EHZ Capacity in Symplectic Topology

• EHZ capacity is a numerical invariant defined via the minimum action of closed characteristics on a symplectic domain, capturing its maximal 'size'.
• It features both variational and combinatorial formulations that reduce complex infinite-dimensional problems to finite, yet NP-hard, computations.
• This capacity is pivotal in understanding symplectic rigidity and periodic orbits, with applications in embedding problems and Hamiltonian dynamics.

The Ekeland–Hofer–Zehnder (EHZ) capacity is a fundamental numerical invariant in symplectic topology and Hamiltonian dynamics, quantifying the maximal “size” of a symplectic manifold or domain in terms of Hamiltonian dynamics. It is defined via action-minimizing closed characteristics and provides essential insight into the existence of periodic orbits and symplectic rigidity. The invariant encapsulates both global symplectic geometry (as in embedding problems) and fine dynamical properties, and comes in several variants, including coisotropic, relative, and generalized settings. The following sections elaborate on formal definitions, key properties, combinatorial and variational formulations, relations to other capacities, computational aspects, and major applications.

1. Formal Definition and Variational Characterization

The Ekeland–Hofer–Zehnder capacity $c_{EHZ}(K)$ of a compact convex domain $K \subset \mathbb{R}^{2n}$ is defined as the minimum action of closed characteristics on $\partial K$. For smooth $K$,

$$c_{EHZ}(K) = \inf \{ A(\gamma) : \gamma \text{ is a closed characteristic on } \partial K \},$$

where $A(\gamma) = \int_\gamma \lambda$, with $\lambda$ the Liouville 1-form.

A fully variational characterization—via a minimax principle—also exists:

• The Ekeland–Hofer capacity for a Hamiltonian $H$ is

$$c_{EH}(H) = \inf \{ c \in \mathbb{R} \mid \text{Fadell–Rabinowitz index of } \{ \Phi_H < c \} \geq 1 \},$$

where $\Phi_H$ is the action functional defined on an appropriate Sobolev space of loops.

• The Hofer–Zehnder capacity for an open set $U \subset \mathbb{R}^{2n}$ with symplectic form $\omega$ is

$$c_{HZ}(U, \omega) = \sup \{ \max H - \min H \mid H \in C^{\infty}_0(U, \mathbb{R}),\ \text{no nonconstant periodic orbits of period} \leq 1 \}.$$

In convex domains, $c_{EHZ}(K) = c_{HZ}(K, \omega_0)$, and both coincide with the minimal action of a closed characteristic on $\partial K$ (Gluskin et al., 2015, Haim-Kislev, 2017, Irie, 2019).

2. Combinatorial Formulas and Computational Complexity

A critical advance is the existence of explicit combinatorial formulas for $c_{EHZ}(K)$, especially when $K$ is a convex polytope. Haim-Kislev’s formula for a convex polytope $K \subset \mathbb{R}^{2n}$ with outer facet normals $\{ n_i \}_{i=1}^N$ and facet heights $h_i$ reads

$$c_{EHZ}(K) = \frac12 \left[ \max_{\sigma \in S_N,\, (\beta_i) \in M(K) } \sum_{1 \leq j < i \leq N} \beta_{\sigma(i)} \beta_{\sigma(j)}\, \omega(n_{\sigma(i)}, n_{\sigma(j)}) \right]^{-1},$$

where $M(K)$ is the set of nonnegative weights satisfying normalization and equilibrium constraints. Alternatively, the minimum is taken over all collections $\{\beta_i, n_i\}$: $$c_{EHZ}(K)^{-1} = \min_{ \substack{ (\beta_i, n_i) \in M_2(K) \ \text{constraints} } } \left( \sum_i \beta_i h_K(n_i) \right)^2.$$ This formulation reduces the infinite-dimensional variational problem to finite combinatorics on facet data (Haim-Kislev, 2017, Shi et al., 2019).

However, the computational problem remains fundamentally hard: computing the $c_{EHZ}$ capacity of a polytope is NP-hard, even in the case of simplices (Leipold et al., 15 Feb 2024). The reduction proceeds by encoding the feedback arc set problem in bipartite tournaments into the maximization problem of the $c_{EHZ}$ capacity via the combinatorial formula, implying the absence of efficient general algorithms.

3. Relationships with Other Symplectic Capacities

In the landscape of symplectic capacities, $c_{EHZ}$ occupies a central role:

• On convex centrally symmetric bodies, $c_{EHZ}$ is asymptotically equivalent (up to dimension-independent constants) to the displacement energy, cylindrical, and linearized capacities:

$$\widehat{c}(K) \leq 4\, c_{EHZ}(K)\,,\quad 1 \leq c_{EHZ}(K) \leq c^\sim(K) \leq \frac{4}{\|J\|_{K^\circ \to K}}$$

where $\|J\|_{K^\circ \to K}$ is the operator norm of the complex structure $J$ from the polar norm (Gluskin et al., 2015).

