Coisotropic Ekeland–Hofer capacities

Abstract

For subsets in the standard symplectic space $(\mathbb {R}^{2n},\omega _0)$ whose closures are intersecting with coisotropic subspace $\mathbb {R}^{n,k}$ we construct relative versions of the Ekeland–Hofer capacities of the subsets with respect to $\mathbb {R}^{n,k}$, establish representation formulas for such capacities of bounded convex domains intersecting with $\mathbb {R}^{n,k}$. We also prove a product formula and a fact that the value of this capacity on a hypersurface $\mathcal {S}$ of restricted contact type containing the origin is equal to the action of a generalized leafwise chord on $\mathcal {S}$.

1. Introduction

1.1 Coisotropic capacity

Recently, Lisi and Rieser [29] introduced the notion of a coisotropic capacity (i.e. a symplectic capacity relative to a coisotropic submanifold of a symplectic manifold), and discussed their motivations and backgrounds. Let $(M,\omega )$ be a symplectic manifold and $N\subset M$ a coisotropic submanifold. (In this paper all manifolds are assumed to be connected without special statements!) An equivalence relation $\sim$ on $N$ was called a coisotropic equivalence relation if $x$ and $y$ are on the same leaf then $x\sim y$ (cf. [29, definition 1.4]). Special examples are the trivial relation defined by $x\sim y$ for every pair $x, y\in N$ and the so-called leaf relation defined by $x\sim y$ if and only if $x$ and $y$ are on the same leaf. For two tuples $(M_0, N_0, \omega _0, \sim _0)$ and $(M_1, N_1, \omega _1, \sim _1)$ as above, a relative symplectic embedding from $(M_0, N_0, \omega _0)$ to $(M_1, N_1, \omega _1)$ is a symplectic embedding $\psi : (M_0, \omega _0)\to (M_1, \omega _1)$ satisfying $\psi ^{-1}(N_1) = N_0$ [29, definition 1.5]. Such an embedding $\psi$ is said to respect the pair of coisotropic equivalence relations $(\sim _0,\sim _1)$ if for every $x, y \in N_0$,

\[ \psi(x) \sim_1 \psi(y)\quad \Longrightarrow \quad x \sim_0 y. \]

The standard symplectic space $(\mathbb {R}^{2n},\omega _0)$ has coisotropic linear subspaces

\[ \mathbb{R}^{n,k}=\{x\in\mathbb{R}^{2n}\,|\,x=(q_1,\ldots,q_n,p_1,\ldots,p_k,0,\ldots,0)\} \]

for $k=0,\ldots,n$, where we understand $\mathbb {R}^{n,0}=\{x\in \mathbb {R}^{2n}\,|\,x=(q_1,\ldots,q_n,0,\ldots,0)\}$. Denote by $\sim$ the leaf relation on $\mathbb {R}^{n,k}$, and by

(1.1)\begin{align} & V_0^{n,k}=\{x\in\mathbb{R}^{2n}\,|\,x=(0,\ldots,0,q_{k+1},\ldots,q_n,0,\ldots,0)\}, \end{align}

(1.2)\begin{align} & V^{n,k}_1=\{x\in\mathbb{R}^{2n}\,|\,x=(q_1,\ldots,q_k,0,\ldots,0,p_1,\ldots,p_k,0,\ldots,0)\}. \end{align}

Hereafter it is understood that $V_0^{n,0}=\{x\in \mathbb {R}^{2n}\,|\,x=(q_{1},\ldots,q_n,0,\ldots,0)\}=\mathbb {R}^{n,0}$, $V^{n,n}_0=\{0\}$ and $V^{n,0}_1=\{0\}$, $V^{n,n}_1=\mathbb {R}^{2n}$. Then $L_0^{n}:=V_0^{n,0}$ is a Lagrangian subspace, and two points $x, y\in \mathbb {R}^{n,k}$ satisfy $x\sim y$ if and only if their difference $x-y$ sits in $V_0^{n,k}$. Observe that $\mathbb {R}^{2n}$ has the orthogonal decomposition $\mathbb {R}^{2n}=J_{2n}V^{n,k}_0\oplus \mathbb {R}^{n,k}=J_{2n}\mathbb {R}^{n,k}\oplus V^{n,k}_0$ with respect to the standard inner product, where $J_{2n}$ denotes the standard complex structure on $\mathbb {R}^{2n}$ given by $(q_1,\ldots,q_n, p_1,\ldots, p_n)\mapsto (p_1,\ldots,p_n, -q_1,\ldots, -q_n)$.

For $a\in \mathbb {R}$ we write ${\bf a}:=(0,\ldots,0,a)\in \mathbb {R}^{2n}$. Denote by $B^{2n}({\bf a}, r)$ and $B^{2n}(r)$ the open balls of radius $r$ centred at ${\bf a}$ and the origin in ${{\mathbb {R}}}^{2n}$ respectively, and by

(1.3)\begin{align} & W^{2n}(R) := \left\{(x_1, \ldots, x_n, y_1, \ldots, y_n) \in {{\mathbb{R}}}^{2n} \,|\, x_n^{2} + y_n^{2} < R^{2}\ \text{or}\ y_n < 0 \right\}, \end{align}

(1.4)\begin{align} & W^{n,k}(R):= W^{2n}(R) \cap {{\mathbb{R}}}^{n,k}\quad\hbox{and}\quad B^{n,k}(r):= B^{2n}(r) \cap {{\mathbb{R}}}^{n,k}. \end{align}

($W^{2n}(R$) was written as $W(R)$ in [29, definition 1.1]).

According to [29, definition 1.7], a coisotropic capacity is a functor $c$, which assigns to every tuple $(M,N,\omega, \sim )$ as above a non-negative (possibly infinite) number $c(M,N,\omega, \sim )$, such that the following conditions hold:

1. (i) Monotonicity. If there exists a relative symplectic embedding $\psi$ from $(M_0, N_0, \omega _0, \sim _0)$ to $(M_1, N_1, \omega _1, \sim _1)$ respecting the coisotropic equivalence relations where $\dim M_0=\dim M_1$, then $c(M_0,N_0,\omega _0, \sim _0) \leq c(M_1,N_1,\omega _1, \sim _1)$.

2. (ii) Conformality. $c(M,N,\alpha \omega, \sim )=|\alpha |c(M,N,\omega, \sim ),\ \forall \ \alpha \in \mathbb {R}\backslash \{0\}$.

3. (iii) Non-triviality. With the leaf relation $\sim$ it holds that for $k=0,\ldots,n-1$,

   (1.5)\begin{equation} c(B^{2n}(1),B^{n,k}(1),\omega_0, \sim ) =\frac{\pi}{2}= c(W^{2n}(1),W^{n,k}(1),\omega_0, \sim ). \end{equation}

As remarked in [29, remark 1.9], any symplectic capacity cannot serve as a coisotropic capacity because of the non-triviality (iii).

From now on, we abbreviate $c(M,N,\omega,\sim )$ as $c(M,N,\omega )$ if $\sim$ is the leaf relation on $N$. In particular, for domains $D\subset \mathbb {R}^{2n}$ we also abbreviate $c(D, D\cap \mathbb {R}^{n,k},\omega _0)$ as $c(D, D\cap \mathbb {R}^{n,k})$ for simplicity.

Given a ($n+k$)-dimensional coisotropic submanifold $N$ in a symplectic manifold $(M, \omega )$ of dimension $2n$ we defined in [26, definition 1.3]

\[ {\it w}_G(N;M,\omega):=\sup\left\{\pi r^{2} \left|\begin{array}{ll} & \exists\ \hbox{a relative symplectic embedding}\\ & (B^{2n}(r), B^{n,k}(r))\to (M,N)\ \hbox{respecting}\\ & \hbox{the leaf relations on}\ B^{n,k}(r)\ \hbox{and}\ N \end{array}\right.\right\} \]

the relative Gromov width of $(M, N, \omega )$. Here we always assume $k\in \{0,1\ldots,n-1\}$. (If $k=n$ then ${\it w}_G(N;M,\omega )$ is equal to the Gromov width ${\it w}_G(N,\omega |_N)$ of $(N,\omega |_N)$.)

When $k=0$, $N$ is a Lagrangian submanifold and this relative Gromov width was introduced by Barraud, Biran and Cornea [6–9]. It is easily seen that ${\it w}_G$ satisfies monotonicity, conformality and

\[ {\it w}_G(B^{2n}(r)\cap\mathbb{R}^{n,k}; B^{2n}(r),\omega_0) =\pi r^{2},\quad\forall\ r>0. \]

In fact ${\it w}_G(N;M,\omega )/2$ is the smallest coisotropic capacity by the nonsqueezing theorem in [29]. Rizell [33] observed that the Lagrangian submanifolds of $\mathbb {C}^{3}$ constructed by Ekholm, Eliashberg, Murphy and Smith [15] have infinite relative Gromov width.

Similar to the construction of the Hofer–Zehnder capacity, Lisi and Rieser [29] constructed an analogue relative to a coisotropic submanifold, called the coisotropic Hofer–Zehnder capacity, and denoted by $c_{\rm LR}$ in this paper. By properties of this coisotropic capacity, they also studied symplectic embeddings relative to coisotropic constraints and got some corresponding dynamical results. The coisotropic capacity $c_{\rm LR}$ also played a key role in the proof of Humiliére–Leclercq–Seyfaddini's important rigidity result that symplectic homeomorphisms preserve coisotropic submanifolds and their characteristic foliations [21].

For the coisotropic capacity $c_{\rm LR}(D, D\cap \mathbb {R}^{n,k})$ of a bounded convex domain $D\subset \mathbb {R}^{2n}$, we [26] proved a representation formula, some interesting corollaries and corresponding versions of a Brunn–Minkowski type inequality by Artstein–Avidan and Ostrover and a theorem by Evgeni Neduv.

1.2 A relative version of the Ekeland–Hofer capacity with respect to a coisotropic submanifold $\mathbb {R}^{n,k}$

Prompted by Gromov's work [17], Ekeland and Hofer [13, 14] constructed a sequence of symplectic invariants for subsets in the standard symplectic space $(\mathbb {R}^{2n},\omega _0)$, the so-called Ekeland and Hofer symplectic capacities. (In this paper, the Ekeland and Hofer symplectic capacity always means the first Ekeland and Hofer symplectic capacity without special statements.) We introduced the generalized Ekeland–Hofer and the symmetric Ekeland–Hofer symplectic capacities and developed corresponding results [24, 25]. The aim of this paper is to construct a coisotropic analogue of the Ekeland–Hofer capacity for subsets in $(\mathbb {R}^{2n},\omega _0)$ relative to a coisotropic submanifold $\mathbb {R}^{n,k}$, the coisotropic Ekeland–Hofer capacity.

Fix an integer $0\le k\le n$. For each subset $B\subset \mathbb {R}^{2n}$ whose closure $\overline {B}$ has nonempty intersection with $\mathbb {R}^{n,k}$, we define a number $c^{n,k}(B)$, called coisotropic Ekeland–Hofer capacity of $B$ (though it does not satisfy the stronger monotonicity as in (i) above (1.5)), which is equal to the Ekeland–Hofer capacity of $B$ if $k=n$. The coisotropic capacity $c^{n,k}$ satisfies $c^{n,k}(B)=c^{n,k}(\overline {B})$ and the following:

Proposition 1.1 Let $\lambda >0$ and $B\subset A\subset \mathbb {R}^{2n}$ satisfy $\overline {B}\cap \mathbb {R}^{n,k}\neq \emptyset$. Then

1. (i) (Monotonicity) $c^{n,k}(B)\le c^{n,k}(A)$.

2. (ii) (Conformality) $c^{n,k}(\lambda B)=\lambda ^{2} c^{n,k}(B)$.

3. (iii) (Exterior regularity) $c^{n,k}(B)=\inf \{c^{n,k}(U_\epsilon (B))\,|\,\epsilon >0\}$ and so $c^{n,k}(B)=c^{n,k}(\overline {B}),$ where $U_\epsilon (B)$ is the $\epsilon$-neighbourhood of $B$.

4. (iv) (Translation invariance) $c^{n,k}(B+ w)=c^{n,k}(B)$ for all $w\in \mathbb {R}^{n,k},$ where $B+w=\{z+w\,|\, z\in B\}$.

The group ${\rm Sp}(2n)={\rm Sp}(2n,\mathbb {R})$ of symplectic matrices in $\mathbb {R}^{2n}$ is a connected Lie group. Kun Shi shows in Appendix A that its subgroup

(1.6)\begin{equation} {\rm Sp}(2n,k):=\{A\in {\rm Sp}(2n)\,|\, Az=z\ \forall\ z\in\mathbb{R}^{n,k}\} \end{equation}

is also connected.

Theorem 1.2 (Symplectic invariance)

Let $B\subset \mathbb {R}^{2n}$ satisfy $\overline {B}\cap \mathbb {R}^{n,k}\neq \emptyset$. Suppose that $\phi \in {\rm Symp}(\mathbb {R}^{2n},\omega _0)$ satisfies for some $w_0\in \mathbb {R}^{n,k},$

\[ \phi(w)=w-w_0\quad \forall\ w\in\mathbb{R}^{n,k}\quad\hbox{and}\quad {\rm d}\phi(w_0)\in {\rm Sp}(2n,k). \]

Then $c^{n,k}(\phi (B))=c^{n,k}(B)$.

Corollary 1.3 For a subset $A\subset \mathbb {R}^{2n}$ satisfying $\overline {A}\cap \mathbb {R}^{n,k}\neq \emptyset,$ suppose that there exists a star-shaped open neighbourhood $U$ of $\overline {A}$ with respect to some point $w_0\in \mathbb {R}^{n,k}$ and a symplectic embedding $\varphi$ from $U$ to $\mathbb {R}^{2n}$ such that

(1.7)\begin{equation} \varphi(w)=w-w_0\quad \forall w\in\mathbb{R}^{n,k}\cap U\quad\hbox{and}\quad {\rm d}\varphi(w_0)\in {\rm Sp}(2n,k). \end{equation}

Then $c^{n,k}(\varphi (A))=c^{n,k}(A)$. In particular, for a subset $A\subset \mathbb {R}^{2n}$ satisfying $\overline {A}\cap \mathbb {R}^{n,k}\neq \emptyset,$ if it is star-shaped with respect to some point $w_0\in \mathbb {R}^{n,k}$ and there exists a symplectic embedding $\varphi$ from some open neighbourhood $U$ of $\overline {A}$ to $\mathbb {R}^{2n}$ such that (1.7) holds, then $c^{n,k}(\varphi (A))=c^{n,k}(A)$.

There exists a natural class of symplectic mappings satisfying the conditions in corollary 1.3. For $\epsilon >0$ small, let $\mathbb {R}^{n,k}_\epsilon =\{(q_1,\ldots, q_n, p_1,\ldots,p_n)\,|\, p_{k+1}^{2}+\cdots + p_n^{2}<\epsilon ^{2}\}$, that is, the tubular open neighbourhood of $\mathbb {R}^{n,k}$ of radius $\epsilon$. Let $U$ be as in corollary 1.3, and let $H:[0,1]\times \mathbb {R}^{2n}\to \mathbb {R}$ be any smooth Hamiltonian that vanishes in $[0, 1]\times (\mathbb {R}^{n,k}_\epsilon \cap U)$. Suppose that $X_H$ can determine a $1$-parameter family of symplectic mappings $\phi _H^{t}$ for $t\in [0, 1]$ as usual (e.g. this can be satisfied if $H$ has compact support). Then $\varphi :=(\psi _{w_0}\circ \phi _H^{1})|_U$ satisfies the conditions in corollary 1.3, where $\psi _{w_0}\in {\rm Symp}(\mathbb {R}^{2n},\omega _0)$ is the translation defined by $\psi _{w_0}(w)=w-w_0$ for $w\in \mathbb {R}^{2n}$.

For a bounded convex domain $D$ in $(\mathbb {R}^{2n},\omega _0)$ with boundary $\mathcal {S}$, recall that a nonconstant absolutely continuous curve $z:[0,T]\to \mathbb {R}^{2n}$ (for some $T>0$) is said to be a generalized characteristic on $\mathcal {S}$ if $z([0,T])\subset \mathcal {S}$ and $\dot {z}(t)\in J_{2n}N_\mathcal {S}(z(t))\ \hbox {a.e.}$, where $N_\mathcal {S}(x)=\{y\in \mathbb {R}^{2n}\,|\, \langle u-x, y\rangle \le 0 \forall \ u\in D\}$ is the normal cone to $D$ at $x\in \mathcal {S}$ [24, definition 1.1]. When $D\cap \mathbb {R}^{n,k}\ne \emptyset$, such a generalized characteristic $z:[0,T]\to \mathcal {S}$ is called a generalized leafwise chord (abbreviated GLC) on $\mathcal {S}$ for $\mathbb {R}^{n,k}$ if $z(0), z(T)\in \mathbb {R}^{n,k}$ and $z(0)-z(T)\in V_0^{n,k}$. (Generalized characteristics and generalized leafwise chords on $\mathcal {S}$ become characteristics and leafwise chords on $\mathcal {S}$ respectively if $\mathcal {S}$ is of class $C^{1}$.) The action of a GLC $z:[0,T]\to \mathcal {S}$ is defined by

\[ A(z)=\frac{1}{2}\int_0^{T}\langle -J_{2n}\dot{z},z\rangle \,{{\rm d}}t, \]

where $\langle \cdot,\cdot \rangle$ denotes the Euclid norm on $\mathbb {R}^{2n}$. As generalizations of representation formulas for the Ekeland–Hofer capacities of bounded convex domains we have:

Theorem 1.4 Let $D\subset \mathbb {R}^{2n}$ be a bounded convex domain with $C^{1,1}$ boundary $S=\partial D$. If $D\cap \mathbb {R}^{n,k}\ne \emptyset$ $($and so $\partial D$ contains at least two points of $\mathbb {R}^{n,k}),$ then there exists a leafwise chord $x^{\ast }$ on $\partial D$ for $\mathbb {R}^{n,k}$ such that

(1.8)\begin{align} A(x^{{\ast}}) & = \min\{A(x)>0\,|\,x\ \hbox{is a leafwise chord on}\ \partial D\ \hbox{for}\ \mathbb{R}^{n,k}\}\nonumber\\ & = c^{n,k}(D) \end{align}

(1.9)\begin{align} & = c^{n,k}(\partial D). \end{align}

Moreover, if $D\subset \mathbb {R}^{2n}$ is only a bounded convex domain such that $D\cap \mathbb {R}^{n,k}\ne \emptyset,$ then the above conclusions are still true after all words ‘leafwise chord’ are replaced by ‘generalized leafwise chord’.

This theorem may be false if the domain is not convex. Consider the following domain

\[ A^{2n}(r_1, r_2):=\{z\in\mathbb{R}^{2n}\,|\,r_1^{2}<|z|< r_2^{2}\}=B^{2n}(r_2)\setminus\overline{B^{2n}(r_1)}, \]

where $0< r_1< r_2<\infty$. Then by monotonicity of coisotropic Ekeland–Hofer capacity and theorem 1.4,

\[ c^{n,k}(\partial B^{2n-1}(r_2))\le c^{n,k}(\overline{A^{2n}(r_1, r_2)}) \le c^{n,k}(\overline{B^{2n}(r_2)})=c^{n,k}(\partial B^{2n-1}(r_2)) \]

and

\[ c^{n,k}(A^{2n}(r_1, r_2))=c^{n,k}(\overline{A^{2n}(r_1, r_2)})= \begin{cases} \dfrac{\pi}{2}r_2^{2}, & k< n,\\ \pi r_2^{2}, & k=n. \end{cases} \]

However, since $\partial A^{2n}(r_1, r_2)=\partial B^{2n-1}(r_1)\cup \partial B^{2n-1}(r_2)$,

\begin{align*} & \min\{A(x)>0\,|\,x\ \hbox{is a leafwise chord on}\ \partial A^{2n}(r_1, r_2)\ \hbox{for}\ \mathbb{R}^{n,k}\}\nonumber\\ & \quad =\min\{\min\{A(x)>0\,|\,x\ \hbox{is a leafwise chord on}\ \partial B^{2n-1}(r_1)\ \hbox{for}\ \mathbb{R}^{n,k}\},\\ & \qquad \min\{A(x)>0\,|\,x\ \hbox{is a leafwise chord on}\ \partial B^{2n-1}(r_2)\ \hbox{for}\ \mathbb{R}^{n,k}\}\}\\ & \quad =\min\{c^{n,k}(\partial B^{2n-1}(r_1)), c^{n,k}(\partial B^{2n-1}(r_2))\}\\ & \quad= \begin{cases} \dfrac{\pi}{2}r_1^{2}, & k< n,\\ \pi r_1^{2}, & k=n. \end{cases} \end{align*}

Hence theorem 1.4 is false for $A^{2n}(r_1, r_2)$.

Theorem 1.4 and [26, theorem 1.5] show that $c^{n,k}(D)=c_{\rm LR}(D,D\cap \mathbb {R}^{n,k})$ for a bounded convex domain $D\subset \mathbb {R}^{2n}$ as in theorem 1.4. It follows from (3.18) and interior regularity of $c_{\rm LR}$ that

(1.10)\begin{equation} c^{n,k}(D)=c_{\rm LR}(D,D\cap \mathbb{R}^{n,k}) \end{equation}

for any convex domain $D\subset \mathbb {R}^{2n}$ such that $D\cap \mathbb {R}^{n,k}\ne \emptyset$. Hence theorems 1.6, 1.12 and corollaries 1.7–1.10 in [26] are still true if $c_{\rm LR}$ is replaced by suitable $c^{n,k}$.

Moser [32] first studied Hamiltonian leafwise chords for understanding perturbations of Hamiltonian dynamical systems, his framework has been extended in many directions, which promotes the research of symplectic topology, see [3, 4, 12, 16, 18, 19, 27, 29, 38, 39] etc. Given two autonomous $C^{1,1}$ Hamiltonians $H, G:\mathbb {R}^{2n}\to \mathbb {R}$ and a regular energy surface $G^{-1}(c')$, one may ask the following natural mechanics problem: Is there a point on $G^{-1}(c')$ from which two particles start, move respectively along Hamiltonian trajectories of $X_H$ and $X_G$ and after some finite time return to an intersection point of these two trajectories?

Suppose for some $c\in \mathbb {R}$ that $D_0:=\{z\in \mathbb {R}^{2n}\,|\, H(z)< c\}$ is a bounded convex domain whose intersection with $G^{-1}(c')$ is a nonempty relative open subset in a $(2n-1)$-dimensional coisotropic subspace $V$ in $\mathbb {R}^{2n}$. Then theorem 1.4 implies an affirmative answer to the problem. In fact, since there exists a linear symplectic transformation $\Psi :(\mathbb {R}^{2n},\omega _0)\to (\mathbb {R}^{2n},\omega _0)$ such that $\Psi (V)=\mathbb {R}^{n,n-1}$, we can replace $H$ and $G$ by $H\circ \Psi ^{-1}$ and $G\circ \Psi ^{-1}$, respectively, and therefore reduce the question to the case $V=\mathbb {R}^{n,n-1}$. The desired conclusion follows from theorem 1.4.

