Symplectic Capacities

• Symplectic capacities are invariants that assign a nonnegative number (or infinity) to a symplectic manifold, based on axioms like monotonicity, conformality, and nontriviality.
• They originated from Gromov’s Nonsqueezing Theorem and have evolved to include Floer-theoretic invariants, providing refined obstructions in embedding and Hamiltonian dynamics.
• Recent studies focus on their equivalence, computability on toric and convex domains, and the impact of normalization conditions on rigidity and embedding phenomena.

A symplectic capacity is a symplectic invariant—an assignment to each symplectic manifold (or subset) of a nonnegative number, possibly infinity, satisfying monotonicity under symplectic embeddings, conformality under scaling, and a nontriviality/normalization condition on standard test domains (balls and cylinders). Symplectic capacities are central to quantitative symplectic topology, capturing non-squeezing phenomena and providing embedding obstructions that refine those given by volume. The study of symplectic capacities originated with Gromov’s Nonsqueezing Theorem and was systematically developed by Ekeland–Hofer, Hofer–Zehnder, Viterbo, and many others. Capacities exist in a rich spectrum, ranging from the classical Gromov width to Floer-theoretic invariants, and are fundamental to rigidity results, embedding questions, and Hamiltonian dynamics.

1. Formal Axioms and Examples of Symplectic Capacities

Let $(\mathbb{R}^{2n},\omega)$ denote the standard symplectic vector space. A symplectic capacity $c$ is a function from subsets of $\mathbb{R}^{2n}$ (or suitable classes of symplectic manifolds) to $[0,\infty]$ satisfying the following axioms (Gluskin et al., 2015):

• Monotonicity: If $U \subset V$ then $c(U) \leq c(V)$.
• Conformality: For any symplectomorphism $\varphi$ with $\varphi^*\omega = a\omega$, $c(\varphi(U)) = |a|\, c(U)$.
• Nontriviality: For any $r > 0$,

$0 < c(B^{2n}(r)), \qquad c(Z^{2n}(r)) < \infty,$

where $B^{2n}(r)$ is the Euclidean ball and $Z^{2n}(r)$ is the standard symplectic cylinder.

A normalized capacity further requires

$c(B^{2n}(r)) = c(Z^{2n}(r)) = \pi r^2.$

Key classical and Floer-theoretic capacities:

Capacity	Definition	Key Property
Gromov width ($c_G$)	$\sup\{\pi r^2 \mid B^{2n}(r)\hookrightarrow U\}$	Smallest normalized
Cylindrical capacity ($c_{cyl}$)	$\inf\{\pi r^2 \mid U\hookrightarrow Z^{2n}(r)\}$	Largest normalized
Ekeland–Hofer–Zehnder ($c_{EHZ}$)	Minimal action of closed characteristic on $\partial K$ for convex $K$	Action-theoretic
Hofer–Zehnder ($c_{HZ}$)	Supremum of energy values for autonomous Hamiltonians without fast periodic orbits	Dynamical
Displacement energy ($c_{disp}$)	Infimum Hofer energy to disjoin $U$ from itself via Hamiltonian diffeomorphisms	Displacement-type

For star-shaped domains the above capacities often coincide, especially for convex sets and ellipsoids (Gluskin et al., 2015, Gutt et al., 2023).

2. Equivalence and Asymptotic Comparison of Capacities

A central conjecture (Viterbo, Hofer–Zehnder, Hermann) states that all normalized symplectic capacities coincide on convex subsets of $\mathbb{R}^{2n}$; equivalently, that for any convex $K$ and any two normalized capacities $c_1$, $c_2$, $c_1(K) = c_2(K)$ (Gluskin et al., 2015).

Asymptotic Equivalence Theorem:

On the class of centrally symmetric convex bodies in $\mathbb{R}^{2n}$, the following holds: $$c_{EHZ}(K) \le C\, c_{cyl}(K) \le C^2\, c_{disp}(K) \le C^3\, c_{EHZ}(K),$$ for an absolute constant $C$ (explicitly, $C=4$ in (Gluskin et al., 2015)). Thus, while the conjecture remains open in full generality, these capacities are universally equivalent up to dimension-independent constants for such domains. In high dimensions, the ratios are $O(n)$ (Gluskin et al., 2015).

Proof strategies exploit geometric dualities (gauge and support function), projection and symplectic linear group actions, action-length estimates tied to convex gauge functions, and classical volume inequalities (Rogers–Shephard, John's theorem).

3. Normalization Conditions: Ball, Cylinder, and Cube

Normalization choices critically impact the behavior and comparison of capacities. In addition to the classical ball normalization (fixing the capacity on the standard ball/cylinder), cube normalization has been introduced, particularly suitable for toric domains modeled on polydisks and cubes (Gutt et al., 2022).

• Ball normalization: $c(B^{2n}(1)) = c(Z^{2n}(1)) = \pi$ (Gromov width, cylindrical capacity).
• Cube normalization: $c(P_n(1)) = c(N_n(1)) = 1$, where $P_n(a)$ is the polydisk and $N_n(a)$ is the union of cylinders of radius $a$. Examples: the cube embedding capacity, the NDUC capacity, and the Lagrangian capacity (Gutt et al., 2022).

