Boundedness of Differential of Symplectic Vortices in Open Cylinder Model

Abstract

Let G be a compact connected Lie group, $(X,\omega ,\mu )$ a Hamiltonian G-manifold with moment map $\mu$, and Z a codimension-2 Hamiltonian G-submanifold of X. We study the boundedness of the differential of symplectic vortices $(A,u)$ near Z, where A is a connection 1-form of a principal G-bundle P over a punctured Riemann surface $\mathring{\Sigma }$, and u is a G-equivariant map from P to an open cylinder model near Z. We show that if the total energy of a family of symplectic vortices on $\mathring{\Sigma }\cong [0,+\infty )\times S^1$ is finite, then the A-twisted differential $d_{A}u(r,\theta )$ is uniformly bounded for all $(r,\theta )\in [0,+\infty )\times S^1$.

1. Introduction

The boundedness of the differential of symplectic vortices is crucial for establishing the compactness theorem for the moduli space of symplectic vortices, which is important for defining Hamiltonian Gromov–Witten invariants [1,2,3,4] for symplectic manifolds with a Hamiltonian G-action. We remark that the symplectic vortex is an important object as well as a special case in the study of mathematical theory of the Gauged Linear Sigma Model (see [5,6,7]). When one studies the relative version of Hamiltonian Gromov–Witten invariants, it is natural to consider the boundedness of the differential of symplectic vortices relative to a Hamiltonian G-submanifold of codimension-2. In this article, we study such a problem for the simpler case of an open cylinder model for symplectic vortices near a Hamiltonian G-submanifold. Note that such an open cylinder model before has been used in the study of constructing relative Gromov–Witten invariants and establishing degeneration formula of Gromov–Witten invariants [8,9,10,11].

Let $(X,\omega )$ be a symplectic manifold, and G be a compact Lie group which acts covariantly on $(X,\omega )$ by symplectomorphisms. This means that there is a smooth group homomorphism from Lie group G to the group of symplectomorphisms $\mathrm{Symp}(X,\omega )$: $g\mapsto {\varphi }_g$. A symplectic manifold $(X,\omega )$ with a Hamiltonian G-action is simply called a Hamiltonian G-manifold. Let $\mathfrak{g}:=\mathrm{Lie}(G)$ be the Lie algebra of G, and ${\mathfrak{g}}^{*}$ the dual of the Lie algebra $\mathfrak{g}$. One can identify ${\mathfrak{g}}^{*}\cong \mathfrak{g}$ via an invariant inner product $⟨·,·⟩$ on the Lie algebra. We remark that our result of this paper does not depend on the choice of such an invariant inner product on $\mathfrak{g}$. The crucial notion for Hamiltonian manifolds is the moment map, which is a G-equivariant smooth map $\mu :X\to {\mathfrak{g}}^{*}$ generating the Hamiltonian G-action, i.e., the following relation holds:

$$⟨d\mu (x)\tau ,\xi ⟩=\omega (V_{\xi }(x),\tau )$$

(1)

for all $\xi \in \mathfrak{g}$, $x\in X$, $\tau \in T_{x}X$, where $V_{\xi }$ is the Hamiltonian vector field generated by the infinitesimal action of $\xi$ defined by

$$V_{\xi }(x):={\displaystyle \frac{d}{dt}}{|}_{t=0}{\varphi }_{exp(t\xi )}x,\forall x\in X.$$

(2)

We denote a Hamiltonian G-manifold by $(X,\omega ,\mu )$.

Given a Riemann surface $\Sigma$, let $\pi :P\to \Sigma$ (simply $P_{\Sigma }$ or P) be a principal G-bundle over $\Sigma$, A a connection 1-form on P which is an equivariant 1-form in ${\Omega }_{ad}^1(P,\mathfrak{g})$, and $F_A\in {\Omega }_{ad}^2(P,\mathfrak{g})$ the curvature form. One can consider a G-equivariant map $u:P\to X$. Let $P{\times }_{G}X$ be the associated fiber bundle over $\Sigma$. The points on $P{\times }_{G}X$ can be denoted by $[p,x]$ as an equivalence class for $p\in P$ and $x\in X$. Note that $[p_1,x]=[p_2,x]$ for ∀$p_1,p_2\in P$, satisfying $\pi (p_1)=\pi (p_2)$ since G acts freely on P. One may equivalently look at G-equivariant map $u:P\to X$ as a section $u:\Sigma \to P{\times }_{G}X$, with the same notation, defined by

$$z\mapsto [p,u(p)]\in P{\times }_{G}X,\pi (p)=z\in \Sigma .$$

Take an $\omega$-compatible and G-invariant almost complex structure J on X and a (fixed) complex structure j on $\Sigma$. A symplectic vortex is a pair $(A,u)$ satisfying the following symplectic vortex equations

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\overline{\partial }}_{J,A}(u)=0,\hfill \\ {\ast }_{\Sigma }{F}_A+\mu (u)=0,\hfill \end{array}\right.\end{array}$$

(3)

where ${\overline{\partial }}_{J,A}(u):={(d_{A}u)}^{0,1}={\displaystyle \frac{1}{2}}(d_{A}u+J\circ d_{A}u\circ j)\in {\Omega }_{J_u}^{0,1}(\Sigma ,u^{*}TX/G)$, $d_{A}u:=du+V_A(u)$ is the A-twisted differential of u. For a fixed A, an equivariant map $u:P_{\Sigma }\to X$ satisfying the first equation of (3) is called a $(J,A)$-holomorphic map.