Theorem 1.4 is also closely related to the famous Arnold's chord conjecture in [5, § 8]. Many cases for this problem have been proved to be true, see [1, 2, 10, 11, 22, 23, 30, 31, 33–35, 40] etc. When $k=0$, the intersection $S\cap \mathbb {R}^{n,0}$ is a closed $C^{1,1}$ Legendrian submanifold of dimension $n-1$ in the contact manifold $S$ with the standard contact form, which is diffeomorphic to the sphere $S^{n-1}$, and theorem 1.4 affirms the conjecture in this case though for the smooth $S$ it was proved by Mohnke [31] with a different method. Clearly, our result also gives the action of this chord.

As the Ekeland–Hofer capacity, $c^{n,k}$ satisfies the following product formulas, which play key roles for computations of $c_{\rm LR}$ and the proof of [26, theorem 1.12].

Theorem 1.5 For convex domains $D_i\subset \mathbb {R}^{2n_i}$ containing the origin, $i=1,\ldots,m\ge 2,$ and integers $0\le l_0\le n:=n_1+\cdots +n_m,$ $l_j=\max \{l_{j-1}-n_j,0\},$ $j=1,\ldots, m-1,$ it holds that

(1.11)\begin{equation} c^{n,l_0}(D_1\times\cdots\times D_m)=\min_ic^{n_i,\min\{n_i,l_{i-1}\}}(D_i). \end{equation}

Moreover, if all these domains $D_i$ are also bounded then

(1.12)\begin{equation} c^{n,l_0}(\partial D_1\times\cdots\times \partial D_m)=\min_ic^{n_i,\min\{n_i,l_{i-1}\}}(D_i). \end{equation}

Hereafter $\mathbb {R}^{2n_1}\times \mathbb {R}^{2n_2}\times \cdots \times \mathbb {R}^{2n_m}$ is identified with $\mathbb {R}^{2(n_1+\cdots +n_m)}$ via

\begin{align*} & \mathbb{R}^{2n_1}\times \mathbb{R}^{2n_2}\times\cdots\times \mathbb{R}^{2n_m} \ni((q^{(1)},p^{(1)}),\ldots, (q^{(m)},p^{(m)}))\\ & \quad \mapsto (q^{(1)},\ldots, q^{(m)},p^{(1)},\ldots, p^{(m)})\in \mathbb{R}^{2n}. \end{align*}

If $l_0=n$ then $l_i=\sum _{j>i}n_j$ and thus $\min \{n_i,l_{i-1}\}=n_i$ for $i=1,\ldots,m$. It follows that theorem 1.5 becomes theorem in [37, § 6.6]. We pointed out in [26, remark 1.11] that theorems 1.4, 1.5 and [26, theorem 1.5] can be combined together to improve some results therein.

Corollary 1.6 Let $S^{1}(r_i)$ be boundaries of discs $B^{2}(0,r_i)\subset \mathbb {R}^{2},$ $i=1,\ldots,n\ge 2,$ and integers $0\le l_0\le n,$ $l_j=\max \{l_{j-1}-1,0\},$ $j=1,\ldots, n-1$. Then

\[ c^{n,l_0}(S^{1}(r_1)\times\cdots\times S^{1}(r_n))=\min_ic^{1,\min\{1,l_{i-1}\}}(B^{2}(0,r_i)). \]

Here $c^{1,1}(B^{2}(0,r_i))=\pi r_i^{2}$ and $c^{1,0}(B^{2}(0,r_i))=\pi r_i^{2}/2$. Precisely,

\begin{align*} & c^{n,0}(S^{1}(r_1)\times\cdots\times S^{1}(r_n))=\min\{\pi r_1^{2}/2, \ldots, \pi r_n^{2}/2\},\\ & c^{n,k}(S^{1}(r_1)\times\cdots\times S^{1}(r_n))=\min\{\min_{i\le k}\pi r_i^{2}, \min_{i>k}\pi r_i^{2}/2\},\quad 0< k< n,\\ & c^{n,n}(S^{1}(r_1)\times\cdots\times S^{1}(r_n))=\min\{\pi r_1^{2}, \ldots, \pi r_n^{2}\}. \end{align*}

Note that corollary 1.6 becomes [37, corollary 6.6] for $l_0=n$.

Define $U^{2}(1)=\{(q_n,p_n)\in \mathbb {R}^{2}\,|\,q_n^{2}+p_n^{2}<1\;\hbox {or} -1< q_n<1 \hbox {and} p_n<0\}$ and

(1.13)\begin{equation} U^{2n}(1)=\mathbb{R}^{2n-2}\times U^{2}(1)\quad\hbox{and}\quad U^{n,k}(1)=U^{2n}(1)\cap\mathbb{R}^{n,k}. \end{equation}

By (1.10) and [26, corollary 1.9] we obtain for $k=0,1,\ldots,n-1$,

(1.14)\begin{equation} c^{n,k}(U^{2n}(1))=c_{\rm LR}(U^{2n}(1),U^{2n}(1)\cap \mathbb{R}^{n,k})=\frac{\pi}{2}. \end{equation}

The proof of theorem 1.5 relies partially on the representation of coisotropic Ekeland–Hofer capacity of convex domains given by theorem 1.4. It is possible that theorem 1.5 is still true for some product of non-convex domains. For integers $0\le l_0\le n:=n_1+ n_2$ and $l_1=\max \{l_0-n_1, 0\}$, as the arguments below theorem 1.5 we can get

\begin{align*} c^{n,l_0}({B^{2n_1}(r_2)}\times B^{2n_2}(r_3))& = c^{n,l_0}(\partial B^{2n_1-1}(r_2)\times \partial B^{2n_2-1}(r_3))\\ & \le c^{n,l_0}(\overline{A^{2n_1}(r_1, r_2)}\times \overline{B^{2n_2}(r_3)})\\ & \le c^{n,l_0}(\overline{B^{2n_1}(r_2)}\times \overline{B^{2n_2}(r_3)}) \end{align*}

and therefore

\begin{align*} c^{n,l_0}({A^{2n_1}(r_1, r_2)}\times B^{2n_2}(r_3)) & = c^{n,l_0} (\overline{A^{2n_1}(r_1, r_2)}\times \overline{B^{2n_2}(r_3)})\\ & = c^{n,l_0}(\overline{B^{2n_1}(r_2)}\times\overline{ B^{2n_2}(r_3)})\\ & = \min\{c^{n_1,\min\{n_1,l_0\}}(B^{2n_1}(r_2)), c^{n_2,\min\{n_2,l_{1}\}}(B^{2n_2}(r_3))\}\\ & = \begin{cases} \min\left\{\dfrac{\pi}{2}r_2^{2}, \dfrac{\pi}{2}r_3^{2}\right\}, & l_0< n_1,\\ \min\{\pi r_2^{2}, c^{n_2, l_0-n_1}(B^{2n_2}(r_3))\}, & l_0\ge n_1. \end{cases} \end{align*}

On the other hand

\begin{align*} & \min\{c^{n_1,\min\{n_1,l_0\}}(A^{2n_1}(r_1, r_2)), c^{n_2,\min\{n_2,l_1\}}(B^{2n_2}(r_3))\}\\ & \quad =\begin{cases} \min\left\{\dfrac{\pi}{2}r_2^{2}, \dfrac{\pi}{2}r_3^{2}\right\}, & l_0< n_1,\\ \min\{\pi r_2^{2}, c^{n_2, l_0-n_1}(B^{2n_2}(r_3))\}, & l_0\ge n_1. \end{cases} \end{align*}

Hence (1.11) is also true for the produce of ${A^{2n_1}(r_1, r_2)}$ and $B^{2n_2}(r_3)$.

Recall that a vector field $X$ defined on an open set $U\subset \mathbb {R}^{2n}$ is called a Liouville vector field if $L_X\omega _0=\omega _0$. A hypersurface $\mathcal {S}\subset \mathbb {R}^{2n}$ is said to be of restricted contact type if there exists a Liouville vector field $X$ globally defined on $\mathbb {R}^{2n}$ which is transversal to $\mathcal {S}$. Corresponding to the representation of the Ekeland–Hofer capacity of a bounded domain in $\mathbb {R}^{2n}$ with boundary of restricted contact type we have:

Theorem 1.7 Let $U\subset (\mathbb {R}^{2n},\omega _0)$ be a bounded domain with $C^{2n+2}$ boundary $\mathcal {S}$ of restricted contact type. Suppose that $U$ contains the origin and that there exists a globally defined $C^{2n+2}$ Liouville vector field $X$ transversal to $\mathcal {S}$ whose flow $\phi ^{t}$ maps $\mathbb {R}^{n,k}$ to $\mathbb {R}^{n,k}$ and preserves the leaf relation of $\mathbb {R}^{n,k}$. Then

(1.15)\begin{equation} \Sigma_{\mathcal{S}}:= \{A(x)>0\,|\,x\ \hbox{is a leafwise chord on}\ \mathcal{S}\ \hbox{for}\ \mathbb{R}^{n,k}\}. \end{equation}

has empty interior and contains $c^{n,k}(U)=c^{n,k}(\mathcal {S})$.

In order to show that $c^{n,k}$ is a coisotropic capacity (with the weaker monotonicity), we need to prove that $c^{n,k}$ satisfies the non-triviality as in (1.5). By theorem 1.4 we immediately obtain

(1.16)\begin{equation} c^{n,k}(B^{2n}(1)) =\frac{\pi}{2},\quad k=0,\ldots,n-1. \end{equation}

Proposition 1.1(i) and (1.14) also lead to $c^{n,k}(W^{2n}(1))\ge c^{n,k}(U^{2n}(1))={\pi }/{2}$ directly. Using the extension monotonicity of $c_{\rm LR}$ in [29, lemma 2.4], Lisi and Rieser proved that

\[ c_{\rm LR}\left(W^{2n}(1), W^{n,k}(1)\right) =c_{\rm LR}\left(U^{2n}(1), U^{n,k}(1)\right) \]

above [29, proposition 3.1]. However, our proposition 1.1 and theorem 1.2 cannot yield such strong extension monotonicity for $c^{n,k}$. Instead, we may use theorems 1.5 and 1.7 (though the latter does not hold for $c_{\rm LR}$ in general), to derive:

Theorem 1.8 For $k=0,\ldots,n-1,$ it holds that

\[ c^{n,k}(W^{2n}(1)) =\frac{\pi}{2}. \]

By this theorem, corollary 1.6 and theorem 1.2 we deduce:

Corollary 1.9 If $\min \{2\min _{i\le k}r_i^{2}, \min _{i>k}r_i^{2}\}>1$ for some $0< k< n,$ then there is no $\phi \in {\rm Symp}(\mathbb {R}^{2n},\omega _0)$ which satisfies $\phi (w)=w-w_0\ \forall \ w\in \mathbb {R}^{n,k}$ and ${\rm d}\phi (w_0)\in {\rm Sp}(2n,k)_0$ for some $w_0\in \mathbb {R}^{n,k},$ such that $\phi$ maps $S^{1}(r_1)\times \cdots \times S^{1}(r_n)$ $= \{ (x_1, \ldots, x_n, y_1, \ldots, y_n) \in {{\mathbb {R}}}^{2n} \,|\, x_i^{2} + y_i^{2}=r_i^{2},\ i=1,\ldots,n\}$ into $W^{2n}(1)$.

Under the assumptions of corollary 1.9 it is easy to see that there always exists a $\phi \in {\rm Symp}(\mathbb {R}^{2n},\omega _0)$ such that $\phi (S^{1}(r_1)\times \cdots \times S^{1}(r_n))\subset W^{2n}(1)$.

Let $\tau _0\in \mathcal {L}(\mathbb {R}^{2n})$ be the canonical involution on $\mathbb {R}^{2n}$ given by $\tau _0(x,y)=(x,-y)$. For a subset $B\subset \mathbb {R}^{2n}$ such that $\tau _0B=B$ and $B\cap L^{n}_0\neq \emptyset$, let $c_{\rm EH,\tau _0}(B)$ be the $\tau _0$-symmetrical Ekeland–Hofer capacity constructed in [25]. We shall prove in § 8:

Theorem 1.10 The $\tau _0$-symmetrical Ekeland–Hofer capacity $c_{\rm EH,\tau _0}(B)$ of each subset $B\subset \mathbb {R}^{2n}$ satisfying $\tau _0B=B$ and $B\cap L^{n}_0\neq \emptyset$ is greater than or equal to $c^{n,0}(B)$.

Structure of the paper. In § 2 we provide necessary variational preparations on the basis of [26, 29]. In § 3 we give the definition of the coisotropic Ekeland–Hofer capacity and proofs of proposition 1.1, theorem 1.2 and corollary 1.3. In § 4 we prove theorem 1.4. In § 5 we prove a product formula, theorem 1.5. In § 6 we prove theorem 1.7 about the representation of the coisotropic capacity $c^{n,k}$ of a bounded domain in $\mathbb {R}^{2n}$ with boundary of restricted contact type. In § 7 we prove theorem 1.8.

2. Variational preparations

We follow [26, 29] to present necessary variational materials. Fix an integer $0\le k< n$. Consider the Hilbert space defined in [29, definition 3.6]

(2.1)\begin{align} L^{2}_{n,k} & = \left\{x\in L^{2}([0,1],\mathbb{R}^{2n})\,\left| x\stackrel{L^{2}}{=}\sum_{m\in\mathbb{Z}} {\rm e}^{m\pi tJ_{2n}}a_m+\sum_{m\in\mathbb{Z}}{\rm e}^{2m\pi tJ_{2n}}b_m \right.\right.\nonumber\\ & \qquad a_m\in V^{n,k}_0,\quad b_m\in V^{n,k}_1, \left.\sum_{m\in\mathbb{Z}}(|a_m|^{2}+|b_m|^{2})<\infty\right\} \end{align}

with $L^{2}$-inner product. We proved in [26, proposition 2.3] that the Hilbert space $L^{2}_{n,k}$ is exactly $L^{2}([0,1],\mathbb {R}^{2n})$. (If $k=n$ this is clear as usual because $V^{n,n}_0=\{0\}$ and $V^{n,n}_1=\mathbb {R}^{2n}$.) For any real $s\ge 0$ we follow [29, definition 3.6] to define

(2.2)\begin{align} & H^{s}_{n,k}=\left\{x\in L^{2}([0,1],\mathbb{R}^{2n})\left|x\stackrel{L^{2}}{=} \sum_{m\in\mathbb{Z}}{\rm e}^{m\pi tJ_{2n}}a_m+ \sum_{m\in\mathbb{Z}}{\rm e}^{2m\pi tJ_{2n}}b_m \right.\right.\nonumber\\ & \qquad\qquad a_m\in V^{n,k}_0,\quad b_m\in V^{n,k}_1, \quad \left.\sum_{m\in\mathbb{Z}}|m|^{2s}(|a_m|^{2}+|b_m|^{2})<\infty\right\}. \end{align}

Lemma 2.1 [29, lemmas 3.8, 3.9]

For each $s\ge 0,$ $H^{s}_{n,k}$ is a Hilbert space with the inner product

\[ \langle\phi,\psi\rangle_{s,n,k}=\langle a_0,a_0'\rangle+\langle b_0,b_0'\rangle + \pi\sum_{m\ne 0}(|m|^{2s}\langle a_m, a_m'\rangle+|2m|^{2s}\langle b_m, b_m'\rangle). \]

Furthermore, if $s>t,$ then the inclusion $\jmath :H^{s}_{n,k}\hookrightarrow H^{t}_{n,k}$ and its Hilbert adjoint $\jmath ^{\ast }:H^{t}_{n,k}\rightarrow H^{s}_{n,k}$ are compact.

Let $\|\cdot \|_{s,n,k}$ denote the norm induced by $\langle \cdot,\cdot \rangle _{s,n,k}$. For $r\in \mathbb {N}$ or $r=\infty$ let $C^{r}_{n,k}([0,1],\mathbb {R}^{2n})$ denote the space of $C^{r}$ maps $x:[0,1]\to \mathbb {R}^{2n}$ such that $x(i)\in \mathbb {R}^{n,k}$, $i=0,1$, and $x(1)\sim x(0)$, where $\sim$ is the leaf relation on $\mathbb {R}^{n,k}$. (Note: $H^{s}_{n,n}$ is exactly the space $H^{s}$ on p. 83 of [20]; $C^{r}_{n,n}([0,1],\mathbb {R}^{2n})$ is $C^{r}(\mathbb {R}/\mathbb {Z}, \mathbb {R}^{2n})$.)

Lemma 2.2 [29, lemma 3.10]

If $x\in H^{s}_{n,k}$ for $s>1/2+r$ where $r$ is an integer, then $x\in C^{r}_{n,k}([0,1],\mathbb {R}^{2n})$.

Lemma 2.3 [29, lemma 3.11]

$\jmath ^{\ast }(L^{2})\subset H^{1}_{n,k}$ and $\|\jmath ^{\ast }(y)\|_{1,n,k}\le \|y\|_{L^{2}}$.

Let

(2.3)\begin{equation} E=H^{1/2}_{n,k}\quad\hbox{and}\quad \|\cdot\|_E:=\|\cdot\|_{1/2,n,k}. \end{equation}

It has an orthogonal decomposition $E=E^{-}\oplus E^{0}\oplus E^{+}$, where

\begin{align*} & E^{-}=\left\{x\in H^{1/2}_{n,k}\left|x\stackrel{L^{2}}{=}\sum_{m<0} {\rm e}^{m\pi tJ_{2n}}a_m+\sum_{m<0}{\rm e}^{2m\pi tJ_{2n}}b_m\right.\right\},\\ & E^{0}=\{x=x_0\in\mathbb{R}^{n,k}\},\\ & E^{+}=\left\{x\in H^{1/2}_{n,k}\left|x\stackrel{L^{2}}{=} \sum_{m>0}{\rm e}^{m\pi tJ_{2n}}a_m+\sum_{m>0}{\rm e}^{2m\pi tJ_{2n}}b_m\right.\right\}. \end{align*}

Let $P^{+}$, $P^{0}$ and $P^{-}$ be the orthogonal projections to $E^{+}$, $E^{0}$ and $E^{-}$ respectively. For $x\in E$ we write

\[ x^{+}=P^{+}x,\quad x^{0}=P^{0}x\quad\hbox{and} \quad x^{-}=P^{-}x. \]

Define a functional $\mathfrak {a}:E\rightarrow \mathbb {R}$ by

\[ \mathfrak{a}(x)=\tfrac{1}{2}(\|x^{+}\|^{2}_E-\|x^{-}\|^{2}_E). \]

Then there holds

\[ \mathfrak{a}(x)=\frac{1}{2}\int_0^{1}\langle -J_{2n}\dot{x},x\rangle {{\rm d}}t,\quad\forall\ x\in C^{1}_{n,k}([0,1],\mathbb{R}^{2n}). \]

(See [29].) The functional $\mathfrak {a}$ is differentiable with gradient $\nabla \mathfrak {a}(x)=x^{+}-x^{-}$.

From now on we assume that for some $L>0$,

(2.4)\begin{equation} H\in C^{1}(\mathbb{R}^{2n}, \mathbb{R})\quad\hbox{and}\quad \|\nabla H(x)-\nabla H(y)\|_{\mathbb{R}^{2n}}\le L\|x-y\|_{\mathbb{R}^{2n}}\ \forall\ x,y\in \mathbb{R}^{2n}. \end{equation}

Then there exist positive real numbers $C_i$, $i=1,2,3,4$, such that

\[ |\nabla H(z)|\le C_1|z|+C_2,\quad |H(z)|\le C_3|z|^{2}+C_4 \]

for all $z\in \mathbb {R}^{2n}$. Define functionals $\mathfrak {b}, \Phi _H: E\rightarrow \mathbb {R}$ by

(2.5)\begin{equation} \mathfrak{b}(x)=\int_0^{1} H(x(t))\,{{\rm d}}t\quad\hbox{and}\quad \Phi_H=\mathfrak{a}-\mathfrak{b}. \end{equation}

Lemma 2.4 [20, § 3.3, lemma 4]

The functional $\mathfrak {b}$ is differentiable. Its gradient $\nabla \mathfrak {b}$ is compact and satisfies a global Lipschitz condition on $E$. In particular, $\mathfrak {b}$ is $C^{1,1}$.

Lemma 2.5 [26, lemma 2.8]

$x\in E$ is a critical point of $\Phi _H$ if and only if $x\in C^{1}_{n,k}([0,1],\mathbb {R}^{2n})$ and solves

\[ \dot{x}=X_H(x)=J_{2n}\nabla H(x). \]

Moreover, if $H$ is of class $C^{l}$ $(l\ge 2)$ then each critical point of $\Phi _H$ on $E$ is $C^{l}$.

Since $\nabla \Phi _H(x)=x^{+}-x^{-}-\nabla \mathfrak {b}(x)$ satisfies the global Lipschitz condition, it has a unique global flow $\mathbb {R}\times E\to E: (u,x)\mapsto \varphi _u(x)$.

Lemma 2.6 [29, lemma 3.25]

$\varphi _u(x)$ has the following form

\[ \varphi_u(x) = {\rm e}^{{-}u} x^{-} + x^{0} + {\rm e}^{u}x^{+} + K(u,x), \]

where $K:\mathbb {R}\times E \to E$ is continuous and maps bounded sets into precompact sets.

This may follow from the proof of lemma 7 in [20, § 3.3] directly.

3. The Ekeland–Hofer capacity relative to a coisotropic subspace

We closely follow Sikorav's approach [37] to the Ekeland–Hofer capacity in [13]. Fix an integer $0\le k\le n$. Let $E=H^{1/2}_{n,k}$ be as in (2.3) and $S^{+}=\{x\in E^{+}\,|\,\|x\|_E=1\}$.

Definition 3.1 A continuous map $\gamma :E\rightarrow E$ is called an admissible deformation if there exists a homotopy $(\gamma _u)_{0\le u\le 1}$ such that $\gamma _0={\rm id},$ $\gamma _1=\gamma$ and satisfies

1. (i) $\forall u\in [0,1],$ $\gamma _u(E\setminus (E^{-}\oplus E^{0}))\subset E\setminus (E^{-}\oplus E^{0}),$ i.e. $\gamma _u(x)^{+}\neq 0$ for any $x\in E$ such that $x^{+}\neq 0$.

2. (ii) $\gamma _u(x)=a(x,u)x^{+}+b(x,u)x^{0}+c(x,u)x^{-}+K(x,u),$ where $(a,b,c,K)$ is a continuous map from $E\times [0,1]$ to $(0,+\infty )^{3}\times E$ and maps any closed bounded sets to compact sets.

Let $\Gamma _{n,k}$ be the set of all admissible deformations on $E$. It is not hard to verify that the composition $\gamma \circ \tilde {\gamma } \in \Gamma _{n,k}$ for any $\gamma, \tilde {\gamma }\in \Gamma _{n,k}$. (If $k=n$, $\Gamma _{n,k}$ is equal to $\Gamma$ in [37].) Corresponding to [37, § 3, proposition 1] or [13, § II, proposition 1] we can easily prove the following intersection property.

Proposition 3.2 $\gamma (S^{+})\cap (E^{-}\oplus E^{0}\oplus \mathbb {R}_+e)\neq \emptyset$ for any $e\in E^{+}\setminus \{0\}$ and $\gamma \in \Gamma _{n,k}$.

