1.1 Overview

A symplectic structure $\omega$ on an even-dimensional manifold $M^{2n}$ can be viewed as a far-reaching generalization of the two-dimensional area on surfaces: by definition, $\omega$ is a closed differential $2$-form whose top wedge power $\omega ^n$ is a volume form. The group $\text {Symp}(M,\omega )$ of all symmetries of a symplectic manifold, i.e. of diffeomorphisms preserving the symplectic structure, contains a remarkable subgroup of Hamiltonian diffeomorphisms $\operatorname {\mathrm {Ham}}(M,\omega )$. When $M$ is closed and its first cohomology vanishes, $\operatorname {\mathrm {Ham}}$ coincides with the identity component of $\text {Symp}$. At the same time, in classical mechanics, where $M$ models the phase space, $\operatorname {\mathrm {Ham}}$ arises as the group of all admissible motions. This group became a central object of interest in symplectic topology in the past three decades. In spite of that, some very basic questions about the algebra, geometry and topology of $\operatorname {\mathrm {Ham}}(M,\omega )$ are far from being understood even when $M$ is a surface.

Let us briefly outline the contents of the paper. First, we obtain new results on Hofer's bi-invariant geometry of $\operatorname {\mathrm {Ham}}(M,\omega )$ in the case where $M=S^2$ is the two-dimensional sphere. We establish, roughly speaking, that $\operatorname {\mathrm {Ham}}(S^2)$ contains flats of arbitrary dimension, thus solving a question open since 2006. A similar result was obtained simultaneously and independently, by using a different technique, in a recent paper by Cristofaro-Gardiner, Humilière, and Seyfaddini [Reference Cristofaro-Gardiner, Humilière and SeyfaddiniCHS21]. Furthermore, our method yields an infinite hierarchy of normal subgroups of area-preserving homeomorphisms of the two-sphere, all of which contain the normal subgroup of homeomorphisms of finite energy discovered in [Reference Cristofaro-Gardiner, Humilière and SeyfaddiniCHS21]. In addition, we find a new constraint on rotationally symmetric homeomorphisms of finite energy.

Second, we extend a number of elementary two-dimensional phenomena taking place on $S^2$ to the stabilized space $S^2 \times S^2$, where the area of the second factor is ‘much smaller’ than that of the first. These phenomena include constraints on packing by circles, which correspond to packings by two-dimensional tori after stabilization, and a version of Poincaré recurrence theorem for sets of zero measure in the context of Hamiltonian diffeomorphisms. Furthermore, we prove a stabilized version of our results on Hofer's geometry presented above.

To illustrate the stabilization paradigm, an elementary area count shows that one cannot fit into the sphere of unit area $k$ pairwise disjoint Hamiltonian images of a circle $L$ bounding a disk of area $> 1/k$. We shall show that the same is true for $L \times \text {equator}$ in the stabilized space. Here the area/volume control miserably fails: a two-dimensional torus does not bound any volume in a four-manifold.

The first instance of such a stabilization was discovered in a recent paper by Mak and Smith [Reference Mak and SmithMS21], who studied constraints on the displaceability of collections of curves on $S^2$. Recall, that a set $X \subset M$ is called displaceable if there exists a Hamiltonian diffeomorphism $\phi$ with $\phi (X) \cap X = \emptyset$. This notion, introduced by Hofer in [Reference HoferHof90], gives rise to a natural ‘small scale’ on symplectic manifolds. Mak and Smith noticed that certain ‘small sets’ on $M$ become more rigid when one looks at them in the symmetric product, a symplectic orbifold $(M)^k/\operatorname {\mathrm {Sym}}_k$ where $\operatorname {\mathrm {Sym}}_k$ stands for the permutation group. On the technical side, our main observation is that a powerful Floer-theoretical tool, Lagrangian spectral invariants with Hamiltonian term, as developed by Leclercq–Zapolsky and Fukaya–Oh–Ohta–Ono, extends to Lagrangian tori in symmetric product orbifolds and can be applied to the study of the above-mentioned questions on Hamiltonian maps. With this language, our paper provides a toolbox for measurements of large energy symplectic effects on small geometric scales by using Floer theory in symmetric products.

Let us note that the idea of looking at configuration spaces of points on a surface, the two-sphere in particular, in order to construct invariants of Hamiltonian diffeomorphisms goes back to Gambaudo and Ghys [Reference Gambaudo and GhysGG04]. Technically, it turns out that for this paper it is beneficial to work with symmetric products, which are certain compactifications of unordered configuration spaces, and instead of the sphere to look at a certain stabilization of it to a four-manifold. Thus, our approach can be considered as a ‘symplectization’ of that by Gambaudo and Ghys. Furthermore, the central objects of interest to us are certain collections of pairwise disjoint Lagrangian submanifolds which in some sense govern Hamiltonian dynamics. These are the Lagrangian configurations appearing in the title of the paper.

1.2 Flats in Hofer's geometry

In [Reference HoferHof90] Hofer has introduced a remarkable bi-invariant metric on the group $\operatorname {\mathrm {Ham}}(M,\omega )$ of Hamiltonian diffeomorphisms of a symplectic manifold $(M,\omega )$. It can intuitively be thought of as the minimal $L^{\infty,1}$ energy required to generate a given Hamiltonian diffeomorphism. For a time-dependent Hamiltonian $H$ in $C^\infty ([0,1]\times M, {\mathbb {R}})$ we denote by $\{ \phi ^t_H \}_{t \in [0,1]}$ the Hamiltonian isotopy generated by the vector field $X^t_H$ given by

\[ \omega(X^t_H, \cdot) ={-} dH_t({\cdot}), \]

where $H_t(\cdot ) = H(t,\cdot )$ for all $t \in [0,1]$. We say that $H$ has zero mean if $\int H_t\, \omega ^n = 0$ for all $t \in [0,1]$. When the symplectic manifold is closed, the Hofer distance of $\phi \in \operatorname {\mathrm {Ham}}(M,\omega )$ to the identity is defined as

\[ d_{\rm{Hofer}}(\phi, \operatorname{\mathrm{id}}) = \inf_{\phi_H^1 = \phi} \int_0^1 \max_M |H(t,-)|\, dt, \]

where the infimum is taken over all zero mean Hamiltonians generating $\phi$. It is extended to arbitrary pairs of diffeomorphisms by the bi-invariance,

\[ d_{\rm{Hofer}}(\psi \phi, \psi \phi') = d_{\rm{Hofer}}( \phi \psi, \phi' \psi) = d_{\rm{Hofer}}(\phi, \phi'), \]

for all $\phi,\phi ',\psi \in \operatorname {\mathrm {Ham}}(M,\omega )$.

The non-degeneracy of $d_{{\mathrm {Hofer}}}$ was studied in many further publications such as [Reference ViterboVit92, Reference PolterovichPol93] and was proven to hold for arbitrary symplectic manifolds in [Reference Lalonde and McDuffLM95]. Let us mention also that Hofer's metric naturally lifts to a (pseudo-)metric on the universal cover $\widetilde {\operatorname {\mathrm {Ham}}}(M,\omega )$, where the question about its non-degeneracy is still open in full generality. We refer to [Reference PolterovichPol01] for a detailed introduction to the Hofer metric and many of its aspects and properties.

The main question related to the Hofer metric, Problem 20 in [Reference McDuff and SalamonMS17], is whether its diameter is infinite for all symplectic manifolds, and when it is, which unbounded groups can be quasi-isometrically embedded into $(\operatorname {\mathrm {Ham}}(M,\omega ),d_{{\mathrm {Hofer}}})$.

This settles a question of the first author and Kapovich from $2006$, cf. Problem 21 [Reference McDuff and SalamonMS17]. A similar result was obtained simultaneously and independently in [Reference Cristofaro-Gardiner, Humilière and SeyfaddiniCHS21] by completely different methods based on periodic Floer homology. As, by a classical theorem [Reference BanachBan93, Théorème 10, p. 187] from a book of Banach, every separable metric space admits an isometric embedding into $C^0_c(I)$, and by smooth approximation, the latter is quasi-isometric to $C^{\infty }_c(I)$, Theorem A implies the following.

For closed surfaces of higher genus, flats of arbitrary dimension were found in [Reference PyPy08]. The proof is based on the fact that such a surface admits an incompressible annulus foliated by non-displaceable closed curves. Symplectic rigidity of these curves yields the result. The lack of such an annulus in the case of $S^2$ requires a new tool, Lagrangian estimators, which we develop by using Lagrangian Floer theory in orbifolds, see §§ 2 and 6.

In fact, our method of proof yields the following statement about Hofer's geometry in dimension four. Throughout the paper $S^2(b)$ stands for the sphere equipped with the standard area form normalized in such a way that the total area equals $b$. We abbreviate $S^2=S^2(1)$.

For $a > 0$, consider a symplectic manifold $M_a = S^2 \times S^2(2a)$. The natural monomorphism

\[ \iota: {\operatorname{\mathrm{Ham}}}(S^2) \to {\operatorname{\mathrm{Ham}}}(M_a),\quad \phi \mapsto \phi \times \operatorname{\mathrm{id}}, \]

satisfies $d_{{\mathrm {Hofer}}}(\iota (\phi ), \iota (\psi )) \leq d_{{\mathrm {Hofer}}}(\phi, \psi )$.

Theorem B Assume that $a>0$ is small enough. The isometric monomorphism $\Phi : {\mathcal {G}}\to \operatorname {\mathrm {Ham}}(S^2)$ from Theorem A can be chosen in such a way that

\[ \Psi = \iota \circ \Phi: ({\mathcal{G}}, d_{C^0}) \hookrightarrow (\operatorname{\mathrm{Ham}}(M_a),d_{\rm{Hofer}}) \]

is a bi-Lipschitz embedding. Furthermore, the monomorphism of the universal covers $\widetilde {\Psi }:{\mathcal {G}} \to \widetilde {\operatorname {\mathrm {Ham}}}(M_a)$ covering $\Psi$ is an isometric embedding.

1.3 Lagrangian packing

Let $K_r$ be a simple closed curve on the sphere $S^2=S^2(1)$ bounding a disk of area $1/{k} > r > 1/(k+1)$, $k \in \mathbb {N}$, $k\geq 2$. Let $S \subset S^2(2a)$ be the equator. Note that the Lagrangian torus $\Lambda _r = K_r \times S$ can be Hamiltonianly $k$-packed into $M_a$, that is, there are $k$ Hamiltonian diffeomorphisms $\phi _1 = \operatorname {\mathrm {id}}, \phi _2,\ldots,\phi _k$ of $M_a$ such that $\{\phi _j (\Lambda _{r}) \}_{1 \leq j \leq k}$ are pairwise disjoint. Indeed, such a packing exists for $K_r \subset S^2$.

