Some quantitative results in ${\mathcal {C}}^0$ symplectic geometry

Abstract

This paper proceeds with the study of the ${\mathcal {C}}^0$-symplectic geometry of smooth submanifolds, as initiated in Humilière et al. (Duke Math J 164(4), 767–799, 2015) and Opshtein (Ann Sci Éc Norm Supér 42(5), 857–864, 2009), with the main focus on the behaviour of symplectic homeomorphisms with respect to numerical invariants like capacities. Our main result is that a symplectic homeomorphism may preserve and squeeze codimension 4 symplectic submanifolds (${\mathcal {C}}^0$-flexibility), while this is impossible for codimension 2 symplectic submanifolds (${\mathcal {C}}^0$-rigidity). We also discuss ${\mathcal {C}}^0$-invariants of coistropic and Lagrangian submanifolds, proving some rigidity results and formulating some conjectures. We finally formulate an Eliashberg-Gromov ${\mathcal {C}}^0$-rigidity type question for submanifolds, which we solve in many cases. Our main technical tool is a quantitative h-principle result in symplectic geometry.

Appendices

A Main Lemmata for Theorem 2.1

The appendix is dedicated to the proof of the following three lemmata which were used in the proof of Theorem 2.1:

Lemma A.1

Let $\epsilon > 0$ be a positive real, $m \geqslant 6$ be an integer, $W \subset \mathbb {R}^m$ be an open set, $u_1, u_2 : \overline{D} \rightarrow W$ be disjoint smoothly embedded discs, and assume that there exists a (continuous) homotopy between $u_1$ and $u_2$ in W , of size less than $\epsilon$ (i.e. a continuous map $F : \overline{D} \times [0,1] \rightarrow W$ such that $F(z,0) = u_1(z)$, $F(z,1) = u_2(z)$, for all $z \in \overline{D}$, and that $\text {size}\,F < \epsilon$). Then there exists a smooth embedded isotopy $\widetilde{F}$ between $u_1$ and $u_2$ in W , of size less than $2 \epsilon$ (i.e. a smooth embedding $\widetilde{F} : \overline{D} \times [0,1] \rightarrow W$, such that $\widetilde{F}(z,0) = u_1(z)$, $\widetilde{F}(z,1) = u_2(z)$, for all $z \in \overline{D}$, and $\text {size}\,\widetilde{F} < 2 \epsilon$). Moreover, if $m > 6$, then the estimate on the size of the isotopy can be improved to $\text {size}\,\widetilde{F} < \epsilon$.

Proof

First, we can slightly perturb our homotopy F between $u_1$ and $u_2$, to obtain a smooth homotopy between $u_1$ and $u_2$ in W , of size less than $\epsilon$. Hence without loss of generality we may assume that the homotopy F is smooth.

When $m > 6$, F can be further perturbed to a smooth embedding, which is an isotopy between $u_1$ and $u_2$ of size less than $\epsilon$. This shows the case $m > 6$.

Now consider the case $m = 6$. Here we can slightly perturb F to obtain a smooth immersion $F : \overline{D} \times [0,1] \rightarrow W$, having only a finite number of double-points which appear at transverse self-intersections, of size less than $\epsilon$, such that the double-points do not lie on the image of the boundary of $\overline{D} \times [0,1]$. Applying one more $\mathcal {C}^0$-small perturbation to F , we may further assume that for any double-point $x = F(z_1,t_1) = F(z_2,t_2)$ (here $z_1,z_2 \in D$, $t_1,t_2 \in (0,1)$), we have $z_1 \ne z_2$. Fix such a double point. Then there exist smooth coordinates (z, t, y) in a neighbourhood V of $F(\{ z_1 \} \times [0,1])$, where

$$\begin{aligned} z \in D(\delta ) = \{ z \in \mathbb {R}^2 \, | \, | z | < \delta \}, \quad \, t \in (-\delta ,1+\delta ) , \quad y \in D(\delta ) \times (-\delta ,\delta ) \subset \mathbb {R}^3 , \end{aligned}$$

and $\delta < 1 - |z_1|, 1 - |z_2 |$, such that

$$\begin{aligned} V \cap F(\overline{D} \times [0,1])= & {} F((D(\delta ) + z_1) \times [0,1]) \cup F((D(\delta ) + z_2)\\&\times (t_2-\delta ,t_2 + \delta )) , \end{aligned}$$

and such that in these coordinates we have $F(z,t) = (z-z_1,t,0,0,0)$ for $(z,t) \in (D(\delta ) + z_1) \times [0,1]$ and $F(z,t) = (0,0,t_1,z-z_2,t-t_2)$ for $(z,t) \in (D(\delta ) + z_2) \times (t_2-\delta ,t_2+\delta )$. Now pick a smooth function $c : [0, \infty ) \rightarrow [0,1]$ such that $c(t) = 1$ for small t , and such that $c(t) = 0$ for $t \geqslant 1$. Then choose a small enough $\eta > 0$, and modify the immersion F on $(D(\delta ) + z_2) \times (t_2-\delta ,t_2+\delta )$ according to:

$$\begin{aligned} F(z,t) := (0,0,t_1 + (1+\eta - t_1)c(|z-z_2|/\eta )c(|t-t_2|/\eta ),z-z_2,t-t_2), \end{aligned}$$

for $(z,t) \in (D(\delta ) + z_2) \times (t_2-\delta ,t_2+\delta )$. After changing F as described above for each double-point of F , we get a smooth embedded isotopy between $u_1$ and $u_2$, of size less than $2 \epsilon$. $\square$

Lemma A.2

Let $(M,\omega )$ be a connected symplectic manifold, let $r > 0$, $G = D(r) \subset \mathbb {C}$, and let $v_1,v_2 : \overline{G} \rightarrow M$ be smoothly embedded symplectic discs, $v_1^* \omega = v_2^* \omega = \omega _{\text {st}}$. Then there exists a compactly supported Hamiltonian isotopy of M , whose time-1 map $\psi$ satisfies $\psi \circ v_1 = v_2$.

Lemma A.3

Let $W \subset \mathbb {C}^n$ be an open subset, diffeomorphic to a ball, endowed with the standard symplectic structure $\omega _{\text {st}}$.

(a):

The absolute case: Let $\gamma : [0,1] \rightarrow \mathbb {C}$ be a smooth embedded curve, let $z_1 = \gamma (0)$, $z_2 = \gamma (1)$, and let $G \supset \gamma ([0,1])$ be an open neighbourhood, $G \subset \mathbb {C}$. Assume that $v_1,v_2 : G \rightarrow W$ are smooth symplectic embeddings, $v_1^* \omega _{\text {st}} = v_2^* \omega _{\text {st}} = \omega _{\text {st}}$, such that $v_1$ and $v_2$ coincide on a neighbourhood of $\{ z_1,z_2 \} \subset G$. Assume moreover that for a 1-form $\lambda$ which is a primitive of $\omega _\text {st}$, i.e. $d\lambda = \omega _{\text {st}}$, we have $\int _{v_1 \circ \gamma } \lambda = \int _{v_2 \circ \gamma } \lambda$. Then there exists a compactly supported Hamiltonian function $H : W \times [0,1] \rightarrow \mathbb {R}$, such that on a neighbourhood of $\{ v_1(z_1),v_1(z_2) \}$ we have that $H(\cdot ,t) = 0$ for every t , and such that for the time-1 map $\psi$ of the flow of H , we have that $\psi \circ v_1 = v_2$ on a neighbourhood of $\gamma ([0,1])$.