Definition 3.3 For $H\in C^{0}(\mathbb {R}^{2n},\mathbb {R}_{\ge 0}),$ the $\mathbb {R}^{n,k}$-coisotropic capacity of $H$ is defined by

(3.1)\begin{equation} c^{n,k}(H):=\sup_{h\in\Gamma_{n,k}}\inf_{x\in h(S^{+})}\Phi_H(x) \end{equation}

where $\Phi _H$ is as in (2.5).

By proposition 1 in [37, § 3.3], for any $H\in C^{0}(\mathbb {R}^{2n},\mathbb {R}_{\ge 0})$ there holds

(3.2)\begin{equation} c^{n,n}(H)\le \sup_{z\in\mathbb{C}^{n}}\left(\pi|z_1|^{2}-H(z)\right), \end{equation}

where $z_1\in \mathbb {C}$ is the projection of $z\in \mathbb {C}^{n}\equiv \mathbb {C}\times \mathbb {C}^{n-1}$ to $\mathbb {C}$. Correspondingly, we have

Proposition 3.4 For any $H\in C^{0}(\mathbb {R}^{2n},\mathbb {R}_{\ge 0})$ there holds

(3.3)\begin{equation} c^{n,k}(H)\le \sup_{z\in\mathbb{C}^{n}}\left(\frac{ \pi}{2}|z|^{2}-H(z)\right),\quad k=0,1,\ldots,n-1. \end{equation}

Proof. Let $e(t)={\rm e}^{\pi J_{2n}t}X$, where $X\in V^{n,k}_0$ and $|X|=1$. For any $x=y+\lambda e$, where $y\in E^{-}\oplus E^{0}$ and $\lambda >0$, it holds that

\[ \mathfrak{a}(x)\le\frac{1}{2}\|\lambda e\|^{2}_E= \frac{ \pi }{2}\lambda^{2} \]

and

\[ \int_0^{1}\langle x(t),{\rm e}^{\pi J_{2n}t}X\rangle \,{{\rm d}}t= \int_0^{1}\langle \lambda \,{\rm e}^{\pi J_{2n}t}X,{\rm e}^{\pi J_{2n}t}X\rangle \,{{\rm d}}t=\lambda. \]

It follows that

\[ \mathfrak{a}(x)\le\frac{\pi }{2}\left(\int_0^{1} \langle x(t),{\rm e}^{\pi J_{2n}t}X\rangle \,{{\rm d}}t\right)^{2}\le\frac{\pi}{2}\int_0^{1}|x(t)|^{2} \,{{\rm d}}t. \]

This and proposition 3.2 lead to

\[ \inf_{x\in\gamma(S^{+})}\Phi_H(x) \le\sup_{x\in E^{-}\oplus E^{0}\oplus\mathbb{R}_+e} \Phi_H(x)\le \sup_{z\in\mathbb{R}^{2n}}\left\{\frac{\pi }{2}|z|^{2}-H(z)\right\} \quad\forall\ \gamma\in\Gamma_{n,k}, \]

and hence (3.3) is proved.

A function $H\in C^{0}(\mathbb {R}^{2n},\mathbb {R}_{\ge 0})$ is called $\mathbb {R}^{n,k}$-admissible if it satisfies:

1. (H1) ${\rm Int}(H^{-1}(0))\ne \emptyset$ and intersects with $\mathbb {R}^{n,k}$,

2. (H2) there exists $z_0\in \mathbb {R}^{n,k}$, real numbers $a, b$ such that $H(z)=a|z|^{2}+ \langle z, z_0\rangle + b$ outside a compact subset of $\mathbb {R}^{2n}$, where $a>\pi$ for $k=n$, and $a>\pi /2$ for $0\le k< n$.

Moreover, a $\mathbb {R}^{n,n}$-admissible $H$ is said to be nonresonant if $a$ in (H2) does not belong to $\pi \mathbb {N}$; and a $\mathbb {R}^{n,k}$-admissible $H$ with $k< n$ is called strong nonresonant if $a$ in (H2) does not sit in $\mathbb {N}\pi /2$.

Clearly, for any $\mathbb {R}^{n,k}$-admissible $H\in C^{2}(\mathbb {R}^{2n},\mathbb {R}_{\ge 0})$, $\nabla H:\mathbb {R}^{2n}\to \mathbb {R}^{2n}$ satisfies a global Lipschitz condition.

Note that $c^{n,k}(H)<+\infty$ if $H\in C^{0}(\mathbb {R}^{2n},\mathbb {R}_{\ge 0})$ satisfies

(3.4)\begin{equation} H(z)\ge a|z|^{2}+C,\quad\forall\ z\in\mathbb{R}^{2n} \end{equation}

for some constant $C$, where $a=\pi$ for $k=n$, and $a=\pi /2$ for $0\le k< n$. In particular, we have $c^{n,k}(H)<+\infty$ for any $H\in C^{0}(\mathbb {R}^{2n},\mathbb {R}_{\ge 0})$ satisfying (H2). In fact, for $k=n$ this can be derived from (3.2) (cf. [37]). For $0\le k< n$, since there exist constants $a>\pi /2, b$ such that $H(z)\ge a|z|^{2}+\langle z,z_0\rangle +b$ for all $z\in \mathbb {R}^{2n}$, using the inequality

\[ |\langle z,z_0\rangle|\le \varepsilon |z|^{2}+ \frac{1}{4\varepsilon}|z_0|^{2} \]

for any $0<\varepsilon < a-\frac {\pi }{2}$, we deduce that

\[ \frac{\pi}{2}|z|^{2}-H(z)\le \left(\varepsilon-\left(a-\frac{\pi}{2}\right)\right)|z|^{2}+ \frac{|z_0|^{2}}{4\varepsilon}-b\le\frac{|z_0|^{2}}{4\varepsilon}-b<\infty. \]

Then proposition 3.4 leads to $c^{n,k}(H)<+\infty$.

It is easy proved that $c^{n,k}(H)$ satisfies:

Proposition 3.5 Let $H, K\in C^{0}(\mathbb {R}^{2n},\mathbb {R}_{\ge 0})$ satisfy (H1) and (H2). Then the following holds$:$

1. (i) (Monotonicity) If $H\le K$ then $c^{n,k}(H)\ge c^{n,k}(K)$.

2. (ii) (Continuity) $|c^{n,k}(H)-c^{n,k}(K)|\le \sup _{z\in \mathbb {R}^{2n}}|H(z)-K(z)|$.

3. (iii) (Homogeneity) $c^{n,k}(\lambda ^{2}H(\cdot /\lambda ))=\lambda ^{2} c^{n,k}(H)$ for $\lambda \ne 0$.

By proposition 2 in [37, § 3.3] the following proposition holds for $k=n$.

Proposition 3.6 Suppose that $H\in C^{0}(\mathbb {R}^{2n},\mathbb {R}_{\ge 0})$ satisfies

(3.5)\begin{equation} H(z_0+z)\le C_1|z|^{2}\quad\hbox{and}\quad H(z_0+z)\le C_2|z|^{3}\quad\forall\ z\in\mathbb{R}^{2n} \end{equation}

for some $z_0\in \mathbb {R}^{n,k}$ and for constants $C_1>0$ and $C_2>0$. Then $c^{n,k}(H)>0$. In particular, $c^{n,k}(H)>0$ for any $\mathbb {R}^{n,k}$-admissible $H\in C^{2}(\mathbb {R}^{2n},\mathbb {R}_{\ge 0})$.

Proof. We assume $k< n$. For a constant $\varepsilon >0$ define $\gamma _\varepsilon \in \Gamma _{n,k}$ by $\gamma _\varepsilon (x)=z_0+\varepsilon x\ \forall \ x\in E$. We claim that

(3.6)\begin{equation} \inf_{y\in\gamma_\varepsilon(S^{+})}\Phi_H(y)>0 \end{equation}

for sufficiently small $\varepsilon$. Since

(3.7)\begin{equation} \Phi_H(z_0+x)=\frac{1}{2}\|x\|^{2}_{E}-\int_0^{1}H(z_0+x)\,{{\rm d}}t\quad\forall\ x\in E^{+}, \end{equation}

it suffices to prove that

(3.8)\begin{equation} \lim_{\|x\|_E\rightarrow 0}\frac{\int_0^{1} H(z_0+x)\,{{\rm d}}t}{\|x\|^{2}_E}=0. \end{equation}

Otherwise, suppose there exists a sequence $(x_j)\subset E$ and $d>0$ satisfying

(3.9)\begin{equation} \|x_j\|_E\rightarrow 0 \quad \hbox{and} \quad \frac{\int_0^{1} H(z_0+x_j)\,{{\rm d}}t}{\|x_j\|^{2}_E} \geq d>0\quad\forall\ j. \end{equation}

Let $y_j={x_j}/{\|x_j\|_E}$ and hence $\|y_j\|_E=1$. Then lemma 2.1 implies that $(y_j)$ has a convergent subsequence in $L^{2}$. By a standard result in $L^{p}$ theory, we have $w\in L^{2}$ and a subsequence of $(y_j)$, still denoted by $(y_j)$, such that $y_j(t)\rightarrow y(t)$ a.e. on $(0,1)$ and that $|y_j(t)|\leq w(t)$ a.e. on $(0,1)$ for each $j$. It follows from (3.5) that

\begin{align*} & \frac{H(z_0+x_j(t))}{\|x_j\|_E^{2}}\leq C_1 \frac{|x_j(t)|^{2}}{\|x_j\|_E^{2}} =C_1|y_j(t)|^{2}\le C_1w(t)^{2},\quad \hbox{a.e. on}\ (0,1),\ \forall\ j,\\ & \frac{H(z_0+x_j(t))}{\|x_j\|_E^{2}}\leq C_2\frac{|x_j(t)|^{3}}{\|x_j\|_E^{2}} =C_2|x_j(t)|\cdot |y_j(t)|^{2} \le C_2|x_j(t)|w(t)^{2},\quad \hbox{a.e. on}\ (0,1),\enspace \forall j. \end{align*}

The first claim in (3.9) implies that $(x_j)$ has a subsequence such that $x_{j_l}(t)\rightarrow 0$, a.e. in $(0,1)$. Hence the Lebesgue dominated convergence theorem leads to

\[ \int_0^{1} \frac{H(z_0+x_{j_l}(t))}{\|x_{j_l}\|_E^{2}} {{\rm d}}t\rightarrow 0. \]

This contradicts the second claim in (3.9).

For any fixed $\mathbb {R}^{n,k}$-admissible $H\in C^{2}(\mathbb {R}^{2n},\mathbb {R}_{\ge 0})$, pick some $z_0\in \mathbb {R}^{n,k}\cap {\rm Int}(H^{-1}(0))$. Since (H1) implies that $H$ vanishes near $z_0$, by (H2) and the Taylor expansion of $H$ at $z_0\in \mathbb {R}^{2n}$, we have constants $C_1>0$ and $C_2>0$ such that $H$ satisfies (3.5).

By (3.2) and propositions 3.4 and 3.6 we see that $c^{n,k}(H)$ is a finite positive number for each $\mathbb {R}^{n,k}$-admissible $H$. The following is a generalization of lemma 3 in [37, § 3.4].

Lemma 3.7 Let $H\in C^{0}(\mathbb {R}^{2n},\mathbb {R}_{\ge 0})$ satisfy (3.4) and (3.5). Then

\[ c^{n,k}(H)=\sup_{F\in\mathcal{F}_{n,k}}\inf_{x\in F}\Phi_H(x), \]

where

(3.10)\begin{equation} \mathcal{F}_{n,k}:=\{\gamma(S^{+})\,|\,\gamma\in\Gamma_{n,k}\ \text{and}\ \inf(\Phi_H|_{\gamma(S^{+})})>0\}. \end{equation}

Moreover, if $H$ is also of class $C^{2}$ and has bounded derivatives of second order, then $\mathcal {F}_{n,k}$ is positive invariant under the flow $\varphi _u$ of $\nabla \Phi _H$ $($which must exist as pointed out above in lemma 2.6).

Proof. Since $c^{n,k}(H)$ is a finite positive number by proposition 3.6, the first claim follows from the arguments above proposition 3.5.

When $H$ has bounded derivatives of second order, (2.4) is satisfied naturally. Then $\nabla \Phi _H$ satisfies the global Lipschitz condition, and thus has a unique global flow $\mathbb {R}\times E\to E: (u,x)\mapsto \varphi _u(x)$ satisfying lemma 2.6, that is, $\varphi _u(x)={\rm e}^{-u}x^{-}+x^{0}+{\rm e}^{u}x^{+}+\widetilde {K}(u,x)$, where $\widetilde {K}:\mathbb {R}\times E\rightarrow E$ is continuous and maps bounded sets into precompact sets. For a set $F=\gamma (S^{+})\in \mathcal {F}_{n,k}$ with $\gamma \in \Gamma _{n,k}$, we have $\alpha :=\inf (\Phi _H|_{\gamma (S^{+})})>0$ by the definition of $\mathcal {F}_{n,k}$. Let $\rho :\mathbb {R}\rightarrow [0,1]$ be a smooth function such that $\rho (s)=0$ for $s\le 0$ and $\rho (s)=1$ for $s\ge \alpha$. Define a vector field $V$ on $E$ by

\[ V(x)=x^{+}-x^{-}-\rho(\Phi_H(x))\nabla \mathfrak{b}(x). \]

Clearly $V$ is locally Lipschitz and has linear growth. These imply that $V$ has a unique global flow, denoted by $\Upsilon _u$. Moreover, it is obvious that $\Upsilon _u$ has the same property as $\varphi _u$ described in lemma 2.6. For $x\in E^{-}\oplus E^{0}$, we have $\Phi _H(x)\le 0$ and hence $V(x)=-x^{-}$, which implies that $\Upsilon _u(E^{-}\oplus E^{0})=E^{-}\oplus E^{0}$ and $\Upsilon _u(E\setminus E^{-}\oplus E^{0})=E\setminus E^{-}\oplus E^{0}$ since $\Upsilon _u$ is a homeomorphism for each $u\in \mathbb {R}$. Therefore, $\Upsilon _u\in \Gamma _{n,k}$ for all $u\in \mathbb {R}$.

Note that $V|_{\Phi _H^{-1}([\alpha,\infty ])}=\nabla \Phi _H(x)$. For each $u\ge 0$ we have $\Upsilon _u(x)=\varphi _u(x)$ for any $x\in \Phi _H^{-1}([\alpha,\infty ])$, and especially $\Upsilon _u(F)=\varphi _u(F)$, that is, $(\Upsilon _u\circ \gamma )(S^{+})=\varphi _u(F)$. Since $\Gamma _{n,k}$ is closed for the composition operation and

\[ \inf(\Phi_H|_{(\Upsilon_u\circ\gamma)(S^{+})})=\inf(\Phi_H|_{\varphi_u(F)})\ge\inf(\Phi_H|_F)>0, \]

we obtain $\varphi _u(F)\in \mathcal {F}_{n,k}$, that is, $\mathcal {F}_{n,k}$ is positively invariant under the flow $\varphi _u$ of $\nabla \Phi _H$.

Clearly, a $\mathbb {R}^{n,k}$-admissible $H\in C^{2}(\mathbb {R}^{2n},\mathbb {R}_{\ge 0})$ satisfies the conditions of lemma 3.7.

Theorem 3.8 If an $\mathbb {R}^{n,k}$-admissible $H\in C^{2}(\mathbb {R}^{2n},\mathbb {R}_{\ge 0})$ is nonresonant for $k=n,$ and strong nonresonant for $k< n,$ then $c^{n,k}(H)$ is a positive critical value of $\Phi _H$.

The case of $k=n$ was proved in [13, § II, proposition 2] (see also [37, § 3.4, proposition 1]). It remains to prove the case $k< n$. By lemma 2.4, the functional $\Phi _H$ is $C^{1,1}$ and its gradient $\nabla \Phi _H$ satisfies a global Lipschitz condition on $E$. By a standard minimax argument, theorem 3.8 follows from lemma 3.7 and the following

Lemma 3.9 If an $\mathbb {R}^{n,k}$-admissible $H\in C^{1}(\mathbb {R}^{2n},\mathbb {R}{\ge 0})$ is strong nonresonant, then each sequence $(x_j)\subset E$ with $\nabla \Phi _{H}(x_j)\to 0$ has a convergent subsequence. In particular, $\Phi _H$ satisfies the $($PS$)$ condition.

Proof. The functional $\mathfrak {b}$ is differentiable. Its gradient $\nabla \mathfrak {b}$ is compact and satisfies a global Lipschitz condition on $E$. Since $\nabla \Phi _{H}(x)=x^{+}-x^{-}-\nabla \mathfrak {b}(x)$ for any $x\in E$, we have

(3.11)\begin{equation} x_j^{+}-x_j^{-}-\nabla \mathfrak{b}(x_j)\rightarrow 0. \end{equation}

Case 1. $(x_j)$ is bounded in $E$. Then $(x^{0}_j)$ is a bounded sequence in the space $\mathbb {R}^{n,k}$ of finite dimension. Hence $(x^{0}_j)$ has a convergent subsequence. Moreover, since $\nabla \mathfrak {b}$ is compact, $(\nabla \mathfrak {b}(x_j))$ has a convergent subsequence, and so both $(x_j^{+})$ and $(x_j^{-})$ have convergent subsequences in $E$. It follows that $(x_j)$ has a convergent subsequence.Case 2. $(x_j)$ is unbounded in $E$. Without loss of generality, we may assume $\lim _{j\rightarrow +\infty }\|x_j\|_E=+\infty$. For $z_0\in \mathbb {R}^{n,k}$ defined as in (H2), let

\[ y_j=\frac{x_j}{\|x_j\|_E}-\frac{1}{2a}z_0. \]

Then $|y_j^{0}|\le \|y_j\|_E\le 1+|{z_0}/{2a}|$, and (3.11) implies

(3.12)\begin{equation} y_j^{+}-y_j^{-}-\jmath^{{\ast}}\left(\frac{ \nabla H(x_j)}{\|x_j\|_E}\right)\rightarrow 0. \end{equation}

Also by (H2) there exist constants $C_1$ and $C_2$ such that

\[ \left\|\frac{\nabla H(x_j)}{\|x_j\|_E}\right\|_{L^{2}}^{2}\leq \frac{8a^{2} \|x_j\|_{L^{2}}^{2}+C_1 }{\|x_k\|_E^{2}}\leq C_2 \]

that is, $(\nabla H(x_j)/\|x_j\|_E)$ is bounded in $L^{2}$. Hence the sequence $\jmath ^{\ast }( \nabla H(x_j)/\|x_j\|_E)$ is compact. (3.12) implies that $(y_j)$ has a convergent subsequence in $E$. Without loss of generality, we may assume that $y_{j}\rightarrow y$ in $E$. Since (H2) implies that $H(z)=Q(z):=a|z|^{2}+ \langle z, z_0\rangle + b$ for $|z|$ sufficiently large, there exists a constant $C>0$ such that $|\nabla H(z)-\nabla Q(z)|\leq C$ for all $z\in \mathbb {R}^{2n}$. It follows that as $j\to \infty$,

\begin{align*} \left\|\frac{\nabla H(x_j)}{\|x_j\|_{E}}-\nabla Q(y)\right\|_{L^{2}} & \leq \left\|\frac{\nabla H(x_j)}{\|x_j\|_{E}}-\nabla Q(y_j)\right\|_{L^{2}}+ \left\|\nabla Q(y_j)-\nabla Q(y)\right\|_{L^{2}}\\ & \leq \left\|\frac{\nabla H(x_j)-\nabla Q(x_j)}{\|x_j\|_E}\right\|_{L^{2}}+ \frac{|z_0|}{\|x_j\|_E}+2a\|y_j-y\|_{L^{2}}\\ & \leq \frac{C}{\|x_j\|_E}+\frac{|z_0|}{\|x_j\|_E}+2a\|y_j-y\|_{L^{2}}\rightarrow 0. \end{align*}

This implies that $\jmath ^{\ast }(\nabla H(x_k)/\|x_k\|_{E})$ tends to $\jmath ^{\ast }(\nabla Q(y))$ in $E$, and thus we arrive at

\[ y^{+}-y^{-}-\jmath^{{\ast}}(\nabla Q (y))=0\quad\hbox{and}\quad \left\|y+\frac{z_0}{2a} \right\|_E=1. \]

Then $y$ is smooth and satisfies

\[ \dot{y}=J_{2n}\nabla Q(y)\quad\hbox{and}\quad y(1)\sim y(0),\enspace y(0),\ y(1)\in\mathbb{R}^{n,k}. \]

Clearly $y(t)$ is given by

\[ y(t)+\frac{1}{2a}z_0={\rm e}^{2aJ_{2n}t}\left(y(0)+\frac{1}{2a}z_0\right). \]

Since $\|y+({1}/{2a})z_0\|_{E}=1$ implies that $y+({1}/{2a})z_0$ is nonconstant, using the boundary condition satisfied by $y$ and the assumption that $z_0\in \mathbb {R}^{n,k}$, we deduce that $2a\in m\mathbb {N}\pi$. This gives rise to a contradiction because $H$ is strong non-resonant.

Corresponding to [37, § 3.5, lemma] we have

Lemma 3.10 Suppose that $H:\mathbb {R}^{2n}\to \mathbb {R}$ is of class $C^{2n+2}$ and that $\nabla H:\mathbb {R}^{2n}\to \mathbb {R}^{2n}$ satisfies a global Lipschitz condition. Then the set of critical values of $\Phi _{H}$ has empty interior in $\mathbb {R}$.

Proof. The method is similar to that of [26, lemma 3.5]. For clearness we give it in details. By lemma 2.4, $\Phi _H$ is $C^{1,1}$. Lemma 2.5 implies that all critical points of $\Phi _{H}$ sit in $C^{2n+2}_{n,k}([0,1],\mathbb {R}^{2n})$. Thus the restriction of $\Phi _{H}$ to $C^{1}_{n,k}([0,1],\mathbb {R}^{2n})$, denoted by $\hat \Phi _{H}$, and $\Phi _{H}$ have the same critical value sets. As in the proof of [24, claim 4.4] we can deduce that $\hat \Phi _{H}$ is of class $C^{2n+1}$.

Let $P_0$ and $P_1$ be the orthogonal projections of $\mathbb {R}^{2n}$ to the spaces $V_0^{n,k}$ and $V_1^{2k}$ in (1.1) and (1.2), respectively. Take a smooth $g:[0,1]\rightarrow [0,1]$ such that $g$ equals $1$ (resp. $0$) near $0$ (resp. $1$). Denote by $\phi ^{t}$ the flow of $X_{H}$. Since $X_{H}$ is $C^{2n+1}$, we have a $C^{2n+1}$ map

\[ \psi:[0,1]\times\mathbb{R}^{n,k}\rightarrow \mathbb{R}^{2n},\quad (t,z)\mapsto g(t)\phi^{t}(z)+(1-g(t))\phi^{t-1}(P_0\phi^{1}(z)+P_1z). \]

For any $z\in \mathbb {R}^{n,k}$, since $\psi (0,z)=\phi ^{0}(z)=z$ and $\psi (1,z)=P_0\phi ^{1}(z)+P_1z$, we have

\[ \psi(1,z),\quad \psi(0,z)\in\mathbb{R}^{n,k}\quad\hbox{and}\quad \psi(1,z)\sim\psi(0,z). \]

These and [24, corollary B.2] show that $\psi$ gives rise to a $C^{2n}$ map

\[ \Omega:\mathbb{R}^{n,k}\to C^{1}_{n,k}([0,1],\mathbb{R}^{2n}),\quad z\mapsto \psi({\cdot},z). \]

Hence $\Phi _{H}\circ \Omega : \mathbb {R}^{n,k}\to \mathbb {R}$ is of class $C^{2n}$. By Sard's theorem we deduce that the critical value sets of $\Phi _{H}\circ \Omega$ is nowhere dense (since $\dim \mathbb {R}^{n,k}<2n$).

