3.1 Shape invariant

Given an exact symplectic manifold $(M^{2n}, \omega = d\lambda )$ with a fixed primitive $\lambda$, the shape invariant of this $(M, \omega = d\lambda )$, denoted by ${\rm Sh}(M, \lambda )$, is defined as the collection of all possible area classes of embedded Lagrangian tori. More explicitly, for any Lagrangian embedding $\phi : \mathbb {T}^n \hookrightarrow M^{2n}$, the pullback $\phi ^*\lambda$ represents a cohomology class in $H^1(\mathbb {T}^n; \mathbb {R})$. By choosing an integral basis $e: = (e_1,\ldots,e_n)$ of $H_1(\mathbb {T}^n; \mathbb {Z})$, the following evaluation

(9)\begin{equation} [\phi^*\lambda](e) = (\lambda(\phi_*(e_1)),\ldots, \lambda(\phi_*(e_n))) \in \mathbb{R}^{n} \end{equation}

induces a map from Lagrangian embeddings to elements in $\mathbb {R}^n$. Of course, the set of values from (9) depends on $\lambda$ and $e$, where a different choice of $\lambda$ results in a uniform shift in $\mathbb {R}^n$ and a different choice of $e$ results in a transformation by an element in ${\rm GL}(n, \mathbb {Z})$. In particular, the action by ${\rm GL}(n, \mathbb {Z})$ provides a certain symmetry of ${\rm Sh}(M, \lambda )$. It is Sikorav's work [Reference SikoravSik89] and Eliashberg's work [Reference EliashbergEli91] that first observed the application of the shape invariant to the study of the rigidity of symplectic or contact embeddings. For further development in this direction, see [Reference MüllerMül19, Reference Rosen and ZhangRZ21, Reference Hind and ZhangHZ21].

In this paper, we consider a restrictive version of the shape invariant, called the Hamiltonian shape invariant, and our $(M^{2n}, \omega = d\lambda ) = (X, \omega _{\rm std} = d\lambda _{\rm std})$ for a toric domain $X$ in $(\mathbb {R}^4, \omega _{\rm std})$. It is defined in (2) as the collection of all possible area classes $(r,s) \in \mathbb {R}_{>0}^2$ that admit $L(r,s) \hookrightarrow X$, and it is denoted by ${\rm Sh}_H(X)$. Observe that in this set-up, there is no dependence of the primitive and we have a canonical choice of the basis $e = (e_1, e_2)$, where $e_1$ is the standard circle in $L(r,s)$, lying in the first $\mathbb {R}^2$-factor of $\mathbb {R}^4$, bounding the disk with area $r$ and $e_2$ is the standard circle in $L(r,s)$, lying in the second $\mathbb {R}^2$-factor of $\mathbb {R}^4$, bounding the disk with area $s$. Then the only symmetry we have is via the reflection $(r,s) \mapsto (s,r)$, which induces a simplified version, the reduced Hamiltonian shape invariant denoted by ${\rm Sh}_H^+(X) := {\rm Sh}_H(X) \cap \{r \leq s\}$.

The explicit computations of ${\rm Sh}_H(X)$ as a subset in $\mathbb {R}_{>0}^2$, even for basic toric domains such as balls, ellipsoids, and polydisks, were carried out quite recently, see [Reference Hind and OpshteinHO20, Reference Hind and ZhangHZ21]. Theorem 2.1 in the introduction summarizes the corresponding results. These results are consequences of a sophisticated analysis of holomorphic curves forming part of symplectic field theory (SFT).

A slightly different version of the shape invariant, which is formulated via the flux of a Lagrangian isotopy (in particular, applied on closed manifolds such as $\mathbb {C} P^n$ and $S^2 \times S^2$), has been studied in [Reference Entov, Ganor and MembrezEGM18, Reference Shelukhin, Tonkonog and ViannaSTV18]. These works point out the intriguing relations between the shape invariant and Poisson-bracket invariants in [Reference Buhovsky, Entov and PolterovichBEP12] and the SYZ fibration in [Reference Strominger, Yau and ZaslowSYZ96], respectively.

3.2 Symplectic field theory

Lagrangian embeddings $\phi : L(s,t) \hookrightarrow X$ can be effectively studied via SFT. It is a modern machinery, originally formulated in the work [Reference Eliashberg, Givental and HoferEGH00] and further developed in [Reference Bourgeois, Eliashberg, Hofer, Wysocki and ZehnderBEH⁺03, Reference HoferHof06, Reference Cieliebak and MohnkeCM18], that can associate a variety of algebraic invariants to a symplectic cobordism. Our work, however, does not rely on this full algebraic machinery, which is indeed still under development. The proofs in the current paper require only compactness and deformation theorems for (somewhere injective) holomorphic curves, modulo some references to ECH and to earlier works of the authors. The input from these earlier works is used to establish existence of a holomorphic curve in Lemma 4.1 (which can be avoided using Remark 4.2) and then in the bubbling analysis of both § 4.1.2 and the main proof in § 4.2, where Lemma 3.7 of [Reference Hind and OpshteinHO20] is a key ingredient.

Topologically, our symplectic cobordism is the complement $W: = X \backslash U_g^*L$ where $L = \phi (L(r,s))$ and $U^*_gL$ is the unit codisk bundle of $L$ with respect to some metric $g$ on $L$; our metrics will always be flat. With a preferred almost complex structure $J$ on $W$, the central objects in SFT are $J$-holomorphic curves $C: (S^2 \backslash \{p_1,\ldots, p_m\}, j) \to (W, J)$, where the asymptotic ends from punctures $p_i$ correspond to Reeb orbits on $\partial X$ (positive ends) and on $S_g^*L : = \partial U_g^*L$ (negative ends). In fact, the Reeb orbits on $S_g^*L$ can readily be classified (see Proposition 3.1 in [Reference Hind and OpshteinHO20]). More concretely, we say a Reeb orbit on $S_g^*L$ is of the type $(-m,-n)$, denoted by $\gamma _{(m, n)}$, if its projection to $L$ lies in the homology class $(-m, -n) \in H^1(\mathbb {T}^2, \mathbb {Z})$. Note that for a $J$-holomorphic plane $C: (S^2 \backslash \{p\}, j) \to (W, J)$ with only one asymptotic end on $\gamma _{(-m, -n)}$, by Stokes’ theorem, its area is

\begin{align*} {\rm area}(C) = 0 - (r(-m) + s(-n)) = rm + sn (>0). \end{align*}

A useful algebraic invariant of $C$ is its Fredholm index. Denote by $\{\gamma _i^+\}_{i=1}^{s_+}$ the collection of positive asymptotic orbits and $\{\gamma _i^-\}_{i=1}^{s_-}$ the collection of negative asymptotic orbits. Without loss of generality, let us assume these Reeb orbits are either non-degenerate or Morse–Bott, that is, they may come in smooth families, and $S_i^+$ and $S_i^-$ are the leaf spaces of the associated Morse–Bott submanifolds. Fix a symplectic trivialization $\tau$ of $C^*TW$ along these Reeb orbits, and $c_1^{\tau }(C^*TW)$ denotes the first Chern number with respect to this trivialization $\tau$. Then