We present an argument, based on asymptotic Hofer's geometry, in § 3.6.

1.4 Lagrangian Poincaré recurrence

Using Lagrangian packing obstructions, we are able to make the following progress on the well-known Lagrangian Poincaré recurrence conjecture in Hamiltonian dynamics [Reference Ginzburg and GürelGG19]. It is a version of the classical Poincaré recurrence theorem, but in the setting of Lagrangian submanifolds instead of sets of positive measure. Note that except for the case of surfaces, this is a purely symplectic question, since Lagrangian submanifolds do not bound and they have zero measure.

Let $\Lambda _r \subset M_a$ be a Lagrangian torus as in § 1.3. For an arbitrary Hamiltonian diffeomorphism $\phi \in \operatorname {\mathrm {Ham}}(M_a)$, consider the recurrence set

\[ \mathcal{R}_\phi := \{n \in \mathbb{N} : \phi^n \Lambda_r \cap \Lambda_r \neq \emptyset\}. \]

We remark that while in [Reference Ginzburg and GürelGG18] a similar statement was proven for specific rigid Hamiltonian diffeomorphisms of complex projective spaces (pseudo-rotations) and arbitrary Lagrangians, we provide the first non-trivial higher-dimensional examples of Lagrangian submanifolds satisfying the recurrence property for all Hamiltonian diffeomorphisms.

1.5 Area-preserving homeomorphisms of $S^2$

It has been established by Cristofaro-Gardiner, Humilière and Seyfaddini [Reference Cristofaro-Gardiner, Humilière and SeyfaddiniCHS21] that the group $G_{S^2}$ of symplectic homeomorphisms of the sphere possesses a non-trivial normal subgroup $G_{S^2}^{F}$ of homeomorphisms of finite energy. These are the homeomorphisms $\phi$ for which there exists a sequence of Hamiltonian diffeomorphisms $\psi _j$ and a constant $C > 0$ such that $d_{C^0}(\psi _j,\phi ) \to 0$ and $d_{{\mathrm {Hofer}}}(\psi _j,\operatorname {\mathrm {id}}) \leq C$. This is achieved by showing that radially symmetric Hamiltonian homeomorphisms of bounded energy satisfying a certain monotone twist condition must, in a precise sense, have finite Calabi invariant. We extend this result as follows.

Let $z:S^2 \to [-1/2,1/2]$ be the moment map for the standard $S^1$-action on $S^2$. It is the height function for the standard embedding of $S^2$ in ${\mathbb {R}}^3$ scaled by a factor of $1/2$. Let $h:[-1/2,1/2) \to {\mathbb {R}}$ be a smooth function that vanishes on $[-1/2,0]$. Consider $\phi \in G_{S^2}$ generated by $H = h \circ z$: it is the $C^0$-limit of $\phi _i = \phi ^1_{H_i} \in \operatorname {\mathrm {Ham}}_c ({\mathbb {D}}^2) \subset \operatorname {\mathrm {Ham}}(S^2)$ for $H_i = h_i \circ z$ for $h_i$ approximations to $h$ constant near $1/2$ which agree with $h$ on $[-1/2,1/2-1/i]$.

The proof is based on a combination of [Reference Cristofaro-Gardiner, Humilière and SeyfaddiniCHS21, Lemma 3.11], an interesting soft result relating $C^0$-smallness, supports, and the Hofer metric, with our Lagrangian spectral estimators. It turns out that suitable linear combination of these invariants are continuous with respect to the $C^0$ topology on $\operatorname {\mathrm {Ham}}(S^2)$ and extend to the group $G_{S^2}$. In fact, our invariants give rise to an infinite series of pairwise distinct normal subgroups of $G_{S^2}$ containing $G^F_{S^2}$, see § 5 for a precise formulation.

1.6 Organization of the paper

In § 2 we introduce and list the properties of Lagrangian estimators, our main technical tool. The detailed construction of the estimators via Lagrangian Floer theory in symmetric products, as well as the proof of their properties is presented in § 6.

Section 3 deals with flats in Hofer geometry. We prove Theorems A and B, and present a result on infinite-dimensional flats in subgroups of Hamiltonian diffeomorphisms of the disk with vanishing Calabi invariant. In addition, we derive an estimate on the asymptotic Hofer norm which is used in the proof of Theorem C on Lagrangian packing. This proof can be found in § 3.6.

In § 4 we deduce Theorem D on Lagrangian recurrence from the packing constraint by a combinatorial argument.

In § 5 we prove the $C^0$-continuity of certain linear combinations of our invariants and present applications to normal subgroups of the group of area-preserving homeomorphisms of $S^2$.

We conclude the paper with a discussion of further directions in § 7.

3.1 Proof of Theorem A

Step 1: construction. We start with a more explicit formulation of the theorem. Consider a symmetric interval $I = (-b, b)$ for $b < 1/6$. Let

\[ \mathcal{F}_I \subset C^{\infty}_c(I) \]

be the space of even compactly supported smooth functions on $I$. In other words, $\mathcal {F}_I$ consists of functions $h \in C^{\infty }_c(I)$ satisfying $h(x) = h(-x)$ for all $x \in I$. Endow $\mathcal {F}_I$ with the $C^0$ norm

\[ |h|_{C^0} = \max_{I} |h|, \]

which induces the distance function $d_{C^0}(h_1,h_2) = |h_1- h_2|_{C^0}$. Observe that the group ${\mathcal {G}} = C^{\infty }_c((0,b))$ with the $C^0$ distance naturally embeds into ${\mathcal {F}}_I$ by even extension.

Consider the standard symplectic sphere $(S^2,\omega )$ of total area $1$. It admits a Hamiltonian $S^1$-action whose zero-mean normalized moment map $z:S^2 \to {\mathbb {R}}$ has image $[-1/2,1/2]$. We define the following embedding

\[ \Phi:{\mathcal{F}}_I \to \operatorname{\mathrm{Ham}}(S^2), \]

keeping in mind that the Hofer metric works with zero-mean Hamiltonians. We first build an embedding of $\mathcal {F}_I$ to the space $\mathcal {I}$ of even functions of integral zero in $C^{\infty }([-1/2,1/2])$. To a function $h \in \mathcal {F}_I$ we associate $h^{\#} \in \mathcal {I}$ defined by $h^{\#}|_{(-b,b)} = h$, $h^{\#}|_{(1/2-b,1/2]}(x) = - h(1/2-x)$, $h^{\#}|_{[-1/2,-1/2+b)}(x) = - h(-1/2-x)$, extended by zero to $[-1/2,1/2]$. Now, for $h \in \mathcal {F}_I$ we consider the mean-zero Hamiltonian $\Gamma (h) \in C^{\infty }(S^2,{\mathbb {R}})$ given by

\[ \Gamma(h) = h^{\#} \circ z, \]

and let

\[ \Phi(h) = \phi^1_{\Gamma(h)} \]

be the time-one map of $\Gamma (h)$. It is immediate by construction that this map $\Phi : \mathcal {F}_I \to \operatorname {\mathrm {Ham}}(S^2)$ is a homomorphism and for all $h_1, h_2 \in {\mathcal {F}}$

(3)\begin{equation} d_{\rm{Hofer}}(\Phi(h_1),\Phi(h_2)) = d_{\rm{Hofer}}(\Phi(h_1-h_2),\operatorname{\mathrm{id}}) \leq |h_1 - h_2|_{C^0}. \end{equation}

We claim that the monomorphism of groups

\[ \Phi: (\mathcal{F}_I, d_{C^0}) \hookrightarrow (\operatorname{\mathrm{Ham}}(S^2),d_{\rm{Hofer}}) \]

is an isometric embedding.

Step 2: proof of the claim. Note that by (3), the main step in the proof of the claim is the inequality

(4)\begin{equation} d_{\rm{Hofer}}(\Phi(h),\operatorname{\mathrm{id}}) \geq |h|_{C^0}\quad \forall h \in \mathcal{F}_I . \end{equation}

As $|(-h)|_{C^0} = |h|_{C^0}$ and

\[ d_{\rm{Hofer}}(\Phi({-}h),\operatorname{\mathrm{id}}) = d_{\rm{Hofer}}(\Phi(h)^{{-}1},\operatorname{\mathrm{id}}) = d_{\rm{Hofer}}(\Phi(h),\operatorname{\mathrm{id}}) \]

for all $h \in \mathcal {F}_I$, it is sufficient to prove (4) under the assumption that $|h|_{C^0} = h(x_0) > 0$. Note that in this case $h(x_0)= h(-x_0)$, and hence either $x_0 = 0$, or we can assume that $x_0 \in (0,b)$. Consider $B = 1/2 - x_0$. ForFootnote ¹ $x_0 \in [0,b)$ we consider $\zeta _i = \zeta ^0_{2,B_i}$, where $B_i = 1/2 - x_i$, and $x_i \in (0,b)$ is a sequence of rational numbers converging to $x_0$. Here we fix a rational parameter $0< a<1/2-3b$ for defining $\zeta ^0_{2,B_i}$ as in Theorem I. By continuity of $h$ we now have

\[ h(x_i) \xrightarrow{i \to \infty} h(x_0) = |h|_{C^0}. \]

Now by the Lagrangian control property of $\zeta _{i}$ we have

\[ \zeta_{i}(\Gamma(h)) = h(x_i), \]

and by the Hofer–Lipschitz and independence of Hamiltonian properties

\[ d_{{\mathrm{Hofer}}}(\Phi(h)) \geq \zeta_{i}(\Gamma(h)) \]

for all $i$. Therefore, taking limits as $i \to \infty$ we obtain

\[ d_{{\mathrm{Hofer}}}(\Phi(h)) \geq |h|_{C^0} \]

as required. This finishes the proof.

3.2 Proof of Theorem B

Let $I = (-b, b)$ for $b<1/6$ and $0< a<1/2-3b$. Arguing as in the proof of Theorem A we get that the monomorphism $\widetilde {\Psi }:{\mathcal {F}}_I \to \widetilde {\operatorname {\mathrm {Ham}}}(M_a)$ covering $\Psi$ is an isometric embedding.

In order to pass from the universal cover to the group itself, we shall restrict ourselves to the subspace $\mathcal {F}^0_I \subset \mathcal {F}_I$ consisting of functions $h$ with $h(0)=0$. Note that the image of ${\mathcal {G}} =C^{\infty }_c ( (0,b))$ in ${\mathcal {F}}_I$ lies in $\mathcal {F}^0_I$.

