2. Notation

We use the following notation through this article.

$\mathscr{FD}(M,\omega)$	the Fukaya–Donaldson category of Lagrangian branes (see Definition 3.1)
${\mathbf{FD}}(M,\omega)$	the set of Floer–Fukaya isomorphism classes (see Definition 3.1)
$(T^*N, d\lambda)$ or $T^*N$	the cotangent bundle of a closed manifold N, with Liouville form $\lambda$ (written $p\,dq$ in local coordinates)
$(T^*N, -d\lambda)$ or $\overline{T^*N}$	the cotangent bundle of a closed manifold equipped with the opposite symplectic form
$\mathrm {gr}(df)$	for $f \in C^1(N, \mathbb{R})$ , the graph of df in $T^*N$ , i.e. $\mathrm {gr}(df) = \{(x, df(x)) \mid x \in N\}$
$(M,\omega)$	an aspherical symplectic manifold satisfying $[\omega]\pi_2(M)=0$ and $c_1(TM)\pi_2(M)=0$ , either closed or convex at infinity
$(M, d\lambda)$	an exact symplectic manifold with Liouville form $\lambda$ , convex at infinity
${{\mathfrak L}}(M,\omega)$	the set of compact Lagrangian submanifolds of $(M,\omega)$ (see Definition 4.1)
$\mathscr{L}(M,d\lambda)$	the set of compact Lagrangian branes $\widetilde{L}$ in $(M,d\lambda)$ (see Definition 4.1)
${{\mathfrak L}}_A(M,\omega)$	a component of ${{\mathfrak L}}(M,\omega)$ in a fixed Floer–Fukaya class
$\mathscr{L}_A(M,d\lambda)$	the subset of $\mathscr{L}(M,d\lambda)$ consisting of branes whose underlying Lagrangian lies in ${{\mathfrak L}}_A(M,\omega)$ (see Definition 4.1)
$c_{+}(\widetilde{L}_1, \widetilde{L}_2),\ c_{-}(\widetilde{L}_1, \widetilde{L}_2)$	spectral numbers associated to pairs of elements in $\mathscr{L}_A(M,\omega)$
cthe spectral metric on $\mathscr{L}_A(M,\omega)$ from Definition 4.1(iv)
$\gamma$	the spectral metric on ${{\mathfrak L}}_A(M,\omega)$
$\widehat {{\mathfrak L}}(M,\omega)$ (respectively $\widehat {{\mathfrak L}}_A(M,\omega)$ )	the Humilière completion of ${\mathfrak L}(M,\omega)$ (resp. ${\mathfrak L}_A(M,\omega)$ ) with respect to $\gamma$
$\widehat {\mathscr{L}}(M,d\lambda)$ (respectively $\widehat {\mathscr{L}}_A(M,d\lambda)$ )	the Humilière completion of $\mathscr{L}(M,d\lambda)$ (resp. ${\mathscr{L}}_A(M,d\lambda)$ ) with respect to the metric c from Definition 4.1(iv)
$\mathfrak{DHam}_c(M,\omega)$	the group of time-one maps of compactly supported Hamiltonian flows, endowed with the $\gamma$ -metric
$\widehat {\mathfrak{DHam}}(M,\omega)$	its Humilière completion with respect to $\gamma$
$\mathscr{DHam}_c(M,\omega)$	the group of compactly supported Hamiltonian diffeomorphisms (see p. 1442)
$\widehat {\mathscr{DHam}}(M,\omega)$	its Humilière completion with respect to the metric c from Definition 4.1(iv)
$\mathscr{H}_\gamma(M,\omega)$	the group of homeomorphisms preserving $\gamma$ (see the discussion preceding Corollary 5.25)

3. Spectral invariants for sheaves and Lagrangians

Readers familiar with the content of this and the next section can jump to § 5.

Let $(M,\omega)$ be a symplectic manifold such that $2c_1(M)=0$ . Let $\Lambda(M)$ be the bundle of Lagrangian subspaces of the tangent bundle to M, with fiber the Lagrangian Grassmannian $\Lambda (T_xM)\simeq \Lambda (n)$ , and denote by $\widetilde \Lambda (M)$ the bundle induced by the universal cover $\widetilde \Lambda (n) \longrightarrow \Lambda(n)$ (see [Reference SeidelSei00]). Given a Lagrangian L, we assume we have a lifting of the Gauss map $G_L: L \longrightarrow \Lambda(M)$ given by $x\mapsto T_xL$ to a map $\widetilde G_L: L \longrightarrow \widetilde\Lambda (M)$ . This is called a grading of L (see [Reference SeidelSei08]). When using coefficients in a field of characteristic different from 2, we assume the Lagrangians have a relative spin structure (see [Reference SeidelSei08]).

We assume the characteristic of the fields is 2 in order to avoid discussions of spin structures, but extending our results to the other characteristics is easy.

Given a graded L, the canonical automorphism of the covering induces a new grading and we denote it as T(L) or L[1], and its qth iteration as $T^q(L)$ or L[q]. The grading yields an absolute grading for the Floer homology of a pair $(L_1,L_2)$ and hence for the complex of sheaves in the theorem stated below. We shall almost never mention explicitly the grading, but notice that for exact Lagrangians in $T^*N$ , a grading always exists since the obstruction to its existence is given by the Maslov class, and for exact Lagrangians in $T^*N$ the Maslov class vanishes, as was proved by Kragh and Abouzaid (see [Reference KraghKra13], and also the sheaf-theoretic proof in [Reference GuillermouGui12, Reference GuillermouGui23]). For $(M, d\lambda)$ , we consider the set $\mathscr L (M,d\lambda)$ of Lagrangian branes; that is, triples $\widetilde L=(L,f_L, \widetilde G_L)$ where L is a compact exact graded Lagrangian, and $f_L$ a primitive of $\lambda_{\mid L}$ . We sometimes talk about an exact Lagrangian, and this is just the pair $(L, f_L)$ .

When $f_L$ is implicit, we only write L. For example, $\mathrm {gr} (df)$ means $(\mathrm {gr} (df), f)$ and, in particular, $0_N$ means $(0_N, 0)$ . In both cases, there is an obvious canonical grading. For $\widetilde L=(L, f_L)$ and c a real constant, we write $\widetilde L+c$ for $(L, f_L+c)$ . Considering compact supported Hamiltonian diffeomorphisms as special correspondences (i.e. Lagrangians in $\overline{M}\times M$ ), we can consider the corresponding branes, and denote this space by $\mathscr{DHam}_{c}(M, \omega)$ .

More generally, we have the following definition.

In fact, two exact Lagrangians $L_0,L_1$ are in the same Floer–Fukaya class if the PSS map yields isomorphisms $H^*(L_0) \longrightarrow FH^*(L_0,L_1) \longrightarrow H^*(L_1)$ . In particular, in $T^*N$ , according to [Reference Fukaya, Seidel and SmithFSS08, Reference KraghKra13], there is a single equivalence class. By abuse of language, if L is in $\mathscr L (M,d\lambda)$ and A is an object in $\mathbf{FD}(M,d\lambda)$ , we write $L\in A$ if L is isomorphic to A.

3.1 Sheaf-theoretic approach in $T^{*}N$

Let N be a closed and k-oriented manifold of dimension n. Let $(L,f_L)$ be an exact Lagrangian in $\mathscr L(T^*N)$ and

\begin{align*}\widehat L=\{(q,\tau p, -f_L(q,p),\tau) \mid (q,p)\in L, \tau \gt 0\}\end{align*}

the homogenized Lagrangian in $T^*(N\times {\mathbb R})$ . We denote by $D^b(N)$ the derived category of bounded complexes of sheaves on N. On $D^b(N\times {\mathbb R} )$ we define $\star$ as follows. First, we set $s: N\times {\mathbb R}\times N \times {\mathbb R} \longrightarrow N\times N \times {\mathbb R} $ given by $s(x_1,t_1, x_2,t_2)=(x_1,x_2,t_1+t_2)$ and $d: N\times {\mathbb R} \longrightarrow N\times N \times {\mathbb R} $ given by $d(x,t)=(x,x,t)$ and now

\begin{align*} {\mathscr F}^{\bullet} \star {\mathscr G}^{\bullet} = d^{-1}(Rs)_!({\mathscr F}^{\bullet} \boxtimes {\mathscr G}^{\bullet}).\end{align*}

Then, $R{\mathscr Hom}^\star$ is the adjoint of $\star$ in the sense that

\begin{align*}\mathrm{Mor}_{D^b} ({\mathscr F}^{\bullet}, R{\mathscr Hom}^\star({\mathscr G}^{\bullet}, {\mathscr H}^{\bullet}))=\mathrm{Mor}_{D^b} ({\mathscr F}^{\bullet} \star {\mathscr G}^{\bullet}, {\mathscr H}^{\bullet}).\end{align*}

According to [Reference GuillermouGui12, Reference GuillermouGui23] (for (i), (ii), (iv)) and [Reference ViterboVit19] (for the other properties), we have the following theorem.

Theorem 3.2. To each $\widetilde L\in \mathscr L(T^*N)$ , we can associate ${\mathscr F}^{\bullet}_L \in D^b(N\times \mathbb R)$ such that:

1. (i) $SS({\mathscr F}^{\bullet}_L)\cap \dot{T}^*(N\times {\mathbb R})=\widehat L$ ;

2. (ii) ${\mathscr F}^{\bullet}_L$ is pure (cf. [Reference Kashiwara and SchapiraKS90, p. 309]), ${\mathscr F}_L^\bullet=0$ near $N\times \{-\infty\}$ and ${\mathscr F}_L^\bullet=k_N$ near $N\times \{+\infty\}$ ;

3. (iii) we have an isomorphism

   \begin{align*}FH^\bullet(L_0,L_1;a,b)=H^*(N\times [a,b[, R{\mathscr Hom}^{\star}({\mathscr F}^{\bullet}_{L_0},{\mathscr F}^{\bullet}_{L_1}));\end{align*}

4. (iv) ${\mathscr F}^{\bullet}_L$ is unique satisfying properties (i) and (ii); and

5. (v) there is a natural product map

   \begin{align*} R{\mathscr Hom}^\star({\mathscr F}^{\bullet}_{L_1},{\mathscr F}^{\bullet}_{L_2}) \otimes R{\mathscr Hom}^\star({\mathscr F}^{\bullet}_{L_2},{\mathscr F}^{\bullet}_{L_3}) \longrightarrow R{\mathscr Hom}^\star({\mathscr F}^{\bullet}_{L_1},{\mathscr F}^{\bullet}_{L_3})\end{align*}
   inducing in cohomology a map
   \begin{gather*}H^*(N\times [\lambda , +\infty [, R{\mathscr Hom}^{\star}({\mathscr F}^{\bullet}_{L_1},{\mathscr F}^{\bullet}_{L_2})) \otimes H^*(N\times [\mu , +\infty [, R{\mathscr Hom}^{\star}({\mathscr F}^{\bullet}_{L_2},{\mathscr F}^{\bullet}_{L_3}))\\ \Big\downarrow \cup_{\star} \\ H^*(N\times [\lambda + \mu , +\infty [, R{\mathscr Hom}^{\star}({\mathscr F}^{\bullet}_{L_1},{\mathscr F}^{\bullet}_{L_3}))\end{gather*}
   that coincides through the above identifications to the triangle product in Floer cohomology.

Remark 3.3. The grading $\widetilde G_L$ defines the grading of ${\mathscr F}^{\bullet}_{L}$ , hence of the Floer cohomology.

Note that, for X open, we denote by $H^*(X\times [\lambda, \mu[,{\mathscr F}^{\bullet})$ the relative cohomology of sections on $X\times {]}{-}\infty, \mu[$ vanishing on $X \times {]}{-}\infty, \lambda[$ and fitting in the exact sequence

\begin{align*}H^*(X\times [\lambda, \mu[, {\mathscr F}^{\bullet}) \longrightarrow H^*(X\times {]}{-}\infty, \mu[, {\mathscr F}^{\bullet}) \longrightarrow H^*(X\times {]}{-}\infty, \lambda[, {\mathscr F}^{\bullet}).\end{align*}

It is also equal to the cohomology associated to the derived functor $R\Gamma_Z$ , where Z is the locally closed set $X\times [\lambda, \mu[$ . We should have written $H^*_{X\times [\lambda, \mu[}(X\times {\mathbb R}, {\mathscr F}^{\bullet})$ and ${\mathscr F}^{\bullet}_{\widetilde L}$ instead of ${\mathscr F}^{\bullet}_{L}$ but this would be too cumbersome and our abuse of notation should be harmless.

Definition 3.4. We denote by $\mathscr T_0(N)$ the set of ${\mathscr F}^{\bullet}$ in $D^b(N \times {\mathbb R} )$ such that $SS({\mathscr F}^{\bullet}) \subset \{\tau \geqslant 0\}$ , ${\mathscr F}^{\bullet}=0$ near $N\times \{-\infty\}$ and ${\mathscr F}^{\bullet}=k_N$ near $N\times \{+\infty\}$ .

Note that the set of ${\mathscr F}^{\bullet}$ such that $SS({\mathscr F}^{\bullet}) \subset \{\tau \geqslant 0\}$ contains the Tamarkin category (see [Reference TamarkinTam08, Reference GuillermouGui23]).