• The symplectic homology capacity $C_{SH}(K)$ coincides with $c_{EHZ}(K)$ for convex subsets of $T^*\mathbb{R}^n$, as established via loop space homology isomorphisms (Irie, 2019).
• A “coisotropic” version localizes the capacity to measure the “symplectic size” relative to a coisotropic submanifold, with corresponding combinatorial formulas and (super)additivity properties in the presence of hyperplane cuts (Lisi et al., 2013, Shi et al., 2019).

4. Additivity, Inequalities, and Extremal Properties

Subadditivity and Brunn–Minkowski Type Inequalities

The $c_{EHZ}$ capacity displays a subadditivity property for polytopes cut by hyperplanes: $c_{EHZ}(K) \leq c_{EHZ}(K_1) + c_{EHZ}(K_2),$ where $K_1, K_2$ are the resulting pieces (Haim-Kislev, 2017, Irie, 2019). This generalizes, via convex hulls, to more general sets. Furthermore, an analogue of the classical Brunn–Minkowski inequality holds for $c_{EHZ}$: $$c_{EHZ}(K_1 + K_2)^{1/2} \geq c_{EHZ}(K_1)^{1/2} + c_{EHZ}(K_2)^{1/2},$$ providing a symplectic counterpart to convex geometry (Kerman et al., 2020). Notably, this property does not extend to higher-index capacities, which can violate such concavity.

Extremal Examples and Minimal Capacity

Among centrally symmetric convex bodies, a sharp lower bound is known (Berezovik, 2023): $$c_{EHZ}(X) \geq 2 + \frac{1}{n},\quad X \subset \mathbb{R}^{2n} \text{ centrally symmetric}.$$ Construction of symplectically self-polar polytopes attaining equality demonstrates optimality.

Local maximizers of the normalized higher Ekeland–Hofer capacities on star-shaped domains in dimension four are precisely rational ellipsoids corresponding to “jump indices” determined by the Reeb flow period multiplicities (Baracco et al., 2023).

5. Generalizations and Relative Capacities

Several significant generalizations stretch the notion of EHZ capacity:

• Ψ-relative capacities: For a symplectomorphism $\Psi \in Sp(2n, \mathbb{R})$, relabeling $x(T) = \Psi x(0)$, one obtains “Ψ-Ekeland–Hofer–Zehnder” capacities with corresponding min–max and representation formulas, detecting generalized characteristics and leafwise intersection points. For convex domains,

$$c_{HZ}^\Psi(D, \omega_0) = \min \{ A(x) > 0 : x \text{ is a generalized }\Psi\text{-characteristic on } \partial D \}$$

(Jin et al., 2019, Shi et al., 2019).

• Coisotropic and Relative Capacities: Modifying admissibility to focus on leafwise return times enables the definition of capacities measuring the symplectic size “relative to” a coisotropic submanifold. These play a role in refined non-squeezing theorems and the existence theory for leafwise chords (Lisi et al., 2013, Shi et al., 2019).
• Symmetry Constraints: For convex domains invariant under an anti-symplectic involution $\tau$, the $\tau$-symmetrical EHZ capacity equals the classical EHZ capacity, as both select the minimal action in the symmetric class (Shi et al., 2020).

6. Applications, Impact, and Computational Directions

The EHZ capacity is instrumental in symplectic embedding problems, providing both upper and lower bounds for the “size” of sets under symplectic transformations; this includes non-squeezing and rigidity phenomena. Combinatorial formulas enable, in principle, explicit computation for polytopes, though NP-hardness places practical limitations on high-dimensional cases (Leipold et al., 15 Feb 2024). The capacity also encodes dynamical information, such as the threshold energy for the existence of (contractible or noncontractible) periodic orbits, connects to systolic inequalities, and informs systolic rigidity for metrics on surfaces (Benedetti et al., 2020, Bimmermann, 2020).

In toric symplectic manifolds, combinatorial invariants (toric width) derived from Delzant polytopes give effective lower bounds for $c_{HZ}$, and hence $c_{EHZ}$ in favorable circumstances (Liu, 2023). These provide computational tools via lattice point enumeration and facet combinatorics.

Most importantly, the EHZ capacity is the unique capacity (up to normalization and asymptotic equivalence) on centrally symmetric convex bodies in high dimensions, coinciding with the displacement energy and cylindrical capacities up to a universal constant (Gluskin et al., 2015).

7. Open Problems and Future Research

Substantial questions remain open regarding:

• The explicit computation of $c_{EHZ}$ beyond low-dimensional or highly symmetric bodies, due to inherent NP-hardness.
• Volume-capacity inequalities and their role in Mahler’s and Viterbo’s conjectures (Berezovik, 2023).
• The algebraic and analytic structure of symplectic capacities, in particular, minimal generating sets and representability (Joksimović et al., 2020).
• Extension to infinite-dimensional (PDE) settings and further interactions with Floer and symplectic homology.

A plausible implication is that, while $c_{EHZ}$ and its relatives encode deep symplectic properties and exhibit powerful rigidity, their full computational and dynamical landscape is governed by complex combinatorial and geometric structures whose complete classification remains open.