A Lower Bound of the Hofer-Zehnder Capacity via Delzant Polytopes

In 1980s, Gromov [8] proved the celebrated non-squeezing theorem which states that a ball $B^{2n}(r)$ of radius $r$ can be symplectically embedded into a cylinder $B^{2}(R)\times{\mathbb{R}}^{2n-2}$ of radius $R$ if and only if $r\leq R$. This led to the notion of the Gromov width of a symplectic manifold $(M,\omega)$, which is the radius of the largest ball of the same dimension that can be symplectically embedded into $M$. The Gromov width is the first example of a symplectic capacity. In [6], Ekeland and Hofer defined symplectic capacities in an axiomatic way and constructed many concrete examples of symplectic capacities. Since then, symplectic capacities play an important role in the study of the symplectic geometry.

Among symplectic capacities, the Hofer-Zehnder capacity, introduced in [10], is based on properties of periodic orbits of Hamiltonian flows on $(M,\omega)$. For a smooth function $H$ on $M$, let $X_{H}$ denote its Hamiltonian vector field. For a compact symplectic manifold $(M,\omega)$, we say $H:M\to{\mathbb{R}}$ is admissible if

• •

  there exist open sets $U,V$ such that $H|_{U}=H_{\max}$ and $H|_{V}=H_{\min}$, and

• •

  $X_{H}$ has no non-constant periodic orbits of period less than $1$.

We denote the set of admissible functions by $\mathcal{H}_{ad}(M,\omega)=\{H\in C^{\infty}(M)$ — $H$ is admissible$\}$. The Hofer-Zehnder capacity of $(M,\omega)$ is defined by

Recall that we say a circle $S^{1}={\mathbb{R}}/{\mathbb{Z}}$ acts on a symplectic manifold $(M,\omega)$ in a Hamiltonian fashion if there exists a map $H:M\to{\mathbb{R}}$ such that $dH=\omega(X,\cdot)$, where $X$ is the fundamental vector field of the $S^{1}$-action. Such an $H$ is called a moment map. Moment maps can be modified to admissible functions (see Lemma 3.4). Hence, by studying $S^{1}$-actions on a symplectic manifold, we can estimate the Hofer-Zehnder capacity using the information from the moment maps. Moreover, with the presence of group actions, one can also study equivariant capacities introduced in [7]. The equivariant technique used in this paper is similar to those in [5], [14], where the authors obtained bounds for equivariant capacities. In fact, the authors computed packing densities, which easily determines the equivariant capacities according to [7, Equation (2)]. For more discussion about equivariant capacities, see [5], [7], [14],[15],[16] and the references therein.

A symplectic toric manifold $(M^{2n},\omega,T^{n},\Phi)$ is a connected, closed $2n$-dimensional symplectic manifold equipped with an effective, Hamiltonian $n$-torus action (see Definition 2.1). Delzant [4] classified symplectic toric manifolds by their moment polytopes which are Delzant polytopes (see Definition 2.2). These manifolds admit many different $S^{1}$-actions and the information of actions is encoded by simple combinatorial data, so it is possible to get a reasonably good estimate of the Hofer-Zehnder capacity from the combinatorial data. The main goal of this paper is to describe an explicit algorithm for how to find a lower bound of the Hofer-Zehnder capacity by simply using the information obtained from the Delzant polytope.

In [13], Lu also gave a lower bound of the Hofer-Zehnder capacity of a toric manifold from the combinatorial data of the Delzant polytope. Lu’s technique is to find symplectic embeddings of balls whose radius is related to the combinatorial data and use the fact that the Hofer-Zehnder capacity is always bounded from below by the Gromov width, while our proof depends on a careful analysis of how to read the maximal (in terms of the cardinality) stabilizer group of non-fixed points from the combinatorial data.

Let $(M^{2n},\omega,T^{n},\Phi)$ be a symplectic toric manifold with the Delzant polytope $\Delta=\Phi(M)$. We can write $\Delta$ as an intersection of half-spaces

where $v_{i}\in{\mathbb{Z}}^{n}$ are primitive outward-pointing normal vectors. (Recall that a vector $v\in{\mathbb{Z}}^{n}$ is primitive if for any $u\in{\mathbb{Z}}^{n}$, the equation $v=nu$ for $n\in{\mathbb{Z}}$ implies that $n=\pm 1$.)