Let Z be a compact codimension-2 Hamiltonian submanifold of X, and $\Sigma$ a Riemann surface with some punctures. Then, the pair $(X,Z)$ can be looked at as a result of symplectic cutting from a Hamiltonian G-manifold $(\widehat{X},\widehat{\omega },\widehat{\mu })$. Let ${\omega }_Z{=\omega |}_Z$ be the restricted symplectic structure on Z and $\pi :SZ\to Z$ the circle (sphere) bundle of the symplectic normal $NZ$. There exists a natural $S^1$-action on the fiber of $SZ$. The product $\mathbb{R}\times SZ$ with a suitable symplectic form and Hamiltonian G-action commuting with the $S^1$-action is called an open cylinder model. One can also consider an open cylinder model for $X\backslash Z$ to be $X_0\cup \{[0,\infty )\times SZ\}$, where $X_0$ are compact symplectic submanifolds with a boundary. Let $\alpha$ be the connection 1-form of $SZ$, and J the almost complex structure on $\mathbb{R}\times SZ$.

For $a,b\in \mathbb{R}$ with $a<b$, denote by ${\mathcal{T}}_{a,b}$ the set of all orientation-preserving diffeomorphisms $\varphi :\mathbb{R}\to (a,b)$. Denote by $\pi :\mathbb{R}\times SZ\to Z$ the projection. For each $\varphi \in {\mathcal{T}}_{a,b}$, we define a closed 2-form on $\mathbb{R}\times SZ$

$${\omega }_{\varphi }:={\pi }^{*}{\omega }_Z+d(\varphi \alpha ),$$

(4)

which is symplectic and tames J if $|a|$ and $|b|$ are sufficiently small. Moreover, for each $\varphi \in {\mathcal{T}}_{a,b}$, one can obtain a moment map ${\mu }_{\varphi }$ on the symplectic manifold $(\mathbb{R}\times SZ,{\omega }_{\varphi })$. Thus, for an open cylinder model, if $|a|$ and $|b|$ are sufficiently small, there exists a ${\mathcal{T}}_{a,b}$-family of Hamiltonian G-manifolds ${\left\{(\mathbb{R}\times SZ,{\omega }_{\varphi },{\mu }_{\varphi })\right\}}_{\varphi \in {\mathcal{T}}_{a,b}}$.

Denote by $\mathring{\Sigma }:=\Sigma \backslash \left\{p\right\}$ a Riemann surface with one puncture carrying a complex structure j. Let J be an almost complex structure compatible with (tamed by) ${\omega }_{\varphi }$ for all $\varphi \in {\mathcal{T}}_{a,b}$. For a fixed $\varphi \in {\mathcal{T}}_{a,b}$, a $\varphi$-symplectic vortex on $\mathring{\Sigma }$ to $(\mathbb{R}\times SZ,{\omega }_{\varphi },{\mu }_{\varphi })$ is a pair $(A,u)=(A_{\varphi },u_{\varphi })$ satisfying the following $\varphi$-symplectic vortex equations

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\overline{\partial }}_{J,A}(u)=0,\hfill \\ {\ast }_{\Sigma }{F}_A+{\mu }_{\varphi }(u)=0.\hfill \end{array}\right.\end{array}$$

(5)

For $\varphi$-symplectic vortices $(A_{\varphi },u_{\varphi })$, we define the energy by

$$E(A_{\varphi },u_{\varphi };\mathring{\Sigma })={\int }_{\mathring{\Sigma }}\left(u_{\varphi }^{*}{\omega }_{\varphi }-d⟨{\mu }_{\varphi }(u_{\varphi }),A_{\varphi }⟩\right).$$

(6)

Then, we define the total energy of the ${\mathcal{T}}_{a,b}$-family of symplectic vortices by

$$E_{a,b}(\mathring{\Sigma }):=\underset{\varphi \in {\mathcal{T}}_{a,b}}{sup}E(A_{\varphi },u_{\varphi };\mathring{\Sigma }),$$

(7)

which may not be finite.

In particular, when we consider vortices on punctured disk $\mathring{D}=D\backslash \left\{0\right\}$, we can use the biholomorphic equivalence $[0,+\infty )\times S^1\to \mathring{D}:(r,\theta )\mapsto e^{-2\pi (r+\mathrm{i}\theta )}$ to just consider vortices on half-infinite cylinder $[0,+\infty )\times S^1$ carrying the metric ${(dr)}^2+{(d\theta )}^2$ and the standard complex structure $\mathfrak{j}({\partial }_r)={\partial }_{\theta }$, with respect to a fixed trivialization of the principal bundle $P_{[0,+\infty )\times S^1}$. It turns out that these reparametrizations have no impact on the energy of the ${\mathcal{T}}_{a,b}$-family of symplectic vortices, that is, $E_{a,b}(\mathring{D})=E_{a,b}([0,+\infty ))$.

In this paper, we consider only this simpler case and study the boundedness of the differential of symplectic vortices on $\Sigma =[0,+\infty )\times S^1$ in the open cylinder model $\mathbb{R}\times SZ$. The following is our main result.