(b):

The proper case: Let now $\gamma :[0,1]\rightarrow \mathbb {C}$ be a smooth embedded curve, G be an open neighbourhood of $\gamma ((0,1))$, and $v_1,v_2:G\rightarrow W$ be two smooth symplectic embeddings which coincide on the intersection of G with a neighbourhood of $\{\gamma (0),\gamma (1)\}$ in $\mathbb {C}$, such that moreover the curve $(0,1) \rightarrow W$, $s \mapsto v_1 \circ \gamma (s)$ is properly embedded. We also assume that the actions of $v_1\circ \gamma$ and $v_2\circ \gamma$ are equal:

$$\begin{aligned}&\int _\delta ^{1-\delta }\lambda (\dot{\overline{v_1\circ \gamma }}(s))ds= \int _\delta ^{1-\delta }\lambda (\dot{\overline{v_2\circ \gamma }}(s))ds, \end{aligned}$$

for all $0 < \delta \ll 1$. Then there exists a compactly supported Hamiltonian $H:W\times [0,1]\rightarrow \mathbb {R}$ whose flow verifies $\psi ^1_H\circ v_1=v_2$ on a neighbourhood of $\gamma ((0,1))$ in $\mathbb {C}$.

Remark A.4

As will be apparent from the proof of this lemma, the hypothesis that W is diffeomorphic to a ball can be replaced by the assumption that there exists a regular relative homotopy between $u_1\circ \gamma$ and $u_2\circ \gamma$ inside W, i.e. a smooth map $F:(0,1)\times [0,1]\rightarrow W$ such that $F(s,0)=u_1\circ \gamma (s)$, $F(s,1)=u_2\circ \gamma (s)$, $F(s,t)=u_1\circ \gamma (s)=u_2\circ \gamma (s)$ for all $t\in [0,1]$ and $s\approx 0,1$, and $s\mapsto F(s,t)$ is embedded for all t. In fact, if $n\geqslant 2$ (in dimension at least 4), this last condition is not a restriction since it can be achieved by perturbing any relative homotopy between $u_1\circ \gamma$ and $u_2\circ \gamma$.

Lemma A.5

Let $n \geqslant 3$, let $W \subset \mathbb {C}^n$ be an open subset, diffeomorphic to a ball, endowed with the standard symplectic structure $\omega _{\text {st}}$, let $r > 0$ and $G = D(r) \subset \mathbb {C}$, and let $v_1,v_2 : \overline{G} \rightarrow W$ be smoothly embedded symplectic discs, $v_1^* \omega _{\text {st}} = v_2^* \omega _{\text {st}} = \omega _{\text {st}}$, which coincide on a neighbourhood of the boundary $\partial G \subset \overline{G}$. Then there exists a compactly supported Hamiltonian function $H : W \times [0,1] \rightarrow \mathbb {R}$, such that on a neighbourhood of $v_1(\partial G)$ we have $H(\cdot ,t) = 0$ for every t , and such that for the time-1 map $\psi$ of the flow of H , we have $\psi \circ v_1 = v_2$ on $\overline{G}$. In other words, $v_1$ can be Hamiltonianly isotoped to $v_2$ inside W , while being kept fixed near the boundary.

Although the Lemmata A.2, A.3, A.5 can be found in [3, 6], we present in this appendix their proofs, for the sake of completeness. The proofs of these lemmata are based on a number of auxiliary lemmata which we describe in Sect. A.1. Let us remark that the proofs in the appendix are not always fully complete, and some of the technical details are left to the reader.

1.1 A.1 Auxiliary lemmata and their proofs

The proofs of Lemmata A.2, A.3 and A.5 use the following lemmata:

Lemma A.6

Let $d \geqslant 4$, let $W \subset \mathbb {R}^d$ be an open subset, diffeomorphic to a ball, and let $\gamma _0, \gamma _1 : [0,1] \rightarrow W$ be smooth embedded curves which coincide on a neighbourhood of the endpoints of [0, 1] . Then there exists a smooth homotopy $F : [0,1] \times [0,1] \rightarrow W$ such that $F(0,t) = \gamma _0(t)$, $F(1,t) = \gamma _1(t)$ for $t \in [0,1]$, such that for some $0< \epsilon < 1/2$ we have $F(s,t) = \gamma _0(t) = \gamma _1(t)$ for every $s \in [0,1]$ and $t \in [0,\epsilon ] \cup [1-\epsilon ,1]$, and such that for every $s \in [0,1]$, the curve $[0,1] \rightarrow W$, $t \mapsto F(s,t)$ is smoothly embedded.

Proof

Without loss of generality we may assume that W is an open ball in $\mathbb {R}^d$. The map $F' : [0,1] \times [0,1] \rightarrow W$ defined by $F'(s,t) = (1-s) \gamma _0(t) + s \gamma _1(t)$, $(s,t) \in [0,1] \times [0,1]$, fulfills all the needed requirements, except maybe for the requirement that for every $s \in [0,1]$, the curve $[0,1] \rightarrow W$, $t \mapsto F'(s,t)$ is smoothly embedded. However, using standard arguments of general position, one can slightly perturb $F'$ and obtain the needed smooth homotopy F .

Lemma A.7

Let $d \geqslant 5$, let $W \subset \mathbb {R}^d$ be an open subset, diffeomorphic to a ball, let $\gamma : S^1 \rightarrow W$ be a smooth closed embedded curve, and let $F : [0,1] \times S^1 \rightarrow W$ be a smooth map such that for every $u \in [0,1]$, the curve $S^1 \rightarrow W$, $t \mapsto F(u,t)$ is smoothly embedded, and such that $F(0,t) = F(1,t) = \gamma (t)$. Then there exists a smooth map $\hat{F} : [0,1] \times [0,1] \times S^1 \rightarrow W$, such that we have $\hat{F}(u,0,t) = \hat{F}(0,s,t) = \hat{F}(1,s,t) = \gamma (t)$, $\hat{F}(u,1,t) = F(u,t)$ for every $u,s \in [0,1]$ and $t \in S^1$, and such that for any $u, s \in [0,1]$, the curve $S^1 \rightarrow W$, $t \mapsto F(u,s,t)$ is smoothly embedded.

Proof

Without loss of generality we may assume that W is an open ball in $\mathbb {R}^d$. The map $\hat{F}' : [0,1] \times [0,1] {\times S^1}\rightarrow W$ defined by $\hat{F}'(u,s,t) = s F(u,t) + (1-s) \gamma (t)$, $(s,t) \in [0,1] \times [0,1]$, fulfills all the needed requirements, except maybe for the requirement that for every $u,s \in [0,1]$, the curve $[0,1] \rightarrow W$, $t \mapsto \hat{F}'(u,s,t)$ is smoothly embedded. However, using standard arguments of general position, one can slightly perturb $\hat{F}'$ and obtain the needed smooth map $\hat{F}$.

Lemma A.8

Let $(M,\omega )$ be a symplectic manifold, and let $F : [0,1] \times [0,1] \rightarrow M$ be a smooth map, such that for every $s \in [0,1]$, the curve $[0,1] \rightarrow M$, $t \mapsto F(s,t)$ is smoothly embedded, such that for some $0< \epsilon < 1/2$, we have $F(s,t) = F(0,t)$ for $(s,t) \in [0,1] \times ( [0,\epsilon ] \cup [1-\epsilon ,1])$, and moreover such that for every $s_0 \in [0,1]$ the symplectic area of the rectangle $[0,s_0] \times [0,1] \rightarrow M$, $(s,t) \mapsto F(s,t)$ is zero. Then there exists a Hamiltonian function $H : M \times [0,1] \rightarrow \mathbb {R}$, $H = H(x,s)$, with a Hamiltonian flow $\psi _{H}^s$, $s \in [0,1]$, such that we have $\psi _{H}^s (F(0,t)) = F(s,t)$, and moreover on some neighbourhood of $\{ F(0,0), F(0,1) \}$ we have $H(\cdot ,s) = 0$ for every $s \in [0,1]$.