Definition 3.5 (See [Reference VicheryVic12, § 8.3]). Let ${\mathscr F}^{\bullet}$ be an element in $\mathscr T_0(N)$ . Let $\alpha \in H^*(N \times {\mathbb R}, {\mathscr F}^{\bullet})\simeq H^*(N)$ be a non-zero class. We define

\begin{align*}c(\alpha; {\mathscr F}^{\bullet})= \sup \{ t \in {\mathbb R} \mid \alpha \in \mathrm{Im} ( H^*(N \times [t, +\infty[, {\mathscr F}^{\bullet}))\} . \end{align*}

We denote $c(\alpha; {\mathscr F}^{\bullet}, {\mathscr G}^{\bullet})=c(\alpha; R{\mathscr Hom}^{\star}({\mathscr F}^{\bullet},{\mathscr G}^{\bullet}))$ and

\begin{align*}c({\mathscr F}^{\bullet}, {\mathscr G}^{\bullet})=\max\{c(\mu_N;{\mathscr F}^{\bullet}, {\mathscr G}^{\bullet}),0\}-\min\{c(1_N; {\mathscr F}^{\bullet}, {\mathscr G}^{\bullet}),0\},\end{align*}

where $1_N, \mu_N$ are the generators of $H^0(N,k)$ and $H^n(N,k)$ , respectively. Finally, we set

\begin{align*}\gamma({\mathscr F}^{\bullet}, {\mathscr G}^{\bullet})=c(\mu_N;{\mathscr F}^{\bullet}, {\mathscr G}^{\bullet})-c(1_N; {\mathscr F}^{\bullet}, {\mathscr G}^{\bullet})= \inf _{c \in \mathbb R} c({\mathscr F}^{\bullet}, T_c{\mathscr G}^{\bullet}). \end{align*}

Note that ${\mathscr F}^{\bullet}_L$ satisfies (ii), so $H^*(N \times {\mathbb R}, {\mathscr F}^{\bullet})\simeq H^*(N)$ and thus we have the following proposition, using the canonical map

\begin{align*}H^*(N \times [t, +\infty[, {\mathscr F}^{\bullet}) \longrightarrow H^*(N \times {\mathbb R}, {\mathscr F}^{\bullet})\end{align*}

and Theorem 3.2(iii).

Proposition 3.6. For ${\mathscr F}^{\bullet}_L$ given by Theorem 3.2, $c(\alpha, {\mathscr F}^{\bullet}_L)$ coincides with the spectral invariant $c(\alpha,L)$ associated to $\alpha$ by using Floer cohomology (see [Reference SchwarzSch00, Reference OhOh05]).

As a consequence, the $c(\alpha, {\mathscr F}^{\bullet})$ satisfy the properties of the Floer homology Lagrangian spectral invariants, and in particular the triangle inequality, since this holds in Floer homology (see [Reference Humilière, Leclercq and SeyfaddiniHLS16a, Theorem 17]). However, we shall sometimes need to extend the triangle inequality to situations where ${\mathscr F}^{\bullet}$ is in $\mathscr T_0(X)$ but does not necessarily correspond to an exact embedded Lagrangian.

Proposition 3.7 (Triangle inequality for sheaves (see [Reference VicheryVic12, Proposition 8.13])). Let ${\mathscr F}^{\bullet}_1, {\mathscr F}^{\bullet}_2, {\mathscr F}^{\bullet}_3$ be sheaves on $X\times {\mathbb R} $ such that ${\mathscr F}^{\bullet}_j \in \mathscr T_0(X)$ . Then, we have

\begin{align*} c(\alpha \cup_\star \beta; {\mathscr F}^{\bullet}_1,{\mathscr F}^{\bullet}_3) \geqslant c(\alpha ; {\mathscr F}^{\bullet}_1,{\mathscr F}^{\bullet}_2) + c( \beta; {\mathscr F}^{\bullet}_2,{\mathscr F}^{\bullet}_3).\end{align*}

Proof. We set ${\mathscr F}^{\bullet}_{i,j}= R{\mathscr Hom}^{\star}({\mathscr F}^{\bullet}_{L_i},{\mathscr F}^{\bullet}_{L_j})$ and we have a product

\begin{align*}\cup_{\star} :{\mathscr F}^{\bullet}_{1,2} \otimes {\mathscr F}^{\bullet}_{2,3} \longrightarrow {\mathscr F}^{\bullet}_{1,3}\end{align*}

inducing the cup-product

\begin{align*}H^*( X\times [s,+\infty[ ; {\mathscr F}^{\bullet}_{1,2}) \otimes H^*( X\times [t,+\infty[ ; {\mathscr F}^{\bullet}_{2,3}) \longrightarrow H^*( X\times [s+t,+\infty[ ; {\mathscr F}^{\bullet}_{1,3})\end{align*}

(see [Reference ViterboVit19, § 9]). Then, we have the following diagram, where the horizontal arrows are cup-products and the vertical arrows are restriction maps.

So, if $\alpha\otimes \beta$ is in the image of the left-hand side, which is equivalent to $s\leqslant c(\alpha, {\mathscr F}^{\bullet}_{1}, {\mathscr F}^{\bullet}_2), t \leqslant c(\beta, {\mathscr F}^{\bullet}_{2},{\mathscr F}^{\bullet}_3)$ , we have $\alpha\cup \beta$ in the image of the right-hand side, so that $ s+t\leqslant c(\alpha\cup \beta , {\mathscr F}^{\bullet}_{1,3})$ .

Similarly, we have the following proposition.

Proposition 3.8 (Lusternik–Shnirelman for sheaves). Let ${\mathscr F}^{\bullet}$ as above. Let $\alpha \in H^*(N\times {\mathbb R}, {\mathscr F}^{\bullet}_{})$ and $\beta \in H^*(N\times {\mathbb R}, {\mathscr G}^{\bullet})$ . We have a product $\alpha \cup_{\star} \beta \in H^*(N\times {\mathbb R}, {\mathscr F}^{\bullet}_{}\star {\mathscr G}^{\bullet})$ . Then,

\begin{align*}c(\alpha \cup_{\star} \beta, {\mathscr F}^{\bullet}\star {\mathscr G}^{\bullet}) \geqslant c(\alpha, {\mathscr F}^{\bullet})\end{align*}

and equality implies that $\beta \neq 0$ in $H^*(\pi(SS({\mathscr F}^{\bullet}) \cap \{t=c\}), {\mathscr G}^{\bullet})$ , where $c=c(\alpha, {\mathscr F}^{\bullet})=c(\alpha \cup_{\star} \beta, {\mathscr F}^{\bullet}\star {\mathscr G}^{\bullet})$ is the common critical value and $\pi: T^*(N\times \mathbb R) \longrightarrow N\times \mathbb R$ the standard projection. In particular, if $c(1; {\mathscr F}^{\bullet})=c(\mu_N;{\mathscr F}^{\bullet})=c$ and ${\mathscr F}^{\bullet}$ is constructible, then $SS({\mathscr F}^{\bullet})\supset 0_N\times T^*_{\{c\}} {\mathbb R}\cap \{\tau \gt 0\}$ .

Proof. By assumption, $\alpha$ vanishes in $H^*(N\times {]}{-}\infty, c- \varepsilon [, {\mathscr F}^{\bullet})$ but not in $H^*(N\times {]}{-}\infty, c+ \varepsilon [, {\mathscr F}^{\bullet})$ . Assume $\beta$ vanishes in $(N\setminus U)\times {]}c- \varepsilon , c+ \varepsilon [$ . Then, $\alpha\cup \beta$ vanishes in $H^*(N\times {]}{-}\infty, c- \varepsilon [ \cup (N\setminus U)\times {]}c- \varepsilon , c+ \varepsilon [, {\mathscr F}^{\bullet}\star {\mathscr G}^{\bullet})$ and by assumption does not vanish in

\begin{align*}H^*(N\times {]}{-}\infty, c+ \varepsilon [ , {\mathscr F}^{\bullet}\star {\mathscr G}^{\bullet}).\end{align*}

But deforming $N\times {]}{-}\infty, c- \varepsilon {[} \cup (N\setminus U) \times {]}{-}\infty, c+ \varepsilon [ $ to $N\times {]}{-}\infty, c+ \varepsilon [$ can be done through a family of hypersurfaces bounding $W_t$ such that

\begin{align*}W_0= N\times {]}{-}\infty, c- \varepsilon {[} \cup (N\setminus U) \times {]}{-}\infty, c+ \varepsilon [\end{align*}

while $W_1= N\times {]}{-}\infty, c+ \varepsilon [$ , and we assume

\begin{align*}SS({\mathscr F}^{\bullet}\star {\mathscr G}^{\bullet})\cap \overline{\Big\{(x,p) \mid x \in \bigcap_{s\gt t} W_{s}\setminus W_{t}\Big\}} \subset 0_{N\times {\mathbb R} } .\end{align*}

According to the microlocal deformation lemma [Reference Kashiwara and SchapiraKS90, Proposition 2.7.2, p. 117 and Corollary 5.4.19, p. 239], this implies that the natural map $H^*(W_1, {\mathscr F}^{\bullet}\star {\mathscr G}^{\bullet}) \longrightarrow H^*(W_0, {\mathscr F}^{\bullet}\star {\mathscr G}^{\bullet})$ is an isomorphism. In our case, this implies that $\alpha \cup_{\star}\beta$ is zero in

\begin{align*}H^*(N\times {]}{-}\infty, c+ \varepsilon [; {\mathscr F}^{\bullet}\star{\mathscr G}^{\bullet}),\end{align*}

hence $c(\alpha\cup \beta; {\mathscr F}^{\bullet}\star {\mathscr G}^{\bullet}) \geqslant c+ \varepsilon $ , a contradiction.

Obviously, when ${\mathscr F}^{\bullet}={\mathscr F}^{\bullet}_{L}$ , the equality $c(1; {\mathscr F}^{\bullet}_{L})=c(\mu_N;{\mathscr F}^{\bullet}_{L})$ implies $\widehat L \supset 0_{N}\times T_{c}^{*}{\mathbb R}$ , hence $L=0_{N}$ .

Corollary 3.9. As a result, $\gamma$ defines a pseudo-metric on $\mathscr T_0(N)$ . It restricts to a metric on the image of $\mathscr L(T^*N)$ by the embedding $L \mapsto {\mathscr F}^{\bullet}_{L}$ , which yields a bi-Lipschitz embedding.

Proof. We have

\begin{align*}c(1;{\mathscr F}^{\bullet}_1,{\mathscr F}^{\bullet}_3) \geqslant c(1;{\mathscr F}^{\bullet}_1,{\mathscr F}^{\bullet}_2)+ c(1;{\mathscr F}^{\bullet}_2,{\mathscr F}^{\bullet}_3)\end{align*}

and

\begin{align*}0=c(\mu; {\mathscr F}^{\bullet}_1,{\mathscr F}^{\bullet}_1)&\geqslant c(1;{\mathscr F}^{\bullet}_1,{\mathscr F}^{\bullet}_2)+ c(\mu;{\mathscr F}^{\bullet}_2,{\mathscr F}^{\bullet}_1)\\c(\mu; {\mathscr F}^{\bullet}_1,{\mathscr F}^{\bullet}_2) &\geqslant c(1;{\mathscr F}^{\bullet}_1,{\mathscr F}^{\bullet}_2) + c(\mu; {\mathscr F}^{\bullet}_2,{\mathscr F}^{\bullet}_2)= c(1;{\mathscr F}^{\bullet}_1,{\mathscr F}^{\bullet}_2)\end{align*}

so that $\gamma({\mathscr F}^{\bullet}_1,{\mathscr F}^{\bullet}_2)\geqslant 0$ .

Remark 3.10. We refer to [Reference Guillermou and ViterboGV24, Reference Asano and IkeAI20] for different definitions of the metric $\gamma$ on $D^{b}(N\times {\mathbb R})$ . As pointed out in [Reference Guillermou and ViterboGV24], the map $Q: L\mapsto\mathscr F_{L}^\bullet$ extends to an isometry $\widehat Q: \widehat {\mathscr L}(T^{*}N) \longrightarrow D^{b}(N\times {\mathbb R})$ . In [Reference Asano, Guillermou, Humilière, Ike and ViterboAGH⁺23] it is proved that $\gamma\text{-}\mathrm{supp}(\widetilde L)=RS(\widehat Q (L))$ , where, for a homogeneous set X in $T^{*}(N\times {\mathbb R})$ , $RS ({X})=X\cap \{\tau=1\}/(t)$ ( $(t,\tau)$ are the canonical coordinates in $T^*{\mathbb R}$ ).

Let $\mu_N\in H^n(N)$ be the fundamental class of N and $1_N\in H^0(N)$ the degree-0 class.

Definition 3.11. We set, for ${\mathscr F}^{\bullet}$ in $\mathscr T_0(N)$ ,

\begin{align*}c_+({\mathscr F}^{\bullet})&=c(\mu_N; {\mathscr F}^{\bullet}),\\ c_-({\mathscr F}^{\bullet})&=c(1_N; {\mathscr F}^{\bullet}),\\ \gamma ({\mathscr F}^{\bullet})&=c_+({\mathscr F}^{\bullet})-c_-({\mathscr F}^{\bullet}).\end{align*}

We set $\mathbb D {\mathscr F}^{\bullet}$ to be the Verdier dual of ${\mathscr F}^{\bullet}$ and $s(x,t)=(x,-t)$ and $\check{{\mathscr F}^{\bullet}}$ is quasi-isomorphic to $0 \to k_{N\times {\mathbb R} } \to s^{-1}(\mathbb D {\mathscr F}^{\bullet})\to 0 $ .