(10)\begin{align} {\rm ind}(C) & = (s_+ + s_- - 2) + 2c_1^{\tau}(C^*TW) \nonumber\\ & \quad + \bigg(\sum_{i=1}^{s^+} {\rm CZ}^{\tau}(\gamma_i^+) + \frac{\dim S_i^+}{2}\bigg) - \bigg(\sum_{i=1}^{s_-} {\rm CZ}^{\tau}(\gamma_i^-) - \frac{\dim S_i^+}{2}\bigg), \end{align}

where ${\rm CZ}^{\tau}$ is the Conley–Zehnder index with respect to $\tau$, which can be computed as the Robbin–Salamon index (see [Reference Robbin and SalamonRS93]). Note that ${\rm ind}(C)$ is independent of the choice of symplectic trivialization $\tau$. When specifying $\gamma _i^{-} = \gamma _{(-m_i, n_i)}$ for $i \in \{1,\ldots, s_-\}$ and taking $\tau$ as the complex trivialization of the contact planes induced by complexifying the trivialization of the 2-torus, the index formula (10) can be computed by

(11)\begin{equation} {\rm ind}(C) = (s_+ + s_- -2) + \bigg(\sum_{i=1}^{s^+} {\rm CZ}^{\tau}(\gamma_i^+) + \frac{\dim S_i^+}{2}\bigg) + 2 \sum_{i=1}^{s_-} (m_i + n_i). \end{equation}

For more detailed elaboration, see Example 4.1 in [Reference Hind and ZhangHZ21]. Sometimes, the toric domain $X$ can be compactified to be a closed symplectic manifold by adding certain curves at infinity. In this paper, we are particularly interested in the following two cases.

(i) Ball $B^4(R)$, where $B^4(R)$ can be compactified to $\mathbb {C} P^2(R)$ with the area of the line at infinity $S_{\infty }$ being $R$. Then we study $C$ without positive ends, but with a topological invariant given by its intersection number with $S_{\infty }$. We denote the intersection by $d$. Then the corresponding formula in (11) is

(12)\begin{equation} {\rm ind}(C) = (s_- -2) + 6d + 2 \sum_{i=1}^{s_-} (m_i + n_i). \end{equation}

(ii) Polydisks $P(c,d)$, where $P(c,d)$ can be compactified to $S^2(c) \times S^2(d)$ with two factors having areas $c$ and $d$, by adding two curves $\{\infty \} \times S^2(b)$ and $S^2(a) \times \{\infty \}$. Then we study $C$ without positive ends, but with a topological invariant given by its intersection with these two curves at infinity. We denote by $(d_1, d_2)$ the bidegree labeling the two intersection numbers. Then the corresponding formula in (11) is

(13)\begin{equation} {\rm ind}(C) = (s_- -2) + 4 (d_1 + d_2) + 2 \sum_{i=1}^{s_-} (m_i + n_i). \end{equation}

A useful technique in SFT is neck-stretching, via a sequence of deformations of almost complex structures on the symplectic cobordism $W$. The standard SFT compactness theorem in [Reference Bourgeois, Eliashberg, Hofer, Wysocki and ZehnderBEH⁺03] promises the existence of a limit curve $C_{\rm lim}$, more precisely, a pseudo-holomorphic building consisting of curves in different levels matched in a possibly complicated way. However, the two invariants introduced above, ${\rm area}(C)$ and ${\rm ind}(C)$, converge in the limit and behave in a rather controllable manner. To be precise, if $C_{\rm lim}$ is obtained by gluing different sub-buildings $\{C_i\}_{i =1}^n$ along matching orbits $\{\gamma _i\}_{i=1}^m$, then

(14)\begin{equation} {\rm area}(C_{\rm lim}) = {\rm area}(C_1) + \cdots + {\rm area}(C_n) \end{equation}

and

(15)\begin{equation} {\rm ind}(C_{\rm lim}) = \sum_{i=1}^n {\rm ind}(C_i) - \sum_{i=1}^m {\rm dim}\,S_i, \end{equation}

where $S_i$ is the leaf space of $\gamma _i$ for $i \in \{1,\ldots, m\}$. For more details, see Proposition 3.3 in [Reference Hind and OpshteinHO20]. In what follows, we will see that the combination of ${\rm area}(C)$ and ${\rm ind}(C)$ actually yield quite strong constraints on the possible configurations of $C_{\rm lim}$. This will be essential to the study of embeddings $L(r,s) \hookrightarrow X$. These ideas were also used in both [Reference Hind and OpshteinHO20, Reference Hind and ZhangHZ21] in the computations of the (Hamiltonian) shape invariant.

4.1 Preparations

4.1.1 Existence of a holomorphic cylinder

The following lemma, Lemma 4.1, guarantees the existence of a holomorphic cylinder which will initiate our main proof.

Recall that $L(r_0, s_0)$ is the product torus in $\mathbb {C}^2$ with area classes $r_0$ and $s_0$. We assume $L(r_0, s_0) \subset E(a,b)$, that is, $kr_0 + s_0 < b$. Recall that $\gamma _{(m,n)}$ denotes the closed Reeb orbits of type $(-m, -n)$ on the unit co-sphere bundle $S^*_gL(r_0, s_0)$ with respect to a preferred flat metric $g$ on $L(r_0, s_0)$. Since $L(r_0, s_0) \subset E(a,b)$, rescaling the metric $g$, we may assume this unit codisk bundle $U_g^*L(r_0, s_0)$ sits inside $E(a,b)$. As discussed in § 3.2, the complement $E(a,b)\backslash U_g^*L(r_0, s_0)$ is a symplectic cobordism and the deformed complex structure, denoted by $J_0$, gives a compatible almost complex structure with cylindrical ends. Denote by $\gamma _b$ the long closed Reeb orbit on $\partial E(a,b)$ with period $b$.

In order to guarantee non-degeneracy, when working with holomorphic curves asymptotic to ellipsoids we will always increase $b$ slightly so that $ {b}/{a} \notin \mathbb {Q}$. As the perturbation can be arbitrarily small this does not affect the statements of any of our results. Here, with our fixed trivialization $\tau$ as in § 3.2, the longer orbit $\gamma _b$ of $E(a,b)$ has its Conley–Zehnder index given by ${\rm CZ}^{\tau }(\gamma _b) = 2\,\lfloor {b}/{a} \rfloor + 3$.

Lemma 4.1 There exists a $J_0$-holomorphic cylinder in the symplectic cobordism

\[ E(a,b)\backslash U_g^*L(r_0, s_0) \]

with a positive end on $\gamma _b$, the longer Reeb orbit of $\partial E(a,b)$, and with a negative end on a Reeb orbit $\gamma _{k,1}$ on $S^*_gL(r_0, s_0)$ (where recall $k = {b}/{a}$).