Let us show that the monomorphism of groups

\[ \Psi = \iota \circ \Phi: (\mathcal{F}^0_I, d_{C^0}) \hookrightarrow (\operatorname{\mathrm{Ham}}(M_a),d_{\rm{Hofer}}) \]

is a bi-Lipschitz embedding. It suffices to show that there exists $K \geq 1$ such that for all $h \in {\mathcal {F}}_I$

(5)\begin{equation} d_{{\mathrm{Hofer}}}(\operatorname{\mathrm{id}}, \Psi(h)) \geq \frac{1}{K} |h|_{C^0}. \end{equation}

Suppose without loss of generality that $\|h\|_{C^0} = h(r)>0$ with $r \in (0,b)$.

Consider the invariant $\mu _{2,B}: \widetilde {\operatorname {\mathrm {Ham}}}(M_a) \to {\mathbb {R}}$ provided by Theorem G for

\[ 1/2 > B = 1/2-r \geq 1/2 - b. \]

In order to proceed further, we have to understand the effect of the fundamental group $\pi _1(\operatorname {\mathrm {Ham}}(M))$. It is a finitely generated abelian group. In fact, by [Reference Abreu and McDuffAM00, Theorem 1.1] we have $\pi _1(\operatorname {\mathrm {Ham}}(M)) \cong {\mathbb {Z}} \oplus {\mathbb {Z}}/2{\mathbb {Z}} \oplus {\mathbb {Z}}/2{\mathbb {Z}}$. The ${\mathbb {Z}}/2{\mathbb {Z}}$ terms appear from the natural maps $\pi _1(\operatorname {\mathrm {Ham}}(S^2)) \to \pi _1(\operatorname {\mathrm {Ham}}(M))$ corresponding to the two components of $M = S^2 \times S^2$. The ${\mathbb {Z}}$ term is the well-known Gromov loop [Reference GromovGro85], investigated in detail by Abreu and McDuff [Reference McDuffMcD87, Reference Abreu and McDuffAM00]. Set ${\mathcal {T}} = {{\mathbb {Z}}/2{\mathbb {Z}}} \oplus {{\mathbb {Z}}/2{\mathbb {Z}}}$ for the torsion part of ${\mathcal {G}} = \pi _1(\operatorname {\mathrm {Ham}}(M))$, and let ${\mathcal {A}} = {\mathcal {G}}/ {\mathcal {T}} \cong {\mathbb {Z}}$ be its free part. Note that $\pi _1(\operatorname {\mathrm {Ham}}(M)) \subset Z(\widetilde {\operatorname {\mathrm {Ham}}}(M))$ is a central subgroup. As in the proof of Theorem A it is easy to see that $\mu _{2,B}$ vanishes on ${\mathcal {T}}$ and by commutative subadditivity descends to $\widetilde {\operatorname {\mathrm {Ham}}}(M)/ {\mathcal {T}}$. However, the same is not clear for ${\mathcal {A}}$. We proceed differently.

Observe that by [Reference OstroverOst06, Theorem 1.2] there exists a homogeneous Calabi quasi-morphism $\rho : \widetilde {\operatorname {\mathrm {Ham}}}(M) \to {\mathbb {R}}$ that is $1$-Lipschitz in the Hofer metric and restricts to a non-trivial homomorphism ${\mathcal {G}} \to {\mathbb {R}}$. It vanishes on the torsion ${\mathcal {T}}$ so we can consider it to be a homomorphism $\rho : {\mathcal {A}} \to {\mathbb {R}}$. Choosing a generator $g$ of ${\mathcal {A}}$, we have $\rho (g^k) = k \cdot \rho (g)$ for a positive constant $\rho (g)$. It is also known that $\rho (\Psi (h)) = h(0)=0$ for $h \in \mathcal {F}^0_I$. This follows from $\rho$ yielding a symplectic quasi-state on $C^\infty (M_a,{\mathbb {R}})$ and $S \times S \subset M_a$, where $S \subset S^2$ is the equator, being a stem (see [Reference Entov and PolterovichEP06, Reference Entov and PolterovichEP09, Reference Polterovich and RosenPR14]).

Define the map

\begin{gather*} \nu_{r}: \widetilde{\operatorname{\mathrm{Ham}}}(M) \to {\mathbb{R}}^2,\\ \nu_{r}(\widetilde{\phi}) = (\mu_{2,B}(\widetilde{\phi}), \rho(\widetilde{\phi})). \end{gather*}

Let $\widetilde {\Psi }:{\mathcal {F}}_I^0 \to \widetilde {\operatorname {\mathrm {Ham}}}(M)$ be the homomorphism covering $\Psi$.

Observe that $\nu _{r}$ is $1$-Lipschitz in Hofer's metric, where ${\mathbb {R}}^2$ is endowed with the $l_{\infty }$ norm.

By the above-mentioned properties of $\rho$, and by Lagrangian control of $\mu _{2,B}$, we can calculate for an arbitrary element $\widetilde {\Psi }(h) g^k f$ covering $\Psi (h)$, where $f \in {\mathcal {T}}$, that

\[ \nu_{r}(\widetilde{\Psi}(h) g^k f) = \nu_{r}(\widetilde{\Psi}(h) g^k) = (\mu_{2,B}(\widetilde{\Psi}(h)g^k), \rho(\widetilde{\Psi}(h)g^k)) = (\mu_{2,B}(\widetilde{\Psi}(h)g^k), k \rho(g)). \]

This implies that

\[ |\nu_{r}(\widetilde{\Psi}(h) g^k f)| \geq \max \{ |h|_{C^0} - |k| d_{{\mathrm{Hofer}}}(g,\operatorname{\mathrm{id}}), |k| \rho(g)\} \geq C_2 |h|_{C^0}, \]

for $C_2 = {\rho (g)}/({\rho (g)) + d_{{\mathrm {Hofer}}}(g,\operatorname {\mathrm {id}})}$, the last step being an easy optimization in $|k|$.

Therefore, we have

\[ C_2 |h|_{C^0} \leq |\nu_{r}(\widetilde{\Psi}(h)g^k f)| \leq d_{{\mathrm{Hofer}}}(\widetilde{\Psi}(h)g^k f,\operatorname{\mathrm{id}}), \]

and, hence,

\[ d_{{\mathrm{Hofer}}}({\Psi}(h),\operatorname{\mathrm{id}}) \geq \frac{1}{K} |h|_{C^0} \]

for $K = 1/C_2 \geq 1$. This finishes the proof.

3.3 Almost flats in the kernel of Calabi

We call a map

\[ f:(X,d_X) \to (Y,d_Y\!) \]

of metric spaces almost isometric if there exists a constant $D \geq 0$ such that

\[ d_X(p,q)-D \leq d_Y(f(p),f(q)) \leq d_X(p,q)+D \]

for all $p,q \in X$.

Fix the interval

\[ I_k = ({-}1/2+1/{(k+1)},-1/2+1/{k}). \]

Consider $\mu _{k,B}$ with $1/k > B > 1/{(k+1)}$. Note that in this case $B>C$, so $\mu _{k,B}$ is well-defined. We prove that the space of all functions ${\mathcal {G}}_k \subset C^{\infty }_c(I_k)$ with zero mean admits an almost isometric embedding into $(\operatorname {\mathrm {Ham}}(S^2), d_{{\mathrm {Hofer}}})$. Recall that for a proper open subset $U \subset S^2$ its symplectic form is exact, and the Calabi homomorphism

\[ {\mathrm{Cal}}_{U}: \operatorname{\mathrm{Ham}}_c(U) \to {\mathbb{R}} \]

is defined as ${\mathrm {Cal}}_{U}(\phi ) = {\mathrm {Cal}}_{U}(\{\phi ^t_H\!\})$ for any $H \in C^\infty _c([0,1] \times U, {\mathbb {R}})$ with $\phi = \phi ^1_H$. Observe that by the natural constant extension we have the inclusion $\operatorname {\mathrm {Ham}}_c(U) \to \operatorname {\mathrm {Ham}}(S^2)$. In particular, $\ker ({\mathrm {Cal}}_U\!)$ can and shall be considered to be a subgroup of $\operatorname {\mathrm {Ham}}(S^2)$. Starting from the following result we require configurations $\mathbb {L}_{k,B}$ for all values of $k$.

Theorem J Let $\Phi _k: ({\mathcal {G}}_k, d_{C^0}) \to (\operatorname {\mathrm {Ham}}(S^2), d_{{\mathrm {Hofer}}})$ be given by

\[ \Phi_k(h) = \phi^1_{H} \]

for $H = k \cdot h\circ z$. Then $\Phi _k$ is a bi-Lipschitz almost isometric group embedding, whose image lies in $\ker ({\mathrm {Cal_{D}}})$, where $D = D_{1/k}$ is the open cap of area $1/k$ around the south pole. Furthermore, for each proper open set $U \subset S^2$ there is a bi-Lipschitz almost isometric embedding of $C^\infty _c(I)$ of an open interval $I$ into $\ker ({\mathrm {Cal_{U}}}\!)$.

A similar result is proved in [Reference Cristofaro-Gardiner, Humilière and SeyfaddiniCHS21].

The proof of this statement is similar to that of Theorem A, with the additional observation that by the conjugation invariance of $\mu _{k,B}$ we may suppose that the open set $U$ contains $D_{1/k}$ for $k$ sufficiently large. Moreover, passing from smooth functions on an interval, say $C^\infty _c((-\frac {1}{2}+ {1}/({k+1}), -\frac {1}{2} + {1}/{2(k+1)}+ {1}/{2k}))$ to functions in ${\mathcal {G}}_k$ can be easily carried out by odd extension about the midpoint ${\upsilon _k = -\frac {1}{2} + {1}/{2(k+1)}+ {1}/{2k}}$ of $I_k$. On the one hand, by use of Lagrangian estimators for the lower bound, and the evident upper bound on the Hofer norm via the Hamiltonian, we obtain

\[ |h_1-h_2|_{C^0} \leq d_{{\mathrm{Hofer}}}(\Phi_k(h_1),\Phi_k(h_2)) \leq k |h_1-h_2|_{C^0} \]

for all $h_1, h_2 \in {\mathcal {G}}_k$. Hence, $\Phi _k$ is bi-Lipschitz. On the other hand, estimating $d_{{\mathrm {Hofer}}}(\Phi _k(h_1), \Phi _k(h_2)) = d_{{\mathrm {Hofer}}}(\Phi _k(h_1-h_2),\operatorname {\mathrm {id}})$ via Sikorav's trick (see § 3.4) we obtain

\[ d_{{\mathrm{Hofer}}}(\Phi_k(h_1),\Phi_k(h_2)) \leq |h_1-h_2|_{C^0} + D_k \]

for a constant $D_k$ depending only on $k$. This proves that $\Phi _k$ is almost isometric. (We remark that one can show that $D_k < 2$ for all $k$.)

