Ekeland–Hofer Capacities

• Ekeland–Hofer Capacities are a series of symplectic invariants defined via the minimal action of closed characteristics on convex domains.
• They employ variational and combinatorial frameworks to extract quantitative measures, linking Hamiltonian dynamics with convex geometry.
• Their symmetry-reduced and equivariant variants provide explicit computational formulas and robust obstructions for symplectic embedding problems.

The Ekeland–Hofer Capacities constitute a hierarchy of symplectic invariants that play a central role in symplectic geometry, Hamiltonian dynamics, and the paper of symplectic embedding problems. This family of capacities extends the classical notion of symplectic capacity to a sequence indexed by positive integers, encoding subtle information about the geometry and dynamics of a domain in symplectic space. Their combinatorial, variational, and homological formulations link dynamical systems, convex geometry, and symplectic topology. Variants adapted to symmetries—such as reversal by anti-symplectic involutions—yield refined tools for distinguishing symplectic embeddings and proving rigidity phenomena.

1. Definition and Variational Formulation

Let $X\subset\mathbb{R}^{2n}$ be a compact, strictly convex domain with smooth boundary $\partial X$. The Ekeland–Hofer capacity (and its Zehnder refinement) is defined via the action of closed characteristics on $\partial X$. Concretely:

• The Ekeland–Hofer–Zehnder capacity is

$$c_{\mathrm{EHZ}}(X) = \inf \left\{\, \mathcal{A}(x) \;\bigm|\; x\text{ is a closed characteristic on }\partial X \,\right\},$$

where each closed characteristic $x:\mathbb{R}/T\mathbb{Z}\to\partial X$ solves $\dot{x}(t)\in\ker(\omega|_{T_{x(t)}\partial X})$, with the action

$\mathcal{A}(x) = \int_0^T \lambda(\dot{x}(t))\,dt,$

for the standard Liouville 1-form $\lambda$ and symplectic form $\omega = d\lambda$ (Shi et al., 2020, Gutt et al., 2017).

The capacity admits a dual variational characterization as the first positive critical value of the action functional

$$\Psi(u) = \int_0^1 H^*(-J\dot{u}(t))\,dt - \int_0^1 \langle u(t),-J\dot{u}(t)\rangle\,dt,$$

where $H$ is the support (gauge) function for $X$, $H^*$ is its Legendre dual, and $u$ ranges over $H^1$-loops in $\mathbb{R}^{2n}$ with zero mean.

Higher capacities $c_k^{\mathrm{EH}}(X)$ constitute a non-decreasing sequence, defined via min–max methods in the loop space using the Fadell–Rabinowitz index or filtered symplectic homology (Gutt et al., 12 Dec 2024, Matijević, 17 Oct 2024).

2. Symmetrized and Equivariant Variants

For a domain $X$ invariant under a linear anti-symplectic involution $\tau$ (such as central symmetry or complex conjugation), the τ-symmetrical Ekeland–Hofer–Zehnder capacity is

$$c_\tau(X) = \inf \left\{ \mathcal{A}(x) \mid x \text{ is a } \tau\text{-symmetric closed characteristic on } \partial X \right\},$$

where $\tau$-symmetric means $x(T-t)=\tau(x(t))$ for all $t$.

The fundamental result is the equality

$c_{\mathrm{EHZ}}(X) = c_\tau(X)$

for strictly convex $\tau$-invariant domains (Shi et al., 2020). This eliminates any “symmetry penalty” at the first capacity level and ensures reversible minimizers exist for the action functional.

More generally, for a linear symplectic map $Y$, one defines $c_{EH}^Y(B)$ as the minimal action among $Y$-characteristics, i.e., closed orbits with $x(T) = Yx(0)$ (Jin et al., 2019).

3. Combinatorial Formulas for Convex Polytopes

For convex polytopes $K\subset\mathbb{R}^{2n}$, the capacity admits a combinatorial representation. Let $F_1,\ldots,F_m$ denote the facets with outward normals $n_i$ and support heights $h_i = h_K(n_i)$. Define

$$M(K) = \left\{ (B_1,\ldots,B_m) \;\middle|\; B_i\geq0,\,\sum B_i h_i=1,\,\sum B_i n_i=0 \right\}.$$

Then

$$c_{\mathrm{EHZ}}(K) = \tfrac{1}{2} \min_{(B_i)\in M(K)} \max_{\sigma\in S_m} \sum_{1\leq j<i\leq m} B_i B_j \,\omega(n_{\sigma(i)},n_{\sigma(j)}),$$

where $S_m$ is the permutation group; $\omega$ is the standard symplectic form (Shi et al., 2019, Haim-Kislev, 2017).