We notice that $SS(\mathbb D {\mathscr F}^{\bullet})=-SS({\mathscr F}^{\bullet})$ where, for $A \subset T^*(N\times {\mathbb R})$ , we set $-A=\{(x,-p, t, -\tau) \mid (x,p,t,\tau)\in A\}$ (see [Reference Kashiwara and SchapiraKS90, Exercise V.13, p. 247]). As a result, $\check{{\mathscr F}^{\bullet}}_L={\mathscr F}^{\bullet}_{-L}$ , where $-L=\{(q,-p) \mid (q,p)\in L\}$ . We then have $\gamma ({\mathscr F}^{\bullet})=\gamma(k_{[0,+\infty[}, {\mathscr F}^{\bullet})$ . The triangle inequality then implies the following.

Proposition 3.12. We have, for ${\mathscr F}^{\bullet}$ constructible in $\mathscr T_0(N)$ :

1. (i) $c_+({\mathscr F}^{\bullet}) \geqslant c_-({\mathscr F}^{\bullet})$ ; and

2. (ii) $c_+({\mathscr F}^{\bullet})=-c_-(\check{{\mathscr F}^{\bullet}})$ $c_-({\mathscr F}^{\bullet})=-c_+(\check{{\mathscr F}^{\bullet}})$ , so that ${\mathscr F}^{\bullet} \longrightarrow \check{{\mathscr F}^{\bullet}}$ is a $\gamma$ -isometry.

And, of course, if $L\in \mathscr L(T^*N)$ we have $c_\pm({\mathscr F}^{\bullet}_L)=c_\pm(L)$ and $\gamma({\mathscr F}^{\bullet}_L)=\gamma(L)$ , where $c_+(L)=c(\mu_N;L), c_-(L)=c(1_N;L), \gamma(L)=c_+(L)-c_-(L)$ .

Proof. Note that, for ${\mathscr F}^{\bullet}$ cohomologically constructible, we have $\check{\check{{\mathscr F}^{\bullet}}}={\mathscr F}^{\bullet}$ .

Of course, if $L\in \mathfrak L (T^*N)$ , the $c_\pm(L)$ are not well-defined (they are only defined up to a constant). However, $\gamma(L)$ is well-defined.

3.2 Persistence modules and barcodes as sheaves on the real line

By a persistence module we mean a constructible sheaf on $( {\mathbb R}, \leqslant)$ , the real line with the topology for open sets $]t,+\infty[$ . We refer to [Reference BarannikovBar94, Reference Chazal, Cohen-Steiner, Glisse, Guibas and OudotCCG⁺09, Reference Edelsbrunner, Letscher and ZomorodianELZ02, Reference Kashiwara and SchapiraKS18, Reference Zomorodian and CarlssonZC05] for the theory and applications. This persistence module is uniquely defined by the spaces $V_t= \mathscr F ({]}{-}\infty, t[)$ and the linear maps $r_{s,t}: V_s \longrightarrow V_t$ defined for $s\leqslant t$ such that:

1. (i) for $s\lt t\lt u$ we have $r_{t,u} \circ r_{s,t}=r_{s,u}$ ;

2. (ii) $\lim_{t\gt s} V_s=V_t$ where the limit is that of the directed system given by the $r_{s,t}$ ;

3. (iii) $r_{t,t}=\mathrm{Id}$ ; and

4. (iv) the spaces $V_t$ are finite-dimensional and the set of s such that $\lim_{t\lt s} V_t\neq V_s$ is finite.

Obviously, $k_{[a,b[}$ is such a persistence module. As a consequence of Gabriel’s theorem on quivers (see [Reference GabrielGab72]), we have the following proposition.

Proposition 3.13 [Reference Crawley-BoeveyCra15]. Any persistence module is isomorphic to a unique sum

\begin{align*}\bigoplus_j k_{[a_j, b_j[}\end{align*}

where $a_j \in {\mathbb R} \cup \{-\infty\}, b_j \in {\mathbb R} \cup \{+\infty\}$ and $a_j\lt b_j$ .

It will be useful to remind the reader of the following lemma.

Lemma 3.14 (See [Reference Kashiwara and SchapiraKS18, 1.10]).

\begin{align*}\mathrm{Mor} (k_{[a,b[},k_{[c,d[})=\bigg\{\begin{array}{l@{\quad}l}k& \text{for } a\leqslant c \lt b\leqslant d, \\[3pt] 0 & \text{otherwise}. \end{array} \end{align*}

There is a graded version, when we consider $D^b( ({\mathbb R}, \leqslant))$ . We denote by $k_{[a,b[}[n]$ the element in $D^b( ({\mathbb R}, \leqslant))$ given by the complex $0 \longrightarrow k_{[a,b[}[n] \longrightarrow 0$ concentrated in degree n. We set $D_{c}^{b}(N)$ to be the category of constructible sheaves on N

Proposition 3.15 (See [Reference GuillermouGui23, Reference Kashiwara and SchapiraKS18]). Any element ${\mathscr F}^{\bullet}$ in $D_{c}^b( ({\mathbb R},\leqslant))$ is isomorphic to a unique sum

\begin{align*}\bigoplus_j k_{[a_j, b_j[}[n_j],\end{align*}

where $a_j \in {\mathbb R} \cup \{-\infty\}, b_j \in {\mathbb R} \cup \{+\infty\}$ , $a_j\lt b_j$ and $n_j\in \mathbb Z$ . If ${\mathscr F}^{\bullet}$ vanishes at $-\infty$ and is equal to $k_X^d$ at $+\infty$ , then all the $a_j$ are different from $-\infty$ and the number of j such that $b_j=+\infty$ is exactly d.

We refer to [Reference Guillermou and ViterboGV24] for extensions to the non-constructible case. It is quite clear that elements of $D_{c}^b( ({\mathbb R}, \leqslant))$ are obtained by considering graded persistence modules $V_t^d$ for d in a finite range (in $\mathbb Z$ ) and with the maps $r_{s,t}$ we have maps $\delta_t : V_t^d \longrightarrow V_t^{d+1}$ so that:

1. (i) $\delta_t^2=0$ ; and

2. (ii) $\delta_s \circ r_{s,t}=r_{s,t}\circ \delta_t$ .

Remark 3.16. One has to be careful. It is not true that $\mathrm{Mor}_{D^b(( {\mathbb R}, \leqslant))}(k_{[a, b[}[m], k_{[c, d[}[n])=0$ for $m\neq n$ .

In particular recall that, given two elements $L_0,L_1 \in \mathscr L (M,\omega)$ , the Floer complex of $L_0,L_1$ is generated by the intersection points of $L_0\cap L_1$ . This complex is filtered by $f_{L_0,L_1}(x)=f_{L_1}(x)- f_{L_0}(x)$ for $x\in L_0\cap L_1$ . Since the coboundary map is given by counting holomorphic strips, it increases the filtration, and we can define $V_t=FC^*(L_0,L_1; t)$ , the subspace of the Floer complex generated by the elements with filtration greater than t. There is a natural map for $s\lt t$ from $V_s=FC^*(L_0,L_1; s)$ to $V_t=FC^*(L_0,L_1; t)$ that defines a persistence module. This is the persistence module associated with $V_t=FC^*(L_0,L_1; t)$ and yields an isomorphism between this persistence module and some $\bigoplus_j k_{[a_j, b_j[}[n_j]$ .

Since $V_t$ vanishes for t small enough, the $a_j$ must all be finite. Since, up to a shift in grading, we have by Floer’s theorem $FH^k(L_0,L_1;t)=H^k(L_0)=H^j(L_1)$ for t large enough, the number of j such that $b_j=+\infty$ and $n_j=k$ is given by $\dim H^k(L_0)=\dim H^k(L_1)$ . Finally, if ${\mathscr F}^{\bullet}_j$ is associated to $L_j$ by Theorem 3.2, we have that the above persistence module is given by the sheaf $(Rt)_*(R{\mathscr Hom}^\star({\mathscr F}^{\bullet}_1, {\mathscr F}^{\bullet}_2))$ . We thus get the following proposition.

Proposition 3.17. Let us set (according to Proposition 3.15) ${\mathscr F}^{\bullet}_j={\mathscr F}^{\bullet}_{L_j}$ and

\begin{align*}(Rt)_*(R{\mathscr Hom}^\star({\mathscr F}^{\bullet}_1, {\mathscr F}^{\bullet}_2))=\bigoplus_j k_{[a_j, b_j[}[n_j] . \end{align*}

Then, there is a unique $j_-$ such that $b_{j_-}=+\infty$ and $n_{j_-}$ is minimal and then $a_{j_-}=c_-(L_0,L_1)$ and a unique $j_+$ such that $b_{j_+}=+\infty$ and $n_{j_+}$ is maximal and then $a_{j_+}=c_+(L_0,L_1)$ . Moreover, $n_{j_+}-n_{j_-}=n=\dim (L_0)=\dim (L_1)$ . In particular, if $\Omega $ is a connected open set with smooth boundary, then we have

\begin{align*}H^{k}(\Omega \times {\mathbb R} , R{\mathscr Hom}^\star({\mathscr F}^{\bullet}_1, {\mathscr F}^{\bullet}_2))= H^{k-n_{j-}}(\Omega)\end{align*}

and

\begin{align*}&H^{k}_c(\Omega \times {\mathbb R}; R{\mathscr Hom}^\star({\mathscr F}^{\bullet}_1, {\mathscr F}^{\bullet}_2))\\&\quad = H^{k}_c(\Omega \times {\mathbb R}, \partial \Omega \times {\mathbb R}; R{\mathscr Hom}^\star({\mathscr F}^{\bullet}_1, {\mathscr F}^{\bullet}_2))= H^{k-n_{j-}}(\Omega). \end{align*}

Remark 3.18. The actual values of $n_j^+,n_j^-$ depend on the grading of $L_1,L_2$ : shifting the grading shifts the values by the same quantity. Note that ${\mathscr F}^{\bullet}=R{\mathscr Hom}^\star({\mathscr F}^{\bullet}_1, {\mathscr F}^{\bullet}_2)$ satisfies $SS({\mathscr F}^{\bullet})\subset \{\tau \geqslant 0\}$ and ${\mathscr F}^{\bullet}$ equals $k_X[n_{j^-}]$ over $X \times \{+\infty\}$ (we shorten this by writing ${\mathscr F}^{\bullet}=k_X[n_{j^-}]$ at $+\infty$ ). Note that we sometimes assume ${\mathscr F}^{\bullet}=k_X$ , which is equivalent to normalizing $n_{j^-}=0$ .

Proof. Only the last statement needs a proof. But by assumption

\begin{align*}SS(R{\mathscr Hom}^\star({\mathscr F}^{\bullet}_1, {\mathscr F}^{\bullet}_2)) \subset \{ \tau \geqslant 0\}\end{align*}

so

\begin{align*}H^{k}(\Omega \times {\mathbb R} , R{\mathscr Hom}^\star({\mathscr F}^{\bullet}_1, {\mathscr F}^{\bullet}_2))= H^{k}(\Omega \times \{+\infty\} , R{\mathscr Hom}^\star({\mathscr F}^{\bullet}_1, {\mathscr F}^{\bullet}_2))= H^{k-n_{j_-}}(\Omega) .\end{align*}

4. Various spaces, metrics, and completions

We remind the reader that, according to [Reference Fukaya, Seidel and SmithFSS08, Reference KraghKra13], $\mathbf {FD}(T^*N,d\lambda)$ has a single object. We shall use the following.

1. (i) Let $\mathscr L_A(M,d\lambda)$ be the set of triples $\widetilde L =(L,f_L, \widetilde G_{L})$ of graded exact closed Lagrangian branes $L\in A$ , where $f_L$ is a primitive of $\lambda_{\mid L}$ (i.e. $df_L=\lambda_L$ ).

2. (ii) Denote by $T_c$ the action of ${\mathbb R}$ on $\mathscr L_A(M, d\lambda)$ given by $(L, f_L,\widetilde G_{L}) \longrightarrow (L, f_L+c,\widetilde G_{L})$ . The action of the generator of the deck transformation $\widetilde \Lambda (M, d\lambda) \longrightarrow \widetilde \Lambda (M, d\lambda)$ is denoted by $\widetilde L \longrightarrow \widetilde L[1]$ .

3. (iii) For $\widetilde L_1,\widetilde L_2\in \mathscr L(T^*N)$ and $\alpha \in H^*(N)$ , denote the spectral invariants obtained using Floer cohomology (see [Reference Humilière, Leclercq and SeyfaddiniHLS16a]) by $c(\alpha, \widetilde L_1, \widetilde L_2)$ . They are usually shortened in $c(\alpha; L_1,L_2)$ if $f_{L_1}, f_{L_2}$ are implicit.

4. (iv) On $\mathscr L(M,d\lambda)$ , define $c_+(\bullet, \bullet) =c(\mu;\bullet, \bullet)$ , $c_-(\bullet, \bullet)=c(1;\bullet, \bullet)$ , and $\gamma (\bullet, \bullet)=c_+(\bullet,\bullet)-c_-(\bullet, \bullet)$ . The metric c is given by

   \begin{align*}c(\widetilde L_1, \widetilde L_2)= \max\{ c_+(\widetilde L_1, \widetilde L_2), 0 \} - \min \{c_-(\widetilde L_1, \widetilde L_2),0\} . \end{align*}

5. (v) Denote by ${\mathfrak L} (M,d \lambda)$ the set of exact closed Lagrangians. There is a forgetful functor $\mathrm{unf}: \mathscr L(M,d\lambda) \longrightarrow {\mathfrak L} (M,d\lambda)$ obtained by forgetting $f_L$ and the grading. The quantity $\gamma$ descends to a metric on ${\mathfrak L} (M, d\lambda)$ , still denoted $\gamma$ .