Outline of the proof of Lemma 4.1. One proof of this is to analyze a neck-stretching argument in [Reference Hind and ZhangHZ21] which produces a union of curves in the cobordism $E(a,b)\backslash U_g^*L(r_0, s_0)$, checking that one of the curves must be a cylinder as required. We also give a direct geometric construction in Remark 4.2 of a cylinder which is holomorphic for some almost complex structure (a result which is actually enough for our purposes).

Proof of Lemma 4.1 There exists a thin and long ellipsoid denoted by $E(\epsilon, \epsilon S)$ which can be embedded inside the unit codisk bundle $U_g^*L(r_0, s_0)$. Here, $\epsilon$ is sufficiently small and we require $S>k+1$. Up to symplectomorphism, we have the following inclusions,

\[ E(\epsilon, \epsilon S) \subset U_g^*L(r_0, s_0) \subset E(a,b). \]

Denote by $\beta$ the short closed Reeb orbit of $\partial E(\epsilon, \epsilon S)$ with period $\epsilon$. By an appropriate deformation of the almost complex structure, we can view $E(a,b) \backslash E(\epsilon, \epsilon S)$ as a symplectic cobordism with respect to an almost complex structure (still) denoted by $J_0$. By Theorem 5.8 in [Reference Hind and ZhangHZ21], there exists a $J_0$-holomorphic cylinder with positive end $\gamma _b$ and negative end $\beta ^{k+1}$.Footnote ¹ Then by a neck-stretching along the boundary $S_g^*L(r_0, s_0)$ as in [Reference Hind and ZhangHZ21], Lemma 6.1, the limiting building $C_{\rm lim}$ contains $(k+2)$-many curves in $E(a,b) \backslash U_g^*L(r_0, s_0)$, denoted by $\{F_1,\ldots, F_{k+2}\}$. Lemma 6.1 in [Reference Hind and ZhangHZ21] also implies that for each $i \in \{1,\ldots, k+2\}$, we have ${\rm ind}(F_i) = 1$. Moreover, since we have only one positive end, the argument in the proof of the ‘only if’ part of Theorem 1.4 in [Reference Hind and ZhangHZ21] implies that all but one of the $F_i$ are planes and the remaining curve is a cylinder with positive end on $\gamma _b$. Without loss of generality, assume $F_1$ is the cylinder. Moreover, suppose the negative end of $F_i$ has type $(-k_i, -l_i)$. Since the negative ends bound a cycle in $T^* L(r_0, s_0)$ we see

(16)\begin{equation} \sum_{i=1}^{k+2} k_i = \sum_{i=1}^{k+2} l_i = 0. \end{equation}

So far, we have implicitly assumed our $J_0$ is generic in order to guarantee curves of non-negative index. However, we would like to work with a $J_0$ such that the $z_1$ and $z_2$ axes are complex (and, hence, are finite energy planes, say $P_1$ and $P_2$, asymptotic to $\gamma _a$ and $\gamma _b$, respectively). As these axes are disjoint from $L(r_0, s_0)$, it is easy to check that they are not covered by any of our limiting curves, and so we may still assume regularity.

Since the axes are complex, positivity of intersection implies that for $i \ge 2$ the $F_i$ are asymptotic to a Reeb orbits representing classes with $k_i \le 0$ and $l_i \le 0$. As the curves have index $1$, the index formula implies that $k_i + l_i =-1$, so the only possibilities are $(k_i, l_i) = (-1,0)$ or $(k_i, l_i) = (0,-1)$. Identifying any matching asymptotic orbits and adding a disk inside $E(\epsilon, \epsilon S)$, our holomorphic building can be compactified to form a homology class in the ellipsoid with boundary $\gamma _b$. Since $P_1$ represents a class with boundary $\gamma _a$, it has intersection number $+1$ with the compactified building. Then by positivity of intersection, we see that at most one $F_i$ is asymptotic to an orbit in the class $(0,-1)$.

Suppose first that exactly one $F_i$ for $i \ge 2$ is asymptotic to an orbit of type $(0,-1)$. Then $F_1$ is asymptotic to an orbit of type $(k,1)$ as required.

Alternatively, all $F_i$ for $i \ge 2$ are asymptotic to orbits of type $(-1,0)$ and $F_1$ is asymptotic to an orbit of type $(k+1,0)$. In this case the limiting building has intersection number at least $k+1$ with $P_2$. However we can compute Siefring's generalized intersection number (see (4-3) in [Reference SiefringSie11]) to be

\[ P_2 \ast P_2 = k. \]

Indeed, trivializing the contact planes in $E(a,b)$ along $\partial P_2$ by using an identification $\Phi$ with the $z_1$ plane, in the notation from [Reference SiefringSie11] we have $i^{\Phi }(P_2, P_2)=0$ and $\Omega ^{\Phi }(P_2, P_2) = k$.

As our limiting holomorphic building has a single unmatched positive end on $\gamma _b$ and (after we compactify with the disk inside $E(\epsilon, \epsilon S)$) is homotopic relative to its compactified positive boundary to $P_2$, Theorem 2.2 from [Reference SiefringSie11] implies that $k$ is an upper bound for the intersections between the curves in our limiting building and $P_2$. This is a contradiction if the building contains $k+1$ planes asymptotic to $(-1,0)$.

Remark 4.2 For an alternative proof of Lemma 4.1, at least for a carefully chosen $J_0$, we observe it is, in fact, possible to write down such a holomorphic cylinder directly. To do this we fix a circle $\Gamma = \{\theta _1 - k \theta _2 =0 \}$ in the torus $\mathbb {T}^2$, and identify all fibers of the moment map $\mu$ with the same $\mathbb {T}^2$, suitably collapsed over the axes. In the moment image $\mu (E(a,b))$ let $\sigma$ be a curve which coincides with the line through $(r_0, s_0)$ of slope $ {1}/{k}$ at one end, and the vertical line through $(0,b)$ at the other. Then $\sigma \times \Gamma$ is a cylinder in $E(a,b)$, which is symplectic provided $\sigma$ never has slope $-k$. We can find such paths exactly because $b > ka$. Moreover, our cylinder coincides with the trivial cylinder over the Reeb orbits at each end, so we can find an almost complex structure $J_0$ making the cylinder holomorphic.

4.1.2 Degenerations of moduli spaces

Here we analyze possible degenerations of a moduli space of genus $0$ curves with Fredholm index bounded below by the number of negative ends. In any degeneration we can isolate a particular limiting curve, and the main Lemma 4.4 shows that areas of curves in the new moduli space are bounded above by the areas of the original moduli space.