Finally, analogously to the proof of Theorem B, we can show that these almost isometric embeddings extend to stabilizations on the level of universal covers and remain bi-Lipschitz embeddings on the level of groups.

Theorem K Let $k \geq 2$, $1/{(k+1)} < B < 1/k$ and $0< a< (({k+1})/({k-1}))B - {1}/({k-1})$. The monomorphism of groups

\[ \Psi_k = \iota \circ \Phi_k: ({\mathcal{G}}_k, d_{C^0}) \hookrightarrow (\operatorname{\mathrm{Ham}}(M_a),d_{\rm{Hofer}}) \]

is a bi-Lipschitz embedding. In fact, there exists $K \geq 1$ such that for all $h_1,h_2 \in {\mathcal {G}}_k$

\[ \frac{1}{K} |h_1-h_2|_{C^0} \leq d_{{\mathrm{Hofer}}}(\Psi_k(h_1),\Psi_k(h_2)) \leq k |h_1-h_2|_{C^0}. \]

At the same time, the monomorphism $\widetilde {\Psi }_k:{\mathcal {G}}_k \to \widetilde {\operatorname {\mathrm {Ham}}}(M_a)$ covering $\Psi _k$ is an almost isometric embedding.

3.4 The asymptotic Hofer norm

Let $(M,\omega )$ be a closed symplectic manifold. The asymptotic Hofer (pseudo-)norm $\nu (\widetilde {\phi })$ of an element $\widetilde {\phi } \in \widetilde {\operatorname {\mathrm {Ham}}}(M,\omega )$ is defined as follows:

\[ \nu(\tilde{\phi}) = \lim_{m\to \infty} \frac{1}{m} \widetilde{d}_{{\mathrm{Hofer}}}(\widetilde{\phi}^m,\operatorname{\mathrm{id}}), \]

where $\widetilde {d}_{{\mathrm {Hofer}}}$ is the Hofer pseudo-norm on $\widetilde {\operatorname {\mathrm {Ham}}}(M,\omega )$. For $\phi \in \operatorname {\mathrm {Ham}}(M,\omega )$ we define its asymptotic Hofer norm $\nu (\phi )$ similarly, or alternatively $\nu (\phi ) = \inf \nu (\widetilde {\phi })$ where the infimum runs over all $\widetilde {\phi } \in \widetilde {\operatorname {\mathrm {Ham}}}(M,\omega )$ covering $\phi$.

The Hofer norm is known to yield quantitative invariants of subsets in symplectic manifolds, for instance the displacement energy [Reference HoferHof90] (see also [Reference PolterovichPol01]). It turns out that the asymptotic Hofer norm produces an invariant of subsets of symplectic manifolds controlling their packing number.

Let $A \subset M$ be a compact subset. Its packing number $k(A) \in {\mathbb {N}} \cup \{\infty \}$ is defined as the maximal $k$ such that there exist $\theta _1,\ldots,\theta _k \in \operatorname {\mathrm {Ham}}(M,\omega )$ with $\theta _1(A),\ldots,\theta _k(A)$ pairwise disjoint ($\theta _i(A) \cap \theta _j(A) = \emptyset$ for all $i \neq j$).

Let $C^{\infty }_0(M;\mathbb {R})$ denote the space of mean-zero smooth functions on $M$. For $f \in C^{\infty }_0(M;\mathbb {R})$, set $\widetilde {\phi }_f \in \widetilde {\mathrm {Ham}}(M,\omega )$ for the class of the time-one Hamiltonian isotopy which it generates.

Definition 11 Let $A \subset M$ be a proper compact set. Its $\alpha$-invariant is

\[ \alpha(A) = \inf_{U \supset A} \sup\{ {\nu}(\widetilde{\phi}_f)\,|\, f \in C^{\infty}_0(M;\mathbb{R}),\ |f|_{C^0} = 1,\ \mathrm{supp}(f) \subset U,\ f|_A \equiv 1 \}, \]

where the infimum runs over all open neighborhoods $U$ of $A$.

Remark 12 A more elaborate alternative invariant $\alpha '(A)$ would be the infimum over open neighborhoods $U$ of $A$ of constants $a>0$ such that the map ${\iota _U: \ker (\operatorname {\mathrm {Cal}}_U) \to \widetilde {\operatorname {\mathrm {Ham}}}(M,\omega )}$ is coarse Lipschitz with constant $a>0$, that is

\[ d_{{\mathrm{Hofer}}}(\iota_U(\widetilde{\phi}),\iota_U(\widetilde{\psi})) \leq a \cdot d_{{\mathrm{Hofer}}}(\widetilde{\phi},\widetilde{\psi}) + b \]

for all $\widetilde {\phi },\widetilde {\psi } \in \ker (\operatorname {\mathrm {Cal}}_U)$ and $b \geq 0$ independent of $\widetilde {\phi },\widetilde {\psi }$. It is easy to see that $\alpha (A) \leq \alpha '(A)$ and Proposition 13 below applies to $\alpha '(A)$ as well.

In view of Sikorav's trick [Reference SikoravSik90, Reference Hofer and ZehnderHZ94], which we recall below in detail for the reader's convenience, we obtain the following.

Proposition 13 The packing number of $A$ satisfies

\[ k(A) \leq \bigg\lfloor \frac{1}{\alpha(A)}\bigg\rfloor, \]

where the right-hand side is by convention $\infty$ if $\alpha (A) = 0$.

Proof of Proposition 13 It is sufficient to prove that $k(A) \leq {1}/{\alpha (A)}$. If $\alpha (A) = 0$, there is nothing to prove. Hence, we suppose that $\alpha (A) > 0$. Then we can prove the equivalent statement that $\alpha (A) \leq 1/k(A)$.

Suppose that $M$ can be packed by $l$ copies of $A$. Namely, let $\theta _1 = \operatorname {\mathrm {id}}, \ldots, \theta _l \in \operatorname {\mathrm {Ham}}(M,\omega )$ yield an $l$-packing of $A$ in $M$: $\{ \theta _j(A) \}_{1 \leq j \leq l}$ are pairwise disjoint. Let $U \supset A$ be a sufficiently small open neighborhood of $A$ so that $\{ \theta _j(U) \}_{1 \leq j \leq l}$ are pairwise disjoint. Set $U_j = \theta _j(U)$ for $1\leq j\leq l$. Let $f \in C^{\infty }_0(M;\mathbb {R})$, $|f|_{C^0} = 1$, $\mathrm {supp}(f) \subset U$, $f|_A \equiv 1$ be as in the definition of $\alpha (A)$. For $1\leq j\leq l$, $f_j = f \circ \theta _j^{-1}$ satisfies ${\mathrm {supp}}(f_j) \subset U_j$.

Let us show that $\nu (\widetilde {\phi }_f,\operatorname {\mathrm {id}}) \leq 1/l$, as calculated in $\widetilde {\operatorname {\mathrm {Ham}}}(M,\omega )$, which implies our claim. Indeed, by the conjugation invariance of Hofer's pseudo-metric on $\widetilde {\operatorname {\mathrm {Ham}}}(M,\omega )$,

\[ d_{{\mathrm{Hofer}}}(\phi_{f}^{lt},\phi^t_{f_1}\cdot\ldots\cdot \phi^t_{f_l}) \leq C, \]

where $C \leq 2(d_{{\mathrm {Hofer}}}(\theta _1,\operatorname {\mathrm {id}}) + \cdots + d_{{\mathrm {Hofer}}}(\theta _l,\operatorname {\mathrm {id}}))$ is independent of $t$. However,

\begin{gather*} \phi^t_{f_1}\cdot\ldots\cdot\phi^t_{f_l} = \phi^t_{F} ,\\ F= f_1 + \cdots +f_l. \end{gather*}

As $|F|_{C^0} = 1$ we obtain that $\nu (\phi _f) = ({1}/{l}) \nu (\phi _F) \leq 1/l$, which proves the proposition.

Next, we study the asymptotic Hofer norm in the following situation. Let $M = S^2$ with a round area form of total area $1$. We write $z$ for the vertical coordinate scaled by a factor of $1/2$. Let us fix $B> C > 0$ with $2B+ (k-1)C=1$. Consider a Lagrangian configuration $\sqcup _{1 \leq j \leq k} \{z = z_j \}$, where

(6)\begin{equation} z_j ={-}1/2 + B+(j-1)C, \quad j=1, \ldots ,k. \end{equation}

Denote by $\sigma _{B,C}$ the measure $ ({1}/{k})\sum _{j=1}^k \delta _{z_j}$. By using Lagrangian estimators $\mu _{B_i,k}$ as in Theorem G, for rational $B_i \xrightarrow {i \to \infty } B$, we get that for every smooth function $h = h(z)$ on $S^2$ with zero mean the asymptotic Hofer norm satisfies

(7)\begin{equation} \nu(\phi_h)= \lim_{t\to+\infty} \frac{{d}_{\rm Hofer}(\phi_h^t,\operatorname{\mathrm{id}})}{t} \geq \int h \,d\sigma_{B,C}. \end{equation}

Let us emphasize that in this definition the flow $\phi _h^t$ naturally lifts to the universal cover $\widetilde {\operatorname {\mathrm {Ham}}}$ and we consider Hofer's metric there.

As a test, we fix small $\delta >0$ and

\[ r \in \bigg(\frac{1}{k+1}, \frac{1}{k}\bigg), \]

and consider $h_{r,\delta }$ to be a smoothing of the indicator function of

\[ [{-}1/2+r-\delta,-1/2+r+\delta], \]

which we arrange to have zero mean by extending it to $[-1/2+r-\delta,-1/2+r+5\delta ]$ as an odd function about the midpoint $-1/2+r+2\delta$ of the interval. Choose $B= r$, put

\[ C = \frac{(1-2B)}{(k-1)}. \]

Note that $h_{r,\delta }$ has $|h_{r,\delta }|_{C^0} = 1$ and equals $1$ on the circle $K_r:= L^{0,0}_{k,B}$ of area $r$ and is supported in its $\delta$-neighborhood. Since $B>C$, we can apply inequality (7) with the measure $\sigma _{B,C}$ and get

\[ u(r) := \liminf_{\delta\to 0}\nu(\phi_{h_{r,\delta}}) \geq \frac{1}{k}. \]

At the same time, note that there exists a packing of $S^2$ by $k$ copies of the support of $h$, so by Proposition 13 $u(r) \leq 1/k$. Thus, $u(r)=1/k$, so the estimate is sharp.