This finite-dimensional convex optimization replaces the a priori infinite-dimensional variational problem, enabling explicit computation in many cases.

For generalized settings:

• Ψ-EHZ capacities: boundary closing condition is $x(T) = \Psi x(0)$ for $\Psi\in Sp(2n)$.
• Coisotropic EHZ capacities: restrict to leafwise chords between coisotropic subspaces.

Combinatorial representation formulas exist for these cases as well (Shi et al., 2019, Jin et al., 2019).

4. Symmetry Reduction, Uniqueness, and Rigidity

A crucial component is the symmetrization lemma: for any closed characteristic $x$, the averaging

$y(t) = \frac{1}{2}(x(t) + \tau(x(T-t)))$

yields a $\tau$-symmetric loop with action no greater than $x$ (Shi et al., 2020). This implies the minimal action is achieved by a symmetric curve when $X$ is strictly convex and $\tau$-invariant.

This conclusion extends to:

• Central symmetry ($\tau=-\mathrm{id}$): the minimal closed characteristic is centrally symmetric on such bodies (Akopyan et al., 2017).
• Other symmetry subgroups (e.g., roots of unity): minimal characteristics inherit these symmetries under analogous arguments.

Thus, invariance under linear (anti-)symplectic group actions forces minimizers to respect the symmetry, with attendant consequences for Mahler geometry and embedding obstructions.

5. Applications and Implications

Ekeland–Hofer capacities provide sharp obstructions for symplectic embeddings and encode dynamical properties such as closed Reeb orbits. Matching of the first capacity with its $\tau$-symmetric version simplifies variational arguments in symmetric settings and establishes sharper distinction tools, especially for centrally symmetric and real-invariant convex bodies. Specific consequences include:

• Mahler geometry: capacity-minimizing curves must be symmetric for centrally symmetric convex bodies, influencing Mahler-type inequalities (Shi et al., 2020, Akopyan et al., 2017).
• Explicit formulas: for real-invariant ellipsoids (e.g., under complex conjugation), the problem reduces to real-symmetric orbits, allowing for direct computations (Shi et al., 2020, Shi et al., 2021).
• Structural results: the subadditivity, superadditivity, and product structures for (coisotropic) capacities under convex decompositions add to the understanding of how these invariants behave under geometric operations (Shi et al., 2019, Haim-Kislev, 2017).

6. Computational Complexity and Explicit Examples

Even with combinatorial formulas, computing the Ekeland–Hofer–Zehnder capacity for polytopes is NP-hard, as recently proved in (Leipold et al., 15 Feb 2024). Special cases, such as ellipsoids and some higher-symmetry bodies, remain tractable with explicit formulas. For instance, the capacity for the product of $K$ with its $J$-twist or with $K$ itself admits elementary upper bounds in terms of facet support distances (Shi, 28 May 2024).

Examples:

• For the polydisk $P=[-1,1]^{2}$, combinatorial weighting yields $c_{EHZ}(P)=2$.
• For symmetric bodies, the minimal action is realized by a centrally symmetric loop (Akopyan et al., 2017).

7. Generalizations and Future Directions

The Ekeland–Hofer capacities have deep connections with Floer and symplectic homology, equivariant versions, and filtered persistence invariants (Gutt et al., 2017, Gutt et al., 12 Dec 2024, Matijević, 17 Oct 2024). Their spectral information characterizes key dynamical classes, such as Besse and Zoll Reeb flows, through the collapse or coincidence of $n$ consecutive capacities (Gutt et al., 23 Nov 2025). The equivalence (proven via filtered $S^1$-equivariant homology) ensures their relevance as spectral invariants for all star-shaped domains, with impact on symplectic rigidity, uniqueness of extremal shapes, and the paper of symplectic embedding problems.

The combinatorial, variational, and homological perspectives on Ekeland–Hofer capacities constitute one of the deepest bridges between Hamiltonian dynamics, convex geometry, and modern symplectic topology. Their properties under symmetry, decomposition, and embedding continue to provide fertile ground for further research.