Cube normalization enables sharper rigidity results:

All cube-normalized capacities agree on monotone toric domains (toric domains with “positively normal” boundary), and a sharp explicit value is given in terms of the diagonal length of the associated moment region (Gutt et al., 2022). However, for general (non-monotone) domains, cube- and ball-normalized capacities can diverge, as explicit polygonal examples show.

4. Computability, Characterization, and Uniqueness on Special Families

Toric domains (subsets of $\mathbb{C}^n$ defined by moment polytope inequalities) provide a broad, computable subclass where symplectic capacities can often be evaluated explicitly.

• Agreement on monotone toric domains:

All normalized capacities coincide for monotone toric domains in any dimension (Cristofaro-Gardiner et al., 2023). For convex (and many concave) toric domains in dimension four, such agreement is established via combinatorial formulas involving “lattice polygons” computed from dual norms associated to the polytope (Landry et al., 2013, Gutt et al., 2017). The sharpness extends to the intersection of ellipsoids and cube embeddings.

• Characterization via values on ellipsoids:

For $k$-normalized capacities—those agreeing with the $k^\text{th}$ Ekeland–Hofer capacity on all ellipsoids—uniqueness holds for $k=n=2$ on four-dimensional convex toric domains but fails on general monotone domains or in higher dimensions ($k=n\ge3$) (Gutt et al., 2023).

• Flat tori/cotangent bundles:

On the unit disc cotangent bundle $D^*T^n$ of a flat torus $T^n$, all normalized capacities coincide with twice the systole of the torus (Benedetti et al., 2023). This phenomenon generalizes to flat reversible Finsler tori and reflects a deep connection between systolic geometry and symplectic capacities.

• Local rigidity near the ball:

In a $C^2$-neighborhood of the Euclidean ball in $\mathbb{C}^n$, all normalized symplectic capacities agree and equal the systole of the domain boundary. This local coincidence fails in $C^1$ (Abbondandolo et al., 2023).

5. Product Structures and Tensor Power Tricks

Symplectic $p$-products:

Given convex bodies $K\subset\mathbb{R}^{2n}$ and $T\subset\mathbb{R}^{2m}$, the $p$-product $K\times_p T$ smoothly interpolates between Cartesian and direct sum products. The behavior of capacities under $p$-products is precisely controlled (Haim-Kislev et al., 2021):

• For $p\ge 2$, $$c_{EHZ}(K\times_p T) = \min\{c_{EHZ}(K),c_{EHZ}(T)\}$$.
• For $1\le p < 2$, the capacity is given by an explicit “mean” of the capacities.

The “tensor power trick”: Iterating $p$-products allows reducing certain volume–capacity-type conjectures to high-dimension asymptotics, notably Viterbo’s conjecture. If the systolic ratio is bounded in high dimension, it is bounded in all dimensions (Haim-Kislev et al., 2021).

6. Role in Embedding Obstructions, Rigidity, and Quantitative Symplectic Topology

Symplectic capacities constitute the foundation for non-squeezing theorems beyond Gromov’s original result, providing obstructions to both local and global symplectic embeddings not detectable by volume. Key applications include:

• Quantitative non-squeezing: For a symplectic embedding to exist between domains, all symplectic capacities (as appropriate) must satisfy the associated inequalities.
• Recognition and classification: The collection of all (generalized) symplectic capacities acts as a complete invariant for large classes of symplectic categories (e.g., simply connected, exact manifolds with negative helicity boundary) (Guggisberg et al., 2022).
• Nonexistence of intermediate capacities: There are no symplectic capacities that interpolate between the Gromov width and volume in higher dimensions, resolving questions posed by Hofer—capacity theory is forced to dichotomize (Pelayo et al., 2012).
• Product and stabilization phenomena: Certain symplectic capacities are stable under Cartesian products (e.g., stabilized embedding obstructions for ellipsoids and polydisks) (Siegel, 2019, McDuff et al., 2021). For coisotropic submanifolds, relative capacities extend non-squeezing and Hamiltonian intersection theory (Lisi et al., 2013).
• Quantum and convex-geometry links: The Hofer–Zehnder capacity relates to quantum uncertainty via the concept of “$\hbar$-polar” pairs, connecting symplectic non-squeezing to classical uncertainty bounds (Gosson, 2013).

7. Capacity Theory: Limitations, Non-Uniqueness, and Open Directions

The set of all (generalized and normalized) symplectic capacities is vast and highly structured, but not all can be generated from a small or canonical family. No countable or continuum-sized generating set suffices except in low complexity situations. “Domain” or “target” representable capacities are exceedingly rare in high-dimensional symplectic categories (Joksimović et al., 2020).

Moreover, while complete agreement holds for all normalized capacities on monotone toric domains or in neighborhoods of the ball, uniqueness can fail for more general domains and under alternative normalization conditions (e.g., higher $k$–normalizations or cube normalizations on non-monotone sets) (Gutt et al., 2022, Gutt et al., 2023).

Significant open problems and research directions include:

• Full resolution of Viterbo’s conjecture for all convex domains.
• Explicit computation of capacities for higher-dimensional and non-toric domains.
• Further understanding the relationships among Floer-theoretic, SFT, and classical capacities in high complexity settings.
• Extension of combinatorial and algebraic methods from toric domains to general symplectic topologies.