Let $z\in \mathbb {R}^{n,k}$ be such that $\phi ^{1}(z)\in \mathbb {R}^{n,k}$ and $\phi ^{1}(z)\sim z$. Then $P_0\phi ^{1}(z)-P_0z=\phi ^{1}(z)-z$ and therefore $P_0\phi ^{1}(z)+P_1z=\phi ^{1}(z)$, which implies $\psi (t,z)=\phi ^{t}(z) \forall \ t\in [0,1]$.

For a critical point $y$ of $\Phi _H$, that is, $y\in C^{2n+2}_{n,k}([0,1],\mathbb {R}^{2n})$ and solves $\dot {y}=J_{2n}\nabla H(y)=X_H(y)$, with $z_y:=y(0)\in \mathbb {R}^{n,k}$ we have $y(t)=\phi ^{t}(z_y)\ \forall \ t\in [0,1]$, which implies that $\phi ^{1}(z_y)\in \mathbb {R}^{n,k}$, $\phi ^{1}(z_y)\sim z_y$ and therefore $y=\psi (\cdot,z_y)=\Omega (z_y)$. Hence $z_y$ is a critical point of $\Phi _{H}\circ \Omega$ and $\Phi _{H}\circ \Omega (z_y)=\Phi _{H}(y)$. Thus the critical value set of $\Phi _{H}$ is contained in that of $\Phi _{H}\circ \Omega$. The desired claim is obtained.

Having this lemma we can prove the following proposition, which corresponds to proposition 3 in [13, § II].

Proposition 3.11 Let $H\in C^{2n+2}(\mathbb {R}^{2n},\mathbb {R}_{\ge 0})$ be $\mathbb {R}^{n,k}$-admissible with $k< n$ and strong nonresonant. Suppose that $[0,1]\ni s\mapsto \psi _s$ is a smooth homotopy of the identity in ${\rm Symp}(\mathbb {R}^{2n},\omega _0)$ satisfying

(3.13)\begin{equation} \psi_s(\mathbb{R}^{n,k})=\mathbb{R}^{n,k},\quad \psi_s(w+ V^{n,k}_0)=\psi_s(w)+ V^{n,k}_0\quad\forall\ w\in\mathbb{R}^{n,k} \end{equation}

and

\[ \psi_s(z)=z+ w_s\quad\forall\ z\in \mathbb{R}^{2n}\setminus B^{2n}(0, R), \]

where $R>0$ and $[0,1]\ni s\mapsto w_s$ is a smooth path in $\mathbb {R}^{n,k}$. Then $s\mapsto c^{n,k}(H\circ \psi _s)$ is constant. Moreover, the same conclusion holds true if all $\psi _s$ are replaced by translations $\mathbb {R}^{2n}\ni z\mapsto z+ w_s,$ where $[0,1]\ni s\mapsto w_s$ is a smooth path in $\mathbb {R}^{n,k}$. In particular, $c^{n,k}(H(\cdot + w))=c^{n,k}(H)$ for any $w\in \mathbb {R}^{n,k}$.

Proof. By the assumptions each $H\circ \psi _s$ is also $\mathbb {R}^{n,k}$-admissible and strong nonresonant. Hence $c(H\circ \psi _s)$ is a positive critical value for each $s$. Let $x\in E$ be a critical point of $\Phi _{H\circ \psi _s}$ with critical value $c(H\circ \psi _s)$. Then $x\in C^{2n+2}_{n,k}([0,1],\mathbb {R}^{2n})$ and solves $\dot {x}=J_{2n}\nabla (H\circ \psi _s)(x)=X_{H\circ \psi _s}(x)$. Let $y_s=\psi _s\circ x$. Then $y_s\in C^{2n+2}_{n,k}([0,1],\mathbb {R}^{2n})$ and satisfies

\[ \dot{y}_s(t)=({\rm d}\psi_s(x(t))\dot{x}(t)= ({\rm d}\psi_s(x(t))X_{H\circ\psi_s}(x(t))= X_{H}(\psi_s(x(t))=J_{2n}\nabla H(y_s(t)) \]

since ${\rm d}\psi _s(z)X_H(z)=X_{H}(\psi _s(z))$ for any $z\in \mathbb {R}^{2n}$ by [20, p. 9]. Therefore $y_s$ is a critical point of $\Phi _H$ on $E$. We claim that

(3.14)\begin{equation} \Phi_H(y_s)=\Phi_{H\circ\psi_s}(x). \end{equation}

Clearly, it suffices to prove the following equality:

(3.15)\begin{equation} A(y_s)=\frac{1}{2}\int^{1}_0 \langle -J_{2n}\dot{y}_s, y_s\rangle \,{{\rm d}}t= \frac{1}{2}\int_0^{1}\langle -J_{2n}\dot{x},x\rangle \,{{\rm d}}t=A(x). \end{equation}

Extend $x$ into a piecewise $C^{2n+2}$-smooth loop $x^{\ast }:[0, 2]\to \mathbb {R}^{2n}$ by setting $x^{\ast }(t)=(2-t)x(1)+(t-1)x(0)$ for any $1\le t\le 2$. We get a piecewise $C^{2n+2}$-smooth loop extending of $y_s$, $y_s^{\ast }=\psi _s(x^{\ast })$. Clearly, we can extend $x^{\ast }$ into a piecewise $C^{2n+2}$-smooth $u:D^{2}\to \mathbb {R}^{2n}$, where $D^{2}$ is a closed disc bounded by $\partial D^{2}\equiv [0,2]/\{0,2\}$. Then $\psi _s\circ u:D^{2}\to \mathbb {R}^{2n}$ is piecewise $C^{2n+2}$-smooth and $\psi _s\circ u|_{\partial D^{2}}=y_s^{\ast }$. Stokes theorem yields

\begin{align*} & \frac{1}{2}\int_0^{2}\langle -J_{2n}\dot{x}^{{\ast}},x^{{\ast}}\rangle \,{{\rm d}}t=\int_{D^{2}}u^{{\ast}}\omega_0,\\ & \frac{1}{2}\int^{2}_0 \langle -J_{2n}\dot{y}^{{\ast}}_s, y^{{\ast}}_s\rangle \,{{\rm d}}t= \int_{D^{2}}(\psi_s\circ u)^{{\ast}}\omega_0=\int_{D^{2}}u^{{\ast}}\omega_0. \end{align*}

Moreover, for any $t\in [1,2]$ we have $\dot {x}^{\ast }(t)=x(0)-x(1)\in V_0^{n,k}$ and $x^{\ast }(t)\in \mathbb {R}^{n,k}$, and therefore $\langle -J_{2n}\dot {x}^{\ast }(t), x^{\ast }(t)\rangle =0$ because $\mathbb {R}^{2n}$ has the orthogonal decomposition $\mathbb {R}^{2n}=J_{2n}V^{n,k}_0\oplus \mathbb {R}^{n,k}$. Then (3.15) follows from these.

Since $s\mapsto c^{n,k}(H\circ \psi _s)$ is continuous by proposition 3.5, and a critical point $x$ of $\Phi _{H\circ \psi _s}$ with critical value $c(H\circ \psi _s)$ yields a critical point $y_s$ of $\Phi _H$ on $E$ satisfying (3.14), we deduce that each $c(H\circ \psi _s)$ is also a critical value of $\Phi _H$. Lemma 3.10 shows that $s\mapsto c^{n,k}(H\circ \psi _s)$ must be constant.

Finally, let $\psi _s(z)=z+ w_s$. It is clear that $H\circ \psi _s$ is $\mathbb {R}^{n,k}$-admissible and strong nonresonant. Thus $c(H\circ \psi _s)$ is a positive critical value. If $x\in E$ is a critical point of $\Phi _{H\circ \psi _s}$ with critical value $c(H\circ \psi _s)$, then $y_s:=\psi _s\circ x$ is a critical point of $\Phi _H$ on $E$ and (3.14) holds. Hence $s\mapsto c^{n,k}(H(\cdot + w_s))$ is constant.

Let $\mathscr {F}_{n,k}(\mathbb {R}^{2n})=\{H\in C^{0}(\mathbb {R}^{2n},\mathbb {R}_{\ge 0})\,|\,H\ \hbox {satisfies (H2)}\}$. For each bounded subset $B\subset \mathbb {R}^{2n}$ such that $\overline {B}\cap \mathbb {R}^{n,k}\neq \emptyset$, we define

(3.16)\begin{equation} \mathscr{F}_{n,k}(\mathbb{R}^{2n},B)=\{H\in \mathscr{F}_{n,k} (\mathbb{R}^{2n})\,|\,H\ \hbox{vanishes near}\ \overline{B}\}. \end{equation}

Definition 3.12 For each bounded subset $B\subset \mathbb {R}^{2n}$ such that $\overline {B}\cap \mathbb {R}^{n,k}\neq \emptyset,$

(3.17)\begin{equation} c^{n,k}(B):=\inf\{c^{n,k}(H)\,|\, H\in \mathscr{F}_{n,k}(\mathbb{R}^{2n},B)\}\in [0, +\infty) \end{equation}

is called the coisotropic Ekeland–Hofer capacity of $B$ $($relative to $\mathbb {R}^{n,k})$. For any unbounded subset $B\subset \mathbb {R}^{2n}$ such that $\overline {B}\cap \mathbb {R}^{n,k}\neq \emptyset,$ its coisotropic Ekeland–Hofer capacity is defined by

(3.18)\begin{equation} c^{n,k}(B)=\sup\{c^{n,k}(A)\,|\, A\subset B, A\ \hbox{is bounded and}\ \overline{A}\cap\mathbb{R}^{n,k}\neq \emptyset\}. \end{equation}

Remark 3.13 When $k=n$ in the above definition, $c^{n,n}(B)$ is the (first) Ekeland–Hofer capacity of $B$.

For each bounded $B\subset \mathbb {R}^{2n}$ such that $\overline {B}\cap \mathbb {R}^{n,k}\neq \emptyset$, we write

\begin{align*} & \mathscr{E}_{n,k}(\mathbb{R}^{2n},B)=\{H\in \mathscr{F}_{n,k}(\mathbb{R}^{2n},B)\,|\,H\ \hbox{is strong nonresonant}\}\quad\hbox{if}\ k< n,\\ & \mathscr{E}_{n,n}(\mathbb{R}^{2n},B)=\{H\in \mathscr{F}_{n,k}(\mathbb{R}^{2n},B)\,|\,H\ \hbox{is nonresonant}\}. \end{align*}

Clearly, each $H\in \mathscr {E}_{n,k}(\mathbb {R}^{2n},B)$ satisfies (H1), and $\mathscr {E}_{n,k}(\mathbb {R}^{2n},B)$ is a cofinal family of $\mathscr {F}_{n,k}(\mathbb {R}^{2n},B)$, that is, for any $H\in \mathscr {F}_{n,k}(\mathbb {R}^{2n},B)$ there exists $G\in \mathscr {E}_{n,k}(\mathbb {R}^{2n},B)$ such that $G\ge H$. Moreover, for each $l\in \mathbb {N}\cup \{\infty \}$ the smaller subset $\mathscr {E}_{n,k}(\mathbb {R}^{2n},B)\cap C^{l}(\mathbb {R}^{2n},\mathbb {R}_{\ge 0})$ is also a cofinal family of $\mathscr {F}_{n,k}(\mathbb {R}^{2n},B)$. By the definition, we immediately get:

Proposition 3.14

1. (i) $c^{n,k}(B)=c^{n,k}(\overline {B})$.

2. (ii) $\mathscr {F}_{n,k}(\mathbb {R}^{2n},B)$ in (3.17) can be replaced by any cofinal subset of it.

3. (iii) Suppose that $\overline {B}\subset B^{2n}(R)$. For each $l\in \mathbb {N}\cup \{\infty \}$ let $\mathscr {E}^{l}_{n,k}(\mathbb {R}^{2n},B)$ consist of $H\in \mathscr {F}_{n,k}(\mathbb {R}^{2n},B)\cap C^{l}(\mathbb {R}^{2n},\mathbb {R}_{\ge 0})$ for which there exists $z_0\in \mathbb {R}^{n,k},$ real numbers $a, b$ such that $H(z)=a|z|^{2}+ \langle z, z_0\rangle + b$ outside the closed ball $\overline {B^{2n}(R)},$ where $a>\pi$ and $a\notin \pi \mathbb {N}$ for $k=n,$ and $a>\pi /2$ and $a\notin \pi \mathbb {N}/2$ for $0\le k< n$. Then each $\mathscr {E}^{l}_{n,k}(\mathbb {R}^{2n},B)$ is a cofinal subset of $\mathscr {F}_{n,k}(\mathbb {R}^{2n},B)$.

Proof. We only prove (iii). By (ii) it suffices to prove that for each given $H\in \mathscr {E}_{n,k}(\mathbb {R}^{2n},B)$ there exists $G\in \mathscr {E}^{l}_{n,k}(\mathbb {R}^{2n},B)$ such that $G\ge H$. We may assume that $H(z)=a|z|^{2}+ \langle z, z_0\rangle + b$ outside a larger closed ball $\overline {B^{2n}(R_1)}$, where $a>\pi$ and $a\notin \pi \mathbb {N}$ for $k=n$, and $a>\pi /2$ and $a\notin \pi \mathbb {N}/2$ for $0\le k< n$. Let $U_\epsilon (B)$ be the $\epsilon$-neighbourhood of $B$. We can also assume that $H$ vanishes in $U_{2\epsilon }(B)$. Since $\overline {B^{2n}(R_1)}$ is compact, we may find numbers $a'>a$, $b'$ such that $a'\notin \pi \mathbb {N}$ for $k=n$, $a'\notin \pi \mathbb {N}/2$ for $0\le k< n$, and $a'|z|^{2}+ \langle z, z_0\rangle + b'\ge H(z)$ for all $z\in \mathbb {R}^{2n}$. Take a smooth function $f:\mathbb {R}^{2n}\to \mathbb {R}_{\ge 0}$ such that it equals to zero in $U_{\epsilon }(B)$ and $1$ outside $U_{2\epsilon }(B)$. Define $G(z):=f(z)(a'|z|^{2}+ \langle z, z_0\rangle + b')$ for $z\in \mathbb {R}^{2n}$. Then $G\ge H$ and $G\in \mathscr {E}^{\infty }_{n,k}(\mathbb {R}^{2n},B)$.

Remark 3.15 Let $\mathscr {H}_{n,k}(\mathbb {R}^{2n},B)$ consist of $H\in C^{\infty }(\mathbb {R}^{2n},\mathbb {R}_{\ge 0})$ which vanishes near $\overline {B}$ and for which there exists $z_0\in \mathbb {R}^{n,k}$ and a real number $a$ such that $H(z)=a|z|^{2}$ outside a compact subset, where $a>\pi$ and $a\notin \pi \mathbb {N}$ for $k=n$, and $a>\pi /2$ and $a\notin \pi \mathbb {N}/2$ for $0\le k< n$. As in the proof of proposition 1.1 it is not hard to prove that $\mathscr {H}_{n,k}(\mathbb {R}^{2n},B)$ is a cofinal subset of $\mathscr {F}_{n,k}(\mathbb {R}^{2n},B)$. When $k=n$ this shows that Sikorav's approach [37] to the Ekeland–Hofer capacity in [13] defines the same capacity.

Proof of proposition 1.1. Proposition 3.5(i)–(iii) lead to the first three claims. Let us prove (iv). We may assume that $B$ is bounded. By (3.17) we have a sequence $(H_j)\subset \mathscr {F}_{n,k}(\mathbb {R}^{2n},B)$ such that $c^{n,k}(H_j)\to c^{n,k}(B)$. Note that $H_j(\cdot -w)\in \mathscr {F}_{n,k}(\mathbb {R}^{2n},B+w)$ for each $j$. Hence

\[ c^{n,k}(B+w)\le\inf_j c^{n,k}(H_j({\cdot}{-}w))= \inf_j c^{n,k}(H_j)=c^{n,k}(B) \]

by the final claim in proposition 3.11. The same reasoning leads to $c^{n,k}(B)=c^{n,k}(B+w+(-w))\le c^{n,k}(B+w)$ and so $c^{n,k}(B+w)=c^{n,k}(B)$.

Proposition 3.16 (Relative monotonicity)

Let subsets $A, B\subset \mathbb {R}^{2n}$ satisfy $\overline {A}\cap \mathbb {R}^{n,k}\neq \emptyset$ and $\overline {B}\cap \mathbb {R}^{n,k}\neq \emptyset$. If there exists a smooth homotopy of the identity in ${\rm Symp}(\mathbb {R}^{2n},\omega _0)$ as in proposition 3.11, $[0,1]\ni s\mapsto \psi _s,$ such that $\psi _1(A)\subset B,$ then $c^{n,k}(A)=c^{n,k}(\psi _s(A))$ for all $s\in [0,1]$, and in particular $c^{n,k}(A)\le c^{n,k}(B)$ by proposition 1.1(i).

Proof. Note that

\[ \mathscr{E}_{n,k}(\mathbb{R}^{2n}, A)\cap C^{\infty}(\mathbb{R}^{2n},\mathbb{R}_{\ge 0})\to \mathscr{E}_{n,k}(\mathbb{R}^{2n}, \psi_s(A))\!\cap\! C^{\infty}(\mathbb{R}^{2n},\mathbb{R}_{\ge 0}),\quad H\!\mapsto\! H\circ\psi_s^{{-}1} \]

is a one-to-one correspondence. Then

\begin{align*} c^{n,k}(\psi_s(A))& = \inf\{c^{n,k}(G)\,|\, G\in \mathscr{E}_{n,k} (\mathbb{R}^{2n}, \psi_s(A))\cap C^{\infty}(\mathbb{R}^{2n},\mathbb{R}_{\ge 0})\}\\ & = \inf\{c^{n,k}(H\circ\psi_s^{{-}1})\,|\, H\in \mathscr{E}_{n,k} (\mathbb{R}^{2n}, A)\cap C^{\infty}(\mathbb{R}^{2n},\mathbb{R}_{\ge 0})\}\\ & = \inf\{c^{n,k}(H)\,|\, H\in \mathscr{E}_{n,k}(\mathbb{R}^{2n}, A) \cap C^{\infty}(\mathbb{R}^{2n},\mathbb{R}_{\ge 0})\}=c^{n,k}(A). \end{align*}

Here the third equality comes from proposition 3.11.

Proof of theorem 1.2. We may assume that $B$ is bounded, and complete the proof in two steps.

Step 1. Prove $c^{n,k}(\Phi (B))=c^{n,k}(B)$ for every $\Phi \in {\rm Sp}(2n,k)$. Take a smooth path $[0, 1]\ni t\mapsto \Phi _t\in {\rm Sp}(2n,k)$ such that $\Phi _0=I_{2n}$ and $\Phi _1=\Phi$. We have a smooth function $[0,1]\times \mathbb {R}^{2n}\ni (t,z)\mapsto G_t(z)\in \mathbb {R}^{2n}$ such that the path $\Phi _t$ is generated by $X_{G_t}$ and that $G_t(z)=0\ \forall \ z\in \mathbb {R}^{n,k}$ (see step 2 below). Since $\cup _{t\in [0, 1]}\Phi _t(\overline {B})$ is compact, it can be contained in a ball $B^{2n}(0, R)$ for some $R>0$. Take a smooth cut function $\rho :\mathbb {R}^{2n}\to [0, 1]$ such that $\rho =1$ on $B^{2n}(0, 2R)$ and $\rho =0$ outside $B^{2n}(0, 3R)$. Define a smooth function $\tilde {G}:[0,1]\times \mathbb {R}^{2n}\to \mathbb {R}$ by $\tilde {G}(t,z)=\rho (z)G_t(z)$ for $(t,z)\in [0,1]\times \mathbb {R}^{2n}$. Denote by $\psi _t$ the Hamiltonian path generated by $\tilde {G}$ in ${\rm Ham}^{c}(\mathbb {R}^{2n},\omega _0)$. Then $\psi _t(z)=\Phi _t(z)$ for all $(t,z)\in [0,1]\times B^{2n}(0, R)$. Moreover each $\psi _t$ restricts to the identity on $\mathbb {R}^{n,k}$ because $\tilde {G}(t,z)=\rho (z)G_t(z)=0$ for all $(t,z)\in [0,1]\times \mathbb {R}^{n,k}$. Hence we obtain $c^{n,k}(\Phi (B))=c^{n,k}(\Phi _1(B))=c^{n,k}(B)$ by proposition 3.16.

Step 2. Prove $c^{n,k}(\phi (B))=c^{n,k}(B)$ in case $w_0=0$. Let $\Phi =({\rm d}\phi (0))^{-1}$. Since $c^{n,k}(\Phi \circ \phi (B))=c^{n,k}(\phi (B))$ by step 1, and $\Phi \circ \phi (w)=w\ \forall \ w\in \mathbb {R}^{n,k}$, replacing $\Phi \circ \phi$ by $\phi$, we may assume ${\rm d}\phi (0)={\rm id}_{\mathbb {R}^{2n}}$. Define a continuous path in ${\rm Symp}(\mathbb {R}^{2n},\omega _0)$,

(3.19)\begin{equation} \varphi_t(z)=\left\{\begin{array}{@{}ll} z & \hbox{if}\ t\le 0,\\ \dfrac{1}{t}\phi(tz) & \hbox{if}\ t> 0, \end{array}\right. \end{equation}

which is smooth except possibly at $t=0$. As in [36, proposition A.1] we can smoothen it with a smooth function $\eta :\mathbb {R}\to \mathbb {R}$ defined by

(3.20)\begin{equation} \eta(t)=\left\{\begin{array}{@{}ll} 0 & \hbox{if}\ t\le 0,\\ { e}^{2}\,{ e}^{{-}2/t} & \hbox{if}\ t> 0, \end{array}\right. \end{equation}

where $e$ is the Euler number. Namely, defining $\phi _t(z):=\varphi _{\eta (t)}(z)$ for $z\in \mathbb {R}^{2n}$ and $t\in \mathbb {R}$, we get a smooth path $\mathbb {R}\ni t\mapsto \phi _t\in {\rm Symp}(\mathbb {R}^{2n},\omega _0)$ such that

(3.21)\begin{equation} \phi_0={\rm id}_{\mathbb{R}^{2n}},\quad \phi_1=\phi,\quad \phi_t(z)=z,\quad \forall\ z\in\mathbb{R}^{n,k},\enspace \forall\ t\in\mathbb{R}. \end{equation}

Define $X_t(z)=(({{{\rm d}}}/{{{\rm d}}t})\phi _t)(\phi _t^{-1}(z))$ and

(3.22)\begin{equation} H_t(z)=\int^{z}_0 i_{X_t}\omega_0, \end{equation}

where the integral is along any piecewise smooth curve from $0$ to $z$ in $\mathbb {R}^{2n}$. Then $\mathbb {R}\times \mathbb {R}^{2n}\ni (t,z)\mapsto H_t(z)\in \mathbb {R}$ is smooth and $X_t=X_{H_t}$. By the final condition in (3.21), for each $(t,z)\in \mathbb {R}\times \mathbb {R}^{n,k}$ we have $X_t(z)=0$ and therefore $H_t(z)=0$. As in step 1, we can assume that $\cup _{t\in [0, 1]}\phi _t(\overline {B})$ is contained a ball $B^{2n}(0, R)$. Take a smooth cut function $\rho :\mathbb {R}^{2n}\to [0, 1]$ as above, and define a smooth function $\tilde {H}:[0,1]\times \mathbb {R}^{2n}\to \mathbb {R}$ by $\tilde {H}(t,z)=\rho (z)H_t(z)$ for $(t,z)\in [0,1]\times \mathbb {R}^{2n}$. Then the Hamiltonian path $\psi _t$ generated by $\tilde {H}$ in ${\rm Ham}^{c}(\mathbb {R}^{2n},\omega _0)$ satisfies

\[ \psi_t(z)=\phi_t(z),\quad \forall\ (t,z)\in [0,1]\times B^{2n}(0, R)\quad\hbox{and}\quad \psi_t(z)=z,\quad \forall\ (t,z)\in [0,1]\times \mathbb{R}^{n,k}. \]

It follows from proposition 3.16 that $c^{n,k}(\phi (B))=c^{n,k}(\psi _1(B))=c^{n,k}(B)$ as above.