Proof

For every $s \in [0,1]$, look at the vector field $X_s(F(s,t)) = \frac{\partial }{\partial s} F(s,t)$, $t \in [0,1]$, along the curve $[0,1] \rightarrow M$, $t \mapsto F(s,t)$. Denote by $\alpha _s(F(s,t)) \in T^*_{F(s,t)} M$ by $\alpha _s|_{F(s,t)}(\cdot ) = \omega (X_s(F(s,t)), \cdot )$. Then $\alpha _s$ is a section of $T^*M$ above the curve $[0,1] \rightarrow W$, $t \mapsto F(s,t)$, we have that $\alpha _s$ vanishes near the endpoints of the curve, and that $\int _{0}^1 \alpha _s(\frac{\partial }{\partial t}F(s,t)) \, dt = 0$. Hence it is easy to see that at least locally, near any given point $s_0 \in [0,1]$, for some neighbourhood $s_0 \in I \subset [0,1]$ of $s_0$ in [0, 1] , we can find a compactly supported function $H_I : W \times \overline{I} \rightarrow \mathbb {R}$, $H_I = H_I(x,s)$, such that $d_x H_I (F(s,t),s) = \alpha _s |_{F(s,t)}$ for every $(s,t) \in \overline{I} \times [0,1]$, and such that $H_I = 0$ on a neighbourhood of $\{ F(0,0), F(0,1) \}$. But then we can produce a “global” Hamiltonian function $H : M \times [0,1] \rightarrow \mathbb {R}$ via a partition of unity. That is, choose a covering of [0, 1] by sufficiently short intervals $\{ I_j \}_{j=1,\ldots ,m}$ which are open in [0, 1] , and compactly supported functions $H_{I_j} : M \times \overline{I} \rightarrow \mathbb {R}$, $H_{I_j} = H_{I_j}(x,s)$, such that $d_{x}H_{I_j} (F(s,t),s) = \alpha _s |_{F(s,t)}$ for every $(s,t) \in \overline{I_j} \times [0,1]$, and such that $H_{I_j} = 0$ on a neighbourhood of $\{ F(0,0), F(0,1) \}$. Then choose a smooth partition of unity $\kappa _j : [0,1] \rightarrow \mathbb {R}$, $j = 1,\ldots ,m$, such that $\kappa _j$ is compactly supported in $I_j$ and such that $\sum _j \kappa _j = 1$ on [0, 1] , and define $H : M \times [0,1] \rightarrow \mathbb {R}$ by $H(x,s) = \sum _j \kappa _j(s) H_{I_j} (x,s)$. Then $d_x H (F(s,t),s) = \alpha _s |_{F(s,t)}$ for every $(s,t) \in [0,1] \times [0,1]$, and $H = 0$ on a neighbourhood of $F([0,1] \times \{ 0,1 \} )$. This means that the Hamiltonian vector field of H at time s , restricted to the curve $[0,1] \rightarrow W$, $t \mapsto F(s,t)$, coincides with $X_s$, and therefore the Hamiltonian flow of H takes the curve $[0,1] \rightarrow W$, $t \mapsto F(0,t)$ exactly through the homotopy F .

Lemma A.9

Let $(M,\omega )$ be a symplectic manifold, and let $\hat{F} : [0,1] \times [0,1] \times S^1 \rightarrow M$ be a smooth map, such that for every $u_0,s_0 \in [0,1]$, the curve $S^1 \rightarrow M$, $t \mapsto \hat{F}(u_0,s_0,t)$ is smoothly embedded, and moreover the symplectic area of the cylinder $[0,s_0] \times S^1 \rightarrow M$, $(s,t) \mapsto \hat{F}(u_0,s,t)$ is zero. Then there exists a smooth family of compactly supported Hamiltonian functions $H_u : M \times [0,1] \rightarrow \mathbb {R}$, $H_u = H_u(x,s)$, $u \in [0,1]$ with Hamiltonian flows $\psi _{H_u}^s$, $s \in [0,1]$, such that we have $\psi _{H_u}^s ( \hat{F}(u,0,t)) = \hat{F}(u,s,t)$, and moreover the following holds: if for some $u_0,s_0 \in [0,1]$ we have $\frac{\partial }{\partial s} \hat{F}(u_0,s_0,t) = 0$ for any $t \in S^1$, then $H_{u_0}(x,s_0) = 0$ for every $x \in M$.

Lemma A.10

Let $(M,\omega )$ be a symplectic manifold, and let $\hat{F} : [0,1] \times [0,1] \times [0,1] \rightarrow M$ be a smooth map, such that for every $u,s \in [0,1]$, the curve $[0,1] \rightarrow M$, $t \mapsto \hat{F}(u,s,t)$ is smoothly embedded. Then there exists a smooth family $\psi _{u,s} \in \text {Ham}(M,\omega )$, $u,s \in [0,1]$ of compactly supported Hamiltonian diffeomorphisms of M , such that we have $\psi _{u,s} (\hat{F}(u,0,t)) = \hat{F}(u,s,t)$.

Lemma A.11

Let $(M,\omega )$ be a symplectic manifold, let $r > 0$ and $G = D(r) \subset \mathbb {C}$, and let $v : [0,1] \times \overline{G} \rightarrow M$, $v(t,z) = v_t(z)$ be a smooth isotopy of symplectic discs, i.e. $v_t^* \omega = \omega _{\text {st}}$ on G for every $t \in [0,1]$. Then there exists a compactly supported smooth Hamiltonian flow $\psi ^t$, $t \in [0,1]$ on M such that $v_t = \psi ^t \circ v_0$ for every $t \in [0,1]$.

Lemma A.12

Let $(M,\omega )$ be a symplectic manifold, let $a > 0$ and let $\Sigma = S^1 \times [0,a]$ be an annulus with the standard symplectic form $\omega _{\text {st}}$. Let $\hat{v} : [0,1] \times [0,1] \times \Sigma \rightarrow M$, $\hat{v}(u,s,z) = \hat{v}_{u,s}(z)$ be a smooth map such that for each $u,s \in [0,1]$, the map $\Sigma \rightarrow M$, $z \mapsto \hat{v}_{u,s}(z)$ is a smooth symplectic embedding, $\hat{v}_{u,s}^* \omega = \omega _{\text {st}}$. Then there exists a smooth family of compactly supported Hamiltonian functions $H_u : M \times [0,1] \rightarrow \mathbb {R}$, $H_u = H_u(x,s)$, $u \in [0,1]$ with Hamiltonian flows $\psi _{H_u}^s$, $s \in [0,1]$, such that we have $\psi _{H_u}^s ( \hat{v}(u,0,z)) = \hat{v}(u,s,z)$, and moreover the following holds: if for some $u_0,s_0 \in [0,1]$ we have $\frac{\partial }{\partial s} \hat{v}(u_0,s_0,z) = 0$ for any $z \in \Sigma$, then $H_{u_0}(x,s_0) = 0$ for every $x \in M$.

The proofs of Lemmata A.9, A.10, A.11 and A.12 are quite similar to the proof of Lemma A.8: first one can find the corresponding Hamiltonian function locally and then use partition of unity to unify these “local” Hamiltonian functions. Therefore we omit the proofs of Lemmata A.9, A.10, A.11 and A.12.

1.2 A.2 Proofs of main lemmata

Let us turn to the proofs of Lemmata A.2, A.3 and A.5.

