Coisotropic Hofer-Zehnder Capacity

• Coisotropic Hofer-Zehnder capacity is a generalization of the classical symplectic capacity that quantifies embedding obstructions and leafwise Hamiltonian chords under coisotropic constraints.
• It satisfies axiomatic properties such as monotonicity, conformality, and nontriviality, ensuring its consistency in relative symplectic topology.
• Variational principles and filtered Floer theory underpin its computation, leading to energy–capacity inequalities and applications in non-squeezing phenomena.

The coisotropic Hofer-Zehnder capacity generalizes the classical Hofer-Zehnder symplectic capacity to situations incorporating coisotropic submanifold constraints, quantifying symplectic embedding obstructions and the existence of leafwise Hamiltonian chords within relative or constrained symplectic topology. It unifies perspectives from Lagrangian Floer theory, convex geometry, and Hamiltonian dynamics, and plays a pivotal role in energy–capacity inequalities, non-squeezing results, and combinatorial formulas for symplectic capacities of convex domains.

1. Definition and Framework

Given a symplectic manifold $(M^{2n},\omega)$ and a coisotropic submanifold $N^{n+k}\subset M$, the characteristic distribution $$T_xN^\omega = \{v\in T_xM \mid \omega(v, T_xN) = 0\}$$ defines integrable leaves. For a chosen coisotropic equivalence relation $\sim$ (most prominently, the leaf relation: $x \sim y \iff x, y$ are on the same leaf), the coisotropic Hofer-Zehnder capacity is defined by considering the supremum of the maximum of admissible Hamiltonians relative to the coisotropic structure.

A (simple) Hamiltonian $H:M\to\mathbb{R}$ is admissible if:

• $H$ is compactly supported and attains maximum $m(H)>0$ away from $N$.
• The flow of $X_H$ starting at $N$ yields either constant orbits or "return times" to $N$ (with respect to $\sim$) strictly greater than 1:

$$T_y = \inf\{ t>0 \mid y(0), y(t) \in N,~ y(0) \sim y(t)\} > 1$$

• This enforces the absence of "short" leafwise chords under the Hamiltonian flow.

The capacity is: $$c_{HZ}^{\rm coiso}(M,N,\omega,\sim) = \sup\{ m(H)~|~ H \text{ is admissible relative to } (N,\sim) \}$$ In the linear model, with $M=\mathbb{R}^{2n}$, $N=D\cap \mathbb{R}^{n,k}$, this is commonly denoted $c_{HZ, \rm coiso}(D; \mathbb{R}^{n,k})$ (Jin et al., 2019, Lisi et al., 30 Oct 2025, Lisi et al., 2013).

2. Axiomatic Properties and Comparison

Coisotropic Hofer–Zehnder capacities satisfy standard axioms for symplectic capacities, specialized to the relative (coisotropic) context:

• Monotonicity: Under relative symplectic embeddings, the capacity is non-decreasing:

$$\phi:(M_0,N_0) \hookrightarrow (M_1,N_1) \implies c(M_0,N_0)\leq c(M_1,N_1)$$

• Conformality: Under symplectic rescaling $\omega \mapsto \lambda^2\omega$, $c_{HZ}^{\rm coiso}$ scales by $\lambda^2$.
• Nontriviality: On standard models, e.g., for the ball $B^{2n}(1), B^{n,k}(1)$, the capacity is computable (e.g., $\pi/2$ for the leafwise model) (Lisi et al., 2013, Jin et al., 2019, Lisi et al., 30 Oct 2025).
• In the special case $k=n-1$, $c_{HZ, \rm coiso}$ coincides with the classical Hofer-Zehnder capacity:

$$c_{HZ, \rm coiso}(D;\mathbb{R}^{n,n-1}) = c_{HZ}(D)$$

Generally, the inequalities

$$c_{HZ}(D) \leq c_{HZ,\rm coiso}(D; \mathbb{R}^{n,k}) \leq c_{HZ,\rm coiso}(D; \mathbb{R}^{n,k+1})$$

hold for convex domains (Jin et al., 2019).

3. Variational and Floer-Theoretic Construction

For convex domains $D$, the coisotropic Hofer–Zehnder capacity admits variational representations:

• The capacity equals the minimal action among generalized leafwise chords $z:[0,T]\to \partial D$ with $z(0),z(T)\in \mathbb{R}^{n,k}$, $z(T)-z(0)\in V_{n,k}$, and tangent conditions determined by the convex geometry and symplectic structure:

$$c_{HZ, \rm coiso}(D; \mathbb{R}^{n,k}) = \min \{\mathcal{A}(z)>0\mid z \text{ a generalized leafwise chord}\}$$

with

$$\mathcal{A}(z) = \int_0^T -\omega_0(J\dot{z}(t),z(t)) dt$$

• Dual (Clarke) variational expressions involve minimizing action integrals over Sobolev loops with coisotropic boundary conditions (Jin et al., 2019, Shi et al., 2019).

Filtered Floer theory underpins the definition and computation for Lagrangian submanifolds $L$ within symplectic manifolds $(W,\omega)$:

• Using admissible autonomous $H$, one considers small nondegenerate perturbations $H_\varepsilon$ and filtered Floer complexes $CF^{(a,b)}(L;H_\varepsilon)$.
• Local Floer homology at the maximum level is isomorphic to a generator in top degree, allowing Morse-theoretic computations for the local contribution, especially over balls in $L$ (Lisi et al., 30 Oct 2025).