Theorem 1.

Take any $\varphi \in {\mathcal{T}}_{a,b}$, given a ϕ-symplectic vortex $(A,u)$ such that $u:[0,+\infty )\times S^1\to \mathbb{R}\times SZ$ and $A\in {\mathcal{A}}^{1,p}(P)$ satisfying (5). Suppose that the following condition on total energy (7) holds:

$$E_{a,b}([0,+\infty )\times S^1)<\infty .$$

(8)

Then, there exists a constant $C>0$ such that

$$|d_{A}u(r,\theta )|\le C$$

(9)

for all $(r,\theta )\in [0,+\infty )\times S^1$.

We remark that the result above implies the property of the boundedness of A-twisted differential of a $(J,A)$-holomorphic curve in the open cylinder model can be verified, provided that the energies of a family of symplectic vortices can be controlled. This phenomenon is similar to the one in the study of boundedness of differential of a J-holomorphic curve in the open cylinder model and relative Gromov–Witten invariants [8]. Such a result might be useful in the study of the compactness of moduli space of the relative version of Hamiltonian Gromov–Witten invariants relating to the Gromov–Witten invariants relative to the codimension-2 symplectic submanifold $Z//G$ in the symplectic reduction $X//G$ (see [12,13]). As a potential application, such relative HGW invariants would be crucial for establishing the degeneration formula of HGW invariants.

2. Symplectic Vortices in Open Cylinder Model

2.1. Energy and Local Description

The Yang–Mills–Higgs energy of a vortex $(A,u)$ (satisfying (3)) on $\Sigma$ is defined by

$$\begin{array}{ccc}\hfill E(A,u;\Sigma )& :=& {\displaystyle \frac{1}{2}}{\int }_D\left(|d_A{u|}_J^2+|F_A{|}_{\mathfrak{g}}^2+{|\mu (u)|}_{\mathfrak{g}}^2\right)d{\mathrm{vol}}_{\Sigma }\hfill \\ & =& {\int }_{\Sigma }\left({\displaystyle \frac{1}{2}}|d_A{u|}_J^2+{|\mu (u)|}_{\mathfrak{g}}^2\right)d{\mathrm{vol}}_{\Sigma }.\hfill \end{array}$$

(10)

This energy is invariant under the action of the gauge group $\mathcal{G}(P)$. From the Proposition 3.1 of [1], we know that for a vortex $(A,u)$ on $\Sigma$

$$E(A,u;\Sigma )={\int }_{\Sigma }\left({(d_{A}u)}^{*}\omega -⟨\mu (u),F_A⟩\right)={\int }_{\Sigma }\left(u^{*}\omega -d⟨\mu (u),A⟩\right),$$

(11)

where $u^{*}\omega -d⟨\mu (u),A⟩$ is a horizontal and G-equivariant 2-form on P and descends to a 2-form on $\Sigma$. In particular, if $\Sigma$ is a closed surface, the energy of a vortex can be written as

$$E(A,u;\Sigma )=⟨[\omega -\mu ],\left[u_G\right]⟩,$$

where $\left[u_G\right]$ is an equivariant homology class in $H_2^G(X,\mathbb{Z})$, $[\omega -\mu ]\in H_G^2(X)$ is the equivariant cohomology class defined by the equivariant closed 2-form $\omega -\mu \in {\Omega }_G^2(X)$, and ${\Omega }_G^{*}(X)$ denotes the space of equivariant polynomials from $\mathfrak{g}$ to ${\Omega }^{*}(X)$. Note that the moment map condition asserts that the polynomial $\mathfrak{g}\to {\Omega }^{*}(X):\xi \mapsto \omega -⟨\mu ,\xi ⟩$ is G-closed and hence defines an equivariant cohomology class.

From [2], we know that one can study the symplectic vortex equations in local coordinates. Let $D\subset \mathbb{C}$ be an open subset and write the complex coordinate as $s+\mathrm{i}t$. For simplicity, we suppose the metric on D is $d{s}^2+d{t}^2$, and the volume form is $d{\mathrm{vol}}_D=ds\wedge dt$. Let $\phi :D\to P$ be a local smooth section, which can be regarded as a lift of a holomorphic coordinate chart $\tilde{\phi }:D\to \Sigma$. A symplectic vortex $(A,u)$ determines a smooth triple $(\Phi ,\Psi ,v)$ on D, satisfying

$${\phi }^{*}A=\Phi ds+\Psi dt,v=u\circ \phi ,$$

(12)

where $\Phi ,\Psi \in C^{\infty }(D,\mathfrak{g})$, $v\in C^{\infty }(D,X)$. The local version symplectic vortex equations for (3) are the following nonlinear partial differential equations on D

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\partial }_{s}v+V_{\Phi }(v)+J\left({\partial }_{t}v+V_{\Psi }(v)\right)=0,\hfill \\ {\partial }_s\Psi -{\partial }_t\Phi +[\Phi ,\Psi ]+\mu (v)=0.\hfill \end{array}\right.\end{array}$$

(13)

For the triple $(\Phi ,\Psi ,v)$ that is a vortex located on D, its Yang–Mills–Higgs energy is expressed as

$$E(\Phi ,\Psi ,v;D)=E(A,u;D)={\int }_D\left(v^{*}\omega -d⟨\mu (v),\Phi ds+\Psi dt⟩\right).$$