Proof of Lemma A.2

By the symplectic neighbourhood theorem, on a neighbourhood $U_1$ of $v_1(\overline{G})$ there exist local symplectic coordinates $(x_1,y_1,\ldots ,x_n,y_n) = (z_1,\ldots ,z_n)$ such that in these coordinates we have

$$\begin{aligned}&U_1 = \left\{ (x_1,y_1,\ldots ,x_n,y_n) \; | \; x_1^2 + y_1^2< (r + \epsilon )^2, x_j^2 + y_j^2 < \epsilon ^2 \quad \text {for} \,\, 2 \leqslant j \leqslant n \right\} ,\\&\quad v_1(x_1,y_1) = (x_1,y_1,0,0,\ldots ,0,0) , \end{aligned}$$

and on a neighbourhood $U_2$ of $v_2(\overline{G})$ there exist local symplectic coordinates

$(x_1,y_1,\ldots ,x_n,y_n)$ (we loosely use the same notation for these coordinates as well) such that in these coordinates we have

$$\begin{aligned}&U_2 = \left\{ (x_1,y_1,\ldots ,x_n,y_n) \; | \; x_1^2 + y_1^2< (r + \epsilon )^2, x_j^2 + y_j^2 < \epsilon ^2 \quad \text {for} \,\, 2 \leqslant j \leqslant n \right\} ,\\&\quad v_2(x_1,y_1) = (x_1,y_1,0,0,\ldots ,0,0) , \end{aligned}$$

where $\epsilon > 0$ is small. In the sequel, by $p = (x_1,y_1,\ldots ,x_n,y_n) \in U_1$ (respectively, by $p = (x_1,y_1,\ldots ,x_n,y_n) \in U_2$) we will always mean that p is written in the local coordinates of $U_1$ (respectively, that p is written in the local coordinates of $U_2$). It easily follows from the Moser’s argument, that for some small $0< \delta < \epsilon$, there exists a Hamiltonian diffeomorphism $\psi '$ of M , such that $\psi '(v_1(0,0)) = v_2(0,0)$, and such that for any $p = (x_1,y_1,\ldots ,x_n,y_n) \in U_1$ with $x_j^2+y_j^2 < \delta ^2$, $j=1,\ldots ,n$, we have $\psi '(p) = (x_1,y_1,\ldots ,x_n,y_n) \in U_2$. Now choose a smooth immersion $f : \overline{G} \rightarrow D(\delta )$ with $f^*\omega = \omega _{\text {st}}$, and define two smooth families of discs $v_{1,t}, v_{2,t} : \overline{G} \rightarrow M$, $t \in [0,\pi / 2)$, by $v_{1,t}(z) = (\cos (t) z, \sin (t) f(z),0,\ldots ,0) \in U_1$ and $v_{2,t}(z) = (\cos (t) z, \sin (t) f(z),0,\ldots ,0) \in U_2$. We have $v_{1,t}^* \omega = v_{2,t}^* \omega = \omega _{\text {st}}$, $t \in [0,\pi /2)$. Moreover we have $v_{1,\pi /2 - \delta /r}(\overline{G}) \subset \{ (x_1,y_1,\ldots ,x_n,y_n) \in U_1 \; | \; x_j^2+y_j^2 < \delta ^2 \,\, \text {for} \,\, 1 \leqslant j \leqslant n \}$, $v_{2,\pi /2 - \delta /r}(\overline{G}) \subset \{ (x_1,y_1,\ldots ,x_n,y_n) \in U_1 \; | \; x_j^2+y_j^2 < \delta ^2 \,\, \text {for} \,\, 1 \leqslant j \leqslant n \}$, and hence $\psi ' \circ v_{1,\pi /2 - \delta /r} = v_{2,\pi /2 - \delta /r}$, so there exists a compactly supported Hamiltonian isotopy of M which takes $v_{1,\pi /2-\delta /r}$ to $v_{2,\pi /2-\delta /r}$. By Lemma A.11, there exists a compactly supported Hamiltonian isotopy of M which takes $v_{1,0} = v_1$ to $v_{1,\pi /2 - \delta /r}$, and there exists another compactly supported Hamiltonian isotopy of M which takes $v_{2,0} = v_2$ to $v_{2,\pi /2 - \delta /r}$. So, finally, the discs $v_1$ and $v_2$ are isotopic via a compactly supported Hamiltonian isotopy of M . $\square$

Proof of Lemma A.3

First of all, without loss of generality we may assume that $\gamma (t) = (t,0)$ for $t \in [0,1]$. We first take care of the absolute situation (a). We divide the construction of the Hamiltonian H into two steps. At the first step we find a Hamiltonian isotopy that brings the map $v_1$ to a map $v_1'$ which coincides with $v_2$ on $\gamma ([0,1])$. The second step provides a Hamiltonian isotopy between $v_1$ and a map $v_1''$ which coincides with $v_2$ on a neighbourhood of $\gamma ([0,1])$. We then explain how the proof of (a) readily implies the proper case (b).

(a) Step 1

By Lemma A.6, we can find a smooth homotopy $F : [0,1] \times [0,1] \rightarrow W$ such that $F(0,t) = v_1 \circ \gamma (t)$, $F(1,t) = v_2 \circ \gamma (t)$ for $t \in [0,1]$, such that for some $0< \epsilon < 1/2$ we have $F(s,t) = v_1 \circ \gamma (t) = v_2 \circ \gamma (t)$ for every $s \in [0,1]$ and $t \in [0,\epsilon ] \cup [1-\epsilon ,1]$, and such that for every $s \in [0,1]$, the curve $[0,1] \rightarrow W$, $t \mapsto F(s,t)$ is smoothly embedded. After making a $C^0$ small perturbation of F on $[0,1]\times (\epsilon /3,\epsilon /2)$, we may assume that for all $s \in [0,1]$, the $\lambda$-actions of the curves $[0,1] \rightarrow W$, $t \mapsto F(s,t)$ are equal, and moreover that we still have that $F(0,t) = v_1 \circ \gamma (t)$, $F(1,t) = v_2 \circ \gamma (t)$ for $t \in [0,1]$, and that the curve $[0,1] \rightarrow W$, $t \mapsto F(s,t)$ is smoothly embedded.

Remark A.13

One possible way to do this is as follows. We have $F(s,t) = v_1 \circ \gamma (t) = v_2 \circ \gamma (t)$ for $t \in [0,\epsilon ]$. Since $v_1 \circ \gamma$ is smoothly embedded, one can find local symplectic coordinates $(x_1,y_1, \ldots , x_n,y_n)$ on a neighbourhood U of $v_1\circ \gamma ((0,\epsilon ))$, such that in these coordinates $U = (0,\epsilon ) \times (-\delta , \delta ) \times D(\delta )^{\times n-1}$, and $v_1 \circ \gamma (t) = (t,0,0,\ldots ,0)$ for $t \in (0,\epsilon )$, where $\delta > 0$ is a small positive number. Now fix a non-negative smooth function $\kappa : (0,\epsilon ) \rightarrow \mathbb {R}$, not identically 0 , such that $\text {Supp}\,(\kappa ) \subset (\epsilon /3, \epsilon /2)$, and such that $\Vert \kappa \Vert _{\infty }$ is small enough. Let $\nu : [0,1] \rightarrow \mathbb {R}$ be a smooth function, which will be specified later. Now we define $F' : [0,1] \times [0,1] \rightarrow M$ by $F'(s,t) = F(s,t)$ for $(s,t) \in [0,1] \times (\{0\} \cup [\epsilon ,1])$, and by $F'(s,t) = (t,0,\kappa (t) \cos (\nu (s) t), \kappa (t) \sin (\nu (s) t), 0,0, \ldots ,0,0)$, for $(s,t) \in [0,1] \times (0,\epsilon )$, where the latter equality is written in the chosen local coordinates on U . We have $\omega (F'|_{[0,s] \times [0,1]}) = \omega (F'|_{[0,s] \times [0,\epsilon ]}) + \omega (F'|_{[0,s] \times [\epsilon ,1]}) = \omega (F'|_{[0,s] \times [0,\epsilon ]}) + \omega (F|_{[0,s] \times [0,1]}) = \frac{1}{2} (\nu (0) - \nu (s)) \int _{0}^{\epsilon } \kappa ^2(t) dt + \omega (F|_{[0,s] \times [0,1]})$, where the latter equality follows from a simple computation. Hence if we define the function $\nu$ to be $\nu (s) = 2 \omega (F|_{[0,s] \times [0,1]})/ (\int _0^{\epsilon } \kappa ^2(t) dt )$, for $s \in [0,1]$, we get $\omega ( F'|_{[0,s] \times [0,1]}) = 0$ for every $s \in [0,1]$. Now replace F by $F'$.