Let $E$ denote an edge of $\Delta$. Delzant’s classification [4] implies that there exists a unique $J_{E}\subset\{1,2,....,d\}$ with $|J_{E}|=n-1$ such that

Let $u\in{\mathbb{Z}}^{n}$ be a primitive vector. To each edge $E$ of $\Delta$, we associate a lattice polytope $P^{u}_{E}=\{tu-\sum_{j\in J_{E}}t_{j}v_{j}\big{|}\,0\leq t,t_{j}\leq 1\}$, i.e. $P^{u}_{E}$ is the convex hull of $\{-v_{j}:j\in J_{E}\}\cup\{u\}$. Let $k^{u}_{E}:=|{\mathbb{Z}}^{n}\cap\textrm{int}(P^{u}_{E})|+1$ denote the number of interior lattice points plus one. Notice that if $u$ is a linear combination of $v_{j}$, then $k^{u}_{E}=1$.

Define

The toric width of a Delzant polytope $\Delta\subset{\mathbb{R}}^{n}$ is

Our main result states that for a symplectic toric manifold, the toric width of its Delzant polytope gives a lower bound of the Hofer-Zehnder capacity.

Recall that two toric actions $\rho_{1},\rho_{2}:T^{n}\times M^{2n}\to M^{2n}$ are equivalent if there exist a symplectomorphism $\Psi:M\to M$ and an automorphism $h:T^{n}\to T^{n}$ such that the following diagram commutes:

Karshon, Kessler and Pinsonnault [12] proved that two toric actions are equivalent if and only if their Delzant polytopes differ by an affine transformation. Indeed, the toric width is invariant under the affine transformations, so equivalent toric actions will give the same bound in Theorem 1.1.

However, a symplectic manifold could admit many inequivalent toric actions. For instance, Karshon, Kessler and Pinsonnault [12] proved that each compact symplectic $4$-manifold admits finitely many inequivalent toric actions, so we immediately have the following corollary.

In Section 2, we review the basic notions of symplectic toric manifolds and study the sub-circle actions. In Section 3, we prove the key lemma that the moment map of a Hamiltonian circle action can be modified to an admissible function. In Section 4, we give an explicit formula for the lower bound of the Hofer-Zehnder capacity of $4$-dimensional symplectic toric manifolds and use the formula to find lower bounds of some familiar manifolds. Furthermore, we compare the lower bound with the result of [11] to show our estimate is sharp in many cases.

In this section, we recall some facts about symplectic toric manifolds and study sub-cirlce actions. Throughout this section, we will use the convention that $T={\mathbb{R}}^{n}/{\mathbb{Z}}^{n}$ and its Lie algebra is $\mathfrak{t}\cong{\mathbb{R}}^{n}$. Furthermore, we fix an inner product on $\mathfrak{t}$ that identifies $\mathfrak{t}$ with $\mathfrak{t}^{*}$.

In this section, we review the definition of the Hofer-Zehnder capacity and prove the key lemma used in the proof of the main theorem. Recall that $H:M\to{\mathbb{R}}$ is admissible if

• •

  there exist open sets $U,V$ such that $H|_{U}=H_{\max}$ and $H|_{V}=H_{\min}$, and

• •

  $X_{H}$ has no non-constant periodic orbits of period less than $1$.

We denote the set of admissible functions by $\mathcal{H}_{ad}(M,\omega)=\{H\in C^{\infty}(M)$ — $H$ is admissible$\}$.

An equivalent description of the Hofer-Zehnder capacity is given below.

For a smooth function $H:M\to{\mathbb{R}}$, let

We first establish a lemma that will be used in the proof of both Proposition 3.2 and a later result.

For a Hamiltonian circle action on a symplectic manifold, its moment map does not attain its max and min on an open set, so it is not an admissible function. We now show that we can always modify the moment map of a Hamiltonian $S^{1}$-action to an admissible function while keeping track of its magnitude, i.e. the difference between its max and its min. In [11], the authors modified the moment map of a semi-free Hamiltonian $S^{1}$-action to get an admissible function. Specifically, they modified the function locally in an $S^{1}$-invariant neighborhood of a minimal fixed point. The similar technique will not work here, as their modification is based on the fact that near a minimal fixed point, the isotropy weights are all $1$. Instead, Lemma 4.0.1 in [2] does the work. We remark that in our proof we need to first divide the moment map by a constant to ensure that the non-constant periodic orbits have periods at least $1$.

In this section, we describe an explicit algorithm to find the lower bound of the Hofer-Zehnder capacity provided in Theorem 1.1 for $4$-dimensional symplectic toric manifolds. The technical difficulty is to find an explicit formula for the number of lattice points in the interior of a simple lattice polytope. The computation in dimension 4 is feasible thanks to the Pick’s Theorem.