Step 3. Prove $c^{n,k}(\phi (B))=c^{n,k}(B)$ in case $w_0\ne 0$. Define $\varphi (w)=\phi (w+w_0)$ for $w\in \mathbb {R}^{2n}$. Then ${\rm d}\varphi (0)={\rm d}\phi (w_0)\in {\rm Sp}(2n,k)$ and $\varphi (w)=\phi (w+w_0)=w \forall \ w\in \mathbb {R}^{n,k}$. By step 2 we arrive at $c^{n,k}(\varphi (B-w_0))=c^{n,k}(B-w_0)$. The desired equality follows because $\phi (B)=\varphi (B-w_0)$ and $c^{n,k}(B-w_0)= c^{n,k}(B)$ by proposition 1.1.

Proof of corollary 1.3. As discussed above the proof is reduced to the case $w_0=0$. Moreover we can assume that both sets $A$ and $U$ are bounded and that $U$ is also star-shaped with respect to the origin $0\in \mathbb {R}^{2n}$.

Next the proof can be completed following [36, proposition A.1]. Now $[0,1]\ni t\mapsto \phi _t(\cdot ):=\varphi _{\eta (t)}(\cdot )$ given by (3.19) and (3.20) is a smooth path of symplectic embeddings from $U$ to $\mathbb {R}^{2n}$ with properties

(3.23)\begin{equation} \phi_0={\rm id}_U,\quad \phi_1=\varphi,\quad \phi_t(z)=z,\quad \forall\ z\in\mathbb{R}^{n,k}\cap U,\enspace \forall\ t\in\mathbb{R}. \end{equation}

Thus $X_t(z):=(({{{\rm d}}}/{{{\rm d}}t})\phi _t)(\phi _t^{-1}(z))$ is a symplectic vector field defined on $\phi _t(U)$, and (3.22) (where the integral is along any piecewise smooth curve from $0$ to $z$ in $\phi _t(U)$) defines a smooth function $H_t$ on $\phi _t(U)$ in the present case. Observe that $H:\cup _{t\in [0,1]}(\{t\}\times \phi _t(U))\to \mathbb {R}$ defined by $H(t,z)=H_t(z)$ is smooth and generates the path $\phi _t$. Since $K=\cup _{t\in [0,1]}\{t\}\times \phi _t(\overline {A})$ is a compact subset in $[0,1]\times \mathbb {R}^{2n}$ we can choose a bounded and relative open neighbourhood $W$ of $K$ in $[0,1]\times \mathbb {R}^{2n}$ such that $W\subset \cup _{t\in [0,1]}(\{t\}\times \phi _t(U))$. Take a smooth cut function $\chi :[0,1]\times \mathbb {R}^{2n}\to \mathbb {R}$ such that $\chi |_K=1$ and $\chi$ vanishes outside $W$. Define $\hat {H}:[0,1]\times \mathbb {R}^{2n}\to \mathbb {R}$ by $\hat {H}(t,z)=\chi (t,z)H(t,z)$. It generates a smooth homotopy $\psi _t$ ($t\in [0,1]$) of the identity in ${\rm Ham}^{c}(\mathbb {R}^{2n},\omega _0)$ such that $\psi _t(z)=\phi _t(z)$ for all $(t,z)\in [0,1]\times A$. Moreover, the final condition in (3.21) implies that $\mathbb {R}^{n,k}\cap U\subset \phi _t(U)$ and $X_t(z)=0$ for any $t\in [0,1]$ and $z\in \mathbb {R}^{n,k}\cap U$. Hence for any $(t,z)\in [0,1]\times \mathbb {R}^{n,k}$ we have $\hat {H}(t,z)=\chi (t,z)H(t,z)=0$ and so $\psi _t(z)=z$. Then proposition 3.16 leads to $c^{n,k}(A)= c^{n,k}(\psi _1(A))=c^{n,k}(\phi _1(A))=c^{n,k}(\varphi (A))$.

4. Proof of theorem 1.4

The case of $k=n$ was proved in [13, 14, 37]. We assume $k< n$ below. By proposition 1.1(iv), $c^{n,k}(D)=c^{n,k}(D+w)$ for any $w\in \mathbb {R}^{n,k}$. Moreover, for each $x\in C^{1}_{n,k}([0,1])$ there holds

\[ A(x)=\frac{1}{2}\int_0^{1}\langle -J_{2n}\dot{x},x\rangle \,{{\rm d}}t= \frac{1}{2}\int_0^{1}\langle -J_{2n}\dot{x},x+w\rangle \,{{\rm d}}t=A(x+w),\quad\forall\ w\in\mathbb{R}^{n,k}. \]

Recalling that $D\cap \mathbb {R}^{n,k}\ne \emptyset$, we may assume that $D$ contains the origin $0$ below.

Let $j_D$ be the Minkowski functional associated to $D$, $H:=j_D^{2}$ and $H^{\ast }$ be the Legendre transform of $H$. Then $\partial D=H^{-1}(1)$, and there exists a constant $R\geq 1$ such that

(4.1)\begin{equation} \frac{|z|^{2}}{R}\leq H(z)\leq R|z|^{2}\quad\hbox{and so}\quad \frac{|z|^{2}}{4R}\leq H^{{\ast}}(z)\leq \frac{R}{4}|z|^{2} \end{equation}

for all $z\in \mathbb {R}^{2n}$. Moreover $H$ is $C^{1,1}$ with uniformly Lipschitz constant.

By [26, theorem 1.5]

\[ \Sigma^{n,k}_{\partial D}:=\{A(x)>0\,|\,x\ \hbox{is a leafwise chord on}\ \partial D\ \hbox{for}\ \mathbb{R}^{n,k}\} \]

contains a minimum number $\varrho$, that is, there exists a leafwise chord $x^{\ast }$ on $\partial D$ for $\mathbb {R}^{n,k}$ such that $A(x^{\ast })=\min \Sigma ^{n,k}_{\partial D}=\varrho$. Actually, the arguments therein shows that there exists $w\in C^{1}_{n,k}([0,1])$ such that

(4.2)\begin{equation} A(w)=1\quad\hbox{and}\quad I(w):=\int_0^{1}H^{{\ast}}({-}J_{2n}\dot{w})\,{{\rm d}}t=A(x^{{\ast}})=\varrho. \end{equation}

Let us prove (1.8) and (1.9) by the following two steps. As done in [24, 25] (see also step 4 below), by approximating arguments we can assume that $\partial D$ is smooth and strictly convex. In this case $\Sigma ^{n,k}_{\partial D}$ has no interior points in $\mathbb {R}$ because of [26, lemma 3.5], and we give a complete proof though the ideas which are similar to those of the proof of [37] (and [25, theorem 1.11] and [24, theorem 1.17]).

Step 1. Prove that $c^{n,k}(D)\ge \varrho$. By the monotonicity of $c^{n,k}$ it suffices to prove $c^{n,k}(\partial D)\ge \varrho$. For a given $\epsilon >0$, consider a cofinal family of $\mathcal {F}_{n,k}(\mathbb {R}^{2n},\partial D)$,

(4.3)\begin{equation} \mathscr{E}^{n,k}_\epsilon(\mathbb{R}^{2n},\partial D) \end{equation}

consisting of $\overline {H}=f\circ H$, where $f\in C^{\infty }(\mathbb {R},\mathbb {R}_{\ge 0})$ satisfies

(4.4)\begin{align} \left.\begin{array}{ll@{}} & f(s)=0\ \hbox{for}\ s\ \hbox{near} 1\in\mathbb{R},\\ & f'(s)\le 0\ \forall\ s\le 1,\quad f'(s)\ge 0\ \forall\ s\ge 1,\\ & f'(s)=\alpha\in\mathbb{R}\setminus\Sigma^{n,k}_{\partial D}\ \hbox{if}\ f(s)\ge\epsilon\quad s>1 \end{array}\right\} \end{align}

and where $\alpha$ is required to satisfy for some constant $C>0$

(4.5)\begin{equation} \alpha H(z)\ge \frac{\pi}{2}|z|^{2}-C\quad\hbox{for}\ |z|\ \hbox{sufficiently large} \end{equation}

because of (4.1) and ${\rm Int}(\Sigma ^{n,k}_{\partial D})=\emptyset$.

Then each $\overline {H}\in \mathscr {E}^{n,k}_\epsilon (\mathbb {R}^{2n},\partial D)$ satisfies all conditions in lemma 3.7. Indeed, it belongs to $C^{\infty }(\mathbb {R}^{2n},\mathbb {R}_{\ge 0})$, restricts to zero near $\partial D$ and thus satisfies (H1). Note that $f(s)=\alpha s+ \epsilon -\alpha s_0$ for $s\ge s_0$, where $s_0=\inf \{s>1\,|\, f(s)\ge \epsilon \}$. (4.5) implies that $\overline {H}(z)\ge ({\pi }/{2})|z|^{2}-C'\ \forall \ z\in \mathbb {R}^{2n}$ for some constant $C'>0$, and therefore $c^{n,k}(\overline {H})<+\infty$ by the arguments above proposition 3.5. Moreover, it is clear that $\mathbb {R}^{n,k}\cap {\rm Int}(\overline {H}^{-1}(0))\ne \emptyset$ and $|\overline {H}_{zz}(z)|$ is bounded on $\mathbb {R}^{2n}$. Then (3.5) is satisfied with any $z_0\in \mathbb {R}^{n,k}\cap {\rm Int}(\overline {H}^{-1}(0))$ by the arguments at the end of proof of proposition 3.6. Hence $c^{n,k}(\overline {H})>0$.

By combining proofs of lemma 3.9 and [26, lemma 3.7] we can obtain the first claim of the following.

Lemma 4.1 For every $\overline {H}\in \mathscr {E}^{n,k}_\epsilon (\mathbb {R}^{2n},\partial D),$ $\Phi _{\overline {H}}$ satisfies the $(PS)$ condition and hence $c^{n,k}(\overline {H})$ is a positive critical value of $\Phi _{\overline {H}}$.

Lemma 4.2 For every $\overline {H}\in \mathscr {E}^{n,k}_\epsilon (\mathbb {R}^{2n},\partial D),$ any positive critical value $c$ of $\Phi _{\overline {H}}$ is greater than $\min \Sigma ^{n,k}_{\partial D}-\epsilon$. In particular, $c^{n,k}(\overline {H})>\min \Sigma ^{n,k}_{\partial D}-\epsilon$.

Proof. For a critical point $x$ of $\Phi _{\overline {H}}$ with positive critical values there holds

\[{-}J_{2n}\dot{x}(t)=\nabla \overline{H}(x(t))=f'(H(x(t)))\nabla H(x(t)),\quad x(1)\sim x(0),\quad x(1), x(0)\in\mathbb{R}^{n,k} \]

and $H(x(t))\equiv c_0$ (a positive constant). Since

\begin{align*} 0<\Phi_{\overline{H}}(x) & = \frac{1}{2}\int^{1}_0\langle J_{2n}x(t),\dot{x}(t)\rangle\,{{\rm d}}t- \int^{1}_0\overline{H}(x(t))\,{{\rm d}}t\\ & = \frac{1}{2}\int^{1}_0\langle x(t),f'(c_0)\nabla H(x(t))\rangle \,{{\rm d}}t-\int^{1}_0f(s)\,{{\rm d}}t\\ & = f'(c_0)c_0-f(c_0), \end{align*}

we deduce $\beta :=f'(c_0)>0$, and so $c_0>1$. Define $y(t)=({1}/{\sqrt {c_0}})x(t/\beta )$ for $0\le t\le \beta$. Then

\[ H(y(t))=1,\quad -J_{2n}\dot{y}=\nabla H(y(t)),\quad y(\beta)\sim y(0),\quad y(\beta), y(0)\in\mathbb{R}^{n,k} \]

and therefore $f'(c_0)=\beta =A(y)\in \Sigma ^{n,k}_{\partial D}$. By the definition of $f$ this implies $f(c_0)<\epsilon$ and so

\[ \Phi_{\overline{H}}(x)=f'(c_0)c_0-f(c_0)> f'(c_0)-\epsilon\ge \min\Sigma^{n,k}_{\partial D}-\epsilon. \]

Since for any $\epsilon >0$ and $G\in \mathcal {F}_{n,k}(\mathbb {R}^{2n},\partial D)$, there exists $\overline {H}\in \mathscr {E}^{n,k}_\epsilon (\mathbb {R}^{2n},\partial D)$ such that $\overline {H}\ge G$, we deduce that $c^{n,k}(G)\ge c^{n,k}(\overline{H})\ge \min \Sigma ^{n,k}_{\partial D}-\epsilon$. Hence $c^{n,k}(\partial D)\ge \min \Sigma ^{n,k}_{\partial D}=\varrho$.

Step 2. Prove that $c^{n,k}(D)\le \varrho$. Denote by $w^{\ast }$ the projections of $w$ in (4.2) onto $E^{\ast }$ (according to the decomposition $E = E^{1/2} =E^{+}\oplus E^{-}\oplus E^{0}$), $\ast =0,-,+$. Then $w^{+}\ne 0$. (Otherwise, a contradiction occurs because $1=A(w) = A(w^{0}\oplus w^{-}) =-\frac {1}{2}\|w^{-}\|^{2}$.) Define $y:=w/\sqrt {\varrho }$. Then $y\in C^{1}_{n,k}([0,1])$ satisfies $I(y)=1$ and $A(y)={1}/{\varrho }$. It follows from the definition of $H^{\ast }$ that for any $\lambda \in \mathbb {R}$ and $x\in E$,

\[ \lambda^{2}=I(\lambda y)=\int^{1}_0H^{{\ast}}(-\lambda J_{2n}\dot{y}(t))\,{{\rm d}}t \ge\int^{1}_0\{\langle x(t), -\lambda J\dot{y}(t)\rangle- H(x(t))\}\,{{\rm d}}t \]

and so

\[ \int^{1}_0H(x(t))\,{{\rm d}}t\ge \int^{1}_0\langle x(t), -\lambda J_{2n}\dot{y}(t)\rangle \,{{\rm d}}t-\lambda^{2} = \lambda \int^{1}_0\langle x(t), - J_{2n}\dot{y}(t)\rangle \,{{\rm d}}t-\lambda^{2}. \]

In particular, taking $\lambda =\frac {1}{2} \int ^{1}_0\langle x(t), - J_{2n}\dot {y}(t)\rangle {\rm d}t$ we arrive at

(4.6)\begin{equation} \int^{1}_0H(x(t))\,{{\rm d}}t\ge\left(\frac{1}{2} \int^{1}_0\langle x(t), - J_{2n}\dot{y}(t)\rangle \,{{\rm d}}t\right)^{2},\quad \forall\ x\in E. \end{equation}

Since $y^{+}=w^{+}/\sqrt {\varrho }\ne 0$ and $E^{-}\oplus E^{0}+\mathbb {R}_+y=E^{-}\oplus E^{0}\oplus \mathbb {R}_+y^{+}$, by proposition 3.2(ii),

\[ \gamma(S^{+})\cap(E^{-}\oplus E^{0}+\mathbb{R}_+y)\ne\emptyset,\quad\forall\ \gamma\in\Gamma_{n,k}. \]

Fixing $\gamma \in \Gamma _{n,k}$ and $x\in \gamma (S^{+})\cap (E^{-}\oplus E^{0}+\mathbb {R}_+y)$, write $x=x^{-0}+ sy=x^{-0}+ sy^{-0}+ sy^{+}$ where $x^{-0}\in E^{-}\oplus E^{0}$, and consider the polynomial

\[ P(t)=\mathfrak{a}(x+ty)=\mathfrak{a}(x)+ t\int^{1}_0\langle x, - J_{2n}\dot{y}\rangle\,{{\rm d}}t + \mathfrak{a}(y)t^{2}=\mathfrak{a}(x^{{-}0}+(t+s)y). \]

Since $\mathfrak {a}|_{E^{-}\oplus E^{0}}\le 0$ implies $P(-s)\le 0$, and $\mathfrak {a}(y)=1/\varrho >0$ implies $P(t)\to +\infty$ as $|t|\to +\infty$, there exists $t_0\in \mathbb {R}$ such that $P(t_0)=0$. It follows that

\[ \left(\int^{1}_0\langle x, - J_{2n}\dot{y}\rangle\,{{\rm d}}t\right)^{2}\ge 4\mathfrak{a}(y)\mathfrak{a}(x). \]

This and (4.6) lead to

(4.7)\begin{equation} \mathfrak{a}(x)\le (\mathfrak{a}(y))^{{-}1} \left(\frac{1}{2}\int^{1}_0\langle x, - J_{2n}\dot{y}\rangle\,{{\rm d}}t\right)^{2} \le\varrho\int^{1}_0H(x(t))\,{{\rm d}}t. \end{equation}

In order to prove that that $c^{n,k}(D)\le \varrho$, it suffices to prove that for any $\varepsilon >0$ there exists $\tilde {H}\in \mathscr {F}_{n,k}(\mathbb {R}^{2n}, D)$ such that $c^{n,k}(\tilde {H})< \varrho +\varepsilon$, which is reduced to prove: for any given $\gamma \in \Gamma _{n,k}$ there exists $x\in h(S^{+})$ such that

(4.8)\begin{equation} \Phi_{\tilde{H}}(x)< \varrho+\varepsilon. \end{equation}

Now for $\tau >0$ there exists $H_\tau \in \mathscr {F}_{n,k}(\mathbb {R}^{2n}, D)$ such that

(4.9)\begin{equation} H_\tau\ge \tau\left(H-\left(1+\frac{\varepsilon}{2\varrho}\right)\right). \end{equation}

For $\gamma \in \Gamma _{n,k}$ choose $x\in h(S^{+})$ satisfying (4.7). We shall prove that for $\tau >0$ large enough $\tilde {H}=H_\tau$ satisfies the requirements.

1. • If $\int ^{1}_0H(x(t))\,{{\rm d}}t\le (1+\frac {\varepsilon }{\varrho })$, then by $H_\tau \ge 0$ and (4.7), we have

   \[ \Phi_{H_\tau}(x)\le \mathfrak{a}(x)\le \varrho\int^{1}_0H(x(t))\,{{\rm d}}t\le \varrho\left(1+\frac{\varepsilon}{\varrho}\right)<\varrho+\varepsilon. \]

2. • If $\int ^{1}_0H(x(t))\,{{\rm d}}t>(1+{\varepsilon }/{\varrho })$, then (4.9) implies

   (4.10)\begin{align} \int^{1}_0H_\tau(x(t))\,{{\rm d}}t & \ge \tau\left(\int^{1}_0H(x(t))\,{{\rm d}}t- \left(1+\frac{\varepsilon}{2\varrho}\right)\right)\nonumber\\ & \ge \tau \frac{\varepsilon}{2a}\left(1+\frac{\varepsilon}{\varrho}\right)^{{-}1} \int^{1}_0H(x(t))\,{{\rm d}}t \end{align}
   because
   \begin{align*} \left(1+\frac{\varepsilon}{2\varrho}\right)& = \left(1+\frac{\varepsilon}{2\varrho}\right)\left(1+\frac{\varepsilon}{\varrho}\right)^{{-}1} \left(1+\frac{\varepsilon}{\varrho}\right)\\ & < \left(1+\frac{\varepsilon}{2\varrho}\right) \left(1+\frac{\varepsilon}{\varrho}\right)^{{-}1}\int^{1}_0H(x(t))\,{{\rm d}}t \end{align*}
   and
   \begin{align*} 1-\left(1+\frac{\varepsilon}{2\varrho}\right)\left(1+\frac{\varepsilon}{\varrho}\right)^{{-}1}&= \left(1+\frac{\varepsilon}{\varrho}\right)^{{-}1}\left[\left(1+\frac{\varepsilon}{\varrho}\right)- \left(1+\frac{\varepsilon}{2\varrho}\right)\right]\\ & = \frac{\varepsilon}{2\varrho}\left(1+\frac{\varepsilon}{\varrho}\right)^{{-}1}. \end{align*}

Choose $\tau >0$ so large that the right side of the last equality is more than $\varrho$. Then

\[ \int^{1}_0H_\tau(x(t))\,{{\rm d}}t\ge \varrho\int^{1}_0H(x(t))\,{{\rm d}}t \]

by (4.10), and hence (4.7) leads to

\[ \Phi_{H_\tau}(x)=\mathfrak{a}(x)-\int^{1}_0H_\tau(x(t))\,{{\rm d}}t\le \mathfrak{a}(x)-\varrho\int^{1}_0H(x(t))\,{{\rm d}}t\le 0. \]

In summary, in the above two cases we have $\Phi _{H_\tau }(x)<\varrho +\varepsilon$. (4.8) is proved.

Step 3. Prove the final claim. By [26, theorem 1.5] we have

\[ c_{\rm LR}(D,D\cap \mathbb{R}^{n,k})=\min\{A(x)>0\,|\,x\ \hbox{is a leafwise chord on}\ \partial D\ \hbox{for}\ \mathbb{R}^{n,k}\}. \]

Using proposition 1.12 and corollary 2.41 in [28] we can choose two sequences of $C^{\infty }$ strictly convex domains with boundaries, $(D^{+}_j)$ and $(D^{-}_j)$, such that

1. (i) $D^{-}_1\subset D^{-}_2\subset \cdots \subset D$ and $\cup ^{\infty }_{j=1}D^{-}_j=D$,

2. (ii) $D^{+}_1\supseteq D^{+}_2\supseteq \cdots \supseteq D$ and $\cap ^{\infty }_{j=1}D^{+}_j=D$,

3. (iii) for any small neighbourhood $O$ of $\partial D$ there exists an integer $N>0$ such that $\partial D^{+}_k\cup \partial D^{-}_k\subset O\ \forall \ k\ge N$.