(14)

In particular, we can consider a cylinder $\Sigma =\mathbb{R}\times S^1$ (or $[0,\infty )\times S^1$, $[-l,l]\times S^1$) carrying the flat metric ${(dr)}^2+{(d\theta )}^2$ and the standard complex structure $\mathfrak{j}({\partial }_r)={\partial }_{\theta }$ with respect to a fixed trivialization of the principal bundle $P_{\mathbb{R}\times S^1}$. Let $(A,u)$ be a symplectic vortex on $\mathbb{R}\times S^1$ in the temporal gauge, i.e.,

$$u\in C_G^{\infty }(P_{\mathbb{R}\times S^1},X),A=d+\xi (r,\theta )d\theta ,$$

(15)

where d denotes the trivial connection and $\xi :\mathbb{R}\times S^1\to \mathfrak{g}$, satisfying the following symplectic vortex equations:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\partial }_{r}u+J\left({\partial }_{\theta }u+V_{\xi }(u)\right)=0,\hfill \\ {\partial }_r\xi +\mu (u)=0.\hfill \end{array}\right.\end{array}$$

(16)

Such a special symplectic vortex is also denoted by $(u,\xi )$.

2.2. $(X,Z)$ from Symplectic Cutting

We consider an operation called symplectic cutting [14]. Given a compact Lie group G and a Hamiltonian G-manifold $(\widehat{X},\widehat{\omega },\widehat{\mu })$, a compact hypersurface $Y:=H^{-1}(0)\subset \widehat{X}$ is obtained as the preimage of the regular value 0 of a (local) Hamiltonian function

$$H:\mathcal{N}(Y)\cong (-l,l)\times Y\to \mathbb{R}$$

which generates a free Hamiltonian $S^1$-action on Y. Assume that H is invariant under the G-action and the moment map $\widehat{\mu }$ is invariant under such Hamiltonian $S^1$-action. Then, (under some conditions) the symplectic cut spaces

$$\left(X^{+}:={\widehat{X}}_{H>0}\cup (H^{-1}(0)/S^1),{\omega }^{+}\right)$$

and

$$\left(X^{-}:={\widehat{X}}_{H<0}\cup (H^{-1}(0)/S^1),{\omega }^{-}\right)$$

have induced Hamiltonian G-actions with moment maps ${\mu }^{+}$ and ${\mu }^{-}$, respectively [14]. So $(X^{+},{\omega }^{+},{\mu }^{+})$ and $(X^{-},{\omega }^{-},{\mu }^{-})$ are also Hamiltonian G-manifolds whose symplectic forms satisfy

$${\omega }^{+}{{|}_{{\widehat{X}}_{H>0}}=\widehat{\omega },{\omega }^{+}|}_Z={\omega }_Z,$$

$${\omega }^{-}{{|}_{{\widehat{X}}_{H<0}}=\widehat{\omega },{\omega }^{-}|}_Z={\omega }_Z,$$

where $Z=\widehat{X}//S^1:=H^{-1}(0)/S^1$, with the induced symplectic form ${\omega }_Z$, is the symplectic $S^1$ reduction of $\widehat{X}$ as well as the common codimension-2 symplectic submanifold of $X^{+}$ and $X^{-}$ with dual normal bundles, denoted by $N_{+}Z$ and $N_{-}Z$, respectively. The moment maps satisfy for some sufficiently small $\delta >0$

$${\mu }^{+}=\widehat{\mu }{{|}_{{\widehat{X}}_{H>\delta }},{\mu }^{-}=\widehat{\mu }|}_{{\widehat{X}}_{H<-\delta }}.$$

Moreover, the descendant Hamiltonian G-actions on ${\overline{X}}_{H\ge 0}$ and ${\overline{X}}_{H\le 0}$ induce the same G-action on Z with an induced moment map denoted by

$${\mu }_Z={\mu }^{+}{{|}_Z={\mu }^{-}|}_Z.$$

We intend to study the geometry for such tuple $(X^{\pm },{\omega }^{\pm },Z,{\mu }^{\pm })$ via an open cylinder model. We remark that this kind of open cylinder models (without G-action) has been used in [8] for the construction of relative Gromov–Witten invariants. Simply denote each such object by $(X,\omega ,Z,\mu )$ as a symplectic cutting of $(\widehat{X},\widehat{\omega },\widehat{\mu })$.

2.3. Circle Bundle over Z and Open Cylinder Model

Denote by ${\displaystyle NZ:=\frac{{TX|}_Z}{TZ}}$ the symplectic normal bundle of Z in $(X,\omega )$, which can be looked as a complex line bundle by taking a complex structure. Denote by ${\omega }_Z{=\omega |}_Z$ the restricted symplectic structure on Z and by

$$\pi :SZ\to Z$$

(17)

the circle (sphere) bundle of $NZ$. We can identify the compact hypersurface Y of $\widehat{X}$ with $SZ$ and a tubular neighborhood $\mathcal{N}(Y)$ with $\mathbb{R}\times SZ$ (or $(-l,l)\times SZ$). The Hamiltonian $S^1$-action on Y is identified with the natural $S^1$-action on the fiber of $SZ$, induced from the complex structure on the normal bundle $NZ$. The restriction of the Hamiltonian vector field $X_H$ to Y corresponds to the characteristic vector field $\xi$ of the $S^1$-action on $SZ$. So we have a local Hamiltonian function, with the same notation,

$$H:(-l,l)\times SZ\to \mathbb{R}.$$

(18)