6. (vi) Denote by $\gamma\text{-}{\rm lim}_k (L_k)$ and $c\text{-}{\rm lim}_k (\widetilde L_k)$ the $\gamma$ -limits and c-limits of the sequences $(L_k)_{k\geqslant 1}$ and $(\widetilde L_k)_{k\geqslant 1}$ .

As mentioned, Proposition 3.6 implies that, for complexes of sheaves of the form ${\mathscr F}^{\bullet}_L$ , the spectral invariants defined by Definition 3.5 coincide with those defined using Floer cohomology. It will be useful to define some modified metrics and completions.

In the following, we fix a class $A \in \mathbf{FD}(M,d\lambda)$ and write ${\mathfrak L} (M,d \lambda)$ and $\mathscr L(M,d\lambda)$ instead of $\mathfrak L_A(M,d\lambda)$ and $\mathscr L_A(M,d\lambda)$ .

We may now state our proposition.

Proof. Proofs of the first three statements are left to the reader. For the fourth, we just notice that we may change $f_{L_1}, f_{L_2}$ to $f_{L_1}+c_1, f_{L_2}+c_2$ for any two constants $c_1,c_2$ . Then, $c_\pm(\widetilde L_1, \widetilde L_2)$ is changed to $c_\pm(\widetilde L_1, \widetilde L_2)+c_1-c_2$ and $c(\widetilde L_1, \widetilde L_2)$ is changed to

\begin{align*}\max\{c_+(\widetilde L_1, \widetilde L_2)+c_1-c_2,0\} - \min \{ c_-(\widetilde L_1, \widetilde L_2)+c_1-c_2, 0\} \end{align*}

and this is minimal when $c_-(\widetilde L_1, \widetilde L_2)\lt c_2-c_1\lt c_+(\widetilde L_1, \widetilde L_2)$ and takes the value

\begin{align*}c_+(\widetilde L_1, \widetilde L_2)-c_-(\widetilde L_1, \widetilde L_2)=\gamma (L_1,L_2).\end{align*}

The last statement is equivalent to $c(\alpha, \varphi_H^t(\widetilde L_1), \widetilde L_2)$ is increasing for $H\geqslant 0$ . But this follows from the formula (see [Reference ViterboVit92, Proposition 4.6 and Lemma 4.7]) for almost all $t\in [0,1]$ , we have

\begin{align*} \frac{d}{dt}c(\alpha, \varphi_H^t(\widetilde L_1), \widetilde L_2)=H(t,z)\end{align*}

for some point z of $\varphi_H^t(L_1)\cap L_2$ .

We may now set the following definition.

The relationship between the two completions $\widehat {\mathscr L}(M, d\lambda)$ and $\widehat {\mathfrak L} (M, d\lambda)$ is clarified by the following.

(2) Let $(L_j)$ be a Cauchy sequence in ${\mathfrak L}(M, d\lambda)$ . We may assume that $\gamma (L_j,L_{j+1})\lt 2^{-j}$ . There is a lift $\widetilde L_j$ of $L_j$ , well determined up to a constant, and we can recursively adjust $\widetilde L_{j+1}$ so that $c(L_j,L_{j+1})\lt 2^{-j}$ . Indeed, assume the $\widetilde L_j$ are defined for $1\leqslant j \leqslant k$ . Since, if $\widetilde L'_{k+1}$ is in the preimage of $L_{k+1}$ we have $\gamma (L_k, L_{k+1})= \inf_{c\in {\mathbb R} } c(\widetilde L_k, \widetilde L'_{k+1}+c)$ , we just choose $\widetilde L_{k+1}=\widetilde L'_{k+1}+c_{k+1}$ , so that $c(\widetilde L_k, \widetilde L_{k+1})\lt 2^{-k}$ . As a result, $(\widetilde L_j)_{j\in \mathbb N}$ is a Cauchy sequence so defines an element in $\widehat {\mathscr L} (T^*N)$ . Moreover, its limit $\widetilde L$ projects on L.

(3) Set $L_j = \widehat {\mathrm{unf}}(\widetilde L_j)$ with $L_2=L_1$ . Let $(\widetilde L_1^k)_{k\geqslant 1}, (\widetilde L_2^k)_{k\geqslant 1}$ be Cauchy sequences in $\mathscr L (T^*N)$ , converging to $\widetilde L_1, \widetilde L_2$ , so that the $L_j^k$ converge to $L_j$ . We showed that $\gamma (L_1^k,L_2^k)= \inf_{c\in {\mathbb R} } c(\widetilde L_1^k, \widetilde L_2^k+c)$ and since, by assumption, $\gamma(L_1^k,L_2^k)$ goes to zero as k goes to infinity, we have a sequence $c_k$ such that $\lim_k c(\widetilde L_1^k, \widetilde L_2^k+c_k)=0$ . But $c_\pm(\widetilde L_1^k, \widetilde L_2^k+c_k)=c_\pm(\widetilde L_1^k, \widetilde L_2^k)+c_k$ , and since $\lim_k c_\pm(\widetilde L_1^k, \widetilde L_2^k)=c_\pm(\widetilde L_1,\widetilde L_2)$ , we must have $\lim_k c_k=-c_\pm(\widetilde L_1,\widetilde L_2)$ . As a result,

\begin{align*}\widetilde L_1=\widetilde L_2-c(\widetilde L_1,\widetilde L_2).\end{align*}

Remark 4.7. Notice that, to a pair $\widetilde L_1,\widetilde L_2$ in $\widehat {\mathscr L}(M,\omega)$ , we may associate a Floer homology as a filtered vector space. Indeed, by the Kislev–Shelukhin inequality (see [Reference Kislev and ShelukhinKS22], or the Appendix of [Reference ViterboVit22]), the bottleneck distance between the persistence module, denoted $V(L_1^k,L_2^k)$ , associated to $FH^*(L^k_1, L^k_2 ; a,b)$ satisfies

\begin{align*}\beta (V(L_1^k,L_2^k), V(L_1^l,L_2^l) \leqslant \gamma (L_1^k, L_1^l)+\gamma (L_2^k, L_2^l),\end{align*}

where $\beta$ is the bottleneck distance (depending on the exact definition of $\beta$ there could be an extra factor 2 on the right-hand side which is, however, irrelevant for our purposes), so we get a Cauchy sequence of persistence modules, and this has a limit as well (however, if we started with persistence modules with finite barcodes, the limit will only have finitely many bars of size $\gt \varepsilon \gt 0$ , but possibly an infinite number in total).

5. Defining the $\gamma$ -support

An element L in $\widehat {{\mathfrak L}}(M, \omega)$ is not a subset of M (and an element in $\widehat {\mathfrak{DHam}} (M, \omega)$ does not define a map), but we may define its $\gamma$ -support as follows.

We can, however, get rid of the second definition.

Proposition 5.3. We have, for $\widehat {\mathrm{unf}}(\widetilde L)=L$ ,

\begin{align*}c\text{-}\mathrm{supp}(\widetilde L)=\gamma\text{-}\mathrm{supp}(L).\end{align*}

Lemma 5.4. Let U be an open set containing the support of the isotopy $(\varphi^t)_{t\in [0,1]}$ and $\psi$ be a Hamiltonian map displacing L outside of U (i.e. $\psi(L)\cap U=\emptyset$ ):

\begin{align*}c(\mu; \varphi^1(\widetilde L), \widetilde L)\leqslant c(\mu, \psi(\widetilde L),\widetilde L)-c(1,\psi(\widetilde L),\widetilde L).\end{align*}

Proof. Since $\psi\varphi^t(L)\cap L=\psi (\varphi^t(L) \cap \psi^{-1}(L))$ does not depend on t, $c(\mu; \psi\varphi^t(\widetilde L),\widetilde L)$ is constant. The triangle inequality implies

\begin{align*}c(\mu;\psi(\widetilde L), \widetilde L)&=c(\mu;\psi\varphi^1(\widetilde L), \widetilde L)= c(\mu;\varphi^1(\widetilde L),\psi^{-1}\widetilde L) \\ &\geqslant c(\mu; \varphi^1(\widetilde L),\widetilde L) +c(1; \widetilde L, \psi^{-1}\widetilde L),\end{align*}

hence

\begin{align*}c(\mu; \varphi^1(\widetilde L),\widetilde L) \leqslant c(\mu;\psi(\widetilde L), \widetilde L)-c(1; \psi(\widetilde L),\widetilde L).\end{align*}

We may now conclude the proof of Proposition 5.3. Assume, by contradiction, that for some $\widetilde L$ the c-support and $\gamma$ -support were different. We would then have a point z and a Hamiltonian H supported in a small ball centered at z such that $c(\varphi_H(\widetilde L),\widetilde L)\gt 0$ while $L=\varphi_H(L)$ . According to Proposition 4.6(iii), this implies that $\varphi_H(\widetilde L)=\widetilde L+c$ . This is impossible for H supported in the ball U, since we would have $\varphi_H^k(\widetilde L) =\widetilde L +k\cdot c$ . Possibly replacing H by $-H$ , we may assume $c\gt 0$ . Then, according to Lemma 5.4, we would have

\begin{align*}k\cdot c=c(\mu; \varphi_H^k(\widetilde L),\widetilde L)\leqslant c(\mu;\varphi_H^k(\widetilde L), \widetilde L) \leqslant c(\mu;\psi(\widetilde L), \widetilde L)-c(1; \psi(\widetilde L), \widetilde L) \lt +\infty, \end{align*}

a contradiction.

We now define compact supported Lagrangians and Hamiltonians in the completion.

Clearly, by Proposition 5.19, a sequence $\widetilde L_{k}$ such that the $\gamma\text{-}\mathrm{supp}(L_{k})$ are contained in a fixed bounded set and $\gamma$ -converge to $\widetilde L_{\infty}$ , $\widetilde L_{\infty}$ is in $\widehat {\mathscr L}_{c}(M, d\lambda)$ . However, the converse is not clear.

We shall make repeated use of the fragmentation lemma.

This implies the following lemma.

Proof. (i) Indeed, if this was not the case, for each $x \in \mathrm{supp} (\varphi)$ there would be an open set $U_x$ such that, for all isotopies $\psi^t$ supported in $U_x$ , we have $\gamma (\psi^1(L),L)=0$ . But by a compactness argument, we may find finitely many $x_j$ and $U_j=U_{x_j}$ ( $1\leqslant j \leqslant k$ ) such that $\mathrm{supp}(\varphi)$ is covered by the $U_j$ . Then, $\varphi^1$ is a product of $\psi_j$ supported in $U_{k(j)}$ but since $\gamma (\psi_j(L),L)=0$ , we get by induction

\begin{align*}\gamma (\psi_1\circ\psi_2\circ\cdots \circ \psi_{j-1}\circ \psi_j(L), L)=0 \end{align*}

and finally $\gamma (\varphi_H(L), L)=0$ , a contradiction.

(ii) The second statement is analogous, since $\varphi_H$ is the product of $\varphi_{H_j}$ with $H_j$ supported in $U_{k(j)}$ . If we had $c(\varphi_{H_j}(\widetilde L), \widetilde L)=0$ for all j, then we would have $\varphi_H(\widetilde L)=\widetilde L$ , a contradiction. Note that here we do not assume that the support of H is small, so we may well have $ \widetilde{\mathrm{supp}} (H) \neq \mathrm{supp} (\varphi_H) $ .

The proof of Proposition 5.11 will make use of the following lemma.

Proof. The conclusion is obvious if L is in $\mathscr L (T^*N)$ , by definition of the action of $\mathfrak{DHam}_c(T^*N)$ on $\mathscr L (T^*N)$ . It is also obvious if $a=0$ , since then H is supported in the complement of $c\text{-}\mathrm{supp}(L)$ (hence $\varphi_H(L)=L$ by Lemma 5.10). Now, fix k such that $c(L,L_k) \lt \varepsilon $ and assume $H_1=a$ on a neighbourhood of $L_k\cup c\text{-}\mathrm{supp}(L)$ . Then,

\begin{align*}c(T_aL_k,\varphi_{H_1}(L))=c(\varphi_{H_1}(L_k),\varphi_{H_1}(L))=c(L_k,L)\lt \varepsilon. \end{align*}

On the other hand, $\varphi_{H_1}^{-1}\varphi_H$ is generated by the Hamiltonian $K(t,z)=H(t,\varphi_H^t(z))-H_1(t,\varphi_H^t(z))$ which vanishes on $c\text{-}\mathrm{supp} (L)$ , so $\varphi_{H_1}^{-1}\varphi_H(L)=\varphi_K(L)=L$ , hence $\varphi_H(L)=\varphi_{H_1}(L)$ . As a result, we have

\begin{align*}c(T_aL_k,\varphi_{H}(L))=c(T_aL_k,\varphi_{H_1}(L)) \lt \varepsilon. \end{align*}

Taking the limit for k we get $c(T_aL,\varphi_{H}(L)=0$ , so $\varphi_{H}(L)=T_aL$ .

