2. The construction

Let M be the one point blow up of $(S^2 \times S^2, \omega_{a,b})$ by weight c, where $\omega_{a,b}$ is the product symplectic form giving areas a and b to the factors with $a \geq b$ and $c< ({b}/{2})$ .

We will first construct a symplectomorphism $\varphi$ and a Lagrangian L such that the Lagrangians $\{\varphi^n(L)\}_{n \in \mathbb{N}}$ are pairwise disjoint. Then, we conclude using Lemma 2.2 that $\varphi$ (if $a \neq b$ ) or $\varphi^2$ (if $a=b$ ) must be a Hamiltonian symplectomorphism.

Let $\mu \colon M \rightarrow \Delta$ be the toric moment map where $\Delta$ is the Delzant polygon in Figure 1(a); $\Delta$ can be written as the non-negative locus of the function

$$\mathcal{F}_\Delta(x_1,x_2) = \min\left\{\frac{a}{2}-\left|x_1\right|, \frac{b}{2} - \left|x_2\right|, x_2 - x_1 +\frac{a+b}{2} - c \right\} .$$

The parameter $\mathcal{F}_\Delta$ measures the integral affine distance to the boundary of $\Delta$ . The level sets of $\mathcal{F}_\Delta$ are also depicted in Figure 1(a). By making a nodal trade [Reference SymingtonSym03, Theorem 6.5] in each corner we obtain an almost toric fibration $\pi_0' \colon X \rightarrow B_0'$ whose base diagram is depicted in Figure 1(b). The solid broken line segment emanating from node $\mathfrak{a}$ in Figure 1(b) is a straight line segment in the integral affine structure on $B_0'$ inherited from the almost toric fibration and is contained in the eigenline of node $\mathfrak{a}$ . Modifying $\pi_0'$ by a nodal slide [Reference SymingtonSym03, Proposition 6.2] along this line segment, we obtain the almost toric fibration $\pi_0 \colon M \rightarrow B_0$ depicted in Figure 1(c).

Figure 1.

Constructing the almost toric fibration $\pi_0$ .

Note that $\max_{x \in\Delta} \{\mathcal{F}_\Delta(x)\} = ({b}/{2})$ . Define the $\mathbb{R}$ -action $t \cdot$ on $\Delta \setminus \mathcal{F}_\Delta^{-1}({b}/{2})$ such that $t\cdot x$ is the translation of x by integral affine distance t along the level set $\mathcal{F}_\Delta^{-1}(\mathcal{F}_\Delta(x))$ in the counter-clockwise direction.

Given $\varepsilon \in (0, \min\{c,({b}/{2})-c\})$ , we will construct $\varphi\in \textrm{Symp}(M)$ such that $\varphi$ acts on the fibres of $\pi_0$ as follows: let $h = \mathcal{F}_\Delta(x)$ . If $h > c+\varepsilon$ , then $\varphi(\pi_0^{-1}(x)) = \pi_0^{-1}(x)$ . If $h < c-\varepsilon$ , then $\varphi(\pi_0^{-1}(x)) = \pi_0^{-1}((c-h)\cdot x)$ . The integral affine length of a level set $\mathcal{F}_\Delta^{-1}(h)$ with $0 \leq h < c$ is $2(a+b)-c-7h$ . So for any $h \in (0,c-\varepsilon)$ with $\frac{c-h}{2(a+b)-c-7h}$ irrational and $x \in \mathcal{F}_\Delta^{-1}(h)$ , the orbit of $\pi_0^{-1}(x)$ under $\varphi$ is dense in the level set $\mathcal{F}_\Delta^{-1}(h)$ and the Lagrangians $\{\varphi^n(\pi_0^{-1}(x))\}_{n \in \mathbb{N}}$ are pairwise disjoint.

2.1.1. Constructing $\varphi$

Note that the node $\mathfrak{n}$ in Figure 2(a) has $\mathcal{F}_\Delta(\mathfrak{n}) = c$ . We modify the fibration $\pi_0$ by a sequence of four nodal slides, illustrated in Figure 2. First, slide the node $\mathfrak{n}$ along the marked path in Figure 2(a) to obtain an almost toric fibration with base diagram Figure 2(b). During the nodal slide, using [Reference EvansEva23, Proof of Theorem 8.10], we may assume that we do not modify the fibration outside of $\mathcal{F}_\Delta^{-1}(c-\varepsilon,c+\varepsilon)$ . Note that the branch cuts in the base diagram Figure 2(b) cut the base $B_0$ into three connected components. Removing the horizontal branch cut drawn with a thick dashed line in Figure 2(b), we get the base diagram Figure 2(c), which has only one connected component. This diagram has a discontinuity in the lower left corner, which can be removed by introducing a branch cut and applying a suitable shear map (given below), giving Figure 2(d). The effect of the two steps (b) $\rightarrow$ (c) $\rightarrow$ (d) is to apply the shear map

(2.1) \begin{equation} (x_1,x_2) \mapsto \Bigl( x_1+\Bigl( c-\frac{b}{2}-x_2\Big), x_2\Big),\end{equation}

to points with $x_2 < c-({b}/{2})$ (the dark shaded region in Figure 2(b)), and the identity everywhere else, giving the base diagram Figure 2(d). Note that the diagram (d) is made identical to diagram (a) by a $90^\circ$ -rotation. Repeat this process three more times as illustrated in Figure 3 until $\mathfrak{n}$ has returned to its original position. By this process we obtain an almost toric fibration $\pi_1 \colon M \rightarrow B_1$ such that, for $x \in \Delta$ , if $h>c+\varepsilon$ the fibres $\pi_1^{-1}(x)$ and $\pi_0^{-1}(x)$ are equal and if $h< c-\varepsilon$ , by combining the effect of the four shear maps (2.1), we have

(2.2) \begin{equation} \pi_1^{-1}(x) = \pi_0^{-1}((c-h)\cdot x) .\end{equation}

Figure 2.