The product $\mathbb{R}\times SZ$ with a suitable symplectic form and Hamiltonian G-action commuting with the $S^1$-action is called an open cylinder model of $\mathcal{N}(Y)$. Also, we can consider an open cylinder model for $X\backslash Z$ as

$$X_0^{+}\cup \{[0,\infty )\times SZ\}\mathrm{or}{X}_0^{-}\cup \{(-\infty ,0]\times SZ\}$$

where $X_0^{\pm }$ are compact symplectic submanifolds with a boundary.

Let $\alpha$ be the connection 1-form of $SZ$, and J the almost complex structure on $\mathbb{R}\times SZ$. For $a,b\in \mathbb{R}$ with $a<b$, denote by

$${\mathcal{T}}_{a,b}:=\{\mathrm{orientation}-\mathrm{preserving}\mathrm{diffeomorphisms}\varphi :\mathbb{R}\to (a,b)\}.$$

We still denote by $\pi :\mathbb{R}\times SZ\to Z$ the projection. For each $\varphi \in {\mathcal{T}}_{a,b}$, define a closed 2-form on $\mathbb{R}\times SZ$

$${\omega }_{\varphi }:={\pi }^{*}{\omega }_Z+d(\varphi \alpha ),$$

(19)

which is symplectic and tames J if $|a|$ and $|b|$ are sufficiently small.

Note that the G-action on $(\mathbb{R}\times SZ,{\omega }_{\varphi })$ is induced from the one on $Y\cong SZ$ such that in the direction of $\mathbb{R}$, the action is trivial. In such an equivariant case, we can take the connection 1-form $\alpha$ on $SZ$ is G-invariant. Since the G-action along $\mathbb{R}$ is trivial, the 1-form $\varphi \alpha$ on $\mathbb{R}\times SZ$ is also G-invariant. By an observation in [15], $\alpha$ can be extended to a G-invariant 1-form ${\alpha }_N$ on $NZ$, then we have an ${\alpha }_N$-moment map ${\mu }_{{\alpha }_N}$ on the normal bundle $NZ$ and so similarly a $\varphi \alpha$-moment map ${\mu }_{\varphi \alpha }$ on $\mathbb{R}\times SZ$. So for sufficiently small $|a|$ and $|b|$ and each of $\varphi \in {\mathcal{T}}_{a,b}$, we have a moment map on the symplectic manifold $(\mathbb{R}\times SZ,{\omega }_{\varphi })$

$${\mu }_{\varphi }:={\mu }_{\mathbb{R}\times SZ}={\mu }_Z\circ \pi +{\mu }_{\varphi \alpha }.$$

(20)

Thus, we have a ${\mathcal{T}}_{a,b}$-family of Hamiltonian G-manifolds ${\left\{(\mathbb{R}\times SZ,{\omega }_{\varphi },{\mu }_{\varphi })\right\}}_{\varphi \in {\mathcal{T}}_{a,b}}$.

2.4. $\varphi$-Symplectic Vortices

Denote by $\mathring{\Sigma }:=\Sigma \backslash \left\{p\right\}$ a Riemann surface with one puncture carrying a complex structure j. Let J be an almost complex structure compatible with (tamed by) ${\omega }_{\varphi }$ for all $\varphi \in {\mathcal{T}}_{a,b}$.

From (1), we see that the moment map $\mu$ depends on the symplectic form $\omega$. Then, a symplectic vortex $(A,u)$ satisfying (3) also depends on $\omega$. For a fixed $\varphi \in {\mathcal{T}}_{a,b}$, a $\varphi$-symplectic vortex on $\Sigma$ to $(\mathbb{R}\times SZ,{\omega }_{\varphi },{\mu }_{\varphi })$ is a pair $(A,u)=(A_{\varphi },u_{\varphi })$ satisfying the following $\varphi$-symplectic vortex equations:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\overline{\partial }}_{J,A}(u)=0,\hfill \\ {\ast }_{\Sigma }{F}_A+{\mu }_{\varphi }(u)=0.\hfill \end{array}\right.\end{array}$$

(21)

When $A=\Phi ds+\Psi dt$, $v=u\circ \phi$, the local version $\varphi$-symplectic vortex equations for (21) is the following:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\partial }_{s}v+V_{\Phi }(v)+J\left({\partial }_{t}v+V_{\Psi }(v\right)=0,\hfill \\ {\partial }_s\Psi -{\partial }_t\Phi +[\Phi ,\Psi ]+{\mu }_{\varphi }(v)=0.\hfill \end{array}\right.\end{array}$$

(22)

When $\Sigma =\mathring{D}$ is a punctured disk, the study for local $\varphi$-symplectic vortex Equation (22) is the same as for (21).