Presumably, progress in the direction outlined in § 7.1 will yield efficient lower bounds on the asymptotic Hofer norm for more general autonomous Hamiltonians.

3.5 Stabilization

Consider now the stabilization of the Hamiltonian diffeomorphism $\phi _{h_{r,\delta }}$ constructed in § 3.4. Fix a closed symplectic manifold $(P,\Omega )$ and put

\[ \Phi_{r,\delta}:= \phi_{h_{r,\delta}} \times \operatorname{\mathrm{id}} \in \operatorname{\mathrm{Ham}}(S^2 \times P). \]

Put

\[ u_P(r) = \lim_{\delta\to 0}\nu(\Phi_{r,\delta}). \]

If the answer is affirmative, it would be interesting to calculate or estimate this quantity. Our method, based on spectral invariants in Lagrangian Floer theory on orbifolds, yields ‘yes’ when $(P,\Omega )$ is a $2$-sphere of area $2a$ with $B+a > C$. In fact, we have in this case $u_P(r) = 1/k$ as above.

Now we discuss another version of the stabilization. Let $S \subset P$ be a closed Lagrangian submanifold, and let $f_W$ be a smoothing of the characteristic function of a small Weinstein neighborhood $W$ of $S$. Consider the Hamiltonian $h_{r,\delta }f_W$. We denote by $u_S(r)$ the lower limit of the corresponding asymptotic Hofer norm when $r \to 0$ and $W$ shrinks to $S$.

For instance, when $P$ is $S^2$ of area $2a$ with $B> C+a$ and $S$ is the equator, Hamiltonians $h_{r,\delta }f_W$, are concentrated near $\Lambda _r:=K_r \times S$, and the same argument based on Theorem G and Proposition 13 yields

(8)\begin{equation} u_S(r) = 1/k. \end{equation}

3.6 Lagrangian packing via Hofer's geometry

Here we use identity (8) for the asymptotic Hofer norm in order to deduce Theorem C.

6.1 Lagrangian Floer homology with bounding cochains and bulk deformation

We briefly discuss the general algebraic properties of the Lagrangian Floer homology theory with weak bounding cochains and bulk. We refer to [Reference Mak and SmithMS21] for a slightly more detailed discussion, and to the original work [Reference Fukaya, Oh, Ohta and OnoFOOO09, Reference Fukaya, Oh, Ohta and OnoFOOO10, Reference Fukaya, Oh, Ohta and OnoFOOO11, Reference Fukaya, Oh, Ohta and OnoFOOO12, Reference Fukaya, Oh, Ohta and OnoFOOO19] for all detailed definitions. We also remark that the Fukaya algebra of a Bohr–Sommerfeld Lagrangian submanifold in a rational symplectic manifold, that is when $[\omega ]$ is contained in the image of $H^2(M;{\mathbb {Q}})$ inside $H^2(M;{\mathbb {R}})$, was constructed in [Reference Charest and WoodwardCW22] by classical transversality techniques. Hence, one could feasibly carry out all constructions in this paper by the above techniques, when restricted to the rational setting, which is sufficient for our purposes. We also note that we establish our result as a rather formal consequence of the methods of [Reference Fukaya, Oh, Ohta and OnoFOOO09, Reference Fukaya, Oh, Ohta and OnoFOOO19, Reference Mak and SmithMS21], hence it applies with whichever perturbation schemes these papers do.

For a subgroup $\Gamma \subset {\mathbb {R}}$, define the Novikov field with coefficients in the field ${\mathbb {K}} = {\mathbb {C}}$ as

\[ \Lambda_{\Gamma} = \bigg\{\sum_j a_j T^{\kappa_j}\bigg | a_j \in {\mathbb{K}},\ \kappa_j \in \Gamma,\ \kappa_j \to +\infty \bigg\}. \]

This field possesses a non-Archimedean valuation $\nu : \Lambda _{\Gamma } \to {\mathbb {R}} \cup \{+\infty \}$ given by $\nu (0) = +\infty$, and

\[ \nu\Big(\sum a_j T^{\kappa_j}\Big) = \min\{\kappa_j\,|\,a_j \neq 0 \}. \]

For now we may assume that $\Gamma = {\mathbb {R}}$, but later it will be important to choose a smaller subgroup. We often omit the subscript $\Gamma$, and write $\Lambda$ for $\Lambda _{\Gamma }$. Set $\Lambda _0 = \nu ^{-1}([0,+\infty )) \subset \Lambda$ to be the subring of elements of non-negative valuation, and $\Lambda _+ = \nu ^{-1}((0,+\infty )) \subset \Lambda _0$ the ideal of elements of positive valuation.

Given a closed connected oriented spin Lagrangian submanifold $L \subset M$, and an $\omega$-tame almost complex structure on $M$, considering the moduli spaces of $J$-holomorphic disks with boundary on $L$ and with boundary and interior punctures, and suitable virtual perturbations required to regularize the problem, as well as suitable homological perturbation techniques, yields the following maps. First, considering only $k+1$ boundary punctures, for $k \geq 0$, we have the maps

\[ m_k: H^*(L;\Lambda)^{{\otimes} k} \to H^*(L;\Lambda). \]

These maps satisfy the relations of a curved filtered $A_{\infty }$ algebra. Furthermore, these maps decompose as $m_k = \sum m_{k,\beta } T^{\omega (\beta )}$, where the sum runs over the relative homology classes $\beta \in H_2(M,L; {\mathbb {Z}})$ of disks in $M$ with boundary on $L$. Moreover, considering also $l$ interior punctures yields maps

\[ q_{l,k}:H^*(M;\Lambda)^{{\otimes} l} \otimes H^*(L;\Lambda)^{{\otimes} k} \to H^*(L;\Lambda). \]

Given a ‘bulk’ class $\mathbf {b} \in H_{\ast }(M; \Lambda _+)$ we can deform the $m_k$ operations to

\[ m^{\mathbf{b}}_k(x_1 \otimes \cdots \otimes x_k) = \sum_{r\geq 0} q_{r,k}(\mathbf{b}^{{\otimes} r} \otimes x_1 \otimes \cdots \otimes x_k). \]

The operations $m^{\mathbf {b}}_k$ for $k \geq 0$ also satisfy the relations of a curved filtered $A_{\infty }$ algebra, and decompose into a sum $m^{\mathbf {b}}_k = \sum m^{\mathbf {b}}_{k,\beta } T^{\omega (\beta )}$ as above, over the relative homology classes $\beta \in H_2(M,L; {\mathbb {Z}})$ of disks in $M$ with boundary on $L$.

Furthermore, given a ‘cochain’ class $b = b_0 + b_+$ for $b_0 \in H^1(L;{\mathbb {C}})$ and $b_+ \in H^1(L;\Lambda _+)$, we define the $b$-twisted $A_{\infty }$ operations with bulk $\mathbf {b} \in H_*(M;\Lambda _+)$ as follows: first set

\[ m^{\mathbf{b}, b_0}_{k, \beta} = e^{{\left\langle b_0, \partial \beta \right\rangle}} \,m^{\mathbf{b}}_{k, \beta}. \]

These operations again satisfy the curved filtered $A_{\infty }$ equations. Now, to further deform by $b_+ \in H^1(L;\Lambda _+)$, we set

\[ m^{\mathbf{b}, b}_{k, \beta}(x_1 \otimes \cdots \otimes x_k) = \sum_{l \geq 0} \sum m^{\mathbf{b}, b_0}_{k+l, \beta}(b_+{\otimes} \cdots \otimes b_+{\otimes} x_1 \otimes b_+{\otimes}\cdots \otimes b_+{\otimes} x_k \otimes b_+{\otimes} \cdots \otimes b_+), \]

where the interior sum runs over all possible positions of the $k$ symbols $x_1,\ldots,x_k$ appearing in this order, and $l$ symbols $b_+$, and finally we set

\[ m^{\mathbf{b}, b}_{k} = \sum_{\beta} T^{\omega(\beta)} m^{\mathbf{b}, b}_{k, \beta}. \]

The operations $m^{\mathbf {b}, b}_{k}$ also satisfy the curved filtered $A_{\infty }$ equations. Moreover, if $b_0 - b'_0 \in H^1(L; 2\pi \sqrt {-1}{\mathbb {Z}})$, then $m^{\mathbf {b}, b}_{k} = m^{\mathbf {b}, b'}_{k}$, $b = b_0 + b_+, b' = b'_0 + b_+$.

Given $\mathbf {b} \in H_*(M;\Lambda _+)$, we say that $b = b_0+b_+ \in H^1(L;\Lambda _0)$ is a weak bounding cochain for $\{m^{\mathbf {b}}_k\}$ if there exists a constant $c \in \Lambda _+$ such that

\[ \sum_k m^{\mathbf{b}, b_0}_k(b_+{\otimes} \cdots \otimes b_+) = c \cdot 1_L, \]

where $1_L \in H^0(L;\Lambda _0)$ is the cohomological unit. We call $c = W^{\mathbf {b}}(b)$ the potential function of the weak bounding cochain $b$. Note that it depends only on the class of $b$ in $H^1(L;\Lambda _0)/H^1(L;2\pi \sqrt {-1} {\mathbb {Z}})$, which is well-defined as there is no torsion in $H^1(L;{\mathbb {Z}})$. We shall henceforth consider bounding cochains as elements of $H^1(L;\Lambda _0)/H^1(L;2\pi \sqrt {-1} {\mathbb {Z}})$.

Now we have the following result regarding Lagrangian tori, which was first proven in the context of Lagrangian torus fibers of toric manifolds in [Reference Fukaya, Oh, Ohta and OnoFOOO10, Theorem 4.10] and [Reference Fukaya, Oh, Ohta and OnoFOOO11, Theorem 3.16].

Theorem N [Reference Fukaya, Oh, Ohta and OnoFOOO12, Theorem 2.3]

If $L \cong T^n$ is a Lagrangian torus and all elements of $H^1(L;\Lambda _0)/H^1(L;2\pi \sqrt {-1} {\mathbb {Z}})$ are weak bounding cochains for $\{m^{\mathbf {b}}_k\}$, then if $b$ is a critical point of the potential function

\[ W^{\mathbf{b}}: H^1(L;\Lambda_0)/H^1(L;2\pi\sqrt{-1} {\mathbb{Z}}) \to \Lambda_+ \]

with $H^1(L;\Lambda _0)/H^1(L;2\pi \sqrt {-1} {\mathbb {Z}}) \cong (\Lambda _0/2\pi \sqrt {-1} {\mathbb {Z}})^n$ identified with $\Lambda _0\setminus \Lambda _+$ by the exponential function, then $m^{\mathbf {b}, b}_1 = 0$ and hence the $(\mathbf {b}, b)$-deformed Floer cohomology of $L$ is isomorphic to $H^*(L;\Lambda )$. Moreover, this implies that $L$ is non-displaceable by Hamiltonian isotopies in $M$.