Now step 1–step 2 and [26, theorem 1.5] give rise to $c_{\rm LR}(D^{+}_j,D\cap \mathbb {R}^{n,k})=c^{n,k}(D^{+}_j)$ and $c_{\rm LR}(D^{-}_j,D\cap \mathbb {R}^{n,k})=c^{n,k}(D^{-}_j)$ for each $j=1,2,\ldots$. We have also that the sequence $c_{\rm LR}(D^{+}_j,D\cap \mathbb {R}^{n,k})$ converges decreasingly to $c_{\rm LR}(D,D\cap \mathbb {R}^{n,k})$ as $j\to \infty$ and that the sequence $c_{\rm LR}(D^{-}_j,D\cap \mathbb {R}^{n,k})$ converges increasingly to $c_{\rm LR}(D,D\cap \mathbb {R}^{n,k})$ as $j\to \infty$. Moreover for each $j$ there holds $c^{n,k}(D^{-}_j)\le c^{n,k}(D)\le c^{n,k}(D^{+}_j)$ by the monotonicity of $c^{n,k}$. These lead to $c^{n,k}(D)=c_{\rm LR}(D,D\cap \mathbb {R}^{n,k})$.

5. Proof of theorem 1.5

Clearly, the proof of theorem 1.5 can be reduced to the case that $m=2$ and all $D_i$ are also bounded. Moreover, by an approximation argument in step 3 of § 4 we only need to prove the following:

Theorem 5.1 For bounded strictly convex domains $D_i\subset \mathbb {R}^{2n_i}$ with $C^{2}$-smooth boundary and containing the origin, $i=1,2,$ and any integer $0\le k\le n:=n_1+n_2$ it holds that

\begin{align*} & c^{n,k}(\partial D_1\times\partial D_2)=c^{n,k}(D_1\times D_2)\\ & \quad =\min\{c^{n_1,\min\{n_1,k\}}(D_1),c^{n_2,\max\{k-n_1,0\}}(D_2)\}. \end{align*}

We first prove two lemmas. For convenience we write $E=H^{{1}/{2}}_{n,k}$ as $E_{n,k}$, and $E^{\ast }$ as $E^{\ast }_{n,k}$, $\ast =+,-, 0$. As a generalization of lemma 2 in [37, § 6.6] we have:

Lemma 5.2 Let $D\subset \mathbb {R}^{2n}$ be a bounded strictly convex domain with $C^{2}$-smooth boundary and containing $0$. Then for any given integer $0\le k\le n,$ function ${H}\in \mathscr {F}_{n,k}(\mathbb {R}^{2n}, \partial D)$ and any $\epsilon >0$ there exists $\gamma \in \Gamma _{n,k}$ such that

(5.1)\begin{equation} \Phi_{{H}}|_{\gamma(B^{+}_{n,k}\setminus\epsilon B^{+}_{n,k})}\ge c^{n,k}(D)-\epsilon \quad\hbox{and}\quad \Phi_{H}|_{\gamma(B^{+}_{n,k})}\ge 0, \end{equation}

where $B_{n,k}^{+}$ is the closed unit ball in $E_{n,k}^{+}$.

Proof. The case $k=n$ was proved in lemma 2 of [37, § 6.6]. We assume $k< n$ below. Let $S_{n,k}^{+}=\partial B_{n,k}^{+}$ and $\mathscr {E}^{n,k}_{\epsilon /2}(\mathbb {R}^{2n},\partial D)$ be as in (4.3). Replacing $H$ by a greater function we may assume $H\in \mathscr {E}^{n,k}_{\epsilon /2}(\mathbb {R}^{2n},\partial D)$. Since $H=0$ near $\partial {D}$, by the arguments at the end of proof of proposition 3.6, the condition (3.5) may be satisfied with any $z_0\in \mathbb {R}^{n,k}\cap {\rm Int}({H}^{-1}(0))$. Fix such a $z_0\in \mathbb {R}^{n,k}\cap {\rm Int}({H}^{-1}(0))$. It follows that there exists $\alpha >0$ such that

(5.2)\begin{equation} \inf \Phi_{H}|_{(z_0+\alpha S^{+}_{n,k})}>0\quad\hbox{and}\quad \Phi_{H}|_{(z_0+\alpha B^{+}_{n,k})}\ge 0, \end{equation}

(see (3.6)–(3.8) in the proof of proposition 3.6). Define $\gamma _\varepsilon :E_{n,k}\to E_{n,k}$ by $\gamma _\varepsilon (z)=z_0+ \alpha z$. It is easily seen that $\gamma _\varepsilon \in \Gamma _{n,k}$. The first inequality in (5.2) shows that $\gamma _\varepsilon (S^{+}_{n,k})$ belongs to the set $\mathcal {F}_{n,k}=\{\gamma (S^{+}_{n,k})\,|\,\gamma \in \Gamma _{n,k} \text{ and}\ \inf (\Phi _H|_{\gamma (S^{+}_{n,k})})>0\}$ in (3.10). Lemma 3.7 shows that

\[ c^{n,k}(H)=\sup_{F\in\mathcal{F}_{n,k}}\inf_{x\in F}\Phi_H(x), \]

and $\mathcal {F}_{n,k}$ is positively invariant under the flow $\varphi _u$ of $\nabla \Phi _H$. Define $S_u=\varphi _u(z_0+\alpha S^{+}_{n,k})$ and ${\rm d}(H)=\sup _{u\ge 0}\inf (\Phi _{H}|_{S_u})$. It follows from these and (5.2) that

\[ 0<\inf \Phi_{H}|_{S_0}\le {\rm d}(H)\le \sup_{F\in\mathcal{F}_{n,k}}\inf_{x\in F}\Phi_H(x)=c^{n,k}(H)<\infty. \]

Since $\Phi _{H}$ satisfies the (PS) condition by lemma 4.1, ${\rm d}(H)$ is a positive critical value of $\Phi _{H}$, and ${\rm d}(H)\ge c^{n,k}(D)-\epsilon /2$ by lemma 4.2. Moreover, by the definition of $d({H)}$ there exists $r>0$ such that $\Phi _{H}|_{S_r}\ge {\rm d}(H)-\epsilon /2$ and thus

(5.3)\begin{equation} \Phi_{H}|_{S_r}\ge c^{n,k}(D)-\epsilon. \end{equation}

Because $\Phi _{H}$ is nondecreasing along the flow $\varphi _u$, we arrive at

(5.4)\begin{equation} \Phi_{H}|_{S_u}\ge \Phi_{H}|_{S_0}\ge \inf (\Phi_{H}|_{S_0})>0,\quad\forall\ u\ge 0. \end{equation}

Define $\gamma :E_{n,k}\to E_{n,k}$ by $\gamma (x^{+}+x^{0}+x^{-})=\widetilde {\gamma }(x^{+})+x^{0}+x^{-}$, where

\begin{align*} & \widetilde{\gamma}(x)=z_0+ 2(\alpha/\epsilon)x \quad \hbox{if}\ x\in E^{+}_{n,k}\ \hbox{and} \|x\|_{E_{n,k}}\le\frac{1}{2}\epsilon,\\ & \widetilde{\gamma}(x)=\varphi_{r(2\|x\|_{E_{n,k}}-\epsilon)/\epsilon} (z_0+\alpha x/\|x\|_{E_{n,k}})\quad\hbox{if}\ x\in E^{+}_{n,k}\ \hbox{and}\ \frac{1}{2}\epsilon<\|x\|_{E_{n,k}}\le \epsilon,\\ & \widetilde{\gamma}(x)=\varphi_{r}(z_0+\alpha\ x/\|x\|_{E_{n,k}})\quad\hbox{if}\ x\in E^{+}_{n,k}\ \hbox{and}\ \|x\|_{E_{n,k}}>\epsilon. \end{align*}

The first and second lines imply $\gamma (({\epsilon }/{2})B^{+}_{n,k})=(z_0+\alpha B^{+}_{n,k})$ and $\gamma (B^{+}_{n,k}\setminus ({\epsilon }/{2})B^{+}_{n,k})=\bigcup _{0\le u\le r}S_u$, respectively, and so

\[ \gamma(B^{+}_{n,k})=(z_0+\alpha B^{+}_{n,k})\bigcup_{0\le u\le r}S_u; \]

the third line implies $\gamma (B^{+}_{n,k}\setminus \epsilon B^{+}_{n,k})=S_r$. It follows from these, (5.2) and (5.3)–(5.4) that $\gamma$ satisfies (5.1).

Finally, we can also know that $\gamma \in \Gamma _{n,k}$ by considering the homotopy

\[ \gamma_0(x)=2(\alpha/\epsilon)x^{+}+x^{0}+x^{-},\quad \gamma_u(x)=\frac{1}{u}{(\gamma(ux)-z_0)}+ z_0,\quad 0< u\le 1. \]

Lemma 5.3 Let integers $n_1, n_2\ge 1,$ $0\le k\le n:=n_1+n_2$. For a bounded strictly convex domain $D\subset \mathbb {R}^{2n_1}$ with $C^{2}$ smooth boundary $\mathcal {S}$ and containing $0,$ it holds that

(5.5)\begin{equation} c^{n,k}(D\times\mathbb{R}^{2n_2})=c^{n_1,\min\{n_1,k\}}(D). \end{equation}

Moreover, if $\Omega \subset \mathbb {R}^{2n_2}$ is a bounded strictly convex domain with $C^{2}$ smooth boundary and containing $0,$ then

\[ c^{n,k}(\mathbb{R}^{2n_1}\times \Omega)=c^{n_2,\max\{k-n_1,0\}}(\Omega). \]

Proof. Let $H(z)=(j_D(z))^{2}$ for $z\in \mathbb {R}^{n_1}$ and define

\[ E_R=\{(z,z')\in\mathbb{R}^{2n_1}\times\mathbb{R}^{2n_2}\,|\,H(z)+ (|z'|/R)^{2}<1\}. \]

By the definition and the monotonicity of $c^{n,k}$ we have

\[ c^{n,k}(D\times\mathbb{R}^{2n_2})=\sup_R c^{n,k}(E_R). \]

Since the function $\mathbb {R}^{2n_1}\times \mathbb {R}^{2n_2}\ni (z,z')\mapsto G(z,z'):=H(z)+ (|z'|/R)^{2}\in \mathbb {R}$ is convex and of class $C^{1,1}$, $E_R$ is convex and $\mathcal {S}_R=\partial E_R$ is of class $C^{1,1}$. By theorem 1.4 we arrive at

\[ c^{n,k}(E_R)=\min\Sigma_{\mathcal{S}_R}^{n,k}. \]

Let $\lambda$ be a positive number and $u=(x,x'):[0,\lambda ]\rightarrow \mathcal {S}_R$ satisfy

(5.6)\begin{equation} \dot{u}=X_G(u)\quad\hbox{and}\quad u(\lambda),u(0)\in\mathbb{R}^{n,k},\quad u(\lambda)\sim u(0). \end{equation}

Namely, $u$ is a leafwise chord on $\mathcal {S}_R$ for $\mathbb {R}^{n,k}$ with action $\lambda$. Let $k_1=\min \{n_1,k\}$ and $k_2=\max \{k-n_1,0\}$. Clearly, $k_1+k_2=k$, and (5.6) is equivalent to the following

(5.7)\begin{align} & \dot{x}=X_H(x)\quad\hbox{and}\quad x(\lambda),x(0)\in\mathbb{R}^{n_1,k_1},\quad x(\lambda)\sim x(0), \end{align}

(5.8)\begin{align} & \dot{x}'=2J_{2n_2}x'/R^{2}\quad\hbox{and}\quad x'(\lambda),x'(0)\in\mathbb{R}^{n_2, k_2},\quad x'(\lambda)\sim x'(0) \end{align}

because $\mathbb {R}^{n,k}\equiv (\mathbb {R}^{n_1,k_1}\times \{0\}^{2n_2})+ (\{0\}^{2n_1}\times \mathbb {R}^{n_2,k_2})$. Note that nonzero constant vectors cannot be solutions of (5.7) and (5.8) and that $H(z)$ and $(|z'|/R)^{2}$ take constant values along solutions of (5.7) and (5.8), respectively. There exist three possibilities for solutions of (5.7) and (5.8):

1. • $x\equiv 0$, $|x'|=R$ and so $2\lambda /R^{2}\in \pi \mathbb {N}$ if $k< n_1+n_2$, and $2\lambda /R^{2}\in 2\pi \mathbb {N}$ if $k=n_1+n_2$ by (5.8).

2. • $x'\equiv 0$, $H(x)\equiv 1$ and so $\lambda \in \Sigma _\mathcal {S}^{n_1,\min \{n_1,k\}}$ by (5.7).

3. • $H(x)\equiv \delta ^{2}\in (0,1)$ and $|x'|^{2}=R^{2}(1-\delta ^{2})$, where $\delta >0$. Then $y(t):=({1}/{\delta })x(t)$ and $y'(t):=x'(t/\delta )$ satisfy respectively the following two lines:

   \begin{align*} & \dot{y}=X_H(y)\quad\hbox{and}\quad y(\lambda), y(0)\in\mathbb{R}^{n_1,k_1},\quad y(\lambda)\sim y(0),\quad H(y)\equiv 1,\\ & \dot{y}'=2J_{2n_2}y'/R^{2}\quad\hbox{and}\quad y'(\lambda), y'(0)\in\mathbb{R}^{n_2, k_2},\quad y'(\lambda)\sim y'(0),\quad|y'|\equiv R. \end{align*}

Hence we have also $\lambda \in \Sigma _\mathcal {S}^{n_1,\min \{n_1,k\}}$ by the first line, and

\[ \lambda\in \frac{R^{2}\pi}{2}\mathbb{N}\quad {\rm if}\ k< n_1+n_2,\quad \lambda\in \pi R^{2}\mathbb{N}\ {\rm if}\ k=n_1+n_2 \]

by the second line.

In summary, we always have

(5.9)\begin{align} & \Sigma_{\mathcal{S}_R}^{n,k}\subset \Sigma_\mathcal{S}^{n_1,\min\{n_1,k\}}\bigcup \frac{R^{2}\pi}{2}\mathbb{N}\quad\hbox{if}\ k< n_1+n_2, \end{align}

(5.10)\begin{align} & \Sigma_{\mathcal{S}_R}^{n,k}\subset \Sigma_\mathcal{S}^{n_1,\min\{n_1,k\}} \bigcup {R^{2}\pi}\mathbb{N}\quad\hbox{if}\ k=n_1+n_2. \end{align}

A solution $x$ of (5.7) siting on $\mathcal {S}$ gives a solution $u=(x,0)$ of (5.6) on $\mathcal {S}_R$. It follows that

\[ \min\Sigma_{\mathcal{S}_R}^{n,k}=\min\Sigma_\mathcal{S}^{n_1,\min\{n_1,k\}} \]

for $R$ sufficiently large. (5.5) is proved.

The second claim can be proved in the similar way.

Proof of theorem 5.1. Since $D_1\times D_2\subset D_1\times \mathbb {R}^{2n_2}$ and $D_1\times D_2\subset \mathbb {R}^{2n_1}\times D_2$, we get

\[ c^{n,k}(D_1\times D_2)\le \min\{c^{n_1,\min\{n_1,k\}}(D_1),c^{n_2,\max\{k-n_1,0\}}(D_2)\} \]

by lemma 5.3. In order to prove the inverse direction inequality it suffices to prove

(5.11)\begin{equation} c^{n,k}(\partial D_1\times \partial D_2)\ge \min\{c^{n_1,\min\{n_1,k\}}(D_1),c^{n_2,\max\{k-n_1,0\}}(D_2)\} \end{equation}

because $c^{n,k}(D_1\times D_2)\ge c^{n,k}(\partial D_1\times \partial D_2)$ by the monotonicity.

We assume $n_1\le k$. (The case $n_1>k$ is similar!) Then (5.11) becomes

(5.12)\begin{equation} c^{n,k}(\partial D_1\times \partial D_2)\ge \min\{c_{\rm EH}(D_1), c^{n_2,k-n_1}(D_2)\} \end{equation}

because $c^{n_1,n_1}(D_1)=c_{\rm EH}(D_1)$ by definition. Note that for each ${H}\in \mathscr {F}_{n,k}(\mathbb {R}^{2n}, \partial D_1\times \partial D_2)$ we may choose $\widehat {H}_1\in \mathscr {F}_{n_1,n_1}(\mathbb {R}^{2n_1}, \partial D_1)$ and $\widehat {H}_2\in \mathscr {F}_{n_2,k-n_1}(\mathbb {R}^{2n_2}, \partial D_2)$ such that

\[ \widehat{H}(z):=\widehat{H}_1(z_1)+\widehat{H}_2(z_2)\ge H(z),\quad\forall\ z. \]

Let $k_1=n_1$ and $k_2=n-k_1$. By lemma 5.2, for any

\[ 0<\epsilon<\min\{c^{n_1,n_1}(D_1),c^{n_2,k-n_1}(D_2), 1/4\} \]

and each $i\in \{1,2\}$ there exists $\gamma _i\in \Gamma _{n_i,k_i}$ such that

(5.13)\begin{equation} \Phi_{\widehat{H}_i}|_{\gamma_i(B^{+}_{n_i,k_i}\setminus\epsilon B^{+}_{n_i,k_i})}\ge c^{n_i,k_i}(D_i)-\epsilon \quad\hbox{and}\quad \Phi_{\widehat{H}_i}|_{\gamma_i(B^{+}_{n_i,k_i})}\ge 0. \end{equation}

Put $\gamma =\gamma _1\times \gamma _2$, which is in $\Gamma _{n,k}$. Since for any $x=(x_1,x_2)\in S^{+}_{n,k}\subset B^{+}_{n_1,k_1}\times B^{+}_{n_2,k_2}$ there exists some $j\in \{1,2\}$ such that

\[ x_j\in B_{n_j,k_j}^{+}\setminus 4^{{-}1}B_{n_j,k_j}^{+}\subset B_{n_j,k_j}^{+}\setminus \epsilon B_{n_j,k_j}^{+}, \]

it follows from this and (5.13) that

\[ \Phi_{\widehat{H}}(\gamma(x))=\Phi_{\widehat{H}_1}(\gamma_1(x_1))+ \Phi_{\widehat{H}_2}(\gamma_2(x_2)) \ge\min\{c^{n_1,n_1}(D_1),c^{n_2,k-n_1}(D_2)\}-\epsilon>0 \]

and hence

\[ c^{n,k}({H})\ge c^{n,k}(\widehat{H})=\sup_{h\in\Gamma_{n,k}} \inf_{y\in h(S^{+}_{n,k})}\Phi_{\widehat{H}}(y)\ge \min\{c^{n_1,n_1}(D_1),c^{n_2,k-n_1}(D_2)\}-\epsilon. \]

This leads to (5.12) because $c^{n_1,n_1}(D_1)=c_{\rm EH}(D_1)$.

6. Proof of theorem 1.7

6.1 The interior of $\Sigma _{\mathcal {S}}$ is empty

Let $\lambda :=\imath _X\omega _0$, and $\lambda _0:=\frac {1}{2}(qdp-pdq)$, where $(q,p)$ is the standard coordinate on $\mathbb {R}^{2n}$.

Claim 6.1 For every leafwise chord on $\mathcal {S}$ for $\mathbb {R}^{n,k},$ $x:[0,T]\rightarrow \mathcal {S},$ there holds

(6.1)\begin{equation} A(x)=\int_x\lambda_0=\int_x\lambda. \end{equation}

Proof. Since $\mathcal {S}$ is of class $C^{2n+2}$, so is $x$. Define $y:[0, T]\rightarrow \mathbb {R}^{n,k}$ by $y(t)= tx(0)+(1-t)x(T)$. As below (3.15) we can take a piecewise $C^{2n+2}$-smooth map $u$ from a suitable closed disc $D^{2}$ to $\mathbb {R}^{2n}$ such that $u|\partial D^{2}$ is equal to the loop $x\cup (-y)$. Now it is easily checked that $\int _y \lambda _0=0$ and hence

(6.2)\begin{equation} \int_x\lambda_0=\int_{x\cup ({-}y)}\lambda_0=\int_{u(D^{2})}{{\rm d}}\lambda_0=\int_{u(D^{2})}\omega_0. \end{equation}

On the other hand, since the flow of $X$ maps $\mathbb {R}^{n,k}$ to $\mathbb {R}^{n,k}$, $X$ is tangent to $\mathbb {R}^{n,k}$ and therefore $\omega _0(X,\dot {y})=0$, i.e. $y^{\ast }\lambda =0$. It follows that

\[ \int_x\lambda=\int_{x\cup({-}y)}\lambda=\int_{u(D^{2})}{{\rm d}}\lambda=\int_{u(D^{2})}\omega_0. \]

This and (6.2) lead to (6.1).

Choosing $\varepsilon >0$ so small that $\mathbb {R}^{2n}\setminus \cup _{t\in (-\varepsilon, \varepsilon )}\phi ^{t}(\mathcal {S})$ has two components, we obtain a very special parameterized family of $C^{2n+2}$ hypersurfaces modelled on $\mathcal {S}$, given by

\[ \psi:(-\varepsilon, \varepsilon)\times\mathcal{S}\ni (s, z)\mapsto \psi(s,z)=\phi^{s}(z)\in \mathbb{R}^{2n} \]

which is $C^{2n+2}$ because both $\mathcal {S}$ and $X$ are $C^{2n+2}$. Define $U:=\cup _{t\in (-\varepsilon, \varepsilon )}\phi ^{t}(\mathcal {S})$ and

\[ K_\psi:U\to\mathbb{R},\quad w\mapsto \tau \]

if $w=\psi (\tau,z)\in U$ where $z\in \mathcal {S}$. This is $C^{2n+2}$. Denote by $X_{K_\psi }$ the Hamiltonian vector field of $K_\psi$ defined by $\omega _0(\cdot,X_{K_\psi })=dK_\psi$. Then it is not hard to prove

\[ X_{K_\psi}(\psi(\tau,z))={\rm e}^{-\tau}\,{{\rm d}}\phi^{\tau}(z)[X_{K_\psi}(z)] \quad\forall\ (\tau,z)\in(-\varepsilon, \varepsilon)\times\mathcal{S}, \]

and for $w=\phi ^{\tau }(z)=\psi (\tau,z)\in U$ there holds

(6.3)\begin{equation} \lambda_w(X_{K_\psi})=(\omega_0)_w(X(w), X_{K_\psi}(w)) = \left.\frac{{{\rm d}}}{{{\rm d}}s}\right|_{s=0}K_\psi(\phi^{s}(w))=1. \end{equation}

Let $\mathcal {S}_\tau :=\psi (\{\tau \}\times \mathcal {S})$. Since $\phi ^{t}$ preserves the leaf of $\mathbb {R}^{n,k}$, $y:[0,T]\to \mathcal {S}_\tau$ satisfies

\[ \dot{y}(t)=X_{K_\psi}(y(t)),\quad y(0), y(T)\in\mathbb{R}^{n,k}\quad\hbox{and}\quad y(T)\sim y(0) \]

if and only if $y(t)=\phi ^{\tau } (x({\rm e}^{-\tau } t))$, where $x:[0, {\rm e}^{-\tau } T]\to \mathcal {S}$ satisfies

\[ \dot{x}(t)=X_{K_\psi}(x(t)),\quad x(0), x({\rm e}^{-\tau} T)\in\mathbb{R}^{n,k} \quad\hbox{and}\quad x({\rm e}^{-\tau} T)\sim x(0). \]

In addition, $y(t)=\phi ^{\tau } (x({\rm e}^{-\tau } t))$ implies $\int _y\lambda ={\rm e}^{\tau }\int _x\lambda$. By (6.1) and (6.3) we deduce

\[ A(y)=\int_y\lambda_0=\int_y\lambda=\int^{T}_0\lambda(\dot y)\,{{\rm d}}t=\int^{T}_0\lambda_w(X_{K_\psi})\,{{\rm d}}t=T \quad\hbox{and}\quad A(x)={\rm e}^{-\tau}T. \]

Fix $0<\delta <\varepsilon$. Let ${\bf A}_\delta$ and ${\bf B}_\delta$ denote the unbounded and bounded components of $\mathbb {R}^{2n}\setminus \cup _{t\in (-\delta, \delta )}\phi ^{t}(\mathcal {S})$, respectively. Then $\psi (\{\tau \}\times \mathcal {S})\subset {\bf B}_\delta$ for $-\varepsilon <\tau <-\delta$. Let $\mathscr {F}_{n,k}(\mathbb {R}^{2n})$ be given by (3.10). We call $H\in \mathscr {F}_{n,k}(\mathbb {R}^{2n})$ adapted to $\psi$ if

(6.4)\begin{equation} H(x)=\left\{\begin{array}{@{}ll} C_0\ge 0 & {\rm if}\ x\in {\bf B}_\delta,\\ f(\tau) & {\rm if}\ x=\psi(\tau,y),\quad y\in\mathcal{S},\enspace \tau\in [-\delta,\delta],\\ C_1\ge 0 & {\rm if}\ x\in {\bf A}_\delta\cap B^{2n}(0,R),\\ h(|x|^{2}) & {\rm if}\ x\in {\bf A}_\delta\setminus B^{2n}(0,R), \end{array}\right. \end{equation}

where $f:(-1,1)\to \mathbb {R}$ and $h:[0, \infty )\to \mathbb {R}$ are smooth functions satisfying

(6.5)\begin{align} & f|_{({-}1,-\delta]}=C_0,\quad f|_{[\delta,1)}=C_1, \end{align}

(6.6)\begin{align} & sh'(s)-h(s)\le 0\quad\forall\ s. \end{align}

Clearly, $H$ defined by (6.4) is $C^{2n+2}$ and its gradient $\nabla H:\mathbb {R}^{2n}\to \mathbb {R}^{2n}$ satisfies a global Lipschitz condition.