4. Explicit Formulas, Inequalities, and Computations

Representation Formulas

For convex polytopes $K\subset \mathbb{R}^{2n}$, the capacity can be computed combinatorially:

• Assign to each facet $F_i$ the outward normal $n_i$, support height $h_i$, and define weight vectors $B=(B_1,\ldots,B_F)$ subject to:

$$\sum B_i h_i = 1, \quad \sum B_i n_i \in V_{n,k}, \quad \sum_{j<i} B_i B_j \omega_0(n_j, n_i) > 0$$

• The coisotropic capacity is:

$$c_{HZ}^{\rm coiso}(K) = 2 \min_{B \in M(K)} \sum_{j<i} B_i B_j \omega_0(n_j, n_i)$$

where the minimization runs over admissible weight vectors (Shi et al., 2019).

Inequalities

Brunn–Minkowski type inequalities extend to coisotropic capacities:

• For convex $D, K$ and Firey $p$-sum $D+_pK$,

$$(c_{HZ, \rm coiso}(D+_pK;C))^2 \geq (c_{HZ, \rm coiso}(D;C))^2 + (c_{HZ, \rm coiso}(K;C))^2$$

with analogous power mean inequalities (Jin et al., 2019).

Energy–Capacity Inequality

For closed, monotone, and displaceable Lagrangian submanifolds $L \subset (W, \omega)$,

$\hat{c}_{HZ}(W, L) \leq d(L)$

where $d(L)$ is the displacement energy (Lisi et al., 30 Oct 2025). The proof combines Floer-theoretic invariants, action filtration, and spectral properties.

5. Applications and Existence Results

Non-Squeezing

Coisotropic capacities provide sharp symplectic embedding obstructions. If a symplectic embedding $\Phi:(B^{2n}(r),B^{n,k}(r)) \hookrightarrow (M,N)$ identifies coisotropic models, then

$$c_{HZ,\rm coiso}(B^{2n}(r); B^{n,k}(r)) \leq c_{HZ,\rm coiso}(M,N)$$

For balls in $\mathbb{R}^{2n}$ cut by $N = \mathbb{R}^{n,k}$, the explicit value is: $$c_{HZ,\rm coiso}(B^{2n}(a,R);B^{n,k}(r)) = \left(\arcsin\frac{r}{R} - \frac{r}{R}\sqrt{1-\frac{r^2}{R^2}}\right) R^2$$ (Jin et al., 2019, Lisi et al., 2013).

Chord Existence

Whenever the coisotropic capacity is finite for a neighborhood of an energy surface transverse to $N$, almost every nearby energy level admits a non-constant leafwise chord of period at most 1. This provides a generalization of the Hamiltonian chord existence theorem to the coisotropic setting (Lisi et al., 2013, Lisi et al., 30 Oct 2025).

Reverse Subadditivity in 2D

For convex domains $D \subset \mathbb{R}^2$ and a "coisotropic" line, hyperplane cuts yield: $$c_{HZ}^{\rm coiso}(D,D\cap\mathbb{R}^{1,0}) \geq c_{HZ}^{\rm coiso}(D_1,D_1\cap\mathbb{R}^{1,0}) + c_{HZ}^{\rm coiso}(D_2,D_2\cap\mathbb{R}^{1,0})$$ in contrast to the classical subadditivity of Ekeland-Hofer-Zehnder capacities (Shi et al., 2019). This is established via count of symplectic areas associated to leafwise chords.

6. Selected Examples

Model	Capacity Value	Reference
Ball $(\mathbb{B}^{2n}(R), \mathbb{R}^n)$	$\pi R^2$	(Lisi et al., 30 Oct 2025)
Ellipsoid $E(r_1, ..., r_n)$ w/ $\mathbb{R}^{n,k}$	$$\min\{2\min_{1\leq i \leq k} r_i^2,~\min_{k<j\leq n} r_j^2\}$$	(Jin et al., 2019)
Cotangent disk $(D^*L(r), L)$	$\frac{1}{2} r^2$ (up to Liouville factor)	(Lisi et al., 30 Oct 2025)
Waffle cylinder $W^{2n}(1)$	$\pi/2$	(Jin et al., 2019)
Polydisk $P(r_1, ..., r_n)$ w/ $\mathbb{R}^{n,k}$	$\min\{2\min_{i>k} r_i^2, \min_{i\leq k} r_i^2\}$	(Jin et al., 2019)

Computational procedures for polytopes reduce to a finite-dimensional minimization over facet weights (see Section 4 above) (Shi et al., 2019).

7. Outlook, Generalizations, and Open Problems

The framework of coisotropic Hofer-Zehnder capacities admits further generalizations:

• Higher codimension and non-autonomous settings: There is an expectation of natural extensions to non-autonomous Hamiltonians or general coisotropic submanifolds, potentially yielding more refined energy–capacity inequalities and rigidity phenomena (Lisi et al., 30 Oct 2025).
• Limit cases and equivalences: For $k=n$, the coisotropic formula reduces to the classical Ekeland-Hofer-Zehnder capacity; for $k=0$, it quantifies the Lagrangian case.
• Subadditivity in higher dimensions: While a reverse subadditivity holds in dimension 2, its full extension to higher codimension remains open except in some special settings (Shi et al., 2019).
• Connections to spectral invariants and persistent homology: The capacity is closely tied to spectral invariants arising from Floer theory, and its calculation often blends Morse-theoretic, variational, and combinatorial techniques (Lisi et al., 30 Oct 2025).
• Technical prerequisites: The use of Floer and quantum homology requires monotonicity and minimal Maslov number constraints ($N_L\geq 2$) for the involved Lagrangians.

Further research continues to explore applications to embedding obstructions, leafwise dynamics, and deeper links to symplectic rigidity phenomena.