Finally, we note that $H^*(L;\Lambda ) \cong H^*(L;{\mathbb {C}}) \otimes _{{\mathbb {C}}} \Lambda$ possesses a non-Archimedean filtration function ${\mathcal {A}}$ determined by requiring that the basis $E = E_0 \otimes 1$, for $E_0 = (e_1,\ldots,e_B)$ a basis of $H^*(L;{\mathbb {C}})$ be orthonormal in the sense that ${\mathcal {A}}(e_j \otimes 1) = 0$ for all $1\leq j \leq B$, ${\mathcal {A}}(0) = -\infty$, and for all $(\lambda _1,\ldots,\lambda _B) \in \Lambda ^{B}$,

\[ {\mathcal{A}}\Big(\sum \lambda_j e_j \otimes 1\Big) = \max \{ {\mathcal{A}}(e_j \otimes 1) - \nu(\lambda_j)\}. \]

It is important to note that the $A_{\infty }$ algebra $\{m_k^{\mathbf {b}, b}\}$ from Theorem N has unit $1_L \in H^*(L;\Lambda )$ of non-Archimedean filtration level ${\mathcal {A}}(1_L) = 0$.

6.2 Lagrangian spectral invariants

In this section we discuss Lagrangian spectral invariants. While essentially going back to Viterbo [Reference ViterboVit92], they were defined in Lagrangian Floer homology by a number of authors in varying degrees of generality, starting with Oh [Reference OhOh97], Leclercq [Reference LeclercqLec08], and Monzner, Vichery, and Zapolsky [Reference Monzner, Vichery and ZapolskyMVZ12]. However, the two main contributions that are relevant to our goals are the paper of Leclercq and Zapolsky [Reference Leclercq and ZapolskyLZ18] in the context of monotone Lagrangians, and of Fukaya, Oh, Ohta, and Ono [Reference Fukaya, Oh, Ohta and OnoFOOO19, Definition 17.15] in the context of Floer homology with bounding cochains and bulk deformation.

6.2.1 Discrete submonoids and their associated subgroups

First we formulate the spaces of possible values of our spectral invariants. Following [Reference Fukaya, Oh, Ohta and OnoFOOO19], consider elements $\mathbf {b} \in H_*(M;\Lambda _0)$, $b \in H^1(L;\Lambda _0)$. We say that they are gapped if they can be written as

\begin{gather*} \mathbf{b} = \sum_{g \in G(\mathbf{b})} \mathbf{b}_g T^{g} ,\quad \mathbf{b}_g \in H_*(M;{\mathbb{C}}),\\ b = \sum_{g \in G(b)} b_g T^{g},\quad b_g \in H^1(L;{\mathbb{C}}), \end{gather*}

where $G(\mathbf {b})$, $G(b)$ are discrete submonoids of ${\mathbb {R}}_{\geq 0}$. In practice, all relevant elements $\mathbf {b}, b$ will be gapped, so we assume that they are for the rest of this section.

Given an $\omega$-tame almost complex structure $J$ on $M$ define the submonoid $G(L,\omega,J)$ to be generated by the areas $\omega ([u])$ of all $J$-holomorphic disks $u$ with boundary on $L$. It is discrete by Gromov compactness. Note that in the orbifold setting below, one includes both areas of smooth holomorphic disks and those of orbifold holomorphic disks.

Definition 24 Let $G(L,\mathbf {b},b) \subset {\mathbb {R}}_{\geq 0}$ be the discrete submonoid generated by the union $G(\mathbf {b}) \cup G(b) \cup G(L,\omega,J)$. Furthermore, let $\Gamma (\mathbf {b})$, $\Gamma (b)$, $\Gamma (L,\omega,J)$, $\Gamma (L, \mathbf {b}, b)$ be the subgroups of ${\mathbb {R}}$ generated by the monoids $G(\mathbf {b}) , G(b) , G(L,\omega,J), G(L,\mathbf {b},b)$, respectively. We call $(L,\mathbf {b}, b)$ rational if $\Gamma (L,\mathbf {b}, b)$ is a discrete subgroup of ${\mathbb {R}}$.

Given a Hamiltonian $H \in C^\infty ([0,1] \times M, {\mathbb {R}})$ we consider the chords $x:[0,1] \to M$ with $x(0),x(1) \in L$, satisfying $\dot {x}(t) = X^t_H(x(t))$ for all $t \in [0,1]$. Furthermore, we restrict attention to only those chords that are contractible relative to $L$. Set $\operatorname {\mathrm {Spec}}({H,L})$ to be the set of all actions of pairs $(x,\overline {x})$ each consisting of a chord $x$ and its contraction $\overline {x}$ to $L$, called a capping, $\overline {x}:{\mathbb {D}} \to M$, $\overline {x}|_{\partial {\mathbb {D}} \cap \{\Im (z) \geq 0\}} = x$, $\overline {x}(\partial {\mathbb {D}} \cap \{\Im (z) \leq 0\}) \subset L$. The action of $(x, \overline {x})$ is given by

\[ \mathcal{A}_{{H,L}}(x,\overline{x}) = \int_0^1 H(t,x(t))\,dt - \int_{\overline{x}} \omega. \]

Definition 25 Define the $(\mathbf {b}, b)$-deformed spectrum ${\mathrm {Spec}}({H,L},\mathbf {b}, b)$ of ${H,L}$ by

\[ {\mathrm{Spec}}({H,L},\mathbf{b}, b) = \operatorname{\mathrm{Spec}}({H,L}) + \Gamma(L,\mathbf{b}, b). \]

We recall that for two subsets $A,B$ of ${\mathbb {R}}$, $A+B$ is the subset of ${\mathbb {R}}$ given by

\[ A+B = \{ a+b \,|\, a\in A,\ b\in B \}. \]

6.2.2 Filtered Floer complex and spectral invariants

Consider a Hamiltonian $H$ and $L \subset M$ a Lagrangian as above. Let $({\mathbf {b}, b})$ be a weak bounding cochain with bulk deformation. Assuming that $({H,L})$ is non-degenerate, that is $\phi ^1_H(L)$ intersects $L$ transversely, and $\{J_t\}_{t \in [0,1]}$ a time-dependent $\omega$-compatible almost complex structure on $M$, following [Reference Fukaya, Oh, Ohta and OnoFOOO09, Reference Fukaya, Oh, Ohta and OnoFOOO19] one constructs a filtered finite rank $\Lambda _0$-complex $CF({H,L},{\mathbf {b}, b}; \Lambda _0)$, where $\Lambda = \Lambda _{\Gamma }$ for $\Gamma = \Gamma (L,\mathbf {b}, b)$.

We set $CF({H,L},{\mathbf {b}, b}) = CF({H,L},{\mathbf {b}, b};\Lambda _0) \otimes _{\Lambda _0} \Lambda$. The module $CF({H,L},{\mathbf {b}, b})$ comes with an non-Archimedean filtration function ${\mathcal {A}}_{{H,L}}$ whose values are contained in

\[ \operatorname {\mathrm {Spec}}({H,L},\mathbf {b}, b) \cup \{-\infty \}. \]

Indeed, as a $\Lambda$-module $CF({H,L},{\mathbf {b}, b})$ is given by the completion with respect to the action functional of the vector space generated by pairs $(x,\overline {x})$ of Hamiltonian $H$-chords from $L$ to $L$ contractible relative to $L$, where we identify between $(x,\overline {x})$, $(x,\overline {x}')$ if $v = \overline {x}' \# \overline {x}^{-}: ({\mathbb {D}},\partial {\mathbb {D}}) \to (M,L)$, defined by gluing suitably reparametrized $\overline {x}'$ and $\overline {x}^{-}(z) = \overline {x}(\overline {z})$ along their common boundary chord, satisfies ${\left \langle [\omega ],[v] \right \rangle } = 0$. In this case the filtration ${\mathcal {A}}$ is defined by declaring any $\Lambda$-basis $[(x_i,\overline {x}_i)]$, for $\{x_i\}$ the finite set of contractible $H$-chords from $L$ to $L$ an orthogonal $\Lambda$-basis of $CF({H,L},{\mathbf {b}, b})$. Finally, the homology $HF({H,L},{\mathbf {b}, b})$ of $CF({H,L},{\mathbf {b}, b})$ is naturally isomorphic to self-Floer homology $HF(L,{\mathbf {b}, b}) = HF((L,{\mathbf {b}, b}),(L,{\mathbf {b}, b}))$ of $L$ deformed by $({\mathbf {b}, b})$. We write $\Phi _H: HF(L,{\mathbf {b}, b}) \to HF({H,L},{\mathbf {b}, b})$ for this isomorphism. In the setting of Theorem N there is a natural isomorphism between $HF({H,L},{\mathbf {b}, b})$ and $H^*(L,\Lambda )$, which, in particular, explains why $L$ is not Hamiltonianly displaceable: if $\phi ^1_H(L) \cap L = \emptyset$ then $CF({H,L},{\mathbf {b}, b}) = 0$ and, hence, one must have $HF({H,L},{\mathbf {b}, b}) = 0$.

For $a \in {\mathbb {R}}$ the subspace $CF({H,L},{\mathbf {b}, b})^a$ generated over $\Lambda _0$ by all $[(x,\bar {x})]$ satisfying $\mathcal {A}_{{H,L}}(x,\bar {x}) < a$ forms a subcomplex of $CF({H,L},{\mathbf {b}, b})$. We denote its homology by $HF({H,L},{\mathbf {b}, b})^a$. It comes with a natural map $HF({H,L},{\mathbf {b}, b})^a \to HF({H,L},{\mathbf {b}, b})$.