Lemma 6.2

1. (i) If $x$ is a nonconstant critical point of $\Phi _H$ on $E$ such that $x(0)\in \psi (\{\tau \}\times \mathcal {S})$ for some $\tau \in (-\delta,\delta )$ satisfying $f'(\tau )>0,$ then

   \[ {\rm e}^{-\tau}f'(\tau)\in \Sigma_{\mathcal{S}}\quad\hbox{and}\quad \Phi_H(x)=f'(\tau)-f(\tau). \]

2. (ii) If some $\tau \in (-\delta,\delta )$ satisfies $f'(\tau )>0$ and ${\rm e}^{-\tau }f'(\tau )\in \Sigma _{\mathcal {S}},$ then there is a nonconstant critical point $x$ of $\Phi _H$ on $E$ such that $x(0)\in \psi (\{\tau \}\times \mathcal {S})$ and $\Phi _H(x)=f'(\tau )-f(\tau )$.

Proof. (i) By lemma 2.5 $x$ is $C^{2n+2}$ and satisfies $\dot {x}=X_H(x)=f'(\tau )X_{K_\psi }(x)$, $x(j)\in \mathbb {R}^{n,k}$, $j=0,1$, and $x(1)\sim x(0)$. Moreover $x(0)\in \psi (\{\tau \}\times \mathcal {S})$ implies $H(x(1))=H(x(0))=f(\tau )$ and therefore $x(1)\in \psi (\{\tau \}\times \mathcal {S})$ by the construction of $H$ above. These show that $x$ is a leafwise chord on $\psi (\{\tau \}\times \mathcal {S})$ for $\mathbb {R}^{n,k}$. By the arguments below (6.3), $[0,1]\ni t\mapsto y(t):=\phi ^{-\tau }(y(t))$ is a leafwise chord on $\mathcal {S}$ for $\mathbb {R}^{n,k}$. It follows from (6.3) and (6.1) that

\begin{align*} f'(\tau)& =\int_0^{1}f'(\tau)\lambda(X_{K_\psi})\,{{\rm d}}t=\int_0^{1}\lambda(X_H)\,{{\rm d}}t= \int_{[0,1]} x^{{\ast}}\lambda=\int_{[0,1]} y^{{\ast}}(\phi^{\tau})^{{\ast}} \lambda\\ & ={\rm e}^{\tau}\int_{[0,1]} y^{{\ast}}\lambda={\rm e}^{\tau} A(y) \end{align*}

These show that ${\rm e}^{-\tau }f'(\tau )=A(y)\in \Sigma _{\mathcal {S}}$. By (6.1) we have

\[ \Phi_H(x)=A(x)-\int_0^{1} H(x(t))\,{{\rm d}}t=\int_{[0,1]} x^{{\ast}} \lambda-\int_0^{1} H(x(t))\,{{\rm d}}t= f'(\tau)-f(\tau). \]

(ii) By the assumption there exists $y:[0, 1]\to \mathcal {S}$ satisfying

\[ \dot{y}(t)={\rm e}^{-\tau}f'(\tau)X_{K_\psi}(y(t)),\quad y(0), y(1)\in\mathbb{R}^{n,k}\quad\hbox{and}\quad y(1)\sim y(0). \]

Hence $x(t)=\psi (\tau, y(t))=\phi ^{\tau }(y(t))$ satisfies

\begin{align*} \dot{x}(t)& ={{\rm d}}\phi^{\tau}(y(t))[\dot{y}(t)]={\rm e}^{-\tau}f'(\tau)\,{{\rm d}}\phi^{\tau}(y(t))[X_{K_\psi}(y)]\\ & =f'(\tau)X_{K_\psi}(\phi^{\tau}(y(t)))=f'(\tau)X_{K_\psi}(x(t))=X_H(x(t)),\\ & x(0, x(1)\in\mathbb{R}^{n,k}, j=0,1, \quad x(1)\sim x(0)\in \phi^{\tau}(\mathcal{S}). \end{align*}

By lemma 2.5, $x$ is a critical point of $\Phi _H$. Moreover $\Phi _H(x)=f'(\tau )-f(\tau )$ as in (i).

Proposition 6.3 Let $\mathcal {S}$ be as in theorem 1.7. Then the interior of $\Sigma _{\mathcal {S}}$ in $\mathbb {R}$ is empty.

Proof. Otherwise, suppose that $T\in \Sigma _{\mathcal {S}}$ is an interior point of $\Sigma _{\mathcal {S}}$. Then for some small $0<\epsilon _1<\delta$ the open neighbourhood $O:=\{{\rm e}^{-\tau }T\,|\,\tau \in (-\epsilon _1,\epsilon _1)\}$ of $T$ is contained in $\Sigma _{\mathcal {S}}$. Let us choose the function $f$ in (6.4) such that $f(u)=Tu+\overline {C}\ge 0\ \forall \ u\in [-\epsilon _1, \epsilon _1]$ (by shrinking $0<\epsilon _1<\delta$ if necessary). By lemma 6.2(ii) we deduce

\begin{align*} & (-\epsilon_1,\epsilon_1)\subset\left\{\tau\in (-\epsilon_1,\epsilon_1)\,|\, {\rm e}^{-\tau}T\in \Sigma_{\mathcal{S}}\right\}\subset \left\{\tau\in (-\epsilon_1,\epsilon_1)\,|\,T\right.\\ & \quad\left. -f(\tau)\ \hbox{is a critical value of}\ \Phi_H\right\} \end{align*}

It follows that the critical value set of $\Phi _H$ has nonempty interior. This is a contradiction by lemma 3.10. Hence $\Sigma _{\mathcal {S}}$ has empty interior.

6.2 $c^{n,k}(U)=c^{n,k}(\mathcal {S})$ belongs to $\Sigma _{\mathcal {S}}$

This can be obtained by slightly modifying the proof of [37, theorem 7.5] (or [25, theorem 1.18] or [24, theorem 1.17]). For completeness we give it in detail. For $C>0$ large enough and $\delta >2\eta >0$ small enough, define $H=H_{C,\eta }\in \mathscr {F}_{n,k}(\mathbb {R}^{2n})$ adapted to $\psi$ as follows:

(6.7)\begin{equation} H_{C,\eta}(x)=\left\{\begin{array}{@{}ll} C\ge 0 & {\rm if}\ x\in {\bf B}_\delta,\\ f_{C,\eta}(\tau) & {\rm if}\ x=\psi(\tau,y),\quad y\in\mathcal{S},\enspace \tau\in [-\delta,\delta],\\ C & {\rm if}\ x\in {\bf A}_\delta\cap B^{2n}(0,R),\\ h(|x|^{2}) & {\rm if}\ x\in {\bf A}_\delta\setminus B^{2n}(0,R) \end{array}\right. \end{equation}

where $B^{2n}(0,R)\supseteq \overline {\psi ((-\varepsilon,\varepsilon )\times \mathcal {S})}$ (the closure of $\psi ((-\varepsilon,\varepsilon )\times \mathcal {S})$), $f_{C,\eta }:(-\varepsilon, \varepsilon )\to \mathbb {R}$ and $h:[0, \infty )\to \mathbb {R}$ are smooth functions satisfying

\begin{align*} & f_{C,\eta}|_{[-\eta,\eta]}\equiv 0,\quad f_{C,\eta}(s)=C\ \hbox{if}\ |s|\ge 2\eta,\\ & f'_{C,\eta}(s)s>0\quad\hbox{if}\ \eta<|s|<2\eta,\\ & f'_{C,\eta}(s)-f_{C,\eta}(s)>c^{n,k}(\mathcal{S})+1\quad\hbox{if}\ s>0\ \hbox{and}\ \eta< f_{C,\eta}(s)< C-\eta,\\ & h_{C,\eta}(s)=a_Hs+b\quad\hbox{for}\ s>0 \hbox{ large enough},\ a_H=C/R^{2}> \frac{\pi}{2},\ a_H\notin \frac{\pi}{2}\mathbb{N},\\ & sh'_{C,\eta}(s)-h_{C,\eta}(s)\le 0\quad\forall\ s\ge 0. \end{align*}

We can choose such a family $H_{C,\eta }$ ($C\to +\infty$, $\eta \to 0$) to be cofinal in $\mathscr {F}^{n,k}(\mathbb {R}^{2n},\mathcal {S})$ defined by (3.16) and also to have the property that

(6.8)\begin{equation} C\le C'\Rightarrow H_{C,\eta}\le H_{C',\eta},\quad \eta\le \eta'\Rightarrow H_{C,\eta}\ge H_{C,\eta'}. \end{equation}

It follows that

\[ c^{n,k}(\mathcal{S})=\lim_{\eta\to 0, C\to+\infty}c^{n,k}(H_{C,\eta}). \]

By proposition 3.5(i) and (6.8), $\eta \le \eta '$ implies that $c^{n,k}(H_{C,\eta })\le c^{n,k}(H_{C,\eta '})$, and hence

(6.9)\begin{equation} \Upsilon(C):=\lim_{\eta\to 0}c^{n,k}(H_{C,\eta}) \end{equation}

exists, and

\[ \Upsilon(C)=\lim_{\eta\to 0}c^{n,k}(H_{C,\eta})\ge \lim_{\eta\to 0}c^{n,k}(H_{C',\eta})=\Upsilon(C'), \]

i.e. $C\mapsto \Upsilon (C)$ is non-increasing. We claim

(6.10)\begin{equation} c^{n,k}(\mathcal{S})=\lim_{C\to+\infty}\Upsilon(C). \end{equation}

In fact, for any $\epsilon >0$ there exists $\eta _0>0$ and $C_0>0$ such that $|c^{n,k}(H_{C,\eta })-c^{n,k}(\mathcal {S})|<\epsilon$ for all $\eta <\eta _0$ and $C>C_0$. Letting $\eta \to 0$ leads to $|\Upsilon (C)-c^{n,k}(\mathcal {S})|\le \epsilon$ for all $C>C_0$. (6.10) holds.

Claim 6.4 Let $\overline {\Sigma _{\mathcal {S}}}$ be the closure of $\Sigma _{\mathcal {S}}$. Then $\overline {\Sigma _{\mathcal {S}}}\subset \Sigma _{\mathcal {S}}\cup \{0\}$.

Proof. In fact, let $\varphi ^{t}$ denote the flow of $X_{K_\psi }$. It is not hard to prove

\[ \Sigma_{\mathcal{S}}=\{T>0\,|\,\exists z\in\mathcal{S}\cap\mathbb{R}^{n,k} \hbox{such that}\ \varphi^{T}(z)\in\mathcal{S}\cap\mathbb{R}^{n,k}\ \& \ \varphi^{T}(z)\sim z\}. \]

Suppose that $(T_k)\subset \Sigma _{\mathcal {S}}$ satisfy $T_k\to T_0\ge 0$. Then there exists a sequence $(z_k)\subset \mathcal {S}\cap \mathbb {R}^{n,k}$ such that $\varphi ^{T_k}(z_k)\in \mathcal {S}\cap \mathbb {R}^{n,k}$ and $\varphi ^{T_k}(z_k)\sim z_k$ for $k=1,2,\ldots$. Define $\gamma _k(t)=\varphi ^{T_kt}(z_k)$ for $t\in [0,1]$ and $k\in \mathbb {N}$. Then $\dot {\gamma }_k(t)=T_kX_{K_\psi }(\gamma _k(t))$. By the Arzelá-Ascoli theorem $(\gamma _k)$ has a subsequence converging to some $\gamma _0$ in $C^{\infty }([0, 1],\mathcal {S})$, which satisfies the following relations

\begin{align*} & \dot{\gamma}_0(t)=T_0X_{K_\psi}(\gamma_0(t))\ \hbox{for all}\ t\in [0, 1],\\ & \gamma_0(0)=\lim_{k\to\infty}\gamma_k(0)=\lim_{k\to\infty}z_k\in \mathcal{S}\cap\mathbb{R}^{n,k},\\ & \gamma_0(1)=\lim_{k\to\infty}\gamma_k(1)=\lim_{k\to\infty} \varphi^{T_k}(z_k)\in \mathcal{S}\cap\mathbb{R}^{n,k},\\ & \gamma_0(1)-\gamma_0(0)=\lim_{k\to\infty} (\gamma_k(1)-\gamma_k(0))\in V_0^{n,k}, \hbox{i.e.} \gamma_0(1)\sim\gamma_0(0). \end{align*}

Hence $\gamma _0(t)=\varphi ^{T_0t}(z_0)$ and $T_0\in \Sigma _{\mathcal {S}}$ if $T_0>0$. It follows that $\overline {\Sigma _{\mathcal {S}}}\subset \Sigma _{\mathcal {S}}\cup \{0\}$.

Note that so far we do not use the assumption $a_H\notin \mathbb {N}\pi /2$.

Claim 6.5 If $a_H\notin \mathbb {N}\pi /2$ then either $\Upsilon (C)\in \overline {\Sigma _{\mathcal {S}}}$ or

(6.11)\begin{equation} \Upsilon(C)+C\in \overline{\Sigma_{\mathcal{S}}}. \end{equation}

Proof. Since $a_H\notin \mathbb {N}\pi /2$, by theorem 3.8 we get that $c^{n,k}(H_{C,\eta })$ is a positive critical value of $\Phi _{H_{C,\eta }}$ and the associated critical point $x\in E$ gives rise to a nonconstant leafwise chord sitting in the interior of $U$. Then lemma 6.2(i) yields

\[ c^{n,k}(H_{C,\eta})=\Phi_{H_{C,\eta}}(x)=f'_{C,\eta}(\tau)-f_{C,\eta}(\tau), \]

where $f'_{C,\eta }(\tau )\in {\rm e}^{\tau }\Sigma _{\mathcal {S}}$ and $\eta <|\tau |<2\eta$. Choose $C>0$ so large that $c^{n,k}(H_{C,\eta })< c^{n,k}(\mathcal {S})+1$. By the choice of $f$ below (6.7) we get either $f_{C,\eta }(\tau )<\eta$ or $f_{C,\eta }(\tau )>C-\eta$. Moreover $c^{n,k}(H_{C,\eta })>0$ implies $f'_{C,\eta }(\tau )>f_{C,\eta }(\tau )\ge 0$ and so $\tau >0$.

Take a sequence of positive numbers $\eta _n\to 0$. By the arguments above, passing to a subsequence we have the following two cases.

Case 1. For each $n\in \mathbb {N}$, $c^{n,k}(H_{C,\eta _n})=f'_{C,\eta _n}(\tau _n)-f_{C,\eta _n}(\tau _n) ={\rm e}^{\tau _n}a_n-f_{C,\eta _n}(\tau _n)$, where $a_n\in \Sigma _\mathcal {S}$, $0\le f_{C,\eta _n}(\tau _n)<\eta _n$ and $\eta _n<\tau _n<2\eta _n$.

Case 2. For each $n\in \mathbb {N}$, $c^{n,k}(H_{C,\eta _n})=f'_{C,\eta _n}(\tau _n)-f_{C,\eta _n}(\tau _n) ={\rm e}^{\tau _n}a_n-f_{C,\eta _n}(\tau _n)={\rm e}^{\tau _n}a_n-C-(f_{C,\eta _n}(\tau _n) -C)$, where $a_n\in \Sigma _\mathcal {S}$, $C-\eta _n< f_{C,\eta _n}(\tau _n)\le C$ and $\eta _n<\tau _n<2\eta _n$.

In case 1, since $c^{n,k}(H_{C,\eta _n})\to \Upsilon (C)$ by (6.9), the sequence $a_n={\rm e}^{-\tau _n}(c^{n,k} (H_{C,\eta _n})+f_{C,\eta _n}(\tau _n))$ is bounded. Passing to a subsequence we may assume $a_n\to a_C\in \overline {\Sigma _{\mathcal {S}}}$. Then

\[ a_C=\lim_{n\to\infty}a_n=\lim_{n\to\infty} \left({\rm e}^{-\tau_n}(c^{n,k}(H_{C,\eta_n})+f_{C,\eta_n}(\tau_n))\right) =\Upsilon(C) \]

because ${\rm e}^{-\tau _n}\to 1$ and $f_{C,\eta _n}(\tau _n)\to 0$.

Similarly, we can prove $\Upsilon (C)+C=a_C\in \overline {\Sigma _{\mathcal {S}}}$ in case 2.

Step 1. Prove $c^{n,k}(\mathcal {S})\in \overline {\Sigma _{\mathcal {S}}}$. Suppose that there exists an increasing sequence $C_n$ tending to $+\infty$ such that $C_n/R^{2}\notin \mathbb {N}\pi /2$ and $\Upsilon (C_n)\in \Sigma _{\mathcal {S}}$ for each $n$. Since $(\Upsilon (C_n))$ is non-increasing we conclude

(6.12)\begin{equation} c^{n,k}(\mathcal{S})=\lim_{n\to\infty}\Upsilon(C_n)\in \overline{\Sigma_\mathcal{S}}. \end{equation}

Otherwise, we have

(6.13)\begin{align} \left.\begin{array}{ll@{}} & \hbox{there exists}\ \bar{C}>0\ \hbox{such that}\ (6.11)\ {\rm holds}\\ & \hbox{for each}\ C\in (\bar{C}, +\infty)\ \hbox{satisfying}\ C/R^{2}\notin\mathbb{N}\pi/2. \end{array}\right\} \end{align}

Claim 6.6 Let $\bar {C}>0$ be as in (6.13). Then for any $C< C'$ in $(\bar {C}, +\infty )$ there holds

\[ \Upsilon(C)+C\ge \Upsilon(C')+C'. \]

Its proof is carried out later. Since $\Xi :=\{C>\bar {C}\,|\, C\ \hbox{satisfying}\ C/R^{2}\notin \mathbb {N}\pi /2\}$ is dense in $(\bar {C}, +\infty )$, it follows from claim 6.6 that $\Upsilon (C')+C'\le \Upsilon (C)+C$ if $C'>C$ are in $\Xi$. Fix a $C^{\ast }\in \Xi$. Then $\Upsilon (C')+C'\le \Upsilon (C^{\ast })+C^{\ast }$ for all $C'\in \{C\in \Xi \,|\, C>C^{\ast }\}$. Taking a sequence $(C_n')\subset \{C\in \Xi \,|\, C>C^{\ast }\}$ such that $C_n'\to +\infty$, we deduce that $\Upsilon (C_n')\to -\infty$. This contradicts the fact that $\Upsilon (C_n')\to c^{n,k}(\mathcal {S})>0$. Hence (6.13) does not hold! (6.12) is proved.

Proof of claim 6.6. By contradiction we assume that for some $C'>C>\overline {C}$,

(6.14)\begin{equation} \Upsilon(C)+C<\Upsilon(C')+C'. \end{equation}

Let us prove that (6.14) implies:

(6.15)\begin{equation} \left.\begin{array}{ll@{}} & \hbox{for any given}\ d\in (\Upsilon(C)+C,\Upsilon(C')+C')\\ & \hbox{there exists}\ C_0\in (C, C')\ \hbox{such that}\ \Upsilon(C_0)+C_0=d. \end{array}\right\} \end{equation}

Clearly, this contradicts the facts that ${\rm Int}(\Sigma _{\mathcal {S}})=\emptyset$ and (6.11) holds for all large $C$ satisfying $C/R^{2}\notin \mathbb {N}\pi /2$.

It remains to prove (6.15). Put $\Delta _d=\{C''\in (C, C')\,|\, C''+\Upsilon (C'')>d\}$. Since $\Upsilon (C')+C'>d$ and $\Upsilon (C')\le \Upsilon (C'')\le \Upsilon (C)$ for any $C''\in (C, C')$ we obtain $\Upsilon (C'')+C''>d$ if $C''\in (C, C')$ is sufficiently close to $C'$. Hence $\Delta _d\ne \emptyset$. Set $C_0=\inf \Delta _d$. Then $C_0\in [C, C')$.

Let $(C_n'')\subset \Delta _d$ satisfy $C_n''\downarrow C_0$. Since $\Upsilon (C_n'')\le \Upsilon (C_0)$, we have $d< C_n''+\Upsilon (C_n'')\le \Upsilon (C_0)+ C_n''$ for each $n\in \mathbb {N}$, and thus $d\le \Upsilon (C_0)+C_0$ by letting $n\to \infty$.

We conclude $d=\Upsilon (C_0)+C_0$, and so (6.15) is proved. By contradiction suppose that

(6.16)\begin{equation} d<\Upsilon(C_0)+C_0. \end{equation}

Since $d>C+\Upsilon (C)$, this implies $C\ne C_0$ and so $C_0>C$. For $\hat {C}\in (C, C_0)$, as $\Upsilon (\hat {C})\ge \Upsilon (C_0)$ we derive from (6.16) that $\Upsilon (\hat {C})+\hat {C}>d$ if $\hat {C}$ is close to $C_0$. Hence such $\hat {C}$ belongs to $\Delta _d$, which contradicts $C_0=\inf \Delta _d$.