Now, by Lemma A.8 one can find a compactly supported Hamiltonian $H' : W \times [0,1] \rightarrow \mathbb {R}$ such that on some neighbourhood of $\{ F(0,0), F(0,1) \}$ we have $H'(\cdot ,s) = 0$ for every $s \in [0,1]$, and such that the time-1 map $\psi '$ of $H'$ satisfies $\psi ' \circ F(0,t) = F(1,t)$, for every $t \in [0,1]$. Hence denoting $v_1' := \psi ' \circ v_1$, we get that $v_1' = v_2$ on $\gamma ([0,1])$ and moreover $v_1' = v_2$ on a neighbourhood of the endpoints $\{ z_1,z_2 \}$.

(a) Step 2

Recall that $\gamma (t) = (t,0)$ for $t \in [0,1]$, and denote by $\hat{\gamma }$ the curve $\hat{\gamma } : [0,1] \rightarrow W$, $\hat{\gamma }(t) = v_1' \circ \gamma (t) = v_2 \circ \gamma (t)$. Define $X_0'(\hat{\gamma }(t)) = \frac{\partial }{\partial y} v_1'(t,0)$, $X_1'(\hat{\gamma }(t)) = \frac{\partial }{\partial y} v_2(t,0)$ and $Y(\hat{\gamma }(t)) = \frac{\partial }{\partial x} v_1'(t,0) = \frac{\partial }{\partial x} v_2(t,0)$ (here $v_1' = v_1'(x,y)$, $v_2 = v_2(x,y)$, where $(x,y) \in \overline{V} \subset \mathbb {R}^2$), so that $X_0',X_1',Y$ are vector fields along the curve $\hat{\gamma }$. Since $v_1',v_2$ are symplectic, we get that $\omega _{\text {st}}(Y,X_0') = \omega _{\text {st}}(Y,X_1') = 1$ at each point $\hat{\gamma }(t)$, for $t \in [0,1]$. Hence if we define $X_s' = (1-s) X_0' + s X_1'$ for $s \in [0,1]$, then $\omega _{\text {st}}(Y,X_s') = 1$ at each point $\hat{\gamma }(t)$, for $t \in [0,1]$. Hence it is not hard to see that for some neighbourhood $G' \supset \gamma ([0,1])$, $G' \subset G$, such that $G'$ is a topological disc bounded by a closed smooth simple curve, there exists a smooth family of smooth symplectic maps $w_s : \overline{G'} \rightarrow W$, $w_s^*\omega _{\text {st}} = \omega _{\text {st}}$, $s \in [0,1]$, such that on the union of $\gamma ([0,1])$ with a neighbourhood of $\{ z_1,z_2 \}$ we have $w_s = v_1' = v_2$ for every $s \in [0,1]$, such that $w_0 = v_1'$ and $w_1 = v_2$ on a neighbourhood of $\gamma ([0,1])$, and such that $\frac{\partial }{\partial y} w_s(t,0) = X_s'(\hat{\gamma }(t))$, $\frac{\partial }{\partial x} w_s(t,0) = Y(\hat{\gamma }(t))$, for each $s,t \in [0,1]$. This family $w_s$ can be realized as follows: write $v_1'(t,y)=\hat{\gamma }(t)+yX_0'(\hat{\gamma }(t))+R'(t,y)$ and $v_2(t,y)=\hat{\gamma }(t)+yX_1'(\hat{\gamma }(t))+R_2(t,y)$, where $R'(t,y),R_2(t,y)\in O(|y|^2)$. Then $\tilde{w}_s(t,y):=\hat{\gamma }(t)+yX_s'(\hat{\gamma }(t))+sR_2(y,t)+(1-s)R'(y,t)$ interpolates between $v_1'$ and $v_2$. Moreover, the restriction of $w_s$ to a sufficiently small neighbourhood $G'$ of $\gamma$ is symplectic in the sense that $w_s^*\omega _\text {st}$ is an area form, but it does not verify yet that $w_s^*\omega _\text {st}=\omega _\text {st}$. This last point is achieved by a source reparametrization, and by changing $G'$ to a smaller neighbourhood if necessary (applying a parametric version of the Moser’s trick).

Now we apply Lemma A.11 to conclude that there exists a compactly supported Hamiltonian function $H'' : W \times [0,1] \rightarrow \mathbb {R}$ whose Hamiltonian flow isotopes $w_0$ to $w_1$ through the family $w_{s}$, $s \in [0,1]$. The points on the image by $v_1'$ of the union of $\gamma ([0,1])$ with a neighbourhood of $\{ z_1,z_2 \}$ stay fixed under the flow of $H''$. This implies that $H''(\cdot ,s)$ is constant and its differential is zero on the image by $v_1'$ of the union of $\gamma ([0,1])$ with a neighbourhood of $\{ z_1,z_2 \}$, at each time $s \in [0,1]$. Without loss of generality we may assume that $H''(\cdot ,s)$ and its differential are zero on the image by $v_1'$ of the union of $\gamma ([0,1])$ with a neighbourhood of $\{ z_1,z_2 \}$, at each time $s \in [0,1]$ (if not, then we can achieve this by adding to $H''$ a function depending solely on time and then making a cutoff outside the union of the images of $w_s$, $s \in [0,1]$). But then if we multiply $H''$ by a function which equals 1 on the complement of a small neighbourhood of $\{ v_1'(z_1),v_1'(z_2) \}$, and equals 0 on a smaller neighbourhood of $\{ v_1'(z_1),v_1'(z_2) \}$, the time-s map of the flow of the resulting Hamiltonian function $\tilde{H}''$ still maps $w_0$ to $w_s$, but we moreover have that $\tilde{H}'' = 0$ on a neighbourhood of $\{ v_1'(z_1),v_1'(z_2) \} = \{ v_1(z_1),v_1(z_2) \} = \{ v_2(z_1),v_2(z_2) \}$. Finally, the concatenation of the flows of $H'$ and $\tilde{H}''$ give us the desired Hamiltonian flow.