Nodal slide and change of branch cut.

Figure 3.

Constructing the almost toric fibration $\pi_1$ .

Now [Reference SymingtonSym03, Corollary 5.4] and [Reference EvansEva23, Proof of Theorem 8.5] give a symplectomorphism $\varphi\in \textrm{Symp}(M)$ such that, outside a small neighbourhood of the nodes,

$$\varphi(\pi_0^{-1}(x)) = \pi_1^{-1}(x).$$

From (2.2) we conclude that, for $h = \mathcal{F}_\Delta(x)<c-\varepsilon$ ,

$$\varphi(\pi_0^{-1}(x)) = \pi_1^{-1}(x) = \pi_0^{-1}((c-h)\cdot x).$$

Remark 2.1. As pointed out by the referee, here is an equivalent manner of seeing the effect of the nodal slide in Figure 3: Figure 4(a) gives an alternative base diagram of the almost toric base $B_0$ pictured in Figure 1(c), which can also be obtained from $(S^2 \times S^2,\omega_{a,b})$ by a non-toric blow up as in [Reference EvansEva23, Section 9.1]. Similarly to [Reference EvansEva23, Section 10.1], we may replace the four branch cuts emanating from the nodes $\mathfrak{a},\mathfrak{b},\mathfrak{c},\mathfrak{d}$ by the tetrapod branch cut T highlighted in Figure 4(a). Denote the branch cut emanating from the node $\mathfrak{n}$ by $\ell$ . Then, $B_0 \setminus (T \cup \ell)$ is homotopy equivalent to $S^1$ , and a base diagram of its universal cover is given by Figure 4(b). The concatenation of all the nodal slides in Figure 3 can be written as one nodal slide marked in Figure 4(a, b) by the solid line emanating from $\mathfrak{n}$ . Performing the nodal slide, we get the base diagram Figure 4(c) and the almost toric fibration $\pi_1 \colon M \rightarrow B_1$ as above. As described in [Reference EvansEva23, Section 7.2], we can rotate the branch cut $\ell$ emanating from $\mathfrak{n}$ until it matches the branch cut in Figure 4(b) to obtain the base diagram of Figure 4(d) for the universal cover of $B_1 \setminus (T \cup \ell)$ . The combined effect of steps (b) $\rightarrow$ (c) $\rightarrow$ (d) is to apply the shear matrix $A = \left(\begin{smallmatrix}1 & 1 \\ 0 & 1\end{smallmatrix}\right)$ to points below the eigenline of $\mathfrak{n}$ : the ‘missing’ triangle below $\mathfrak{n}$ is spanned by vectors in directions $(a,-1), (a+1,-1)$ for some value of $a \in \mathbb{R}$ , so when a point x touches the sliding or rotating branch cut, in the base diagram the point jumps over the gap given by the missing triangle according the shear matrix A. As above we use [Reference EvansEva23, Proof of Theorem 8.10] to ensure the nodal slide only modifies fibres near the eigenline of $\mathfrak{n}$ , and [Reference SymingtonSym03, Corollary 5.4] and [Reference EvansEva23, Proof of Theorem 8.5] to get the desired symplectomorphism.

Figure 4.

An alternative picture of the nodal slides used in the construction of the symplectomorphism $\varphi$ .

We conclude the construction by showing that $\varphi$ or $\varphi^2$ is a Hamiltonian diffeomorphism.

Lemma 2.2. Let $\psi$ be a symplectomorphism of M.

1. (1) If $a \neq b$ , then $\psi$ is a Hamiltonian diffeomorphism.

2. (2) If $a=b$ , then $\psi^2$ is a Hamiltonian diffeomorphism.

Proof. It is shown in [Reference Li, Li and WuLLW15, Corollary 1.3] that the symplectic mapping class group of M is a finite reflection group generated by Dehn twists along embedded Lagrangian spheres, and that a generating set is given by the classes $\xi \in H_2(M;\mathbb{Z})$ which can be represented by smoothly embedded spheres and satisfy

$$\xi \cdot \xi = -2, \quad c_1(M) (\xi) = 0, \quad [\omega] (\xi) = 0.$$

Already, the conditions $\xi \cdot \xi = -2$ and $c_1(M) (\xi) = 0$ imply that $\xi$ is the anti-diagonal class $-D$ , see e.g. [Reference EvansEva10, Lemma 4.1.4]. Since the symplectic area of $-D$ vanishes only for $a=b$ , and since in this case the Dehn twist along the anti-diagonal has order of at most 2, the symplectic mapping class group is trivial in case $a \neq b$ and has at most two elements in case $a=b$ . Finally, note that the symplectomorphisms in the component of the identity are Hamiltonian, since the first homology of M vanishes.

Alternatively [Reference Li, Li and WuLLW22, Corollary 4.6] yields a shorter but slightly weaker argument. See Remark 3.3.