Step 2. Prove $c^{n,k}(U)=c^{n,k}(\mathcal {S})$. Note that $c^{n,k}(U)=\inf _{\eta >0, C>0}c^{n,k}(\hat {H}_{C,\eta })$, where

\[ \hat{H}_{C,\eta}(x)=\left\{\begin{array}{@{}ll} 0 & {\rm if}\ x\in {\bf B}_\delta,\\ \hat{f}_{C,\eta}(\tau) & {\rm if}\ x=\psi(\tau,y),\quad y\in\mathcal{S},\enspace \tau\in [-\delta,\delta],\\ C & {\rm if}\ x\in {\bf A}_\delta\cap B^{2n}(0,R),\\ \hat{h}(|x|^{2}) & {\rm if}\ x\in {\bf A}_\delta\setminus B^{2n}(0,R) \end{array}\right. \]

where $B^{2n}(0,R)\supseteq \overline {\psi ((-\varepsilon,\varepsilon )\times \mathcal {S})}$, $\hat {f}_{C,\eta }:(-\varepsilon, \varepsilon )\to \mathbb {R}$ and $\hat {h}:[0, \infty )\to \mathbb {R}$ are smooth functions satisfying the following conditions

\begin{align*} & \hat{f}_{C,\eta}|_{(-\infty,\eta]}\equiv 0,\quad \hat{f}_{C,\eta}(s)=C\quad\hbox{if}\ s\ge 2\eta,\\ & \hat{f}'_{C,\eta}(s)s>0\quad\hbox{if}\ \eta< s<2\eta,\\ & \hat{f}'_{C,\eta}(s)-\hat{f}_{C,\eta}(s)>c^{n,k}(\mathcal{S})+1\quad\hbox{if}\ s>0\quad \hbox{and}\quad \eta<\hat{f}_{C,\eta}(s)< C-\eta,\\ & \hat{h}_{C,\eta}(s)=a_Hs+b\quad\hbox{for}\ s>0 \ \hbox{large enough},\ a_H=C/R^{2}>\frac{\pi}{2},\ a_H\notin \frac{\pi}{2}\mathbb{N},\\ & s\hat{h}'_{C,\eta}(s)-\hat{h}_{C,\eta}(s)\le 0\quad\forall\ s\ge 0. \end{align*}

For $H_{C,\eta }$ in (6.7), choose an associated $\hat {H}_{C,\eta }$, where $\hat {f}_{C,\eta }|_{[0,\infty )}={f}_{C,\eta }|_{[0,\infty )}$ and $\hat {h}_{C,\eta }={h}_{C,\eta }$. Consider $H_s=sH_{C,\eta }+(1-s)\hat {H}_{C,\eta }$, $0\le s\le 1$, and put $\Phi _s(x):=\Phi _{H_s}(x)$ for $x\in E$.

It suffices to prove $c^{n,k}(H_0)=c^{n,k}(H_1)$. If $x$ is a critical point of $\Phi _s$ with $\Phi _s(x)>0$, as in lemma 6.2, we have $x([0,1])\in \mathcal {S}_\tau =\psi (\{\tau \}\times \mathcal {S})$ for some $\tau \in (\eta,2\eta )$. The choice of $\hat {H}_{C,\eta }$ shows $H_s(x(t))\equiv {H}_{C,\eta }(x(t))$ for $t\in [0,1]$. This implies that each $\Phi _s$ has the same positive critical value as $\Phi _{H_{C,\eta }}$. By the continuity in proposition 3.5(ii), $s\mapsto c^{n,k}(H_s)$ is continuous and takes values in the set of positive critical value of $\Phi _{H_{C,\eta }}$ (which has measure zero by Sard's theorem). Hence $s\mapsto c^{n,k}(H_s)$ is constant. We get $c^{n,k}(\hat {H}_{C,\eta })=c^{n,k}(H_0)=c^{\Psi }_{\rm EH}(H_1)=c^{n,k}(H_{C,\eta })$.

Summarizing the above arguments we have proved that $c^{n,k}(\mathcal {S})=c^{n,k}(U)\in \overline {\Sigma _{\mathcal {S}}}$. Noting that $c^{n,k}(U)>0$, we deduce $c^{n,k}(\mathcal {S})=c^{n,k}(U)\in \Sigma _{\mathcal {S}}$ by claim 6.4.

7. Proof of theorem 1.8

For $W^{2n}(1)$ in (1.3), note that $W^{2n}(1)\equiv \mathbb {R}^{2n-2}\times W^{2}(1)\supseteq \mathbb {R}^{2n-2}\times U^{2}(1)$ via the identification under (1.12). For each integer $0\le k< n$, (1.14) and (1.11) yield

\[ c^{n,k}(W^{2n}(1))\ge\min\{c^{n-1,k}(\mathbb{R}^{2n-2}), c^{1,0}(U^{2}(1))\}=\frac{\pi}{2}. \]

We only need to prove the inverse direction of the inequality.

Fix a number $0<\varepsilon <\frac {1}{100}$. For $N>2$ define

\[ W^{2}(1,N):=\left\{(x_n,y_n)\in W^{2}(1)\,|\,|x_n|< N,\ |y_n|< N\right\}. \]

Let us smoothen $W^{2}(1)$ and $W^{2}(1,N)$ in the following way. Choose positive numbers $\delta _1, \delta _2\ll 1$ and a smooth even function $g:\mathbb {R}\to \mathbb {R}$ satisfying the following conditions:

1. (i) $g(t)=\sqrt {1-t^{2}}$ for $0\le t\le 1-\delta _1$,

2. (ii) $g(t)=0$ for $t\ge 1+\delta _2$,

3. (iii) $g$ is strictly monotone decreasing, and $g(t)\ge \sqrt {1-t^{2}}$ for $1-\delta _1\le t\le 1$.

Denote by

\[ W^{2}_g(1):=\{(x_n,y_n)\in\mathbb{R}^{2}\,|\, y_n< g(x_n)\}, \]

and by $W^{2}_{g}(1,N)$ the open subset in $\mathbb {R}^{2}(x_n,y_n)$ surrounded by curves $y_n=g(x_n)$, $y_n=-N$, $x_n=N$ and $x_n=-N$ (see figure 2). Then $W^{2}_{g}(1,N)$ contains $W^{2}(1,N)$, and we can require $\delta _1,\delta _2$ so small that

(7.1)\begin{equation} 0<{\rm Area}(W^{2}_{g}(1,N))-{\rm Area}(W^{2}(1,N))<\frac{\varepsilon}{2}. \end{equation}

Take another smooth function $h:[0, \infty )\to \mathbb {R}$ satisfying the following conditions:

1. (iv) $h(0)={\varepsilon }/{2}$ and $h(t)=0$ for $t>{\varepsilon }/{2}$,

2. (v) $h'(t)<0$ and $h''(t)>0$ for any $t\in (0, {\varepsilon }/{2})$,

3. (vi) the curve $\{(t, h(t))\,|\, 0\le t\le {\varepsilon }/{2}\}$ is symmetric with respect to line $s=t$ in $\mathbb {R}^{2}(s,t)$.

Let $\triangle _1$ be the closed domain in $\mathbb {R}^{2}(x_n,y_n)$ surrounded by curves $y_n=h(x_n)$, $y_n=0$ and $x_n=0$ (see figure 1). Denote by

\[ \triangle_2=\{(x_n,y_n)\in \mathbb{R}^{2}\,|\,({-}x_n,y_n)\in \triangle_1\},\quad \triangle_3={-}\triangle_1,\quad \triangle_4={-}\triangle_2. \]

Let $p_1=(N,0),\ p_2=(-N,0),\ p_3=(-N,-N),\ p_4=(N,-N)$. Define

\[ W^{2}_{g,\varepsilon}(1,N)=W^{2}_{g}(1,N)\setminus ((p_1+ \triangle_3)\cup(p_2+ \triangle_4)\cup(p_3+ \triangle_1)\cup(p_4+ \triangle_2)). \]

Then $W^{2}_{g,\varepsilon }(1,N)$ has smooth boundary (see figure 2) and

\[ 0<{\rm Area}(W^{2}_g(1,N))-{\rm Area}(W^{2}_{g,\varepsilon}(1,N))=4{\rm Area} (\triangle_1)<4\left(\frac{\varepsilon}{2}\right)^{2}=\varepsilon^{2}<\frac{\varepsilon}{2}. \]

FIG. 1. Domains $\triangle _i$, $i=1,2,3,4$.

FIG. 2. Domain $W^{2}_{g,\varepsilon }(1,N)$.

For $n>1$ and $N>2$ we define

\begin{align*} & W^{2n}_{g}(1):=\{(x,y)\in\mathbb{R}^{2n}\,|\, (x_n,y_n)\in W^{2}_{g}(1)\}= \mathbb{R}^{2n-2}\times W^{2}_{g}(1) ,\\ & W^{2n}(1,N):=\left \{(x,y)\in W^{2n}(1)\,|\,|x_n|< N,\ |y_n|< N\right\}= \mathbb{R}^{2n-2}\times W^{2}(1,N),\\ & W^{2n}_{g}(1,N):=\{(x,y)\in\mathbb{R}^{2n}\,|\, (x_n,y_n)\in W^{2}_{g}(1, N)\}= \mathbb{R}^{2n-2}\times W^{2}_g(1,N),\\ & W^{2n}_{g,\varepsilon}(1,N):=\{(x,y)\in\mathbb{R}^{2n}\,|\, (x_n,y_n)\in W^{2}_{g,\varepsilon}(1,N)\}=\mathbb{R}^{2n-2}\times W^{2}_{g,\varepsilon}(1,N). \end{align*}

Clearly, $W^{2n}_{g,\varepsilon }(1, N)\subset W^{2n}_{g,\varepsilon }(1,M)$ for any $M>N>2$, and each bounded subset of $W^{2n}_g(1)$ can be contained in $W^{2n}_{g,\varepsilon }(1,N)$ for some large $N>2$. It follows that

(7.2)\begin{equation} c^{n,k}(W^{2n}_g(1))=\sup_{N>2}\{c^{n,k}(W^{2n}_{g,\varepsilon}(1,N))\}= \lim_{N\to+\infty}c^{n,k}(W^{2n}_{g,\varepsilon}(1,N)). \end{equation}

Let us estimate $c^{n,k}(W^{2n}_{g,\varepsilon }(1,N))$ with theorem 1.7. Regrettably, $W^{2}_{g,\varepsilon }(1,N)$ is not star-shaped with respect to the origin. Fortunately, it can be approximated arbitrarily by star-shaped domains with respect to the origin and with smooth boundary. Indeed, for a very small $0<\eta <\varepsilon$ the set

\[ W^{2}_{g,\varepsilon}(1,N,\eta):=W^{2}_{g,\varepsilon}(1,N)\cup (W^{2}_{g,\varepsilon}(1,N)+ (0,\eta)) \]

is the desired one.

Define $j_{g,N,\epsilon, \eta }:\mathbb {R}^{2}\to \mathbb {R}$ by

\[ j_{g,N,\epsilon,\eta}(z_n):=\inf\left\{\lambda>0\left|\frac{z_n}{\lambda} \in W^{2}_{g,\varepsilon}(1,N,\eta)\right.\right\},\quad\forall\ z_n=(x_n,y_n)\in\mathbb{R}^{2}. \]

Then $j_{g,N,\epsilon,\eta }$ is positively homogeneous, and smooth in $\mathbb {R}^{2}\setminus \{0\}$. For $(x,y)\in \mathbb {R}^{2n}$ we write $(x,y)=(\hat {z},z_n)$ and define

\[ W^{2n}_{g,\varepsilon,R}(1,N,\eta):=\left\{\frac{|\hat{z}|^{2}}{R^{2}}+j_{g,N,\epsilon,\eta}^{2}(z_n)<1\right\}, \quad \forall\ R>0. \]

Then we have $W^{2n}_{g,\varepsilon,R_1}(1,N,\eta )\subset W^{2n}_{g,\varepsilon, R_2}(1,N,\eta )$ for $R_1< R_2$, and

\[ W^{2n}_{g,\varepsilon}(1,N,\eta)=\bigcup_{R>0}W^{2n}_{g,\varepsilon,R}(1,N,\eta), \]

which implies by (3.18) that

(7.3)\begin{equation} c^{n,k}(W^{2n}_{g,\varepsilon}(1,N,\eta))=\lim_{R\to+\infty}c^{n,k}(W^{2n}_{g,\varepsilon,R}(1,N,\eta)). \end{equation}

Observe that for arbitrary $N>2$ and $R>0$ we can shrink $0<\eta <\varepsilon$ so that there holds

\[ W^{2n}_{g,\varepsilon,R}(1,N,\eta)\subset W^{2n}_{g,\varepsilon}(1,N,\eta)\subset U^{2n}(N), \]

where for $r>0$,

\[ U^{2n}(r):=\{(x,y)\in\mathbb{R}^{2n}\,|\,x_n^{2}+y_n^{2}< r^{2}\}\cup \{(x,y)\in\mathbb{R}^{2n}\,|\,|x_n|< r\ \hbox{and}\ y_n<0\}. \]

We obtain

(7.4)\begin{equation} c^{n,k}(W^{2n}_{g,\varepsilon,R}(1,N,\eta))\le c^{n,k}(W^{2n}_{g,\varepsilon}(1,N,\eta)) \le c^{n,k}(U^{2n}(N))=\frac{\pi}{2}N^{2}. \end{equation}

Note that $W^{2n}_{g, \varepsilon, R}(1,N,\eta )$ is a star-shaped domain with respect to the origin and with smooth boundary $\mathcal {S}_{N, g, \varepsilon,R,\eta }$ transversal to the globally defined Liouville vector field $X(z)=z$. Since the flow $\phi ^{t}$ of $X$, $\phi ^{t}(z)={\rm e}^{t}z$, maps $\mathbb {R}^{n,k}$ to $\mathbb {R}^{n,k}$ and preserves the leaf relation of $\mathbb {R}^{n,k}$, by theorem 1.7 we obtain

\[ c^{n,k}(W^{2n}_{g,\varepsilon,R}(1,N,\eta))\in \Sigma_{\mathcal{S}_{N, g, \varepsilon,R,\eta}} \]

where

\[ \Sigma_{\mathcal{S}_{g,N,\varepsilon, R,\eta}}=\{A(x)>0\,|\,x\ \hbox{is a leafwise chord on}\ \mathcal{S}_{N, g, \varepsilon,R,\eta}\ \hbox{for}\ \mathbb{R}^{n,k}\}. \]

Arguing as in the proof of (5.9) we get that

\[ \Sigma_{\mathcal{S}_{g,N,\varepsilon,R,\eta}}\subset \Sigma _{\partial W^{2}_{g,\varepsilon}(1,N,\eta)}\bigcup \frac{\pi R^{2}}{2}\mathbb{N}. \]

Hence for $R>N$, by (7.4) we have

(7.5)\begin{equation} c^{n,k}(W^{2n}_{g,\varepsilon,R}(1,N,\eta))\in \Sigma _{\partial W^{2}_{g,\varepsilon}(1,N,\eta)}. \end{equation}

Let us compute $\Sigma _{\partial W^{2}_{g,\varepsilon }(1,N,\eta )}$. Note that the part of $\partial W^{2}_{g,\varepsilon }(1,N)$ over the line $y_n=-\frac {\varepsilon }{2}$ and between lines $x_n=-N$ and $x_n=N$ is $\{(x_n, f(x_n))\in \mathbb {R}^{2}\,|\, |x_n|\le N\}$, where

\[ f(x_n)=\left\{\begin{array}{@{}ll} -h(x_n+N) & {\rm if}\ -N\le x_n\le -N+\dfrac{\varepsilon}{2},\\ g(x_n) & {\rm if}\ -N+\dfrac{\varepsilon}{2}< x_n< N-\dfrac{\varepsilon}{2},\\ - h({-}x_n+N) & {\rm if}\ N-\dfrac{\varepsilon}{2}< x_n\le N. \end{array}\right. \]

Let $t_0\in (0, \varepsilon /2)$ be the unique number satisfying $h(t_0)=\eta$. Then there only exist two leafwise chords on $\partial W^{2}_{g,\varepsilon }(1,N,\eta )$ for $\mathbb {R}^{1,0}$. One is the curve in $\mathbb {R}^{2}(x_n,y_n)$,

\[ \gamma_1:=\{(x_n, \eta+ f(x_n))\in\mathbb{R}^{2}\,|\, t_0-N\le x_n\le N-t_0\}, \]

and the other is $\gamma _2:=\partial W^{2}_{g,\varepsilon }(1,N,\eta )\setminus \gamma _1$. Then $A(\gamma _1)$ is equal to the area of the domain in $\mathbb {R}^{2}(x_n,y_n)$ surrounded by curves $\gamma _1$ and $x_n$-axis, that is,

(7.6)\begin{align} A(\gamma_1)& =\int^{N-t_0}_{t_0-N}(\eta+ f(x_n))\,{{\rm d}}x_n\nonumber\\ & = 2(N-t_0)\eta+ {\rm Area}(W^{2}_{g}(1,N))-2N^{2}-2 \int_{t_0}^{{\varepsilon}/{2}}h(t)\,{{\rm d}}t, \end{align}

and

(7.7)\begin{align} A(\gamma_2) & = 2N^{2}- 2{\rm Area}(\triangle_1)-2\int^{t_0}_0h(t)\,{{\rm d}}t \nonumber\\ & \ge 2N^{2}- 4{\rm Area}(\triangle_1) \nonumber\\ & > 2N^{2}-\varepsilon. \end{align}

Hence $\Sigma _{\partial W^{2}_{g,\epsilon }(1,N,\eta )}=\{A(\gamma _1), A(\gamma _2)\}$. Let us choose $N>2$ so large that $({\pi }/{2})N^{2}<2N^{2}-\varepsilon$. Then (7.4), (7.5) and (7.7) lead to

(7.8)\begin{equation} c^{n,k}(W^{2n}_{g,\varepsilon,R}(1,N,\eta))=A(\gamma_1). \end{equation}

Note that $2N^{2}- 4{\rm Area}(\triangle _1)>2N^{2}-\varepsilon$ and that (7.1) implies

\[ {\rm Area}(W^{2}_{g}(1,N))-2N^{2} < {\rm Area}(W^{2}(1,N))+ \frac{\varepsilon}{2}-2N^{2}= \frac{\pi}{2}+\frac{\varepsilon}{2}. \]

It follows from this, (7.6) and (7.8) that

\[ c^{n,k}(W^{2n}_{g,\varepsilon,R}(1,N,\eta))=A(\gamma_1)< \frac{\pi}{2}+ \frac{\varepsilon}{2}+2(N-t_0)\eta-2\int_{t_0}^{{\varepsilon}/{2}}h(t)\,{{\rm d}}t. \]

For fixed $N$ and $\varepsilon$ we may choose $0<\eta <\varepsilon$ so small that $2(N-t_0)\eta <{\varepsilon }/{2}$. Then

\[ c^{n,k}(W^{2n}_{g,\varepsilon,R}(1,N,\eta))< \frac{\pi}{2}+ \varepsilon. \]

From this and (7.2)–(7.3) we derive

\[ c^{n,k}(W^{2n}(1))\le c^{n,k}(W^{2n}_g(1))\le \frac{\pi}{2}+ \varepsilon \]

and hence $c^{n,k}(W^{2n}(1))\le {\pi }/{2}$ by letting $\varepsilon \to 0+$.

8. Comparison to symmetrical Ekeland–Hofer capacities

For each $i=1,\ldots,n$, let $e_i$ be the vector in $\mathbb {R}^{2n}$ with $1$ in the $i$th position and $0$s elsewhere. Then $\{e_i\}_{i=1}^{n}$ is an orthonormal basis for $L_0^{n}:=V_0^{n,0}=\{x\in \mathbb {R}^{2n}\,|\,x=(q_{1},\ldots,q_n,0,\ldots,0)\}=\mathbb {R}^{n,0}$. It was proved in [26, corollary 2.2] that $L^{2}([0,1],\mathbb {R}^{2n})$ has an orthogonal basis

\[ \{{\rm e}^{m\pi tJ_{2n}}e_i\}_{1\le i\le n,\ m\in\mathbb{Z}}, \]

and every $x\in L^{2}([0,1],\mathbb {R}^{2n})$ can be uniquely expanded as form $x=\sum _{m\in \mathbb {Z}}{\rm e}^{m\pi tJ_{2n}}x_m$, where $x_m\in L_0^{n}$ for all $m\in \mathbb {Z}$ and satisfies $\sum _{m\in \mathbb {Z}}|x_m|^{2}<\infty$. Noting that $V^{n,0}_1=\{0\}$, the spaces in (2.1) and (2.2) become, respectively,

\begin{align*} L^{2}_{n,0} & = \left\{x\in L^{2}([0,1],\mathbb{R}^{2n}) \left| x\stackrel{L^{2}}{=}\sum_{m\in\mathbb{Z}}{\rm e}^{m\pi tJ_{2n}}a_m, a_m\in L^{n}_0,\ \sum_{m\in\mathbb{Z}}|a_m|^{2}<\infty\right.\right\}\\ & = L^{2}([0,1],\mathbb{R}^{2n}) \end{align*}

and

\[ H^{s}_{n,0}=\left\{x\in L^{2}([0,1],\mathbb{R}^{2n}) \left|x\stackrel{L^{2}}{=}\sum_{m\in\mathbb{Z}}{\rm e}^{m\pi tJ_{2n}}a_m,\ a_m\in L^{n}_0, \sum_{m\in\mathbb{Z}}|m|^{2s}|a_m|^{2}<\infty\right.\right\} \]

for any real $s\ge 0$. It follows that the space $\mathbb {E}$ in [25, § 1.2] is a subspace of $E=H^{1/2}_{n,0}$ in (2.3). Denote by $\widehat {\Gamma }$ the set of the admissible deformations on $\mathbb {E}$ (see [25, § 1.2]) and $\widehat {S}^{+}$ the unit sphere in $\mathbb {E}$. Then $\Gamma _{n,0}|_{\mathbb {E}}\subset \widehat {\Gamma }$ and $\widehat {S}^{+}\subset S^{+}_{n,0}$. Note that each function $H\in C^{0}(\mathbb {R}^{2n},\mathbb {R}_{\ge 0})$ satisfying the conditions (H1), (H2) and (H3) below [25, definition 1.4] is naturally $\mathbb {R}^{n,0}$-admissible. Then

\begin{align*} c^{n,0}(H) & = \sup_{\gamma\in\Gamma_{n,0}}\inf_{x\in \gamma(S^{+}_{n,0})}\Phi_H(x)\\ & \le \sup_{\gamma\in\Gamma_{n,0}}\inf_{x\in \gamma(\widehat{S}^{+})}\Phi_H(x)\\ & \le \sup_{\gamma\in\widehat{\Gamma}}\inf_{x\in \gamma(\widehat{S}^{+})}\Phi_H(x) =c_{\rm EH,\tau_0}(H). \end{align*}

It follows that $c^{n,0}(B)\le c_{\rm EH,\tau _0}(B)$ for each $B\subset \mathbb {R}^{2n}$ intersecting with $\mathbb {R}^{n,0}$.