(b) First, choose $0< \epsilon < 1/2$ such that $v_1 = v_2$ on a neighbourhood of $\gamma ((0,\epsilon ] \cup [1-\epsilon ,1))$. Then, similarly as in the proof of the absolute case (a), by applying Lemma A.6 we can find a smooth homotopy $F : [0,1] \times [\epsilon ,1-\epsilon ] \rightarrow W$, such that $F(0,t) = v_1 \circ \gamma (t)$, $F(1,t) = v_2 \circ \gamma (t)$ for $t \in [\epsilon ,1-\epsilon ]$, such that for some $0< \delta < 1/2 - \epsilon$ we have $F(s,t) = v_1 \circ \gamma (t) = v_2 \circ \gamma (t)$ for $s \in [0,1]$ and $t \in [\epsilon ,\epsilon +\delta ] \cup [1-\epsilon -\delta ,1-\epsilon ]$, and such that for each $s \in [0,1]$ the curve $[\epsilon ,1-\epsilon ] \rightarrow W$, $t \mapsto F(s,t)$ is smoothly embedded. Then, since we are in dimension $2n \geqslant 4$, after slightly perturbing F we may assume that $\text {Im}\,F \cap v_1 \circ \gamma ((0,\epsilon ) \cup (1-\epsilon ,1)) = \emptyset$. Moreover, similarly as in the proof of the absolute case (a), after performing one more ${\mathcal {C}}^0$-small perturbation of F near $t = \epsilon , 1-\epsilon$, we may in addition assume that the actions of curves $[\epsilon ,1-\epsilon ] \rightarrow W$, $t \mapsto F(s,t)$ are all equal, when $s \in [0,1]$. Now, by the proof of the absolute case (a), we can find a compactly supported Hamiltonian function $\tilde{H} : W \times [0,1] \rightarrow \mathbb {R}$, such that on a neighbourhood of $\{ v_1 \circ \gamma (\epsilon ), v_1 \circ \gamma (1-\epsilon ) \}$ we have $\tilde{H}(\cdot ,s) = 0$ for all $s \in [0,1]$, and such that $\varphi _{\tilde{H}}^1 \circ v_1 = v_2$ on a neighbourhood of $\gamma ([\epsilon ,1-\epsilon ])$, where $\varphi _{\tilde{H}}^1$ is the time-1 map of the Hamiltonian flow generated by $\tilde{H}$. Choose sufficiently small neighbourhoods $W' \Subset W'' \Subset W$ of $\text {Im}\,F$, such that $\tilde{H}(\cdot ,s) = 0$ on a neighbourhood of $v_1 \circ \gamma ((0,\epsilon ] \cup [1-\epsilon ,1)) \cap W''$, for all $s \in [0,1]$. Then any $(W',W'')$ cut-off of $\tilde{H}$ gives a desired Hamiltonian function. $\square$

Proof of Lemma A.5

Choose a 1-form $\lambda$ which is a primitive of $\omega$ on W , $d\lambda = \omega$. We divide the proof into two steps. In the first step we find a Hamiltonian flow that takes the disc $v_1$ to the disc $v_2$ while the boundary of the disc is being kept fixed. In the second step we correct the Hamiltonian flow from the first step so that now a neighbourhood of the boundary is being kept fixed during the isotopy.

Step 1

By Lemma A.2, there exists a compactly supported Hamiltonian isotopy $\varphi ^u$, $u \in [0,1]$ of W , such that $\varphi ^1 \circ v_1 = v_2$. Define a map $F : [0,1] \times S^1 \rightarrow W$ by $F(u,t) = \varphi ^u(v_1(t))$, while here we identify $S^1 \cong \partial G$. Denote $\gamma (t) := F(0,t) = F(1,t)$ for every $t \in S^1$. Our aim is to find a smooth path $\Phi ^{u}$, $u \in [0,1]$ of compactly supported Hamiltonian diffeomorphisms of W such that $\Phi ^u ( \gamma (t)) = F(u,t)$ for every $u \in [0,1]$, $t \in S^1$, such that we moreover have $\Phi ^0 = \Phi ^1 = id$. Provided that we have such $\Phi ^u$, $u \in [0,1]$, we can define the Hamiltonian flow $\psi _1^u = (\Phi ^u)^{-1} \circ \varphi ^u$, and then we have $\psi _1^1 \circ v_1 = v_2$ and $\psi _1^u (v_1(z)) = v_1(z) = v_2(z)$ for every $u \in [0,1]$ and $z \in \partial G$.

First, use Lemma A.7 to find a smooth map $\hat{F} : [0,1] \times [0,1] \times S^1 \rightarrow W$ such that we have $\hat{F}(u,0,t) = \hat{F}(0,s,t) = \hat{F}(1,s,t) = \gamma (t)$, $\hat{F}(u,1,t) = F(u,t)$ for every $u,s \in [0,1]$ and $t \in S^1$, and such that for any $u, s \in [0,1]$, the curve $S^1 \rightarrow W$, $t \mapsto F(u,s,t)$ is smoothly embedded. After slightly perturbing $\hat{F}$, we may assume that we moreover have that for every $u,s \in [0,1]$, the $\lambda$-action of the curve $S^1 \rightarrow W$, $t \mapsto \hat{F}(u,s,t)$, equals the $\lambda$-action of $\gamma$, or in other words that for every $u_0,s_0 \in [0,1]$, the symplectic area of the cylinder $[0,s_0] \times S^1 \rightarrow W$, $(s,t) \mapsto \hat{F}(u_0,s,t)$ equals zero.

Remark A.14

One possible way to obtain such a perturbation is as follows. Choose a closed arc $I \subset S^1$, and apply Lemma A.10 to the restriction of $\hat{F} |_{ [0,1] \times [0,1] \times I }$. We obtain a smooth family $\psi _{u,s} \in \text {Ham}(W,\omega )$, $u,s \in [0,1]$ of compactly supported Hamiltonian diffeomorphisms of W , such that we have $\psi _{u,s}(\gamma (t)) = \psi _{u,s}(\hat{F}(u,0,t)) = \hat{F}(u,s,t)$ for every $(u,s,t) \in [0,1] \times [0,1] \times I$. If we define $\hat{F}' : [0,1] \times [0,1] \times I \rightarrow W$ by $\hat{F}' (u,s,t) = \psi _{u,s}^{-1} \circ \hat{F}(u,s,t)$, then we have $\hat{F}'(u,s,t) = \gamma (t)$ for every $(u,s,t) \in [0,1] \times [0,1] \times I$. But now, using the fact that all the curves $S^1 \rightarrow W$, $t \mapsto \hat{F}'(u,s,t)$ coincide with the curve $\gamma$ on I , we can perturb $\hat{F}'$ similarly to a perturbation scheme which was done in Remark A.13, and obtain a new map $\hat{F}'' : [0,1] \times [0,1] \times S^1 \rightarrow W$ such that for every $u,s \in [0,1]$, the curve $S^1 \rightarrow W$, $t \mapsto \hat{F}''(u,s,t)$ is smoothly embedded and has the same $\lambda$-action as $\gamma$, and moreover such that for any $u,s \in [0,1]$ for which the $\lambda$-action of the curve $S^1 \rightarrow W$, $t \mapsto \hat{F}'(u,s,t)$ already equals the $\lambda$-action of $\gamma$, we have $\hat{F}''(u,s,t) = \hat{F}'(u,s,t)$ for all $t \in [0,1]$. Then the map $[0,1] \times [0,1] \times S^1 \rightarrow W$, $(u,s,t) \mapsto \psi _{u,s} \circ \hat{F}''(u,s,t)$ is a desired perturbation of $\hat{F}$.