Proof of Proposition 5.11. Let $\widetilde L_1, \widetilde L_2$ be elements in $\widehat {\mathscr L}(M,d\lambda)$ having image by $\widehat {\mathrm{unf}}$ equal to $L_1,L_2$ . Assume their $\gamma$ -supports are disjoint and the $\gamma$ -support of $L_2$ is compact. Let H be a compact supported Hamiltonian, and assume H has support in the complement of $\gamma\text{-}\mathrm{supp}(L_1)$ and equals $a\lt 0$ in a neighbourhood of $\gamma\text{-}\mathrm{supp}(L_2)$ . Then,

\begin{align*} c_+ (\varphi_H^1(\widetilde L_1),\widetilde L_1)&\geqslant c_+(\varphi_H^1(\widetilde L_1), \widetilde L_2) -c_+(\widetilde L_1, \widetilde L_2) \\&= c_+(\widetilde L_1,\varphi_H^{-1}(\widetilde L_2)) - c_+(\widetilde L_1, \widetilde L_2)=-a,\end{align*}

since $\varphi_H^{-1}(\widetilde L_2)=\widetilde {L_2}-a$ by the previous lemma. As a result, $\mathrm{supp} (H)$ intersects $\gamma\text{-}\mathrm{supp} (L_1)$ and this implies that $\gamma\text{-}\mathrm{supp}(L_1)\cap \gamma\text{-}\mathrm{supp} (L_2)\neq \emptyset$ .

It is easy to prove that, for $L \in {\mathfrak L}_c(T^*N)$ , we have that $\gamma\text{-}\mathrm{supp}(L)$ intersects any vertical fiber $T_x^*N$ . Indeed, if this was not the case, we could find a small ball $B(x, \varepsilon )$ such that $T^*(B(x, \varepsilon )) \cap \gamma\text{-}\mathrm{supp}(L) =\emptyset$ . Now, if f is a smooth function such that all critical points of f are in $B(x, \varepsilon )$ , then for any bounded set W contained in the complement of $T^*(B(x, \varepsilon ))$ , we have $\mathrm {gr} (t df) \cap W=\emptyset $ for t large enough, so $\mathrm {gr}(tdf) \cap L=\emptyset$ . But this contradicts our first statement.

One of the goals of this article is to partially answer this question.

In [Reference HumilièreHum08b, § 2.3.1], a different notion of support was presented.

We first prove that our definition of the support yields an a priori smaller set than Humilière’s. We actually prove something slightly more general.

Definition 5.18. Let $X_{k}$ be a sequence of subsets in the topological space Z. We define its topological upper limit as

\begin{align*} \limsup_{j} X_{j}&=\bigcap_n \overline{\bigcup_{j\geqslant n}X_{j}} \\ &=\Big\{x\in Z \mid \exists (x_j)_{j\geqslant 1}, \;x_j\in X_j\; \text{for infinitely many}\; j, \; \lim_j x_j=x{\kern-1pt}\Big\}\end{align*}

and its topological lower limit as

\begin{align*}\liminf_j X_j=\Big\{x\in Z \mid \exists(x_j)_{j\geqslant 1}, x_j\in X_j, \lim_j x_j=x{\kern-1pt}\Big\}.\end{align*}

Note that, if $\liminf_j X_j=\limsup_j X_j$ , then this is the topological limit, which for a compact metric space coincides with the Hausdorff limit (see [Reference KechrisKec95, pp. 25–26]), but we shall not need this here. Also, $x\in \liminf_jX_j$ if and only if $\lim_j d(x,X_j)=0$ . Note that this limit could, in principle, be empty, even if the $X_j$ are non-empty. An easy result is now the following.

Proposition 5.19. Let $(L_k)_{k\geqslant 1}$ be a sequence in $\widehat {{\mathfrak L}}(M, \omega)$ of Lagrangians such that $\gamma\text{-}{\rm lim}_k (L_k)=L$ . Assume $\gamma\text{-}\mathrm{supp} (L_k)\cap U=\emptyset$ for infinitely many k. Then,

\begin{align*}\gamma\text{-}\mathrm{supp}(L)\cap U=\emptyset.\end{align*}

In other words,

\begin{align*} \gamma\text{-}\mathrm{supp}(L) \subset \liminf_k (\gamma\text{-}\mathrm{supp} (L_k)).\end{align*}

Now, if $x\notin \liminf_k (\gamma\text{-}\mathrm{supp} (L_k))$ , there must be a subsequence such that $d(x, \liminf_k (\gamma\text{-}\mathrm{supp} (L_k)))\gt \varepsilon _0\gt 0$ . But then, on this subsequence, $B(x, \varepsilon_0/2)\cap \gamma\text{-}\mathrm{supp}(L_k)=\emptyset$ and applying the previous result, we have $B(x, \varepsilon_0/2)\cap \gamma\text{-}\mathrm{supp}(L)=\emptyset$ , so $x \notin \gamma\text{-}\mathrm{supp}(L)$ .

For the case where L and the $L_{k}$ are smooth, this remark was previously made by Seyfaddini and the author (see [Reference VicheryVic13]) and is also a consequence of Lemma 7 in [Reference Humilière, Leclercq and SeyfaddiniHLS15]. From Proposition 5.19 we immediately conclude the following.

Corollary 5.20. We have, for $L \in \widehat {{\mathfrak L}}(M, \omega)$ , the inclusion

\begin{align*} \gamma\text{-}\mathrm{supp}(L) \subset H\text{-}\mathrm{supp} (L).\end{align*}

Our definition of support has one advantage compared to Humilière’s definition: if $W\cap H\text{-}\mathrm{supp}(L)=\emptyset$ , we do not know whether there is a sequence $(L_k)_{k\geqslant 1}$ $\gamma$ -converging to L and such that $L_k\cap W= \emptyset$ , we only know that W can be covered by sets $W_j$ such that, for each j, there is a sequence $(L_k^j)_{k\geqslant 1}$ such that $\gamma\text{-}{\rm lim}_k L_k^j=L$ and $L_k^j\cap W_j=\emptyset$ . On the other hand, if $W \cap \gamma\text{-}\mathrm{supp}(L)=\emptyset$ , we know that for all $\varphi$ supported in W we have $\gamma (\varphi(L),L)=0$ . This still leaves open the following.

Note that the Hamiltonians we need to consider to determine whether a point is in the support can be restricted to a rather small family. Let $\chi(r)$ be a non-negative smooth function equal to 1 on $[-1/2, 1/2]$ and supported in $[-1,1]$ . We set $H^\chi_{z_0, \varepsilon }(z)=\rho (d(z,z_0)/\varepsilon$ ) and $\varphi_{z_0, \varepsilon }^t$ to be the flow associated to $H_{z_0, \varepsilon }^\chi$ . Since, for any $\varphi$ supported in $B(z_0, \varepsilon/2 )$ , we can find a positive constant c so that $\varphi \preceq \varphi_{z_0, \varepsilon }^c$ (i.e. $c_-(\varphi_{z_0, \varepsilon }^c\circ \varphi^{-1})=0$ ), we have, for a lift $\widetilde L$ of L, $c_+(\varphi(\widetilde L),\widetilde L) \leqslant c_+(\varphi_{z_0, \varepsilon }^c(\widetilde L), \widetilde L)$ , so $0\lt c_+(\varphi(\widetilde L),\widetilde L)$ implies $0 \lt c_+(\varphi_{z_0, \varepsilon }^c(\widetilde L), \widetilde L)$ . Note that, if $c_-(\varphi(\widetilde L),\widetilde L)\lt 0$ , we have $0\lt c_+(\widetilde L, \varphi(\widetilde L))= c_+(\varphi^{-1}(\widetilde L),\widetilde L)$ , so applying the same argument to $\varphi^{-1}$ we almost get the following.

Proposition 5.22 (Criterion for the $\gamma$ -support). A point z is in the $\gamma$ -support of $\widetilde L \in \widehat {\mathscr {L}} (M, \omega)$ if and only if we have

\begin{align*} \forall \varepsilon \gt 0,\; \; c_+(\varphi_{z_0, \varepsilon }(\widetilde L),\widetilde L) )\gt 0,\end{align*}

where $\varphi_{z_0, \varepsilon }$ is the time-one map associated to $H_{z_0, \varepsilon }^\chi$ .

Proof. The only point missing is to show that we can take $c=1$ . But let n be an integer such that $c\lt n$ . Assume we had $c_+(\varphi_{z_0, \varepsilon }(\widetilde L), \widetilde L)=0$ . Then, since $\varphi_{ z_0, \varepsilon }^{n-c}\succeq \mathrm{Id}$ , we have

\begin{align*} c_+(\varphi_{z_0, \varepsilon }^c(\widetilde L), \widetilde L) \leqslant c_+(\varphi_{z_0, \varepsilon }^n(\widetilde L), \widetilde L) \leqslant \sum_{j=1}^n c_+(\varphi_{ z_0, \varepsilon }^j(\widetilde L), \varphi_{ z_0, \varepsilon }^{j-1}(\widetilde L)) =n c_+(\varphi_{z_0, \varepsilon }(\widetilde L), \widetilde L)=0.\end{align*}

This contradicts our assumption that $ c_+(\varphi_{z_0, \varepsilon }^c(\widetilde L), \widetilde L)\gt 0$ .

Let $A\subset M_1\times \overline{M_2}, B\subset M_2\times \overline{M_3}$ . We set $A\circ B$ to be the projection of $( A\times B) \cap (M_1\times \Delta_{M_2}\times M_3)$ on $M_1\times M_3$ . We then have the following.

For (ii), this follows from Proposition 5.11.

For (iii), we have $\gamma (\varphi \psi(L), \psi(L))\gt 0 \Longleftrightarrow \gamma (\psi^{-1}\varphi\psi (L), L ) \gt 0$ and since $\mathrm{supp} (\psi^{-1}\varphi\psi)=\psi^{-1}( \mathrm{supp} (\varphi))$ , this proves our statement.

For (iv), we note that if $\gamma (\varphi(L),L)\gt 0$ we either have $c_+(\varphi(L),L)\gt 0$ or $c_-(\varphi(L),L)\gt 0$ . Changing $\varphi$ to $\varphi^{-1}$ , we can always assume $c_+(\varphi(L),L)\gt 0$ . Then, suppose there is a $\psi$ supported in $U_1\times M$ such that $c_+(\psi(L_1\times L_2), L_1\times L_2)\gt 0$ . We can then find $\varphi_1$ supported in $U_1$ such that

\begin{align*}\varphi_1\times \mathrm{Id} \succeq \psi.\end{align*}

Just replace the Hamiltonian generating $\psi$ , $K(z_1,z_2)$ , by $H_1(z_1)$ such that $K(z_1,z_2) \leqslant H(z_1)$ (note that K is compact supported). Then,

\begin{align*}&c_+(\psi(L_1\times L_2), L_1\times L_2) \leqslant c_+((\varphi_1\times \mathrm{Id})(L_1\times L_2), L_1\times L_2)\\&\quad =c_+(\varphi_1(L_1),L_1)+ c_+(L_2,L_2)=c_+(\varphi_1(L_1),L_1)=0,\end{align*}

a contradiction. So, we proved that $U_1\cap \mathrm{supp}(L_1)=\emptyset$ implies $U_1\times M \cap \mathrm{supp} (L_1\times L_2)=\emptyset$ . The same holds by exchanging the two variables, so we get

\begin{align*}\mathrm{supp} (L_1\times L_2) \subset \mathrm{supp} (L_1)\times \mathrm{supp} (L_2).\end{align*}

For (v), let $z\in \gamma\text{-}\mathrm{supp}(L_{K})$ so there exists $\varphi\in \mathfrak{DHam}_{c}(B(z, \varepsilon ))$ such that $\gamma(\varphi(L_K),L_K)\gt 0$ . We want to prove that $z\in (\gamma\text{-}\mathrm{supp}(L))_K$ .

Let $U_{ \varepsilon }$ be the preimage of $B(z, \varepsilon )$ in K and $\tilde\varphi\in \mathfrak{DHam}(M,\omega)$ be a lift of $\varphi$ (i.e. $\widetilde\varphi$ preserves K and projects to $\varphi$ ). If $\varphi$ is the flow of H, we set $\widetilde H$ to be an extension of $H\circ \pi$ (where $\pi: K \longrightarrow K/\mathscr K$ is the projection) and set $\widetilde \varphi$ to be the flow of $\widetilde H$ . Now, the neighbourhood of the leaf $I=\mathscr K_{z}$ can be identified to a neighbourhood of $I\times \{z\}$ in $T^{*}I\times B(z, \varepsilon )$ . Then, by the reduction inequality [Reference ViterboVit21, Proposition 7.60] and the equality $(\widetilde\varphi(L))_{K}=\varphi(L_{K})$ , we have

\begin{align*}\gamma (\widetilde\varphi(L),L) \geqslant \gamma (\varphi(L_{K}),L_{K}).\end{align*}

This inequality holds a priori in the smooth case, but extends to the completion by continuity. Indeed, if $L^{\nu}$ $\gamma$ -converges to L, the sequence $L_{K}^{\nu}$ $\gamma$ -converges to $L_{K}$ and $ \gamma (\varphi(L^{\nu}_{K}),L^{\nu}_{K})\gt \delta_{0}\gt 0$ implies

\begin{align*} \gamma (\widetilde\varphi(L^{\nu}),L^{\nu})\geqslant \gamma (\varphi(L^{\nu}_{K}),L^{\nu}_{K})\gt \delta_{0}\gt 0\end{align*}

(i.e. $\gamma ((\widetilde\varphi(L),L)\gt \delta_{0}\gt 0$ ). Therefore, by Lemma 5.10, this implies $U_\varepsilon\cap \gamma\text{-}\mathrm{supp}(L)=\mathrm{supp}(\widetilde \varphi) \cap \gamma\text{-}\mathrm{supp}(L)\neq \emptyset$ , which implies $\gamma\text{-}\mathrm{supp}(L)_K \cap B(z, \varepsilon )\neq \emptyset$ and finally $z\in \gamma\text{-}\mathrm{supp}(L)_K$ .