For fixed sufficiently small a, b and a family of Hamiltonian G-manifolds $(\mathbb{R}\times SZ,{\omega }_{\varphi },{\mu }_{\varphi })$, we define the energy for a $\varphi$-symplectic vortices $(A_{\varphi },u_{\varphi })$ by

$$E(A_{\varphi },u_{\varphi };\mathring{\Sigma })={\int }_{\mathring{\Sigma }}\left(u_{\varphi }^{*}{\omega }_{\varphi }-d⟨{\mu }_{\varphi }(u_{\varphi }),A_{\varphi }⟩\right).$$

(23)

In particular, when we consider the unit disk $\Sigma =D\subset \mathbb{C}$, $\mathring{D}=D\backslash \left\{0\right\}$, the energy is

$$\begin{array}{c}\hfill E({\Phi }_{\varphi },{\Psi }_{\varphi },v_{\varphi };\mathring{D})={\int }_{\mathring{D}}\left(v_{\varphi }^{*}{\omega }_{\varphi }-d⟨{\mu }_{\varphi }(v),{\Phi }_{\varphi }ds+{\Psi }_{\varphi }dt⟩\right).\end{array}$$

(24)

Then, we define a number, called total energy of the ${\mathcal{T}}_{a,b}$-family of symplectic vortices,

$$E_{a,b}(\mathring{\Sigma }):=\underset{\varphi \in {\mathcal{T}}_{a,b}}{sup}E(A_{\varphi },u_{\varphi };\mathring{\Sigma }),$$

(25)

which may not be finite.

Remark 1.

When we consider vortices on punctured disk $\mathring{D}=D\backslash \left\{0\right\}$, we can use the biholomorphic map

$$\tau :C_{+}:=[0,+\infty )\times S^1\to \mathring{D}:(r,\theta )\mapsto e^{-2\pi (r+\mathrm{i}\theta )}$$

(26)

to just consider vortices on half-infinite cylinder $C_{+}$. Since $s+\mathrm{i}t$ is the coordinate on D, we have a change in variables

$$s=e^{-2\pi r}cos2\pi \theta ,t=-e^{-2\pi r}sin2\pi \theta .$$

if $E_{a,b}(\mathring{D})<\infty$. Denote the principal G-bundle over $\mathring{D}$ by $P_{\mathring{D}}$. Let $A\in \mathcal{A}(P_{\mathring{D}})$ and $u:\mathring{D}\to \mathbb{R}\times SZ$ be a symplectic vortex over a punctured disk. Then, denote $A^{\prime }={\tau }^{*}A\in \mathcal{A}(P_{C_{+}})$ and $v:=u\circ \tau :C_{+}\to \mathbb{R}\times SZ$. Note that these reparametrizations have no impact on the energy of the ${\mathcal{T}}_{a,b}$-family of symplectic vortices, that is,

$$E_{a,b}(\mathring{D})=E_{a,b}(C_{+}).$$

3. Asymptotic Property of $\mathbf{J}$-Holomorphic Planes and Cylinders

We consider symplectic vortices for a family of Hamiltonian G-manifolds $(\mathbb{R}\times SZ,{\omega }_{\varphi },{\mu }_{\varphi })$, where

$${\omega }_{\varphi }={\pi }^{*}{\omega }_Z+d(\varphi \alpha ),{\mu }_{\varphi }={\mu }_Z\circ \pi +{\mu }_{\varphi \alpha }$$

for ∀$\varphi \in {\mathcal{T}}_{a,b}$ (Recall (17), (19) and (20)).

For a principal bundle $P_{\Sigma }$ over $\Sigma$ (may be $\mathbb{R}\times S^1$ or not), an equivariant smooth map in $C_G^{\infty }(P_{\Sigma },\mathbb{R}\times SZ)$ can be also denoted as $u=(u_{\mathbb{R}},u_{SZ})$, where $u_{\mathbb{R}}:P_{\Sigma }\to \mathbb{R}$ is a smooth function and the map $u_{SZ}:P_{\Sigma }\to SZ$ is smooth and G-equivariant.

Let $D\subset \mathbb{C}$ be the unit disk, $\mathring{D}=D\backslash \left\{0\right\}$.

Let $\alpha$ be the connection 1-form of $SZ$, and J the almost complex structure on $\mathbb{R}\times SZ$. For $a,b\in \mathbb{R}$ with $a<b$, the energy of a J-holomorphic curve $u:\Sigma \to \mathbb{R}\times SZ$ is defined by

$$\begin{array}{c}\hfill {\tilde{E}}_{a,b}(u;\Sigma ):=\underset{\varphi \in {\mathcal{T}}_{a,b}}{sup}{\int }_{\mathring{\Sigma }}{u}^{*}{\omega }_{\varphi },\end{array}$$

(27)

where ${\mathcal{T}}_{a,b}=\{\varphi \in C^{\infty }(\mathbb{R},(a,b))|{\varphi }^{\prime }>0\}$, ${\omega }_{\varphi }={\pi }^{*}{\omega }_Z+d(\varphi \alpha )$ for ∀$\varphi \in {\mathcal{T}}_{a,b}$ (recall (17) and (19)). Note that for sufficiently small $a,b$ and each $\varphi \in {\mathcal{T}}_{a,b}$, ${\omega }_{\varphi }$ is a symplectic form on $\mathbb{R}\times SZ$. Also, we can define another number for this J-holomorphic curve

$$\begin{array}{c}\hfill {\tilde{E}}_Z(u;\Sigma )={\int }_{\mathring{\Sigma }}{u}^{*}{\pi }^{*}{\omega }_Z.\end{array}$$

(28)

Note that both ${\tilde{E}}_{a,b}$ and ${\tilde{E}}_Z$ may not be finite.