Given an isotopy of Lagrangian submanifolds $\{L_t\}_{t \in [0,1]}$, corresponding compatible almost complex structures $J_t$ on the complement of $L_t$, and a sequence of $J_{t_n}$-holomorphic curves $\{C_n\}_{n \in \mathbb {N}}$ with (negative) ends on the Lagrangian submanifolds $\{L_{t_n}\}_{n \in \mathbb {N}}$, the standard SFT-compactness implies that if $t_n \to t_*$ and the $C_n$ have bounded area, for example if they appear in the same moduli space, then the $C_n$ converge to a $J_{t_*}$-holomorphic building $C_{\rm lim}$. To be precise, the standard SFT-compactness applies to a sequence of ${J_n}$-holomorphic curves $C_n$ in a fixed cobordism, however we can use Fukaya's trick (see § 2.1 in [Reference Shelukhin, Tonkonog and ViannaSTV18]) to transfer the moving boundary conditions on $L_{t_n}$ to a sequence of almost complex structures on a fixed cobordism as required. Hence, we still obtain compactness as in [Reference Bourgeois, Eliashberg, Hofer, Wysocki and ZehnderBEH⁺03]. A careful study of $C_{\rm lim}$ plays an important role in the proof of obstruction (I) in Theorems 2.11 and 2.12. In particular, the following technical lemma, Lemma 4.3, is useful to us.

Let $C$ denote a curve with ${\rm ind}(C) = e$, one positive end on $\gamma _b$, and $e$-many negative ends on the Lagrangian torus $L_0$. For instance, the cylinder provided by Lemma 4.1 satisfies this condition with $L_o = L(r_0, s_0)$, since by (11)

\[ {\rm ind}(C) = (2k+3) - (2k+2) = 1 = \# \,\mbox{negative ends of } C. \]

Denote by $\mathcal {M}_{C}(t)$ for $t \in [0,1]$ the moduli space corresponding to the curve $C$ with boundary conditions on $L_t$. Suppose the universal moduli space of the $\mathcal {M}_C(t)$ is not compact, that is, there exist a degeneration at $t_* \in [0,1]$ with a limiting building $C_{\rm lim} \notin \mathcal {M}_C(t_*)$. The proof of Lemma 3.7 in [Reference Hind and OpshteinHO20, p. 552] shows there exists a curve of $C_{\rm lim}$ in $E(a,b) \backslash U_g^*L_{t_*}$, which we denote by $C_0$, having ${\rm ind}(C_0) \geq \# \,\mbox {negative ends of } C_0$. We denote by $C_1,\ldots, C_k$ the components (as connected sub-buildings of $C_{\rm lim}$ consisting of curves in $E(a,b) \backslash U_g^*L_{t_*}$ and in the symplectization $T^*L_{t_*}\backslash 0_{L_{t_*}}$) of the complement of $C_0$ in $C_{\rm lim}$ such that each $C_i$ matches with $C_0$ at only one negative end of $C_0$. See Figure 9 for an example of $C_{\rm lim}$. Suppose $C_i$ for $i \in \{1,\ldots, k\}$ admits $e_i$-many negative ends, while the cardinality of the unmatched ends of $C_0$ is denoted by $e_0$ (hence, the total number of the negative ends of $C_{\rm lim}$ is $\sum _{i=0}^k e_i$). Finally, as shown in Figure 9, suppose the negative ends of the curves in $E(a,b) \backslash U_g^*L_{t_*}$ from the component $C_i$ have type $\{(-m_i^{j}, -n_i^{j})\}_{j = 1,\ldots, l_i}$. Then we have the following result.

Lemma 4.3 Suppose $C_0$ contains the positive end $\gamma _b$. Then, with $(-m_i^j, -n_i^j)$ defined as above, we have $\sum _{i=1}^k \sum _{j=1}^{l_i} (m_i^j + n_i^j) \leq 0$.

Figure 9. A limit curve in $\mathcal {M}_C(t_*)$.

Outline of the proof of Lemma 4.3. We will compare the indices of the various components $C_i$ with the index of the limiting building using (15). Together with an analysis of matching ends we will derive the result.

Proof of Lemma 4.3 First, by the index matching formula (15), we have

(17)\begin{equation} e = {\rm ind}(C_{\rm lim}) = \bigg(\sum_{i=0}^k {\rm ind}(C_i) \bigg) - k, \quad \mbox{which is} \quad \sum_{i=0}^k {\rm ind}(C_i) = e+k. \end{equation}

Since $e = \#\,\mbox {negative ends of } C_{\rm lim} = e_0 + \cdots + e_k$, by regrouping these terms, (17) is equivalent to the relation $\big ({\rm ind}(C_0) - (e_0+k)\big ) + \sum _{i=1}^k ({\rm ind}(C_i) - e_i) =0.$ Moreover, since $e_0 +k$ equals the $\# \, \mbox {negative ends of } C_0$, by assumption, ${\rm ind}(C_0) \geq e_0+k$, so

(18)\begin{equation} \sum_{i=1}^k ({\rm ind}(C_i) - e_i) \leq 0. \end{equation}

Second, let us focus on a component $C_i$ for $i \in \{1,\ldots, k\}$. Since $C_0$ already occupies the only positive end $\gamma _b$, each such $C_i$ only has ends on $L_{t_*}$. By a further decomposition, suppose $C_i$ consists of $Q_i$-many sub-components lying entirely in the symplectization $T^*L_{t_*} \backslash 0_{L_{t_*}}$, denoted by $\{u_i^j\}_{j=1,\ldots, Q_i}$. Each $u_i^j$ has $e_i^j$-many negative ends and $s_i^j$-many positive ends. In particular, $\sum _{j=1}^{Q_i} e_i^j = e_i$. Similarly, $C_i$ consists of $R_i$-many curves in $E(a,b) \backslash U_g^*L_{t_*}$, denoted by $\{v_i^j\}_{j=1,\ldots, R_i}$. Each $v_i^j$ has $r^j_i$-many negative ends on $L_{t_*}$, of type $\{(-m_i^{j,l}, -n_i^{j, l})\}_{l = 1,\ldots, r_i^j}$. See Figure 10 for an example of this decomposition of $C_i$. Note that $\sum _{j=1}^{Q_i} s_i^j = (\sum _{j=1}^{R_i} r_i^j) + 1$ since there is one extra end matching with $C_0$. By Proposition 3.4(b) in [Reference Hind and OpshteinHO20], curves $u_i^j$ in the symplectization have index ${\rm ind}(u_i^j) = 2s_i^j + e_i^j -2$. In addition, curves $v_i^j$ in $E(a,b) \backslash U_g^*L_{t_*}$ have index ${\rm ind}(v_i^j) = r_i^j -2 + 2 \sum _{l=1}^{r_i^j} (m_i^{j,l} + n_i^{j, l})$ by the formula (11). Then by the index matching formula (15) again,

\begin{align*} {\rm ind}(C_i) - e_i & = \bigg(\sum_{j=1}^{R_i} {\rm ind}(v_i^j) + \sum_{j=1}^{Q_i} {\rm ind}(u_i^j) - \sum_{j=1}^{R_i} r_i^j \bigg) - e_i\\ & = \sum_{j=1}^{R_i} \bigg(r_i^j -2 + 2 \sum_{l=1}^{r_i^j} (m_i^{j,l} + n_i^{j, l})\bigg) \\ &\quad + \sum_{j=1}^{Q_i} \big(2s_i^j + e_i^j -2 \big) - \sum_{j=1}^{R_i} r_i^j - \sum_{j=1}^{Q_i} e_i^j\\ & = 2 \bigg(- R_i + \sum_{j=1}^{R_i} \sum_{l=1}^{r_i^j}(m_i^{j,l} + n_i^{j, l}) - Q_i + \sum_{j=1}^{Q_i} s_i^j \bigg). \end{align*}