For (vi), we notice that $\Lambda_1\circ \Lambda_2$ is the composition of the product $\Lambda_1\times \Lambda_2$ and a reduction of $M_1\times \overline{M_2}\times M_2 \times \overline{M_3}$ by $M_1\times \Delta_{M_2}\times \overline{M_3}$ , where $\Delta_{M_2}$ is the diagonal of $M_2$ .

Thus, it is enough to prove that the $\gamma$ -support of the reduction is contained in the reduction of the $\gamma$ -support, which we just proved.

Let us consider the group $ \mathscr {H}_\gamma (M, \omega)$ of homeomorphisms preserving $\gamma$ . In particular, since $\gamma$ is $C^0$ -continuous (see [Reference Buhovsky, Humilière and SeyfaddiniBHS21]), this group contains the group of homeomorphisms which are $C^0$ -limits of symplectic diffeomorphisms, denoted ${\mathrm{Homeo}} (M, \omega)$ , usually called the group of symplectic homeomorphisms. We have $ {\mathrm{Homeo}}(M, \omega)\subset \widehat {\mathscr{DHam}}(M, \omega)$ .

Corollary 5.25. Let $\psi \in \mathscr {H}_\gamma (M, \omega)$ . Then, we have

\begin{align*}\gamma\text{-}\mathrm{supp} (\psi(L))=\psi( \gamma\text{-}\mathrm{supp}(L)).\end{align*}

Figure 1.

The sequence $(L_k)_{k\geqslant 1}$ and the $\gamma$ -support of the limit $\Lambda$ of the sequence $(L_k)_{k\geqslant 1}$ .

Figure 2.

The sequence $(\overline L_k)_{k\geqslant 1}$ .

Finally note that we can define a kind of density of $L\in {\mathfrak L}(M, \omega)$ as follows.

Definition 5.28. Let $L \in {\mathfrak L}(M, \omega)$ . We define

\begin{align*}\rho_{\varepsilon}(z,L)=\sup \{\gamma(\varphi(L),L) \mid \varphi \in \mathfrak{DHam}_{c} (B(z, \varepsilon))\}\end{align*}

and $\rho(z,L)$ to be the limit

\begin{align*}\limsup_{ \varepsilon \to 0} \frac{1}{-2\log (\varepsilon )}\log \rho_{ \varepsilon }(z,L).\end{align*}

It is easy to show that, for $\psi \in \mathfrak{DHam}_{c} (M, \omega)$ , we have $\rho (\psi(z),\psi(L))=\rho(z,L)$ so that, for a smooth Lagrangian, $\rho(z,L)=1$ , since this is the case for a Lagrangian vector space in $ {\mathbb R}^{2n}$ . We leave the proof of the following to the reader.

6. The $\gamma$ -coisotropic subsets in a symplectic manifold: definitions and first properties

We now define the notion of a $\gamma$ -coisotropic subset in a symplectic manifold. Of course, this notion will coincide with the usual coisotropy in the case of smooth submanifolds. We start with a new definition that will play a central role in this article. After this article was written, we realized that the analogue of $\gamma$ -coisotropic, with $\gamma$ replaced by the Hofer distance (and under the name of ‘locally rigid’), had already been defined in [Reference UsherUsh22]. Using such a notion, the property of, for a submanifold, the equivalence between local rigidity and being coisotropic was stated by Usher; he proved that the Humilière–Leclercq–Seyfaddini theorem [Reference Humilière, Leclercq and SeyfaddiniHLS15] follows from the definition and its invariance by symplectic homeomorphism.

6.1 Basic definitions

Definition 6.1. Let V be a subset of $(M, \omega)$ and $x\in V$ . We shall say that V is non- $\gamma$ -coisotropic at x if, for any ball $B(x, \varepsilon )$ , there exists a ball $B(x,\eta) \subset B(x,\varepsilon )$ and a sequence $(\varphi_k)_{k\geqslant 1}$ of Hamiltonian maps supported in $B(x, \varepsilon )$ such that $\gamma\text{-}{\rm lim} \varphi_k=\mathrm{Id}$ and we have $\varphi_k(V)\cap B(x, \eta )=\emptyset$ .

In other words, a set V is $\gamma$ -coisotropic at $x\in V$ if there exists $ \varepsilon \gt 0$ such that, for any ball $B(x,\eta)$ with $0\lt \eta\lt \varepsilon $ , there is a $\delta (\eta)\gt 0$ such that, for all $\varphi \in \mathfrak{DHam}_{c}(B(z, \varepsilon ))$ such that $\varphi(V) \cap B(x, \eta)=\emptyset$ , we have $\gamma (\varphi)\gt \delta(\eta)$ .

We shall say that V is $\gamma$ -coisotropic if it is non-empty and $\gamma$ -coisotropic at each $x\in V$ . It is nowhere $\gamma$ -coisotropic if each point $x\in V$ is non- $\gamma$ -coisotropic.

Remark 6.2.

1. (i) A set is $\gamma$ -coisotropic at a point if ‘locally displacing it away from itself’ in a neighbourhood of that point cannot be achieved by an arbitrarily small (as measured by $\gamma$ ) Hamiltonian diffeomorphism.

2. (ii) A variation of this definition assumes, in the definition of non- $\gamma$ -coisotropic, that $\varphi(V)=V'$ is fixed. This would be better in some instances, but the locally hereditary property, explained below, becomes non-obvious (if at all true). Note that our definition should make the empty set be $\gamma$ -coisotropic, however, we explicitly excluded this case.

3. (iii) Note that we do not need to define the spectral distance on $(M,\omega)$ to define $\gamma$ -coisotropic subsets. Indeed, we only need to consider elements in $\mathfrak{DHam}_{c}(B(x, \varepsilon))$ for $ \varepsilon $ small enough. But for this, using a Darboux chart, it is enough to have $\gamma$ defined in $\mathfrak{DHam}_{c}( {\mathbb R}^{2n})$ .

Example 6.3.

1. (i) A point in $ {\mathbb R}^{2}$ is non- $\gamma$ -coisotropic. Let H(q,p) be such that ${\partial H}/{\partial q_{1}}(0,p)$ is sufficiently large, but H is $C^{0}$ -small. Then, the flow of H takes the origin outside of $B(0, \eta)$ . By truncation, we can do this while being compact supported in $B(0, \varepsilon )$ .

2. (ii) If $I=[0,1]$ in $ {\mathbb R}^{2}$ it is easy to see that interior points of the interval are $\gamma$ -coisotropic, while the boundary points are not. Thus, I is not $\gamma$ -coisotropic.

3. (iii) It is not hard to see that $\{(0,0)\} \times {\mathbb R}^{2n-2}$ is not $\gamma$ -coisotropic, and neither is $I\times \{0\}\times {\mathbb R}^{2n-2}$ . However, this last set is $\gamma$ -coisotropic at the point $(x,0,\ldots,0)$ if and only if $0\lt x\lt 1$ . More generally, $D^{n+r}(1)\times \{0\}$ is $\gamma$ -coisotropic for $1\leqslant r\leqslant n$ but not for $r=0$ .

4. (iv) A smooth Lagrangian submanifold without boundary is $\gamma$ -coisotropic.

Question 6.4. Let V be a $\gamma$ -coisotropic subset. Does V have positive displacement energy? Even better, is it true that $\gamma(V)=\inf\{\gamma (\psi) \mid \psi(V)\cap V=\emptyset \} \gt 0$ ? (See [Reference BolleBol98, Reference GinzburgGin07, Reference TonnelierTon10] for results in this direction in the smooth case.) Note that the answer is in general negative (see [Reference UsherUsh22]).

One of the goals of this article is to show a number of natural occurrences of $\gamma$ -coisotropic sets, starting from the $\gamma$ -support of elements of $\widehat {{\mathfrak L}}(M,\omega)$ (see Theorem 6.12 and [Reference Guillermou and ViterboGV24]).

We denote by $\mathscr L_\gamma(M, \omega)$ the set of images of smooth Lagrangians by elements of $ \mathscr {H}_\gamma (M, \omega)$ and by $\mathscr S_{2}(M, \omega)$ the set of images of symplectic submanifolds (not necessarily closed) of codimension $\geqslant 2$ by elements of $ \mathscr {H}_\gamma (M, \omega)$ . Note that these are subsets (and even topological submanifolds) of M, which is not the case for elements in $\widehat {{{\mathfrak L}}} (M,\omega)$ .

The following is quite easy.

Proposition 6.5. We have the following properties.

1. (i) Being $\gamma$ -coisotropic is invariant by $ \mathscr {H}_\gamma (M, \omega)$ , hence by ${\mathrm{Homeo}} (M, \omega)$ (and also by conformally symplectic homeomorphisms).

2. (ii) Being $\gamma$ -coisotropic is a local property in M. It only depends on a neighbourhood of V in $(M, \omega)$ .

3. (iii) If X,Y are $\gamma$ -coisotropic, then $X\cup Y$ is $\gamma$ -coisotropic.

4. (iv) Being $\gamma$ -coisotropic implies locally rigid (in the sense of [Reference UsherUsh22]).

5. (v) Being $\gamma$ -coisotropic is locally hereditary in the following sense: if, through every point $x \in V$ , there is a $\gamma$ -coisotropic subset $V_x\subset V$ containing x, then V is $\gamma$ -coisotropic. In particular if, through any point of V, there is a coisotropic or a local Lagrangian submanifold, then V is $\gamma$ -coisotropic. If any point in V has a neighbourhood contained in an element of $\mathscr S_2(M, \omega)$ , then V is non- $\gamma$ -coisotropic.

Proof. The first two statements, as well as (iii), are obvious from the definition. The assertion (iv) follows immediately from the inequality $\gamma \leqslant d_H$ between Hofer distance and spectral norm. For (v), the only non-obvious statement is that a smooth Lagrangian germ is coisotropic at any interior point, while a positive codimension symplectic submanifold is everywhere non- $\gamma$ -coisotropic. For a Lagrangian, if $B(x_0, 1 )$ is a ball and $L\cap B( x_{0},1 )= {\mathbb R}^n \times \{0\}$ and $L'\cap B(x_{0},1)= \emptyset$ , then $\gamma (L, L') \geqslant \pi/2$ . Indeed, it is easy to construct a Hamiltonian isotopy supported in $B(x_{0},1)$ such that $\gamma (L_t,L)= \pi/2$ and $L_t=L$ outside B(1) (see Figure 3).

Figure 3.

L and $L'=\varphi(L)$ with $\varphi$ supported in the interior of the disc. The hatched region has area $ \varepsilon^{2}$ .

As a result,

\begin{align*}\pi/2=c_+ (L_1, L')\leqslant c_+ (L_1, L')+c_+(L',L) \leqslant c_+ (L, L')-c_-(L,L')=\gamma (L,L'). \end{align*}

But this implies $ \gamma (\varphi) \geqslant \gamma (L',L) \geqslant \pi/2$ ; this concludes the proof in the Lagrangian case, since by a conformal map we can always reduce to this case.

Now, let S be symplectic of codimension 2. We want to prove that S is nowhere coisotropic. Up to taking a subset of S and after a symplectic change of coordinates (and possibly a dilation), we may assume that S, U are identified locally to

\begin{align*}S_0=\{(0,0, \bar q, \bar p) \in D^{2}(1) \times D^{2n-2}(1)\} \subset D^{2}(1) \times D^{2n-2}(1)=U,\end{align*}

where $D^{2}(r)$ (respectively $D^{2n-2}(r)$ ) are the symplectic balls of radius r in $({\mathbb R}^2, \sigma_{2})$ (respectively in $({\mathbb R}^{2n-2}, \sigma_{2n-2})$ ). Consider the isotopy

\begin{align*}t \mapsto ( t a(\bar q, \bar p), 0, \bar q, \bar p),\end{align*}

where $a(\bar q, \bar p)= A( \vert \bar q \vert ^2+ \vert \bar p \vert ^2)$ and A is a compactly supported function bounded by 1, and equal to 1 in $D^{2n-2}(1)$ . This is a Hamiltonian isotopy generated by $H( q_1,p_1, \bar q, \overline p)=\chi (\overline q, \overline p)a(\widehat q, \widehat p)$ , where $(\overline q, \overline p)\in {\mathbb R}^{2k}$ and $(\widehat q, \widehat p)\in {\mathbb R}^{2n-2k}$ , sending $S_0$ to $S_1$ , where

\begin{align*}S_1 =\{(1,0, \bar q, \bar p) \in D^{2k}(1) \times D^{2n-2k}(1) \}\end{align*}

and, in particular, $S_0\cap S_1=\emptyset$ . Let

\begin{align*}H(q_1,\bar q, p_1,\bar p)= \chi(q_1,p_1)\cdot a(\bar q, \bar p)\end{align*}

be such that $({\chi(q_1,0)})/({\partial q_1}) =0$ and $({\chi(q_1,0)})/({\partial p_1}) =1$ for $ \vert q_1\vert \leqslant 1$ and such that $ \Vert \chi \Vert_{C^0} \leqslant \varepsilon $ . The flow of H is given by

\begin{align} \left \{ \begin{array}{l} \dot q_1 = \dfrac{\partial \chi}{\partial p_1}(q_1(t), p_1(t)) A (\vert \bar q(t)\vert ^2+ \vert \bar p (t) \vert ^2),\\[12pt] \dot p_1= - \dfrac{\partial \chi}{\partial q_1}(q_1(t), p_1(t)) A (\vert \bar q(t)\vert ^2+ \vert \bar p (t) \vert ^2),\\[12pt] \dot {\bar q} (t) = 2\chi(q_1(t),p_1(t)) A'(\vert \bar q(t)\vert ^2+ \vert \bar p (t) \vert ^2)\bar p (t),\\[6pt] \dot {\bar p}(t)=-2\chi(q_1(t),p_1(t)) A'(\vert \bar q(t)\vert ^2+ \vert \bar p (t) \vert ^2)\bar q (t).\end{array}\right . \end{align}

The last two equations imply that $ \vert \bar q \vert ^2+ \vert \bar p \vert ^2$ is constant, hence $a( \bar q(t), \bar p(t))$ is constant. If we start from $p_1=q_1=0$ , we have $p_1(t)=0$ and $q_1(t)= t A (\vert \bar q(0) \vert ^2+ \vert \bar p (0) \vert ^2)$ . As a result, we have $\varphi_H^1(S_0)=S_1$ . Since $ \Vert H \Vert _{C^0} \leqslant \varepsilon $ , this proves our claim.