Recall that Z is a codimension-2 symplectic submanifold of $(X,\omega )$, $NZ$ is its symplectic normal bundle, and $\pi :SZ\to Z$ is the unit circle bundle over $(Z,{\omega }_Z)$ with a connection 1-form $\alpha$. Let $\xi =ker\alpha$ be the horizontal vector field of $SZ$. The complex structure on $NZ$ induces an $S^1$-action on $SZ$. Let $\mathcal{Y}$ be the characteristic vector field of the $S^1$-action, which corresponds to the Hamiltonian vector field $X_H$ on $Y\subset \widehat{X}$. Denote by $\mathcal{J}({\omega }_Z,\alpha )$ the space of smooth almost complex structures J on $\mathbb{R}\times SZ$ with the following properties:

(1)

J is invariant under the $\mathbb{R}$-action on $\mathbb{R}\times SZ$ by translation of the first factor;

(2)

$J{\partial }_r=\mathcal{Y}$ and $J\mathcal{Y}=-{\partial }_r$, where r denotes the natural coordinate on the first factor;

(3)

$J(\xi )=\xi$ and ${J|}_{\xi }$ is compatible with the symplectic vector bundle structure ${\pi }^{*}{\omega }_Z$.

The following result is taken from ([8], Lemma 3.5) and ([11], Lemma 6.4), which is originally proved in the SFT setting ([16], Theorem 31). It will be used to prove our main result.

Lemma 1.

Let $J\in \mathcal{J}({\omega }_Z,\alpha )$, $(\mathbb{R}\times SZ,J)$ be an almost complex manifold with a local Hamiltonian function H on $(-ϵ,ϵ)\times SZ\cong \mathcal{N}(Y)$, and $u:\Sigma \to \mathbb{R}\times SZ$ be a J-holomorphic curve satisfying for sufficiently small a, b,

$${\tilde{E}}_{a,b}(u;\Sigma )<\infty \mathrm{and}{\tilde{E}}_Z(u;\Sigma )=0.$$

(29)

(1)

If $\Sigma =\mathbb{C}$, then u is constant.

(2)

If $\Sigma =\mathbb{R}\times S^1$, then either u is constant or there exists an integer $m\in \mathbb{Z}$ and $c\in \mathbb{R}$ such that u is biholomorphically equivalent to a trivial cylinder over (the m-fold cover of) some 1-periodic Hamiltonian orbit $\gamma :S^1\to SZ\cong Y$ of H such that

$$u(r,\theta )=(mr+c,\gamma (m\theta )).$$

We remark that the assumption ${\int }_{\Sigma }{u}^{*}{\pi }^{*}{\omega }_Z=0$ in (29) implies that the image of u lies in $\mathbb{R}\times S_{z}Z$ for some $z\in Z$. This means that the image $\mathrm{im}(u)$ is contained in the image of some J-holomorphic plane of the form $u_{\gamma }:\mathbb{C}\to \mathbb{R}\times SZ$, $s+t\mathrm{i}\mapsto (s,\gamma (t))$, where $\gamma :\mathbb{R}\to SZ$ is a (not necessarily periodic) Hamiltonian orbit of H. Under the condition ${\tilde{E}}_{a,b}(u)<\infty$, the integer m arises as the mapping degree of the holomorphic extension (removal of singularity) of a unique holomorphic map $\Phi :\Sigma \to \mathbb{C}$, which expresses the decomposition $u=u_{\gamma }\circ \Phi$,

4. Boundedness of Differential

Fix the standard metric on $\mathbb{R}\times SZ$ that is translation invariant and an almost complex structure J. For a map $u:[0,+\infty )\times S^1\to \mathbb{R}\times SZ$, the norm of the linear map $d_{A}u(z):T_z([0,+\infty )\times S^1)\to T_{u(z)}(\mathbb{R}\times SZ)$ is defined as

$$|d_{A}u|:=(|d_A{u(\zeta )|}_J^2+|d_{A}u(j\zeta ){{|}_J^2)}^{{\displaystyle \frac{1}{2}}}$$

for any unit vector $\zeta \in T_z([0,+\infty )\times S^1)$ (independent of the choice of $\zeta$). Since for a punctured disk $\Sigma \cong \mathring{D}$, the study for local $\varphi$-symplectic vortex Equation (22) is the same as that for (21). We restate Theorem 1 as follows.

Theorem 2.

Take any $\varphi \in {\mathcal{T}}_{a,b}$. Given a ϕ-symplectic vortex $(A,v)$ such that $v:[0,+\infty )\times S^1\to \mathbb{R}\times SZ$ and $A\in {\mathcal{A}}^{1,p}(P)$ satisfying (22). Suppose that the following condition holds:

$$E_{a,b}([0,+\infty )\times S^1)<\infty .$$

(30)

Then, there exists a constant $C>0$ such that

$$|d_{A}v(r,\theta )|\le C$$

(31)

for all $(r,\theta )\in [0,+\infty )\times S^1$.

Proof. 