Given a class $z \in HF(L,{\mathbf {b}, b})$, we define its spectral invariant with respect to $H$ as in [Reference Fukaya, Oh, Ohta and OnoFOOO19, Definition 17.15] by

\[ c(L,{\mathbf{b}, b}; z, H) = \inf \{a \in {\mathbb{R}}\,|\, \Phi_H(z) \in \operatorname{\mathrm{im}}(HF({H,L},{\mathbf{b}, b})^a \to HF({H,L},{\mathbf{b}, b}))\}. \]

In the following, we will primarily work with $z = 1_L$, the unit of the algebra on $HF(L,{\mathbf {b}, b})$ induced by the $m_2^{{\mathbf {b}, b}}$ operation. Following the arguments of [Reference Fukaya, Oh, Ohta and OnoFOOO19, Theorem 7.2], [Reference Leclercq and ZapolskyLZ18, Theorem 35] together with the argument in [Reference Polterovich and RosenPR14, Remark 4.3.2] for rational spectrality, it is straightforward to show the following properties of the spectral invariant $c(L,{\mathbf {b}, b}; z, H)$.

1. (1) Non-degenerate spectrality: for each $z \in HF(L,{\mathbf {b}, b}) \setminus \{0\}$, and Hamiltonian $H$ such that ${H,L}$ is non-degenerate,

   \[ c(L,{\mathbf{b}, b};z,H) \in \operatorname{\mathrm{Spec}}({H,L},{\mathbf{b}, b}). \]

2. (2) Rational spectrality: if $\Gamma (H,{\mathbf {b}, b})$ is rational, then for $z \in HF(L,{\mathbf {b}, b}) \setminus \{0\}$, the condition

   \[ c(L,{\mathbf{b}, b};z,H) \in \operatorname{\mathrm{Spec}}({H,L},{\mathbf{b}, b}) \]
   holds for each Hamiltonian $H$.

3. (3) Non-Archimedean property: for each Hamiltonian $H$, $c(L,{\mathbf {b}, b};-,H)$ is a non-Archimedean filtration function on $HF(L,{\mathbf {b}, b})$ as a module over the Novikov field $\Lambda$ with its natural valuation.

4. (4) Hofer–Lipschitz: for each $z \in HF(L,{\mathbf {b}, b}) \setminus \{0\}$,

   \[ |c(L,{\mathbf{b}, b};z,F) - c(L,{\mathbf{b}, b};z,G)| \leq \max\{\mathcal{E}_+(F-G), \mathcal{E}_{-}(F-G)\}, \]
   where for a Hamiltonian $H$, we set $\mathcal {E}_+(H) = \int _{0}^{1} \max _M(H_t)\,dt$ and $\mathcal {E}_{-}(H) = \mathcal {E}_+(-H)$. Note that $\mathcal {E}_{\pm }(H) \leq \int _0^1 \max _M |H_t|\,dt$.

5. (5) Normalization: For each $H \in C^\infty ([0,1] \times M, {\mathbb {R}})$ and $b\in C^\infty ([0,1],{\mathbb {R}})$,

   \[ c(L,{\mathbf{b}, b};z,H+b) = c(L,{\mathbf{b}, b};z,H) + \int_{0}^{1} b(t)\,dt. \]

6. (6) Monotonicity: if two Hamiltonians $F,G$ satisfy $F_t \leq G_t$ for all $t \in [0,1]$, then $c(L,{\mathbf {b}, b}; z, F) \leq c(L,{\mathbf {b}, b}; z, G)$ for each $z \in HF(L,{\mathbf {b}, b})$.

7. (7) Lagrangian control: if $\Gamma (H,{\mathbf {b}, b})$ is rational and $(H_t)|_L = c(t) \in \mathbb {R}$ for all $t \in [0,1]$ then setting $c_+(H) = c(L,{\mathbf {b}, b}; 1_L, H)$ we have

   \[ c_+(H)=\int_0^1 c(t) \,dt, \]
   hence for all $H \in C^\infty ([0,1] \times M,{\mathbb {R}})$, $\int _0^1 \min _L H_t \,dt \leq c_+(H) \leq \int _0^1 \max _L H_t \,dt$.

8. (8) Homotopy invariance: for $H_t$ of mean-zero for all $t \in [0,1]$, $c(L,{\mathbf {b}, b};z,H)$ depends only on the class $\widetilde {\phi }_H \in \widetilde {\operatorname {\mathrm {Ham}}}(M,\omega )$ of the Hamiltonian isotopy $\{\phi ^t_H\!\}_{t \in [0,1]}$.

9. (9) Triangle inequality: for each $z,w \in HF(L,{\mathbf {b}, b})$, and Hamiltonians $F,G$,

   \[ c(L,{\mathbf{b}, b}; m^{{\mathbf{b}, b}}_2(z,w),F\#G) \leq c(L,{\mathbf{b}, b};z ,F) + c(L,{\mathbf{b}, b};w ,G), \]
   where $F \# G(t,x) = F(t,x) + G(t,(\phi ^t_F)^{-1}x)$ generates the flow $\{ \phi ^t_F \phi ^t_G\}_{t \in [0,1]}$.

6.3 Orbifold setting

We shall consider only a very simple kind of a symplectic orbifold $X$. It is called a global quotient symplectic orbifold, and consists of the data of a closed symplectic manifold $\widetilde {M}$ and an effective symplectic action of a finite group $G$ on it.

In fact, we shall only consider $\widetilde {M} = M^k = M \times \cdots \times M$ for a symplectic manifold $M$ and the symmetric group $G = \operatorname {\mathrm {Sym}}_k$ acting on $\widetilde {M}$ by permutations of the coordinates. The corresponding global quotient orbifold is called the symmetric power $X = Sym^k(M)$ of $M$.

The inertia orbifold $IX$ of a global quotient orbifold $X$ is itself a global quotient orbifold given by the action of $G$ on the disjoint union $\bigsqcup _{g \in G} \widetilde {M}^g$, where for $g\in G$, $\widetilde {M}^g$ is the fixed point submanifold of $g$, and $f \in G$ acts by $\widetilde {M}^g \to \widetilde {M}^{fgf^{-1}}$, $x \mapsto fx$. See [Reference Adem, Leida and RuanALR07] for further details.

6.3.1 Orbifold Hofer metric

While on a symplectic orbifold one can define smooth functions and, hence, Hamiltonian diffeomorphisms, in the case of a global quotient orbifold $X$, the data of a smooth Hamiltonian $H \in C^\infty ([0,1] \times X, {\mathbb {R}})$ is equivalent to the data of a $G$-invariant Hamiltonian $\widetilde {H} \in C^\infty ([0,1] \times \widetilde {M}, {\mathbb {R}})$, that is $H(t,g\cdot x) = H(t,x)$ for all $t \in [0,1], x\in \widetilde {M}, g\in G$. In our particular case $\widetilde {M} = M^k$ and our Hamiltonians satisfy $H(t,x_1,\ldots, x_k) = H(t,x_{\sigma ^{-1}(1)},\ldots, x_{\sigma ^{-1}(k)})$ for all $t \in [0,1], x_1,\ldots,x_k \in M$, and $\sigma \in \operatorname {\mathrm {Sym}}_k$. Given an orbifold Hamiltonian diffeomorphism $\phi$ of $X$, which is equivalently a $G$-equivariant Hamiltonian diffeomorphism $\widetilde {\phi }$ of $\widetilde {M}$ that is generated by a $G$-invariant Hamiltonian $H$ on $\widetilde {M}$, we define its orbifold Hofer distance to the identity by

\[ d_{{\mathrm{Hofer}}}(\phi, \operatorname{\mathrm{id}}) = \inf_{\phi_H^1 = \widetilde{\phi}} \int_0^1 \max_{\widetilde{M}} H(t,-) - \min_{\widetilde{M}} H(t,-) \, dt, \]

where the infinimum runs over all such $G$-invariant Hamiltonians generating $\widetilde {\phi }$. Finally, we remark that orbifold Hamiltonian diffeomorphisms form a group $\operatorname {\mathrm {Ham}}(X)$, it is isomorphic to the identity component $\operatorname {\mathrm {Ham}}(\widetilde {M})^G$ of the subgroup of $\operatorname {\mathrm {Ham}}(\widetilde {M})$ consisting of $G$-equivariant Hamiltonian diffeomorphisms, and $d_{{\mathrm {Hofer}}}$ extends to a bi-invariant (non-degenerate) metric on $\operatorname {\mathrm {Ham}}(X)$. We denote by $\widetilde {\operatorname {\mathrm {Ham}}}(X)$ the universal cover of this group with basepoint at the identity. This is also a group.

6.3.3 Lagrangians in symmetric products and their spectral invariants

Consider $S^2$ with the symplectic form $\omega$ of total area $1$. Under this normalization consider the Lagrangian configuration $L^{0,j}_{2,B}$, $j = 0,1$, see (1). In [Reference Mak and SmithMS21] Mak and Smith show, as translated to our normalizations, that if

\[ M = (S^2 \times S^2, \omega \oplus 2a \omega), \]

with $0 < a < 3B - 1 = B-C$, then the Lagrangian link $\mathbb {L}_{2,B} = \mathbb {L}^0_{2,B} \times S^1$ in $M$ is Hamiltonianly non-displaceable: for all $\phi \in \operatorname {\mathrm {Ham}}(X)$, $\phi (\mathbb {L}_{2,B}) \cap \mathbb {L}_{2,B} \neq \emptyset$.

The key observation of this paper is that, in fact, the proof of [Reference Mak and SmithMS21] gives strictly stronger information. We recall their approach. The two connected components of $\mathbb {L}_{2,B}$ are

\[ L^j_{2,B} = L^{0,j}_{2,B} \times S, \]

where $S \subset S^2(2a)$ is the equator. Consider the product Lagrangian

\[ {\mathcal{L}}'_{2,B} = {L}^0_{2,B} \times L^1_{2,B} \subset \widetilde{M} = M \times M. \]

Since ${\mathcal {L}}'_{2,B}$ is disjoint from the diagonal $\Delta _M \subset M \times M$, it descends to a smooth Lagrangian submanifold ${\mathcal {L}}_{2,B}$ in the regular locus ${X}^{\rm {reg}}$ of the symplectic global quotient orbifold ${X} = (M \times M)/({\mathbb {Z}}/2{\mathbb {Z}})$, where ${\mathbb {Z}}/2{\mathbb {Z}}$ acts by exchanging the factors. Similarly, in the situation of configurations for arbitrary values $(k,B)$ of parameters, one defines

\[ {\mathcal{L}}'_{k,B} = {L}^0_{k,B} \times \cdots \times L^{k-1}_{k,B} \subset M^k, \]

and it descends to a Lagrangian ${\mathcal {L}}_{k,B}$ in the regular locus of $X^{{\mathrm {reg}}}$ the symplectic orbifold $X = Sym^k(M) = M^k/\operatorname {\mathrm {Sym}}_k$, where the symmetric group $\operatorname {\mathrm {Sym}}_k$ acts on $M^k$ by permutations of the coordinates.