A first consequence is the following.

Proposition 6.6. Let V be a smooth submanifold. Then, it is $\gamma$ -coisotropic if and only if it is coisotropic in the usual sense (i.e. for all points $x\in V$ we have $(T_xV)^\omega \subset T_xV$ ).

Proof. Assume C is coisotropic. Locally, C can be identified to

\begin{align*} \{(x_1,\ldots,x_n, p_1,\ldots,p_k,0,\ldots,0) \mid x_j, p_j \in {\mathbb R}.\}\end{align*}

This contains the Lagrangian $p_1=\cdots=p_n=0$ , which we proved to be $\gamma$ -coisotropic, hence C is $\gamma$ -coisotropic.

Conversely, assume V is smooth but not coisotropic. Then locally, we can embed V in a codimension-2 symplectic submanifold as follows: according to the decomposition theorem [Reference ViterboVit21, Theorem 2.14], we can write $T_xV=I\oplus S$ , where I is isotropic, S symplectic and there exists K, uniquely defined by the choice of S, such that $(K\oplus I)$ is symplectic and contained in $S^\omega$ . Moreover, $K\oplus I \oplus S=D(x)$ is symplectic, so choosing a continuously varying S, the same will hold for D. If V is not coisotropic, then $D(x)\neq T_xM$ . We thus have a symplectic distribution D near $x_0$ such that $T_xV \subset D(x)$ . Since being symplectic is an open condition, D(x) will be symplectic in a neighbourhood of $x_0$ . Then, using the exponential map, we may find a symplectic manifold W defined in a neighbourhood of V such that $T_xW=D(x)$ . Since we proved that a codimension-2 symplectic submanifold is non-coisotropic, and being $\gamma$ -coisotropic is locally hereditary, we conclude that V is not $\gamma$ -coisotropic.

6.2 Links with other notions of coisotropy

There are other definitions of coisotropic subsets.Footnote ⁴

Definition 6.7 (Poisson coisotropic). We shall say that the set V is Poisson coisotropic if $\mathscr P_V=\{f \in C^\infty(M) \mid f=0\; \text{on}\; V\}$ is closed for the Poisson bracket. In other words, if f,g vanish on V, so does $\{f,g\}$ .

Finally, following Bouligand (see [Reference BouligandBou32]), for a subset V in a smooth manifold, we define two cones.

Definition 6.8. The paratingent cone of a set V at x is

\begin{align*}C^+(x,V)= \Big\{\! \lim_n c_n(x_n-y_n) \mid x_n, y_n\in V, c_n \in {\mathbb R}\; \lim_nx_n=\lim_n y_n=x, \lim_nc_n=+\infty\!\Big\}.\end{align*}

The contingent cone of a set V at x is

\begin{align*}C^-(x,V)= \Big\{\! \lim_n c_n(x_n-x)\mid x_n \in V, c_n\in {\mathbb R}, \; \lim_nx_n=x, \lim_n c_n=+\infty\!\Big\}.\end{align*}

Clearly, $C^-(x,V) \subset C^+(x,V)$ . Note that $C^+(x,V)$ is invariant by $v \mapsto -v$ , while it is not necessarily the case for $C^-(x,V)$ . We then have the following definition, for which we refer to Kashiwara and Schapira

Definition 6.9 (Cone-coisotropic, see [Reference Kashiwara and SchapiraKS90, Theorem 6.5.1 p. 271]). We shall say that V is cone-coisotropic if, whenever a hyperplane H is such that $C^+(x,V) \subset H$ , then the symplectic orthogonal of H, $H^\omega$ , is contained in $ C^-(x,V)$ .

We refer to Figure 4 for an illustration.

Figure 4.

Two cones $C \subset H$ . The one on the left is coisotropic, the one on the right is not.

Note that what we call here cone-coisotropic is called involutivity in [Reference Kashiwara and SchapiraKS90].

It is an elementary fact that in both cases, a smooth submanifold is Poisson coisotropic or cone coisotropic if and only if it is coisotropic in the usual sense. We proved in [Reference Guillermou and ViterboGV24] the first part of the following proposition.

Proposition 6.10. For a subset V in $(M, \omega)$ , we have the following implications:

\begin{align*} {\gamma-coisotropic} \Longrightarrow \textit{cone-coisotropic} \Longrightarrow {Poisson\,coisotropic}. \end{align*}

Proof of the second implication of the proposition. We argue by contradiction. Assume there are two functions f,g vanishing on V such that $\{f,g\}\neq 0$ . Let $x_0$ be a point such that $\{f,g\}(x_0)\neq 0$ . Consider the set $S=f^{-1}(0)\cap g^{-1}(0)$ near $x_0$ . Then, since $df(x_0), dg(x_0)$ are non-zero and linearly independent, S is a codimension-2 submanifold near $x_0$ . Furthermore, since $X_f, X_g$ are normal to S and have non-zero symplectic product, we see that $(T_{x_0}S)^\omega = \langle X_f(x_0), X_g(x_0) \rangle$ is symplectic, so S is symplectic. We may conclude that V is not cone-coisotropic since, in local coordinates, S is given by $\{q_n=p_n=0\}$ and $T_{x_{0}}S \subset \{q_n=0\}$ . However, ${\partial}/{\partial p_n} \notin T_{x_{0}}S$ and since, in this case, $T_{x_{0}}S=C^+(x_{0},S)=C^-(x_{0},S)$ , this proves that S is not cone-coisotropic.

There are Poisson-coisotropic sets which are not cone-coisotropic. In [Reference Guillermou and ViterboGV24] we give an example of a cone-coisotropic set which is not $\gamma$ -coisotropic.

Example 6.11. Let $V=\{(q,p) \mid p=0, q\geqslant 0\}$ . It is Poisson coisotropic, as this is obvious on $V\cap \{q\gt 0\}$ and the set of points where V is Poisson coisotropic is closed. But V is not cone-coisotropic at (0,0) because $C^+(0,V)= {\mathbb R} ({\partial }/{\partial q})$ while $C^-(0,V)= {\mathbb R}_+ ({\partial }/{\partial q})$ , so the cone condition is violated.

Our main result in this section is the following theorem.

Theorem 6.12. The $\gamma$ -support satisfies the following.

1. (i) For any $L \in \widehat {{\mathfrak L}}(M,\omega)$ , $\gamma\text{-}\mathrm{supp}(L)$ is $\gamma$ -coisotropic.

2. (ii) (Peano Lagrangian) For any r in $\{1,\ldots, n\}$ , we may find $L \in\widehat {{\mathfrak L}}_c(T^*N)$ such that $\gamma\text{-}\mathrm{supp}(L)$ is equal to the $\gamma$ -coisotropic set $0_N\cup K_r$ , where

   \begin{align*}K_r= \{ (q,p) \mid \vert q \vert \leqslant 1, \vert p\vert \leqslant 1, p_{r+1}=\cdots=p_n=0\}.\end{align*}

3. (iii) There exists $L\neq L'$ in $\widehat {\mathfrak L}(T^*N)$ such that $\gamma\text{-}\mathrm{supp}(L)=\gamma\text{-}\mathrm{supp}(L')$ .

Proof of Theorem 6.12(i). Assume $V=\mathrm{supp}(L)$ is non- $\gamma$ -coisotropic at $z\in V$ . Then, there exists a sequence of Hamiltonian maps $\varphi_k$ going to Id (again for $\gamma$ ), supported in $B(z, \varepsilon )$ and such that $\varphi_k(V)\cap B(z,\eta) = \emptyset$ . Now, on the one hand $\gamma\text{-}{\rm lim} \varphi_{k} (L)= L$ since $\gamma\text{-}{\rm lim}_{ k } \varphi_ k = \mathrm{Id}$ . Thus, $\gamma\text{-}{\rm lim}_k \varphi_{k} (L_k)=L$ has support $\mathrm{supp}(L)$ . On the other hand, $\varphi_k(L)$ has support $\varphi_k (\mathrm{supp} (L))$ and since $\varphi_k (\mathrm{supp} (L))\cap B(z,\eta) = \emptyset$ , Proposition 5.19 implies that $\mathrm{supp} (L) \cap B(z,\eta)=\emptyset$ . We thus get a contradiction.

For (ii), we call such Lagrangians ‘Peano Lagrangians’ because their supports, like the Peano curve, exhibit surprising dimensional properties while being associated to a Lagrangian. Let us consider the ‘cube’ $K_r$ as above. Any embedding of $D^n$ in N yields such a cube. Our goal is first to construct $L \in \widehat {{\mathfrak L}}(T^*D^n)$ such that $\mathrm{supp}(L) \supset K_r$ .

Theorem 6.13. There exists an element $L \in {{\mathfrak L}} (T^*N)$ with support satisfying $\gamma\text{-}\mathrm{supp} (L)= K_n\cup 0_N $ .

Proof. Consider $L=\psi (0_N)$ , where $\psi$ is supported in $K=K_n$ . Let A be a finite set in $\mathring K\setminus L$ , z a point in $\mathring K\setminus L$ not contained in A, and let $U \subset K\setminus (L\cup A)$ be the symplectic image of the product of two Lagrangian balls $\sigma : B^n(\varepsilon)\times B^n( \frac{1}{3}) \longrightarrow K\setminus {L\cup A}$ . Notice that $B^n(\varepsilon)\times B^n( \frac{1}{3})$ is symplectically isotopic to $B^{n}(\sqrt {\varepsilon/3})\times B^{n}(\sqrt {\varepsilon/3})$ . We assume $z=\sigma (0)$ and set $\Delta=\sigma (B^n ( \varepsilon )\times \{0\})$ to be the image of the first Lagrangian ball.

Lemma 6.14. Let $L,K,A, U, \Delta$ be as above. Then, there is a constant c such that for all $0\lt \varepsilon \lt c$ , there is a symplectic isotopy $\rho$ such that:

1. (i) $\rho$ is supported in $\mathring K\setminus A$ ;

2. (ii) $\rho(L)\cap U=\Delta$ ;

3. (iii) $\gamma (\rho) \lt \varepsilon $ , hence $\gamma (\rho(L), L) \lt \varepsilon$ ; and

4. (iv) there exists $\varphi$ supported in U such that $\gamma (\varphi(\rho(L)),\rho(L)) \gt {\varepsilon/5}$ .

We refer to Figures 5 and 6 for an illustration.

Figure 5.

Illustration of Lemma 6.14.

Figure 6.

The Lagrangian $\rho(0_N)\cap U$ and the path $\rho_{k}(\Delta)$ . The hatched region has area less than $ \varepsilon_k $ .

Proof. First of all, we may apply an isotopy sending L to the zero section. Then, since A is discrete, we may push all its points by a symplectic isotopy in a neighbourhood of the boundary of K and at the same time move z to $q_1=\cdots=q_n=0, p_1= \frac{1}{2},\ldots,p_n= \frac{1}{2}$ , since $K\setminus 0_N$ is connected (except in dimension 2, in which case we have two connected components, and z is either $(0, \frac{1}{2})$ or $(0, - \frac{1}{2})$ ). As a result, we may assume that z is the center of a translate of $B^n(\varepsilon)\times B^n( \frac{1}{3})$ contained in K and avoiding both A and $0_N$ . We claim that we can move the zero section by a Hamiltonian isotopy $\tau_ \varepsilon $ generated by a Hamiltonian supported in K, with norm $ \Vert H \Vert _{C^{0}} \leqslant \varepsilon $ and such that

\begin{align*}\tau_ \varepsilon (0_N)\cap (B^n(\varepsilon)\times B^n( \tfrac{1}{3}))=z+ (B^n(\varepsilon)\times \{0\}). \end{align*}

Taking $\rho=\tau_ \varepsilon$ , this proves (i), (ii) and (iii). Finally, to prove (iv), it is enough to construct a Hamiltonian isotopy supported in $B^n(\varepsilon)\times B^n( \frac{1}{3})$ , as is done in the proof of Proposition 5.23 (see Figure 3), such that if $L\cap ( B^n(\varepsilon)\times B^n( \frac{1}{3} ))=B^n(\varepsilon)\times \{0\}$ , we have $\gamma (\varphi(L), L) \geqslant \varepsilon/5$ . For this it is enough to replace $B^n( \varepsilon )\times \{0\}$ by the graph of df, where f is supported in $B^n( \varepsilon )$ and $ \vert df \vert \lt \frac{1}{3} $ and $\mathrm{osc} f \gt \varepsilon/5$ . This induces a Hamiltonian isotopy such that $\gamma (\varphi(L),L) \gt \varepsilon/5$ .