Now we apply Lemma A.9 to obtain a smooth family of Hamiltonian functions $H_u : W \times [0,1] \rightarrow \mathbb {R}$, $H_u = H_u(x,s)$, $u \in [0,1]$ with Hamiltonian flows $\psi _{H_u}^s$, $s \in [0,1]$, such that we have $\psi _{H_u}^s ( \hat{F}(u,0,t)) = \hat{F}(u,s,t)$, and moreover the following holds: if for some $u_0,s_0 \in [0,1]$ we have $\frac{\partial }{\partial s} \hat{F}(u_0,s_0,t) = 0$ for any $t \in S^1$, then $H_{u_0}(x,s_0) = 0$ for every $x \in M$. But this implies that $\psi _{H_0}^1 = \psi _{H_1}^1 = id$, and if we set $\Phi ^u := \psi _{H_u}^1$, and define the Hamiltonian flow $\psi _1^u = (\Phi ^u)^{-1} \circ \varphi ^u$, then we get $\psi _1^1 \circ v_1 = v_2$ on $\overline{G}$, and $\psi _1^u (v_1(z)) = v_1(z) = v_2(z)$ for every $u \in [0,1]$ and $z \in \partial G$.

Step 2

This step is analogical to the step 2 in the proof of Lemma A.3. Consider the coordinates $\rho \in [0,r)$, $t \in S^1 \cong \mathbb {R} / 2\pi \mathbb {Z}$ on $\overline{G} \setminus \{ 0 \}$ such that for $z \in \overline{G} \setminus \{ 0 \}$ we have $z = (r-\rho )t$, and look at $v_1 = v_1(\rho ,t)$, $v_2 = v_2(\rho ,t)$. Define vector fields $X_{u,0}(\gamma (t)) = \frac{\partial }{\partial \rho } \psi _1^u \circ v_1(0,t)$, $X_{u,1}(\gamma (t)) = \frac{\partial }{\partial \rho } v_1(0,t) = \frac{\partial }{\partial \rho } v_2(0,t)$ and $Y(\gamma (t)) = \frac{\partial }{\partial t} v_1(0,t) = \frac{\partial }{\partial t} v_2(0,t)$, so that $X_{u,0},X_{u,1},Y$ are vector fields along the curve $\gamma$. Since $\psi _1^u \circ v_1,v_2$ are symplectic, we get that $\omega _{\text {st}}(Y,X_{u,0}) = \omega _{\text {st}}(Y,X_{u,1}) = 1$ at each point $\gamma (t)$ for $t \in S^1$. If we define $X_{u,s} = (1-s) X_{u,0} + s X_{u,1}$ for $s \in [0,1]$, then $\omega _{\text {st}}(Y,X_{u,s}) = 1$ at each point $\gamma (t)$ for $t \in S^1$, and moreover $X_{0,s} = X_{1,s} = X_{0,0}$ for every $s \in [0,1]$. Hence it is not hard to see that for some neighbourhood $G'$ of $\partial G$, $G' \subset \overline{G}$, such that $G'$ is a topological annulus bounded by a smooth simple curve in G and $\partial G$, there exists a smooth family of smooth symplectic embeddings $\hat{v}_{u,s} : \overline{G'} \rightarrow W$, $\hat{v}_{u,s}(z) = \hat{v}(u,s,z)$, $\hat{v}_{u,s}^*\omega _{\text {st}} = \omega _{\text {st}}$, $u, s \in [0,1]$, such that $\hat{v}_{u,s}(0,t) = \gamma (t)$ for all $t \in S^1$, $u,s \in [0,1]$, such that $\hat{v}_{0,s} = \hat{v}_{1,s} = v_1 = v_2$ on $\overline{G'}$ for each $s \in [0,1]$, and such that $\frac{\partial }{\partial \rho } \hat{v}_{u,s}(0,t) = X_{u,s}(\gamma (t))$ for each $t \in S^1$. Now applying Lemma A.12, we find a smooth family of compactly supported Hamiltonian functions $H_u : W \times [0,1] \rightarrow \mathbb {R}$, $H_u = H_u(x,s)$, $u \in [0,1]$ with Hamiltonian flows $\psi _{H_u}^s$, $s \in [0,1]$, such that we have $\psi _{H_u}^s ( \hat{v}(u,0,z)) = \hat{v}(u,s,z)$, and moreover the following holds: if for some $u_0,s_0 \in [0,1]$ we have $\frac{\partial }{\partial s} \hat{v}(u_0,s_0,z) = 0$ for any $z \in \overline{V'}$, then $H_{u_0}(x,s_0) = 0$ for every $x \in W$. If we now define the Hamiltonian flow $\psi _2^u : W \rightarrow W$ by $\psi _2^u = \psi _{H_u}^1 \circ \psi _1^u$, then we get that $\psi _2^1 \circ v_1 = v_2$ on $\overline{G}$, and $\psi _2^u (v_1(z)) = v_1(z) = v_2(z)$ for every $u \in [0,1]$ and $z \in \overline{G'}$. Let $H_2 : W \times [0,1] \rightarrow \mathbb {R}$, $H_2 = H_2(x,u)$ be the compactly supported Hamiltonian function of the flow $\psi _2^u$. Since the points of $v_1(\overline{G'})$ stay fixed under the flow $\psi _2^u$, the differential of $H_2(\cdot ,u)$ vanishes at each point of $v_1(\overline{G'})$, in particular $H_2(\cdot ,u)$ is constant on $v_1(\overline{G'})$, for each $u \in [0,1]$. After adding to $H_2$ a function depending solely on time, and making a cutoff outside $\cup _{t \in [0,1]} \psi _2^t(v_1(\overline{G}))$, if necessary, we may moreover assume without loss of generality that $H_2(\cdot ,u) = 0$ on $v_1(\overline{G'})$, for each $u \in [0,1]$. Finally, if we make a cutoff of the Hamiltonian function $H_2$, multiplying it by a function on W which equals 1 outside a small neighbourhood of $u_1(\partial G)$ and equals 0 on a smaller neighbourhood of $u_1(\partial G)$, we obtain a Hamiltonian function $H : W \times [0,1] \rightarrow \mathbb {R}$, $H = H(x,u)$ with a Hamiltonian flow $\psi ^u$, $u \in [0,1]$, such that on a neighbourhood of $v_1(\partial G)$ we have $H(\cdot ,u) = 0$ for every $u \in [0,1]$, and such that for the time-1 map $\psi ^1$ of the flow of H , we have $\psi ^1 \circ v_1 = v_2$ on $\overline{G}$. $\square$

B Relations between questions $\ldots$

Lemma B.1

Every smooth submanifold of a symplectic manifold has a dense relatively open subset which is a union of symplectically homogeneous relatively open subsets.

Proof

Let $X \subset M$ be a smooth submanifold of a symplectic manifold $(M,\omega )$. It is enough to show that for every relatively open $U \subset X$ there exists a relatively open symplectically homogeneous subset $V \subset U$. To show the latter, choose a point $x_0 \in U$ such that the dimension of the kernel of $\omega$ in $T_{x_0} X$ is minimal. Since for $x \in U$, the dimension of the kernel of $\omega$ in $T_x X$ is an upper semi-continuous function of x , it is enough to take V to be a small relatively open neighbourhood of $x_0$ in U . $\square$

Lemma B.2

We have an affirmative answer to Question 3 for given $X \subset M$ provided that for each relatively open $Y \subset X \times S^2$ we have a positive answer to Question 1’ for Y inside $M \times S^2$. In particular, the same is true for Question 1 instead of Question 1’.