Now, for such a decomposition of $C_i$ having genus $0$, we can associate a tree to it by adding a vertex for each curve and each asymptotic end, and an edge between the vertex for a curve and each of its asymptotic end. We identify the two vertices corresponding to matching asymptotic ends. Since the Euler characteristic of a tree is always $1$, we have

\[ R_i + Q_i - \bigg(\sum_{j=1}^{Q_i} s_i^j -1\bigg) = 1, \quad \mbox{which implies} \ Q_i + R_i - \sum_{j=1}^{Q_i} s_i^j = 0. \]

Therefore, we obtain the following relation,

\[ {\rm ind}(C_i) - e_i = 2 \bigg(\sum_{j=1}^{R_i} \sum_{l=1}^{r_i^j}(m_i^{j,l} + n_i^{j, l}) \bigg). \]

Finally, summing over all $i = \{1,\ldots, k\}$, we have

\begin{align*} 2 \sum_{i=1}^k \sum_{j=1}^{l_i} (m_i^j + n_i^j) & = 2 \sum_{i=1}^k \sum_{j=1}^{R_i} \sum_{l=1}^{r_i^j}(m_i^{j,l} + n_i^{j, l}) \\ & = 2 \sum_{i=1}^k ({\rm ind}(C_i) - e_i) \leq 0, \end{align*}

where $l_i = \sum _{j=1}^{R_i} r_i^j$ and the last step comes from (18). Thus, we complete the proof.

Figure 10. A decomposition of $C_i$.

A corollary of Lemma 4.3 is the following useful result. Denote by $\mathcal {M}_{C_0}(t)$ the moduli space of the curve $C_0$ in Lemma 4.3 (so, in particular, we are assuming the curves have a single positive end $\gamma _b$) with moving boundary conditions on $L_t$. Recall that $L_t$ is a Lagrangian isotopy which will cover a path $\gamma (t) = (r(t), s(t))$ in the shape of $E(a,b)$. Denote by ${\rm area}_{C_0}(t)$ the (time-dependent) area of a curve in $\mathcal {M}_{C_0}(t)$ with boundary on $L_t$, and similarly define ${\rm area}_C(t)$ where $C$ is the curve we started from whose moduli space degenerated into $C_{\rm lim}$. We note that these area formulas only depend on the moduli space of $C$ or $C_0$, and actually only on the homology class of $C_0$ and $C$. Thus, they are well defined even if the corresponding moduli space is empty and, in particular, ${\rm area}_C(t)$ is defined when $t > t_*$. Recall that $t_* \in [0,1]$ is the moment where $C$ degenerates.

Proposition 4.4 Suppose $r(t) / s(t)$ is non-increasing with respect the parameter $t \in [0,1]$ and the moduli space of $C$ degenerates at time $t_*$. Then ${\rm area}_{C_0}(t) \leq {\rm area}_{C}(t)$ for all $t \in [t_*,1]$.

Proof. We know that ${\rm area}_{C_0}(t_*) \leq {\rm area}_{C_{\rm lim}}(t_*) = {\rm area}_{C}(t_*)$ since $C_0$ is a component of $C_{\rm lim}$, it suffices to prove the conclusion for $t \in (t_*,1]$. Arguing by contradiction, suppose there exists some $t' \in (t_*, 1]$ such that ${\rm area}_{C_0}(t') > {\rm area}_{C}(t')$. Note that for any $t \in [t_*,1]$, we have

\begin{align*} {\rm area}_C(t) & = \sum_{i=0}^k {\rm area}_{C_i}(t) \\ & = {\rm area}_{C_0}(t) + \bigg(\sum_{i=1}^k\sum_{j=1}^{l_i} m_i^j\bigg) r_t + \bigg(\sum_{i=1}^k\sum_{j=1}^{l_i} n_i^j\bigg) s_t, \end{align*}

since, for homological reasons, curves in the symplectization contribute $0$ to the area. For brevity, let $M := \sum _{i=1}^k\sum _{j=1}^{l_i} m_i^j$ and $N: = \sum _{i=1}^k\sum _{j=1}^{l_i} n_i^j$.

Arguing by contradiction, the relation ${\rm area}_{C_0}(t') > {\rm area}_{C}(t')$ above implies that $Mr_{t'} + Ns_{t'} <0$. Meanwhile, since at $t= t_*$ each $C_i$ for $i \in \{1,\ldots, k\}$ is a holomorphic curve, we have $\sum _{i=1}^k {\rm area}_{C_i}(t_*) >0$, which is equivalent to $Mr_{t_*} + Ns_{t_*} >0$. Hence, since the function $Mr_t + Ns_t$ is continuous with respect to time $t$, the intermediate value theorem implies that there exists $t_{{\dagger} } \in (t_*, t')$ such that

(19)\begin{equation} Mr_{t_{{\dagger}}} + N s_{t_{{\dagger}}} = 0. \end{equation}

There are two cases as follows, see Figure 11, depending on the relative position between $(r_{t_*}, s_{t_*})$ and the line $Mr + Ns =0$. In the left picture, the condition $Mr_{t_*} + N s_{t_*} >0$ implies that the lower half plane given by the division of the line $Mr + Ns = 0$ is the positive region. In particular, for any $r = s(>0)$, we have

\[ Mr + Nr >0 \quad \mbox{which implies that } M + N >0. \]

This contradicts Lemma 4.3. In the right picture, the condition $Mr_{t_*} + N s_{t_*} >0$ implies that the upper half plane given by the division of the line $Mr + Ns = 0$ is the positive region. By our hypothesis that $ {r(t)}/{s(t)}$ is non-increasing, $\gamma |_{t >t_*}$ remains above the line $Mr + Ns = 0$, contradicting the existence of a $t_{{\dagger} }$.

Figure 11. Relative position between $(r_{t_*}, s_{t_*})$ and $Mr + Ns = 0$.

4.2 Proof of obstruction (I) in Theorems 2.11 and 2.12

Outline. We will trace the moduli space of the cylinder constructed in Lemma 4.1 under our Lagrangian isotopy. The idea is that if an isotopy as in obstruction (I) existed, then the area of any curves would decrease to $0$, and hence as holomorphic curves have positive area we deduce there must be a degeneration to a non-trivial building. But by Proposition 4.4, we can then work instead with a moduli space of curves from the limiting building whose areas would also decrease to $0$, giving a contradiction.

Proof. Suppose by contradiction that a path $\gamma = \{(r_t, s_t)\}_{t \in [0,1]}$ as in obstruction (I) admits a lift $\{L_t\}_{t \in [0,1]}$.