Proof of Theorem 6.13. Now, let $(z_j)_{j\geqslant 1}$ be a dense subset in $\mathring K$ . We apply the above proposition inductively with $L=L_k, A_k=\{z_1,\ldots,z_k\}$ , and $\varepsilon = \varepsilon _k$ . We shall determine the sequence $ (\varepsilon _k)_{k\geqslant 1}$ later. We thus get sequences $(\rho_k)_{k\geqslant 1},(\varphi_k)_{k\geqslant 1}$ such that properties (i)–(iv) of Lemma 6.14 hold. In other words, we have:

1. (i) $\rho_k$ is supported in $\mathring K\setminus A_k$ ;

2. (ii) $\gamma (\rho_k(L_k), L_k) \lt \varepsilon_k$ ; and

3. (iii) there exists $\varphi_k$ supported in $U_k$ such that $\gamma (\varphi_k(\rho_k(L_k)),\rho_k(L_k)) \gt {\varepsilon_k}/{5}$ .

We then set $L_{k+1}=\rho_k(L_k)$ . According to property (iv), this sequence will be $\gamma$ -Cauchy if the series $\sum_j \varepsilon _j$ converges. More precisely, $\gamma (L_k, L_\infty) \lt \sum_{j=k}^{+\infty} \varepsilon_j$ . Now, assume the sequence $ (\varepsilon _j)_{j\geqslant 1}$ satisfies $\varepsilon _{j+1} \lt {\varepsilon _j}/{20}$ . Then, denoting by $L_\infty$ the limit of $L_k$ , we have

\begin{align*}\gamma(L_{k+1},L_\infty)\lt \varepsilon_{k+1} \sum_{j=0}^{+\infty} \frac{1}{20^j} = \frac{20}{19} \varepsilon_{k+1} \lt \frac{1}{19} \varepsilon_{k}. \end{align*}

We claim the following lemma.

Lemma 6.15. Consider the sequence $L_k$ just defined and let $L_\infty$ be its $\gamma$ -limit. Then, for all k large enough, we have $\gamma(\varphi_k(L_\infty), L_\infty) \gt \frac{1}{11} \varepsilon _k$ .

Proof. Let us indeed use the triangle inequality to compute

\begin{align}\gamma(\varphi_k(L_{k+1}),L_{k+1}) \leqslant \gamma(\varphi_k(L_{k+1}), \varphi_k(L_\infty)) +\gamma(\varphi_k(L_\infty), L_\infty)+ \gamma( L_\infty, L_{k+1})\end{align}

so that

\begin{align}\gamma(\varphi_k(L_\infty), L_\infty)\geqslant \gamma(\varphi_k(L_{k+1}),L_{k+1}) -\gamma(\varphi_k(L_{k+1}), \varphi_k(L_\infty))-\gamma( L_\infty, L_{k+1}) \end{align}

and since $\varphi_k$ is an isometry for $\gamma$ , we have

\begin{align*}\gamma(\varphi_k(L_\infty), L_\infty)&\geqslant\gamma(\varphi_k(L_{k+1}),L_{k+1}) -2\gamma( L_\infty, L_{k+1}) \\&\geqslant\frac{\varepsilon _k}{5}- \frac{2}{19} \varepsilon_k\gt \frac{9}{95} \varepsilon_k \gt \frac{1}{11}\varepsilon_k.\end{align*}

End of the proof of Theorem 6.13. We now consider the case where the sequence $(z_k)_{k\geqslant 1}$ is just the sequence of points with rational coordinates, and $U_k$ are neighbourhoods of $z_k$ making a basis of open sets for the topology of K. Then, $\mathrm{supp} (L_\infty)$ is a closed set, meeting all the $\mathrm{supp}(\varphi_k)$ , and since $\gamma (\varphi_k(L_\infty), L_\infty) \gt 0$ , it meets all the $U_k$ . This implies that $\mathrm{supp}(L_\infty)$ is dense in K, hence contains K.

End of the proof of Theorem 6.12. We start with (ii). To conclude the proof in the case $1\leqslant r\lt n$ , we first use the following proposition.

Proposition 6.16. Let $L_{1} \in \widehat {\mathfrak L}(M_{1},\omega_{1})$ and $L_{2}\in {\mathfrak L}(M_{2},\omega_{2})$ . Then, we have

\begin{align*}\gamma\text{-}\mathrm{supp} (L_{1}\times L_{2})=\gamma\text{-}\mathrm{supp} (L_{1})\times L_{2}.\end{align*}

Proof. We already proved the inclusion $\gamma\text{-}\mathrm{supp}(L_{1}\times L_{2})\subset \gamma\text{-}\mathrm{supp} (L_{1})\times \gamma\text{-}\mathrm{supp}(L_{2})=\gamma\text{-}\mathrm{supp}(L_{1})\times L_{2}$ . The opposite inclusion follows from the following two claims:

1. (i) $\gamma\text{-}\mathrm{supp} (L_{1}\times L_{2})\cap \{z_{1}\}\times L_{2 } \neq \emptyset$ for all $z_{1}\in \gamma\text{-}\mathrm{supp}(L_{1})$ ; and

2. (ii) $\gamma\text{-}\mathrm{supp} (L_{1}\times L_{2})$ is invariant by $\mathrm{id}\times \varphi$ for any $\varphi \in \mathfrak{DHam} (M_{2},\omega_{2})$ such that $\varphi(L_{2})=L_{2}$ .

Let us first show that these two claims imply the proposition. Indeed, for $(z_{1},z_{2})\in \gamma\text{-}\mathrm{supp}(L_{1})\times L_{2}$ , we have by (i) that there exists $z'_{2}$ such that $(z_{1},z'_{2})\in\gamma\text{-}\mathrm{supp} (L_{1}\times L_{2})$ . Now, by (ii), choosing $\varphi$ such that $\varphi(z'_{2})=z_{2}$ (which is possible since the Hamiltonian maps preserving $L_{2}$ act transitively on $L_{2}$ ), we get that $(z_{1},z_{2})\in \gamma\text{-}\mathrm{supp} (L_{1}\times L_{2})$ . For (i), let $\varphi_{1}$ be supported in $B(z_{1}, \varepsilon )$ such that $\varphi(L_{1},L_{1})\gt 0$ . Then, we have $\gamma (\varphi_{1}\times \mathrm{Id})(L_{1}\times L_{2}), L_{1}\times L_{2})=\gamma(\varphi_{1}(L_{1})\times L_{2},L_{1}\times L_{2})=\gamma(\varphi(L_{1}),L_{1})\gt 0$ , where the last equality follows from corollary 7.53 of [Reference ViterboVit21]. This implies that $\mathrm{supp} (\varphi\times \mathrm{Id})$ intersects $\gamma\text{-}\mathrm{supp}(L_{1}\times L_{2})$ and, letting $ \varepsilon $ go to 0, we get

\begin{align*}\gamma\text{-}\mathrm{supp}(L_{1}\times L_{2})\cap \{z_1\}\times M_2\neq \emptyset.\end{align*}

Since $\gamma\text{-}\mathrm{supp} (L_{1}\times L_{2})\subset \gamma\text{-}\mathrm{supp} (L_{1})\times L_{2}$ , we have

\begin{align*} \gamma\text{-}\mathrm{supp}(L_1\times L_2)\cap \{z_1\}\times M_2\subset \gamma\text{-}\mathrm{supp} (L_{1})\times L_{2}\cap \{z_1\}\times M_2 =\{z_1\}\times L_2.\end{align*}

Therefore, $\gamma\text{-}\mathrm{supp}(L_1\times L_2)\cap \{z_1\}\times L_2\neq \emptyset$ .

Claim (i) then follows from the fact that $\mathrm{id}\times \varphi$ preserves $L_1\times L_2$ , and hence preserves $\gamma\text{-}\mathrm{supp} (L_1\times L_2)$ .

We may now conclude the proof of the case $1\leqslant r\lt n$ . Consider $L \subset T^*N$ , where N has dimension r such that $\gamma\text{-}\mathrm{supp}(L_1)=0_N\cup [-1,1]^{2r}$ . Let M be $(n-r)$ -dimensional and containing an embedding of $[-1,1]^{n-r}$ (this is, of course, always the case). Then, $\gamma \text{-}\mathrm{supp} (L_1\times 0_M)=\gamma\text{-}\mathrm{supp}(L)\times 0_M=(0_L\cup [-1,1]^{2r})\times 0_M \supset 0_{L\times M} \cup K_r$ .

As for (iii), we can construct two sequences $(L_k)_{k\geqslant 1}$ and $(L'_k)_{k\geqslant 1}$ having $\gamma$ -limits L, L’ as above so that $\gamma\text{-}\mathrm{supp}(L)=\gamma\text{-}\mathrm{supp} (L')$ and $L\neq L'$ . Indeed, we may create the first ‘tongue’ in different directions, so that $\gamma (L_1, L'_1) \geqslant \varepsilon_0$ , and if we choose the $ \varepsilon_k$ such that $\varepsilon_0 \gt \sum_{j=1}^{+\infty} \varepsilon _j$ , we get $\gamma (L,L') \gt \varepsilon _0- \sum_{j=1}^{+\infty} \varepsilon_j$ . Thus, the support does not, in general, determine a unique element in $\widehat {\mathfrak L}(M,\omega)$ .

Remark 6.17.

1. (i) It is easy to see that $\mathrm{supp}(L_\infty)\subset 0_N\cup K$ , so that $\mathrm{supp}(L_\infty)=0_N\cup K$ .

2. (ii) Similarly, we can find $L \in \widehat {{\mathfrak L}}(T^*N)$ such that $\gamma\text{-}\mathrm{supp}(L) = 0_N \cup T^*D^n(1)$ .

Question 6.18. Can a coisotropic submanifold ‘containing no Lagrangian’ (e.g. such that the coisotropic foliation has a dense leaf) be the $\gamma$ -support of an element $L \in \widehat {{\mathfrak L}}(T^*N)$ ? One should be careful about the meaning of ‘Lagrangian’ as this should be understood in a weak sense since, for a $C^0$ function f, the graph $\mathrm {gr} (df)$ will usually not contain a smooth Lagrangian.

We consider a sequence of $\gamma$ -coisotropic sets $V_{k}$ . If the $V_{k}$ are compact and contained in a bounded set, then, up to taking a subsequence, they have a Hausdorff limit and setting $V= \lim_{k }V_{k}$ , we may ask whether V is $\gamma$ -coisotropic. The answer is obviously negative: take $V_{k}$ to be a sphere of center 0 and radius $1/k$ . Then, $V_{k}$ has limit $\{0\}$ , which is not coisotropic. However, the following question is more sensible.

Question 6.19. Let $L_{k}$ be a sequence in $\widehat {\mathfrak L}(M, \omega)$ and $V_{k}=\gamma\text{-}\mathrm{supp}(L_{k})$ . Assume $V=\lim_{k} V_{k}$ where the limit is a Hausdorff limit. Does V contain a $\gamma$ -coisotropic set?

Note that Figure 8 gives an example of a sequence such that the limit is not $\gamma$ -coisotropic. Also, the sequence cannot collapse to a set of Hausdorff dimension less than n because of the intersection properties of the $\gamma$ -supports (Proposition 5.11).

Figure 7.

The Lagrangian from Example 6.21.

Figure 8.

A Hausdorff limit (dotted line) of Lagrangians that is not $\gamma$ -coisotropic.

Remark 6.20.

1. (i) For the connection between Hausdorff convergence and $\gamma$ -convergence (for Lagrangians) in the presence of Riemannian constraints, we refer to [Reference ChasséCha21].

2. (ii) Note that, if all $\gamma$ -supports contain an exact Lagrangian, then Proposition 5.11 becomes obvious. Of course, this cannot be true in the usual sense (i.e. all $\gamma$ -supports do not contain a smooth Lagrangian): if f is a $C^{0}$ function, then $\mathrm {gr} (df)$ cannot contain a smooth Lagrangian. On the other hand, if this is not the case, then we get a really new class of subsets, invariant by symplectic isotopy, having intersection properties.

Note that one could hope that if $u_L$ is the graph selector associated to L, we have $\gamma\text{-}\mathrm{supp}(\mathrm {gr}(du_L)) \subset \gamma\text{-}\mathrm{supp}(L)$ . This is, however, not the case, as we see from the following example.

Example 6.21. Let $u(x)= \vert x \vert $ . It is then easy to see, using smooth approximations of u, that $\gamma\text{-}\mathrm{supp}(\mathrm {gr}(du)) $ is the union of $\{(x,-1) \mid x\leqslant 0\}$ , $\{(x,1) \mid x\geqslant 0\}$ and $\{0\}\times [-1,1]$ . However, u is the graph selector for the Lagrangian represented in Figure 7 (see [Reference ChaperonCha91, Reference Ottolenghi and ViterboOV94, Reference Viterbo, Paul Biran and LalondeVit06]).

However, it is not difficult to prove that

\begin{align*}\gamma\text{-}\mathrm{supp}(\mathrm {gr}(du_L)) \subset {\rm Conv}_p(\gamma\text{-}\mathrm{supp}(L)),\end{align*}

where ${\rm Conv}_p$ is the p-convex hull, that is,

\begin{align*}{\rm Conv}_p(X)= \bigg\{(q,p)\in T^*N \mid \exists(p_1,\ldots,p_r)\in T_q^*N, t_j\geqslant 0, \sum t_j=1 , p= \sum_{j=1}^n t_j p_j\bigg\}.\end{align*}

Question 6.22. Does $\gamma\text{-}\mathrm{supp} (du_L)\cap \gamma\text{-}\mathrm{supp}(L)$ contain the extremal points of $ {\rm Conv}_p(\gamma\text{-}\mathrm{supp}(L))$ ?