Proof

First of all, it is enough to show that for given $X \subset M$, if for every relatively open $Y \subset X$ we have a positive answer to Question 1’ for Y inside M , then for any symplectic homeomorphism $h : M \rightarrow M'$ such that $X' = h(X) \subset M'$ is a smooth submanifold and such that the restriction $h_{|X} : X \rightarrow X'$ is a diffeomorphism, the restriction is either symplectic or antisymplectic. Indeed, assume that we have shown this, and let $X \subset M$ satisfy the assumptions of the lemma. Let $h : M \rightarrow M'$ be a symplectic homeomorphism, such that $h(X) = X' \subset M'$ is a smooth submanifold and such that $h_{|X} : X \rightarrow X'$ is a diffeomorphism. Define the symplectic homeomorphism

$$\begin{aligned} \begin{array}{rcccl} \hat{h} &{} : &{} M \times S^2 &{} \longrightarrow &{}M' \times S^2 \\ &{} &{} (x,z) &{} \longmapsto &{}(h(x),z). \end{array} \end{aligned}$$

Since the assumptions of the lemma are satisfied, $\hat{h}(X \times S^2) = X' \times S^2 \subset M' \times S^2$ is a smooth submanifold, and the restriction $\hat{h}_{|X \times S^2} : X \times S^2 \rightarrow X' \times S^2$ is a diffeomorphism, we conclude that the restriction is either symplectic or antisymplectic. However, since $\hat{h}_{|X \times S^2}$ acts as the identity on the $S^2$-factor, the only possibility is that $\hat{h}_{|X \times S^2}$ is symplectic. Therefore $h_{|X} : X \rightarrow X'$ is symplectic.

By Lemma B.1, X has a dense open subset which is a union of symplectically homogeneous relatively open subsets. Since it is enough to show that h is either symplectic or antisymplectic on this dense open subset, we may assume without loss of generality that X is itself a symplectically homogeneous submanifold. By our assumptions, X and $X'$ are symplectomorphic, hence $X'$ is also symplectically homogeneous, having the same dimension and symplectic co-rank as X . Now, for given $x_0 \in X$ and for $x_0' = h(x_0) \in X'$ we need to show that the differential of $h_{|X} : X \rightarrow X'$ at $x_0$ is either symplectic or antisymplectic. There is a symplectic embedding $i : B \rightarrow M$ of a small ball $B \subset \mathbb {R}^{2n}$ centred at the origin, and a symplectic embedding $i' : B' \rightarrow M$ of a small ball $B' \subset \mathbb {R}^{2n}$ centred at the origin, such that $i(0) = x_0$, $i'(0) = x_0'$, and such that $i^{-1}(X) = B \cap L$ and $i'^{-1}(X') = B' \cap L$, where $L = \{ (x_1,y_1,\ldots ,x_n,y_n) \; | \; x_{m+1} = \ldots = x_{n} = y_{m+r+1} = \ldots = y_n = 0 \} \subset \mathbb {R}^{2n}$. Let us identify B with i(B) via i and identify $B'$ and $i'(B')$ via $i'$. Denote the differential of $h : X \rightarrow X'$ at $x_0$ by A , and we may think of A as a linear map $A : L \rightarrow L$. Now fix some smooth strictly convex bounded open set $K \subset L$. Take $\epsilon > 0$ small enough, and consider $Y = \epsilon K \subset X$. By our assumption, Y is symplectomorphic to $Y' = h(Y) \subset X'$, i.e. there exists a symplectic diffeomorphism $f : Y \rightarrow Y'$. Moreover, Y is strictly convex, and since $\epsilon$ is small and $h_{|X} : X \rightarrow X'$ is a diffeomorphsim, it follows that the intersection of each of the leaves of the characteristic foliation of X with Y is connected, and that the intersection of each of the leaves of the characteristic foliation of $X'$ with $Y'$ is connected. Hence it follows that f on Y has the form $f(z,t) = (\varphi (z),\psi (z,t))$, where $z = (x_1,y_1,\ldots ,x_m,y_m)$, $t = (y_{m+1},\ldots ,y_{m+r})$, and $\varphi$ is a symplectomorphism from $\pi _z (Y)$ onto $\pi _z(Y')$, where $\pi _z : L \rightarrow \mathbb {R}^{2m} \times \{ 0_{r} \}$ is the orthogonal projection. In particular, $\pi _z(Y)$ and $\pi _z(Y')$ are symplectomorphic. If we choose any symplectic capacity c , it follows that $c(\pi _z(Y)) = c(\pi _z(Y'))$. But we have $\lim _{\epsilon \rightarrow 0} \frac{c(\pi _z(Y))}{\epsilon ^2} = c(\pi _z(K))$ and $\lim _{\epsilon \rightarrow 0} \frac{c(\pi _z(Y'))}{\epsilon ^2} = c(\pi _z(A(K)))$. Hence it follows that $c(\pi _z(K)) = c(\pi _z(A(K)))$ for any strictly convex bounded open body with smooth boundary $K \subset L$. By continuity, we conclude that $c(\pi _z(K)) = c(\pi _z(A(K)))$ for any convex bounded open body $K \subset L$. From here it is easy to show that $A : L \rightarrow L$ is either a symplectic or an antisymplectic linear map, and we leave this to the reader. $\square$

Lemma B.3

For given $X \subset M$, a positive answer to question 4’ implies positive answer to Question 3. In particular, for given $X \subset M$, a positive answer to Question 4 implies positive answer to Question 3.

Proof

It is clearly enough to show the following generalisation of the Eliashberg-Gromov theorem: if $X \subset M$, $X' \subset M'$ are smooth submanifolds, and $f_k : X \rightarrow X'$, $k = 1,2, \ldots$ is a sequence of symplectic diffeomorphisms, that $\mathcal {C}^0$-converge to a diffeomorphism $f : X \rightarrow X'$, then f is symplectic. Let us show this. By Lemma B.1, X and $X'$ have dense subsets, each of which is a union of symplectically homogeneous relatively open subsets. Hence it is enough to show that if $U \subset X$, $U' \subset X'$ are relatively open connected symplectically homogeneous subsets, such that $f(U) \Subset U'$, then the restriction $f_{|U} : U \rightarrow U'$ is a symplectic embedding. We have $f_k(U) \Subset U'$ for large k , hence U and $U'$ has the same symplectic co-rank. It is enough to show that for given $x_0 \in U$, f is symplectic near $x_0$. Denote $x_0'= f(x_0)$. There is a symplectic embedding $i : B \rightarrow M$ of a small ball $B \subset \mathbb {R}^{2n}$ centred at the origin, and a symplectic embedding $i' : B' \rightarrow M$ of a small ball $B' \subset \mathbb {R}^{2n}$ centred at the origin, such that $i(0) = x_0$, $i'(0) = x_0'$, and such that $i^{-1}(U) = B \cap L$ and $i'^{-1}(U') = B' \cap L$, where $L = \{ (x_1,y_1,\ldots ,x_n,y_n) \; | \; x_{m+1} = \ldots = x_{n} = y_{m+r+1} = \ldots = y_n = 0 \} \subset \mathbb {R}^{2n}$. Let us identify B with i(B) via i and identify $B'$ and $i'(B')$ via $i'$. Since $f_k$ are symplectic, on some small neighbourhood $0 \in V \subset L$ they have the form $f_k(z,t) = (\varphi _k(z),\psi _k(z,t))$, where $z = (x_1,y_1,\ldots ,x_m,y_m)$, $t = (y_{m+1},\ldots ,y_{m+r})$, and $\varphi _k$ are symplectic embeddings from $\pi _z (V)$ into $\mathbb {R}^{2m}$, where $\pi _z : L \rightarrow \mathbb {R}^{2m} \times \{ 0_{r} \}$ is the orthogonal projection. Since the sequence $f_k$ $\mathcal {C}^0$-converges to f , it follows that f on V also has the form $f(z,t) = (\varphi (z),\psi (z,t))$, where the sequence of embeddings $\varphi _k$ $\mathcal {C}^0$ converges to $\varphi$. Now the statement follows from the proof of the Eliashberg-Gromov theorem. $\square$