Denote by ${\mathcal {M}}_1(t)$ the moduli space consisting of cylinders with positive end $\gamma _b$ and negative end $\gamma _{k,1}$ on the moving boundary $L_t$. By Lemma 4.1, we know that ${\mathcal {M}}_1(0) \neq \emptyset$, that is, there exists a $C^{(1)} \in {\mathcal {M}}_1(0)$. As before, the area of curves in ${\mathcal {M}}_1(t)$ will be denoted by ${\rm area}_{C^{(1)}}(t)$.

By (11) each $C^{(1)}_t \in \mathcal {M}_1(t)$ has its index equal to

\begin{align*} {\rm ind}(C^{(1)}_t) & = (1+1-2) + 0 + (2 + 2k + 1) - \big (2(k+1) + \tfrac{1}{2} - \tfrac{1}{2}\big) \\ & = (2k+3) - (2k+2) = 1. \end{align*}

Hence, since ${\rm ind}(C^{(1)}_t) = 1> -1 = 2 \cdot {\rm genus} (C^{(1)}_t) - 2 + \# \Gamma _{\rm even}$, where $\# \Gamma _{\rm even}$ is the number of asymptotic ends with even CZ indices, curves in ${\mathcal {M}}_1(t)$ are automatically regular (see Theorem 1 in [Reference WendlWen10]). Thus, ${\mathcal {M}}_1(t)$ is non-empty for an open subset of $t \in [0,1]$.

Now suppose that the moduli space is compact. Then $\mathcal {M}_1(0) \neq \emptyset$ together with automatic regularity implies that ${\mathcal {M}}_1(1) \neq \emptyset$. However, any curve $C^{(1)}_1 \in \mathcal {M}_1(1)$ has

(20)\begin{equation} {\rm area}(C^{(1)}_1) = b - kr_1 - s_1 >0, \quad \mbox{which means} \ \frac{r_1}{a} + \frac{s_1}{b} < 1. \end{equation}

This is in contradiction to our assumption that $(r_1, s_1) \notin \mu (E(a,b))^+$. Hence, the moduli space is not compact and we have a degeneration at some time $t_* \in [0,1]$, that is, we have a holomorphic building $C^{(1)}_{\rm lim}$ which is a limit of curves $C_n \in \mathcal {M}_1(t_n)$ with $t_n \to t_*$. The argument (20) implies that the degeneration point $(r_{t_*}, s_{t_*})$ in fact occurs inside ${\rm int}(\mu (E(a,b)^+)$.

By § 3.3 in [Reference Hind and OpshteinHO20], $C^{(1)}_{\rm lim}$ contains a top-level curve, denoted by $C^{(2)}$, of index at least $e$ and with $e$ negative ends. We claim that $C^{(2)}$ must have a positive end; we can then check that for index reasons it must be asymptotic to $\gamma _b$. Indeed, suppose not, then $C^{(2)}$ can be thought of as a curve in $\mathbb {C}^2 \setminus L_{t^*}$ and by Lemma 3.7 in [Reference Hind and OpshteinHO20] the curve $C^{(2)}$ has area at least $r_{t_*}$. As any curve in ${\mathcal {M}}_1(t_*)$ has area $b - kr_{t_*} - s_{t_*}$, we must then have $r_{t_*} \leq {\rm area}_{C^{(1)}}(t_*) < b - kr_{t_*} - s_{t_*}$, which is equivalent to $(k+1)r_{t_*} + s_{t_*} < b$, contradicting our assumption (I) on the path.

Denote by $\mathcal {M}_2(t)$ the moduli space corresponding to the curve $C^{(2)}$ above with the moving boundary condition on $L_t$ for $t \in [t_*,1]$ and again we define ${\rm area}_{C^{(2)}}(t)$ to be the area of curves. If needed, consider $\mathcal {M}_2(t)$ with extra marked points, so the resulting index is exactly equal to $e$ (instead of at least $e$). Then Proposition 4.4 implies that curves in the moduli space of $C^{(2)}$ will not survive until $t =1$. In fact, there must be a degeneration before time $t_1$, where $\gamma |_{t=t_1}$ lies outside $\mu (E(a,b))$. Indeed, otherwise there will exists some curve denoted by $C^{(2)}_{t_1} \in \mathcal {M}_2(t_1)$ that results in the following contradiction,

\[ {\rm area}(C_{t_1}^{(2)})>0 \quad \mbox{but} \ {\rm area}_{C^{(2)}}(t_1) \leq {\rm area}_{C^{(1)}}(t_1) \le 0. \]

Therefore, the moduli space $\mathcal {M}_2(t)$ is not compact over $[t_*,1]$ and it will degenerate again at some $t_{**} \in (t_*, t_1)$. We repeat the argument above by considering the limit building $C^{(2)}_{\rm lim}$ from a degeneration of $C^{(2)}$ and pick the preferred top curve $C^{(3)}$ as above. Next by Proposition 4.4 again, we will get a further degeneration, now within the time interval $(t_{**}, t_1)$, and so on.

We note that this process must terminate, and hence the desired contradiction is given by an inductive argument. To see this, suppose we have a sequence of $C^{(k)}$ asymptotic to $L_k \to L_{\infty }$. Using Fukaya's trick again there exists a family of global diffeomorphisms mapping our $L_k$ to an $\tilde {L}_{\infty }$, where $\tilde {L}_{\infty }$ lies very close to $L_{\infty }$ but has rational area class, that is, $\tilde {L}_{\infty }$ is Hamiltonian isotopic to an $L(\delta m, \delta n)$ with $m,n \in \mathbb {N}$. We may assume that the almost complex structures on the complement of the $L_k$ push forward to tame almost complex structures on the complement of $\tilde {L}_{\infty }$. By construction, our $C^{(k)}$ have a positive end, and in a degeneration of $C^{(k)}$ the curve $C^{(k+1)}$ is the only curve in the limiting building with an end on $\partial E(a,b)$. Therefore, the other top-level curves in the limit, when pushed forward to the complement of $\tilde {L}_{\infty }$, have area at least $\delta$. Hence, in the complement of $\tilde {L}_{\infty }$, the areas of the $C^{(k)}$ decrease by at least $\delta$ at each step, and so indeed our recursion is finite.

4.3 Proof of obstruction (I) in Theorem 2.13

Outline. The proof of the obstruction to the path lifting in the polydisks $P(c,d)$ is similar to the proof in balls and integral ellipsoids. The only variation is that we start from a curve that is different from that from Lemma 4.1 or Remark 4.2.

Proof. Suppose by contradiction that path $\gamma = \{(r_t, s_t)\}_{t \in [0,1]}$ admits a lift $\{L_t\}_{t \in [0,1]}$. Compactify $P(c,d)$ to $S^2(c) \times S^2(d)$ with two factors having areas $c$ and $d$. Denote by $\mathcal {M}_1(t)$ the moduli space consisting of finite energy planes that intersect $S^2(c) \times \{\infty \}$ only once (i.e. it is of bidegree $(0,1))$ and has its negative asymptotic end $\gamma _{(0,1)}$ on the moving boundary $L_t$ for $t \in [0,1]$. The moduli space $\mathcal {M}_1(0)$ is non-empty since we can simply write out (the image of) a curve $C^{(1)}$ explicitly as follows:

\[ C^{(1)} = \{(z,w) \in \mathbb{C}^2 \,| \, z = {\rm constant}, \ \pi|w|^2 >d\}, \]

and by (13) ${\rm ind}(C^{(1)}) = -1 + 2\cdot 2 +2\cdot (-1) = 1$, plus the automatic regularity can be verified.