In [Reference Mak and SmithMS21] the authors have proved for $k=2$, and we show in § 6.3.5 that the same extends to arbitrary values of parameters $(k,B)$ and $a < ({(k+1)B -1})/({k-1})$ the following statement.

Theorem O There exists an integrable almost complex structure $J_M$ on $M$ such that with respect to the complex structure $J = Sym^k(J_M)$ on $X = Sym^k(M)$, the Lagrangian ${\mathcal {L}}_{k,B} \subset X$ has a well-defined Fukaya algebra in the sense of [Reference Fukaya, Oh, Ohta and OnoFOOO09, Reference Cho and PoddarCP14, Reference Fukaya, Oh, Ohta and OnoFOOO19]. Moreover, this Fukaya algebra admits gapped orbifold bulk deformation $\mathbf {b}$ and weak bounding cochain $b$ with the bulk-deformed Floer homology

\[ HF( ({\mathcal{L}}_{k,B},\mathbf{b},b), ({\mathcal{L}}_{k,B},\mathbf{b},b)) \cong HF( {\mathcal{L}}_{k,B}, {\mathbf{b}, b} ) \]

of $({\mathcal {L}}_{k,B},\mathbf {b},b)$ with itself well-defined and isomorphic to $H^*({\mathcal {L}}_{k,B};\Lambda )$ with coefficients in the Novikov field $\Lambda = \Lambda _{\Gamma }$, where $\Gamma = \Gamma (L,\mathbf {b}, b)$. Moreover, for $B \in (1/(k+1),1/2)$,

\[ a \in (0, ({(k+1)B -1})/({k-1})) \]

rational, ${\mathbf {b}, b}$ can be chosen in such a way that $\Gamma$ is rational. Furthermore, $HF({\mathcal {L}}_{k,B},{\mathbf {b}, b})$ is an associative unital algebra, and we denote its unit by $1_{{\mathcal {L}}_{k,B}}$.

The proof of this theorem appears in § 6.3.5. It implies, in particular, that the unit $1_{{\mathcal {L}}_{k,B}} \in HF({\mathcal {L}}_{k,B},\mathbf {b},b)$ does not vanish. This enables us to use spectral invariants associated to this element. For $H\in C^\infty ([0,1] \times X, {\mathbb {R}})$ we set

\[ \sigma_{k,B}(H) = c({\mathcal{L}}_{k,B},{\mathbf{b}, b}; 1_{{\mathcal{L}}_{k,B}}, H). \]

Then $\sigma _{k,B}$ satisfies the properties of Lagrangian spectral invariants, including rational spectrality, from § 6.2.

1. (1) Spectrality: $\Gamma (H,{\mathbf {b}, b})$ being rational, the condition

   \[ \sigma_{k,B}(H) \in \operatorname{\mathrm{Spec}}({H,L},{\mathbf{b}, b}) \]
   holds for each Hamiltonian $H$.

2. (2) Hofer–Lipschitz:

   \[ |\sigma_{k,B}(F) - \sigma_{k,B}(G)| \leq \max\{\mathcal{E}_+(F-G), \mathcal{E}_{-}(F-G)\}. \]

3. (3) Normalization: For each $H \in C^\infty ([0,1] \times M, {\mathbb {R}})$ and $b\in C^\infty ([0,1],{\mathbb {R}})$,

   \[ \sigma_{k,B}(H+b) = \sigma_{k,B}(H) + \int_{0}^{1} b(t)\,dt. \]

4. (4) Monotonicity: if two Hamiltonians $F,G$ satisfy $F_t \leq G_t$ for all $t \in [0,1]$, then $\sigma _{k,B}(F) \leq \sigma _{k,B}(G)$.

5. (5) Lagrangian control: $\Gamma (H,{\mathbf {b}, b})$ being rational, if $(H_t)|_{{\mathcal {L}}_{k,B}} = c(t) \in \mathbb {R}$ for all $t \in [0,1]$, then we have

   \[ \sigma_{k,B}(H)=\int_0^1 c(t) \,dt, \]
   hence for each Hamiltonian $H$, $\int _0^1 \min _{{\mathcal {L}}_{k,B}} H_t \,dt \leq \sigma _{k,B}(H) \leq \int _0^1 \max _{{\mathcal {L}}_{k,B}} H_t \,dt$.

6. (6) Homotopy invariance: for $H_t$ of mean-zero for all $t \in [0,1]$, $\sigma _{k,B}(H)$ depends only on the class ${\phi }_H \in \widetilde {\operatorname {\mathrm {Ham}}}(X,\omega )$ of the Hamiltonian isotopy $\{\phi ^t_H\}_{t \in [0,1]}$.

7. (7) Triangle inequality: for each two Hamiltonians $F,G$,

   \[ \sigma_{k,B}(F\#G) \leq \sigma_{k,B}(F) + \sigma_{k,B}(G). \]

6.3.4 Proof of Theorems F and G

We now prove Theorems F and G. For a Hamiltonian $F \in C^\infty (M_a,{\mathbb {R}})$ let $H \in C^\infty (X,{\mathbb {R}})$ be the Hamiltonian on $X$ determined by the $\operatorname {\mathrm {Sym}}_k$-invariant Hamiltonian $\widetilde {H} = F \oplus \cdots \oplus F$ in $C^\infty ([0,1] \times M^k, {\mathbb {R}})$, that is $\widetilde {H}(t,x_1,\ldots, x_k) = F(t,x_1) + \cdots + F(t,x_k)$. Observe that if $F$ is mean-zero, then so is $H$. Furthermore, if $F_1 \mapsto H_1$, $F_2 \mapsto H_2$, then $F_1 \# F_2 \mapsto H_1 \# H_2$ and the map $F \mapsto H$ induces a map $\widetilde {\operatorname {\mathrm {Ham}}}(M_a) \to \widetilde {\operatorname {\mathrm {Ham}}}(X)$.

We set

\begin{gather*} c_{k,B}(F) = \frac{1}{k} \sigma_{k,B}(H),\\ \mu_{k,B}(F) = \lim_{m\to \infty} \frac{1}{m} c_{k,B}(F^{\#m}). \end{gather*}

The limit exists by Fekete's lemma by the subadditivity property of $\sigma _{k,B}$. The Hofer–Lipschitz property of $c_{k,B}$ and $\mu _{k,B}$ follows from that of $\sigma _{k,B}$ by the subadditivity of Hofer's energy functional. The monotonicity, normalization, Lagrangian control, and independence of Hamiltonian properties are immediate consequences of those for $\sigma _{k,B}$. Note the factor $1/k$ in the definition of $c_{k,B}$: it serves for instance to obtain the Hofer–Lipschitz property with coefficient $1$, as ${\mathcal {E}}_+(H) \leq k {\mathcal {E}}_+(F)$ for $F \mapsto H$. Similarly, if $F_t|_{L^j_{k,B}} \equiv c_j(t)$, then $H_t|_{{\mathcal {L}}_{k,B}} \equiv \sum _{0 \leq j < k} c_j(t)$.

Subadditivity for $c_{k,B}$ and commutative subadditivity for $\mu _{k,B}$ are direct consequences of the subadditivity of $\sigma _{k,B}$. Positive homogeneity of $\mu _{k,B}$ is immediate from the definition of $\mu _{k,B}$.

It remains to prove the conjugation invariance of $\mu _{k,B}$, controlled additivity of $c_{k,B}$ and the Calabi properties of $c_{k,B}$ and $\mu _{k,B}$.

To prove conjugation invariance of $\mu _{k,B}$, we note that by the subadditivity of $\sigma _{k,B}$ and, hence, of $c_{k,B}$, we have for all $m \in {\mathbb {Z}}_{>0}$,

(10)\begin{equation} c_{k,B}(\phi^m) - q_{k,B}(\psi) \leq c_{k,B}(\psi\phi^m\psi^{{-}1}) \leq c_{k,B}(\phi^m) + q_{k,B}(\psi), \end{equation}

where $q_{k,B}(\psi ) = c_{k,B}(\psi ) + c_{k,B}(\psi ^{-1})$. Hence, dividing by $m$ and taking the limit as $m \to \infty$, we obtain

\[ \mu_{k,B}(\phi) = \mu_{k,B}(\psi\phi\psi^{{-}1}) \]

as required.

Now we prove the Calabi property. The proofs for $c_{k,B}$ and $\mu _{k,B}$ are identical, so we focus on $\mu _{k,B}$ for example. Suppose that $U \subset M_a$ is disjoint from $\mathbb {L}_{k,B}$. Given a Hamiltonian $H \in C^\infty ([0,1] \times M,{\mathbb {R}})$ supported in $[0,1] \times U$, consider the Hamiltonian

(11)\begin{equation} G(t,x) = H(t,x) + b(t) \end{equation}

such that for all $t \in [0,1]$,

\[ b(t) ={-} \frac{1}{\operatorname{vol}(M_a)} \int_U H(t,x)\, \omega^2. \]

In that case $\mu _{k,B}(H) = 0$. Hence, in view of the normalization property of $\mu _{k,B}$ we obtain

\[ \mu_{k,B}(G) ={-} \int_{0}^1 b(t)\,dt ={-} \frac{1}{\operatorname{vol}(M_a)} \int_{0}^1 \int_U H(t,x)\, \omega^2\,dt ={-} \frac{1}{\operatorname{vol}(M_a)} \operatorname{\mathrm{Cal}}_U(\{\phi^t_H\}). \]

Finally, to prove controlled additivity it suffices to observe that for $H$ supported in $[0,1] \times U$ and each Hamiltonian $F \in C^\infty ([0,1] \times M_a, {\mathbb {R}})$, there is a bijective correspondence between the generators of the Lagrangian Floer complex of $F \# H$ and that of $F$, and, furthermore,

\[ \operatorname{\mathrm{Spec}}({\mathcal{L}}_{k,B},{\mathbf{b}, b};1_{{\mathcal{L}}_{k,b}},\widetilde{F\#H}) = \operatorname{\mathrm{Spec}}({\mathcal{L}}_{k,B},{\mathbf{b}, b};1_{{\mathcal{L}}_{k,b}},\widetilde{F}). \]

The proof now proceeds by the spectrality and Hofer–Lipschitz axioms for the family $\widetilde {F \# sH}$ of Hamiltonians where $s \in [0,1]$, as that of the Lagrangian control property. We obtain that $c_{k,B}(F \# sH)$ is constant in $s \in [0,1]$ and, hence, $c_{k,B}(F\#H) = c_{k,B}(F)$. The proof is now concluded by the normalization axiom to pass to the mean-zero Hamiltonian $G$ as in (11) instead of $H$.