gap closing property, mirroring the algebraic filtration structure of classical ECH.

3. Asymptotics, Volume Recovery, and Weyl Laws

ECH spectral invariants exhibit striking asymptotic behavior, producing volume recovery formulas. For a closed contact manifold $(Y, \lambda)$, the asymptotic law is

$$\lim_{k \rightarrow \infty} \frac{(c_k(Y, \lambda))^2}{k} = 2 \cdot \mathrm{vol}(Y, \lambda), \quad \text{where } \mathrm{vol}(Y, \lambda) = \int_Y \lambda \wedge d\lambda.$$

An analogous law holds for Liouville domains $(X, \omega)$:

$$\lim_{k \rightarrow \infty} \frac{(c_k(X, \omega))^2}{k} = 4 \cdot \mathrm{vol}(X, \omega), \quad \mathrm{vol}(X, \omega) = \int_X \omega \wedge \omega.$$

These laws link the combinatorics of ECH orbits and holomorphic curves to global symplectic invariants and undergird applications to quantitative symplectic geometry (Cristofaro-Gardiner et al., 2012, Irie, 2015).

4. Dynamical Characterizations: Closing Lemmas and Reeb Flows

ECH spectral invariants underlie robust dynamical results, notably the existence and density of periodic Reeb orbits. The spectral gap closing bound,

$$\mathrm{Close}^L(Y, \lambda) \leq c_k(Y, \lambda) - c_{k-1}(Y, \lambda),$$

implies that when the gap vanishes, every point is intersected by a periodic orbit of action at most $L$. This yields a quantitative closing lemma: for generic contact forms, periodic Reeb orbits are dense (Irie, 2015). These invariants also characterize special classes of contact forms:

• Zoll forms on $S^3$: $c_1 = c_2 = \mathcal{A}_{\min}$ precisely characterizes Zoll forms (all Reeb orbits with same minimal period).
• Besse forms: $c_k = c_{k+1}$ for some $k$ indicates every point lies on a closed Reeb orbit.

Lens spaces $L(p,1)$ admit similar ECH-based criteria, though with subtle distinctions arising from the topology and the chain complex structure (Fernandes et al., 7 Oct 2025).

5. Obstructions to Symplectic Embeddings and Algebraic Connections

ECH spectral invariants produce sharp obstructions to symplectic embeddings, especially in four dimensions. ECH capacities, defined via filtered counts (or combinatorial models), yield precise criteria for embeddings between ellipsoids, polydisks, toric domains, etc. In cases with more complicated domains, refined invariants such as convex generator counts augment classical ECH capacities for stronger obstructions (Hutchings, 2014).

Recent works establish deep connections to algebraic geometry by interpreting ECH spectral invariants and capacities as solutions to optimization problems over nef divisors or lattice point counts associated to toric surfaces. Algebraic capacities $c_k^{\mathrm{alg}}(Y,A)$ coincide with ECH capacities in toric settings and satisfy axioms guaranteeing their effectiveness as symplectic embedding obstructions (Chaidez et al., 2020, Wormleighton, 2021).

6. Comparative Structures, Generalizations, and Limitations

ECH spectral invariants are part of a broader landscape encompassing spectral invariants from Hamiltonian Floer theory, Rabinowitz Floer homology, and periodic Floer homology (PFH). Extensions to Rabinowitz Floer homology follow analogous minimax constructions and admit Lipschitz continuity and robustness under global Hamiltonian perturbations, remaining effective even in degenerate or non-Morse scenarios (Albers et al., 2010). Comparisons between ECH and Seiberg-Witten theory via Taubes' isomorphism reveal instances where spectral invariants reduce to previously known invariants or become trivial in certain topological settings (e.g., homotopy 4-spheres (Gerig, 2019), Lagrangian torus invariants (Gerig, 2019)).

Recent advances provide elementary alternatives to PfH/ECH spectral invariants, enabling proofs of closing lemmas and simplicity results without recourse to Floer-theoretic machinery (Edtmair, 2022, Hutchings, 2022), and analogues over various coefficient rings illuminate subtle phenomena such as unbounded spectral norms over $\mathbb{Z}$ (Kawamoto et al., 2023).

7. Future Directions and Open Problems

Ongoing research focuses on:

• Refinement and computation of spectral invariants for broader classes of symplectic and contact manifolds, including higher dimensions, non-toric domains, and singular surfaces.
• Understanding the subleading asymptotics and error terms, governed by algebraic or combinatorial "balance" conditions, which encode finer embedding obstructions and dynamical rigidity (Wormleighton, 2021).
• Development of new invariants or chain-level structures in ECH to capture more intricate dynamical or topological features, with conjectural extensions in progress (Hutchings, 2014).
• Applications to quantitative dynamics, rigidity phenomena, and the geometric group theory of symplectomorphism groups, exemplified by results in Hofer’s geometry and dynamical stability (Kawamoto et al., 2023).

Summary Table: Key Spectral Invariant Formulas

Invariant Type	Defining Formula	Main Properties/Applications
ECH spectral invariant $c_\sigma$	$$c_\sigma(Y, \lambda) = \inf\{L | \sigma \in \mathrm{ECH}^L(Y,\lambda)\}$$	Volume recovery, Reeb orbit existence
Max-min invariant $c_k$	$c_k(Y,\lambda) = \sup \inf \mathcal{A}(\alpha_+)$ (via $J$-hol. curves)	Closing lemmas, dynamical characterizations
ECH capacity $c_k$ (toric domain)	Minimum action over convex generators with lattice point constraints	Embedding obstructions, monotonicity
Algebraic capacity $c_k^{\mathrm{alg}}$	Optimization over nef divisors: $\inf_{D}(D\cdot A\,|\,I(D)\geq 2k)$	Algebraic–symplectic embedding obstructions

ECH spectral invariants, in both their classical and elementary forms, constitute a cornerstone in contemporary symplectic topology—linking analytic, algebraic, and dynamical invariants through holomorphic curve theory, and extending their utility to the most refined questions about Reeb dynamics, symplectic embeddings, and rigidity phenomena.