Now, suppose that the moduli space is compact, then $\mathcal {M}_1(1) \neq \emptyset$. Any curve $C^{(1)}_1 \in \mathcal {M}_1(1)$ satisfies ${\rm area}(C_1^{(1)}) = d - s_1 >0$, which contradicts our assumption that $(r_1, s_1) \notin \mu (P(c,d))^+$. Hence, we have a degeneration at some time $t_* \in [0,1]$ with a limit curve a holomorphic building $C_{\rm lim}^{(1)}$. By § 3.3 in [Reference Hind and OpshteinHO20], $C_{\rm lim}^{(1)}$ contains a top-level curve $C^{(2)}$ of index at least $e$ with $e$ negative ends. This curve $C^{(2)}$ intersects $S^2(c) \times \{\infty \}$ since otherwise, by [Reference Hind and OpshteinHO20, Lemma 3.7], it has area at least $r_{t_*}$ and then

\[ r_{t_*} \leq {\rm area}_{C^{(2)}}(t_*) < d - s_{t_*} \]

which contradicts our assumption (I) on the path.

Denote by $\mathcal {M}_2(t)$ the moduli space corresponding to the curve $C^{(2)}$ above with moving boundary on $L_t$ for $t \in [t_*,1]$. It is readily checked that the same conclusion in Lemma 4.3 holds if we replace the assumption ‘contains the positive end $\gamma _b$’ with ‘intersects $S^2(c) \times \{\infty \}$’. Then Proposition 4.4, whose argument only involves areas, implies that $\mathcal {M}_2(t)$ is not compact, so it will degenerate again at $t_{**} \in (t_*, t_1)$ where $t_1$ is the first time that $\gamma$ intersects the line segment $\{(r, d) \in \mathbb {R}_{>0}^2 \,| \, 0< r< {c}/{2}\}$. The rest of the proof goes exactly the same as that in § 4.2, and thus we get the desired conclusion.

5.1 Path lifting criterion

In this section we construct our path lifts. The main result is the following theorem giving a Hamiltonian isotopy with controlled support between an inclusion and a rolled up Lagrangian. The domains of interest are the $Q(a,b)$ defined here.

Definition 5.1 The domain $Q(a,b) \subset \mathbb {C}^2$ is defined by

\[ Q(a,b) = \Omega_{q(a,b)} := \{(z,w) \in \mathbb{C}^2 \,|\, (\pi|z|^2, \pi|w|^2) \in q(a,b)\} \]

where

(21)\begin{equation} q(a,b) = \bigg\{(x,y) \in \mathbb{R}^2 \, \bigg| \, 0 \leq x \leq 2a - \frac{ay}{a+b}, 0 \leq y \leq a+b \bigg\}, \end{equation}

a quadrilateral region in $\mathbb {R}^2$.

We use coordinates $(z,w)$ on $\mathbb {C}^2$. We define a product Lagrangian torus $\tilde {L}(a,b)$ in $\mathbb {C}^2$ by

\[ \tilde{L}(a,b) :=\partial D(a) \times \partial ([0,1] \times [0,b]), \]

where $[0,1] \times [0,b]$ is a rectangle in $w$-plane. Let $w = x + \sqrt {-1} y$. Note that a smoothing of $\tilde {L}(a,b)$ is symplectomorphic to our standard product by a symplectomorphism on $\mathbb {C}$ that interchanges the disk $D(b)$ with the rectangle $[0,1] \times [0,b]$ in the $w$-plane.

Consider a product Hamiltonian function

\[ H(z,w) = \chi(y) \cdot G(z), \]

where $\chi : [0,a+\epsilon ] \to [0,1]$ is a smooth function, extended by zero outside $[0, a+\epsilon ]$, such that for a sufficiently small $\delta >0$ we have:

1. (i) $\chi |_{[0,\delta ]} = 1$, $\chi |_{[a+\epsilon -\delta, a+\epsilon ]} = 0$, and $\chi |_{[2\delta, a+\epsilon -2\delta ]}$ is linear of slope $-1/(a + \epsilon + \delta$);

2. (ii) $\chi |_{[\delta,2\delta ]}$ is concavely decreasing and $\chi |_{[a+\epsilon -2\delta, a+\epsilon -\delta ]}$ is convexly decreasing.

See Figure 12 for the pictures of $\chi (y)$ and $-\chi '(y)$. The function $G(z)$ is defined by Lemma 5.4.

Figure 12. Graphs of $\chi (y)$ and $-\chi '(y)$.

By the product rule we have $dH(z,w) = \chi '(y) G(z) \,{{d}y} + \chi (y) \,dG(z)$. Using the standard symplectic structure on $\mathbb {C}^2$, that is, $\omega _{\rm std} = \omega _{\rm std,\mathbb {C}}+{{d}x} \wedge {{d}y}$, we can write down the Hamiltonian vector field $X_H$ (via the Hamiltonian equation $-dH = \iota _{X_H} \omega _{\rm std}$), namely

\[ X_H(z,w) = - \chi'(y) G(z) \frac{\partial}{\partial x} + \chi(y) X_{G(z)}(z), \]

where $X_{G(z)}$ is the Hamiltonian vector field generated by the function $G(z)$ above with respect to $\omega _{\rm std,\mathbb {C}}$. Therefore, the resulting Hamiltonian diffeomorphism of $H$ is

(22)\begin{equation} \phi^t_H(z,w) = (\phi_{\chi \cdot G}^t(z), (x- t\chi'(y)G(z), y)). \end{equation}

Now, apply the Hamiltonian diffeomorphism $\phi ^1_H$ to $\tilde {L}(a,b)$, and Figure 13 shows a schematic picture of how the projection to the $w$-plane changes (indicated by the shaded part). For later use, we label the rectangle $S: = [0,1] \times [a+\epsilon, b]$. By our choice of $\epsilon$ and $\delta$ as above,

\[ \max_{\tilde{L}(a,b)} \{-\chi'(y) G(z)\} \leq \frac{a + \epsilon}{a+\epsilon + \delta} <1. \]

Therefore, the bottom part of the rectangle which is moved by $\phi _H^1$ is contained in the rectangle $[0,2] \times [0, a+\epsilon ]$ and touches neither the line $x = 1$ nor $x=2$.

Figure 13. Behavior of $\phi _H^1(L(a,b))$ on the $w$-plane.

On the other hand, for the behavior on the $z$-plane, there are two extreme cases.

1. (a) For $w \in I$, i.e. $w = (x,0)$ for $x \in [0,1]$, since $\chi (0) = 1$, the first factor in (22) is simply $\phi _G^1(z)$ which by definition displaces $\partial D(a)$ (since it displaces $D(a)$).

2. (b) For $w$ near $S$, where $S$ is the upper part of the rectangle as in Figure 13, since $\chi (y) = 0$, the second factor in (22) is identity, so $\partial D(a)$ stays the same.

Next, we also consider a Hamiltonian isotopy $\{\psi _t\}_{t \in [0,1]}$ on the $w$-plane that wraps the region $S$ around the deformed $w$-projection to overlap the rectangle $[0,1] \times [0,a+\epsilon ]$, see Figure 14. Although on the $w$-plane, there are intersection points of $\psi _t|_{S}(S)$ with the line segment $I$, by our construction, viewed in four dimensions there will be no extra intersections. Indeed, for any point $(z,w)$ with $w$ near $S$, the corresponding $z \in D(a)$, but by case (a) above for any point $(z,w)$ with $w \in I$, the corresponding $z$ lies in $\phi _G^1(D(a))$ which is disjoint from $D(a)$. Continuing this wrapping construction as shown in Figure 15, we can wrap all of $S$ into the shaded region Figure 15, which is a subset of $[0,1] \times [0,a+\epsilon ]$, union with an arbitrarily small neighborhood, say of width $\delta \ll \epsilon$, of the sides $[0,1] \times \{0\}$, $\{0\} \times [0, a+ \epsilon ]$ and $[0,1] \times \{a+\epsilon \}$.

Figure 14. Wrap the region $S$.

Figure 15. More wrappings.

The Hamiltonian isotopy of interest is the image, under a symplectomorphism ${\mathbb 1} \times \Phi$, of the concatenation of the two constructions above,

(23)\begin{equation} \{\phi_H^t(\tilde{L}(a,b))\}_{t \in [0,1]} \#\{\psi_t(\phi_H^1(\tilde{L}(a,b)))\}_{t \in [0,1]}. \end{equation}

Here $\Phi$ is a symplectomorphism of the $w$-plane that maps

1. (i) $[-\delta,2] \times [-\delta,a+\epsilon + \delta ] \cup [0,1] \times [a,y]$ into $D(a+y+ 3\epsilon )$ for $y \geq a+\epsilon$; and

2. (ii) each horizontal rectangle $[-\delta,2] \times [-\delta,y]$ into $D(2y+\epsilon )$ for $y \in [0,a+\epsilon ]$.

See Figure 16 for a layer-by-layer picture of such a map $\Phi$ for part (ii). The existence of such symplectomorphisms follows Lemma 3.1.5 in the book [Reference SchlenkSch05].

Figure 16. The map $\Phi$ on $[0,2] \times [0, a+\epsilon ]$.

Note that both $\Phi ( [0,1] \times [0,b] )$ and $D(b)$ lie inside $D(a+b+ 3\epsilon )$. Therefore, without loss of generality, we may assume $\Phi ( [0,1] \times [0,b] ) = D(b)$, that is, our isotopy starts at the standard product $L(a,b)$.

We need to keep track of the moment image $(\pi |z|^2, \pi |w|^2)$ along the Hamiltonian isotopy and have the following observations.

Case $\alpha$. Let $(z,w) = (z, (x,y)) \in \tilde {L}(a,b)$ with $y \in [0, a+ \epsilon ]$. We denote the image of $(z,w)$ under $\phi ^t_H$ by $(z', w')$, and this point is described by (22). In particular, the $y$ coordinate is invariant under the flow. Hence, under the symplectomorphism ${\mathbb 1} \times \Phi$, we get the relation

(24)\begin{equation} \pi|\Phi(w')|^2 \leq 2y + \epsilon \quad \mbox{and}\quad z' \in D(2a - y + 4\epsilon). \end{equation}

For the bound on $\pi |\Phi (w')|^2$ we are applying condition (ii) on $\Phi$. For the condition on $z'$, by (22) we note that $z' = \phi ^t_{\chi G}(z) = \phi ^{t\chi }_G(z)$ where $z \in D(a)$, since $G$ is independent of time. But

\begin{align*} \chi(y) \le \frac{a+\epsilon - y}{a+\epsilon + \delta} < 1 - \frac{y}{a+ \epsilon + \delta} \end{align*}

and so by Lemma 5.4

\begin{align*} z' \in D\left(a + a - \frac{ay}{a+\epsilon + \delta} + \epsilon\right) \subset D(2a - y + 4\epsilon) \end{align*}

as required.

The relation (24) yields

\begin{align*} 2a - y + 4 \epsilon \leq 2a - \frac{\pi |\Phi(w')|^2 - \epsilon}{2} + 4 \epsilon \leq 2a - \frac{\pi |\Phi(w')|^2}{2} + 5 \epsilon, \end{align*}

and then, importantly,

\begin{align*} \pi|z'|^2 \leq 2a - y + 4 \epsilon \leq 2a - \frac{\pi |\Phi(w')|^2}{2} + 5 \epsilon. \end{align*}

Such points $(z', w')$ are then fixed by the flow $\psi _t$.

Case $\beta$. Let $(z,w) = (z, (x,y)) \in \tilde {L}(a,b)$ with $y \in [a +\epsilon, b]$. Such points are fixed by $\phi ^t_H$, and the flow of $\psi _t$ fixes the $z$ coordinate and leaves the $w$ coordinate inside an $\epsilon$ neighborhood of $[0,1] \times [0,b]$. This region is mapped close to $D(a+b)$ by $\Phi$, using property (i).

Therefore, we obtain a Lagrangian isotopy with its moment image lying in regions corresponding to either case $\alpha$ or case $\beta$ as above. The following graph illustrates these regions, for brevity, when $\epsilon \to 0$.

The union of these regions is contained in an arbitrarily small neighborhood of the quadrilateral region defined by $q(a,b) \subset \mathbb {R}^2$ in our hypothesis. We note in ${\rm Case}\,\beta$ the points $\psi _1(z,w)$ lie in an $\epsilon$ neighborhood of $D(a) \times [0,1] \times [0,a]$. Therefore, by property (i) of $\Phi$, the image under ${\mathbb 1} \times \Phi$ lies close to $D(a) \times D(2a)$ and the moment image is described only by the lower trapezium in the graph above. In other words, at $t=1$ the moment image of the Lagrangian isotopy (23) lies in an arbitrarily small neighborhood of $q(a,a)$. Hence, we complete the proof.

In constructing lifts, our strategy will be to divide $\gamma$ into segments, and on each segment lift either by inclusions or by the rolled up Lagrangian embeddings described in Theorem 5.2. The following lemma says that a Hamiltonian isotopy is enough to piece these lifts together. In other words, we are free to concatenate paths with endpoints lying in the same path component of a fiber.

Let $X_{\Omega }$ be a toric domain in $\mathbb {R}^4$ with moment image $\Omega$. Recall that a Type-I path is a path that entirely lies in $\Omega ^+: = \Omega \cap \{r \leq s\}$ and any other path with starting point in $\Omega ^+$ is a Type-II path. The following corollary of Theorem 5.2 provides our key sufficient condition to lift a general path.

5.2 Proof of criterion (II) in Theorems 2.11, 2.12, and 2.13