We argue by contradiction. Suppose there exists a sequence

$$z_k=(r_k,{\theta }_k)\to z^{*}\in [0,+\infty )\times S^1$$

with

$$|d_{A}v(z_k)|=|dv(z_k)+V_A(v(z_k))|\to \infty ,\mathrm{as}k\to \infty .$$

Denote by $R_k:=|dv(z_k)|$. Then, ${lim}_{k\to \infty }{R}_k=+\infty$, since $SZ$ is compact, and the G-action along the $\mathbb{R}$-direction is trivial, and so $|V_A|$ is bounded on $\mathbb{R}\times SZ$. For each $z_k$, we can choose a sequence ${ϵ}_k>0$ such that ${ϵ}_k\to 0$ and ${ϵ}_k{R}_k\to \infty$, and such that for all z in the ${ϵ}_k$-disk centered at $z_k$, denoted by $D_{{ϵ}_k}(z_k)$ (we can assume that all of these ${ϵ}_k$-disks are disjoint), we assume without loss of generality

$$|dv(z)|\le 2|dv(z_k)|=2R_k$$

for all $z\in D_{{ϵ}_k}(z_k)\subset [0,+\infty )\times S^1$. We use the biholomorphic transformation

$$\tau :[0,\infty )\times S^1\to D\backslash \left\{0\right\}:(r,\theta )\mapsto (s=e^{-2\pi r}cos2\pi \theta ,t=-e^{-2\pi r}sin2\pi \theta ).$$

Then, $(s_k,t_k)=\tau (r_k,{\theta }_k)$ is the holomorphic coordinates of $z_k$, and we obtain a symplectic vortex $({({\phi }^{-1})}^{*}A,w)$ with

$$w:D\backslash \left\{0\right\}\to \mathbb{R}\times SZ,$$

$$w(s,t):=v\circ {\tau }^{-1}(s,t),$$

such that

$$|dw(s,t)|\le 2R_k$$

(32)

for all $(s,t)\in D_{{ϵ}_k}(\tau (z_k)):=\tau (D_{{ϵ}_k}(z_k))\subset D\backslash \left\{0\right\}$.

Denote in local coordinate ${({\tau }^{-1})}^{*}A=\Phi ds+\Psi dt$. Then, consider a rescaled sequence of maps with domains as ${ϵ}_k{R}_k$-disk

$$w_k:D_{{ϵ}_k{R}_k}(0)\to \mathbb{R}\times SZ,$$

$$w_k(s,t):=w(s_k+{\displaystyle \frac{s}{R_k}},t_k+{\displaystyle \frac{t}{R_k}}),$$

$${\Phi }_k(s,t):=\Phi (s_k+{\displaystyle \frac{s}{R_k}},t_k+{\displaystyle \frac{t}{R_k}}),{\Psi }_k(s,t):=\Psi (s_k+{\displaystyle \frac{s}{R_k}},t_k+{\displaystyle \frac{t}{R_k}}).$$

Note that $(s_k,t_k)\to (s^{*},t^{*})=\tau (z^{*})$. Then,

$${\Phi }_k(s,t)\to \Phi (s^{*},t^{*}),{\Psi }_k(s,t)\to \Psi (s^{*},t^{*}).$$

By the first equation of (22), this sequence satisfies

$${\partial }_s{w}_k+J(w_k){\partial }_t{w}_k=-{{\displaystyle \frac{1}{R}}}_k\left(V_{{\Phi }_k}(w_k)+J(w_k)V_{{\Psi }_k}(w_k)\right).$$

(33)

We can see that $|d{w}_k(0,0)|={\displaystyle \frac{1}{R_k}}|dv(z_k)|=1$ and $|d{w}_k|={\displaystyle \frac{1}{R_k}}|dv|\le 2$ on $D_{{ϵ}_k{R}_k}(0)$, and so bounded is $|d_A{w}_k|$. But they are not necessarily $C^1$-bounded since their images may escape to infinity. We consider the following three possibilities, at least one of which must hold:

• Case 1: $w_k(0,0)=w(s_k,t_k)$ has a bounded subsequence.

Then, the corresponding subsequence of $w_k:D_{{ϵ}_k{R}_k}(0)\to \mathbb{R}\times SZ$ is uniformly $C^1$-bounded on every compact subset. By elliptic regularity and Equation (33), there exists a subsequence convergent in $C_{loc}^{\infty }(\mathbb{C})$ to a J-holomorphic plane

$$w_{\infty }:\mathbb{C}\to \mathbb{R}\times SZ$$

satisfying

$$|d{w}_{\infty }(0,0)|=\underset{k\to \infty }{lim}|d{w}_k(0,0)|=\underset{k\to \infty }{lim}{\displaystyle \frac{1}{R_k}}|dw(s_k,t_k)|=\underset{k\to \infty }{lim}{\displaystyle \frac{1}{R_k}}|dv(z_k)|=1.$$

(34)

Now

$$\begin{array}{ccc}\hfill \sum _k{\int }_{D_{{ϵ}_k{R}_k}(0)}\left(w_k^{*}{\omega }_{\varphi }-d⟨{\mu }_{\varphi }(w_k),A_k⟩\right)& =& \sum _k{\int }_{D_{{ϵ}_k}(s_k,t_k)}\left(w^{*}{\omega }_{\varphi }-d⟨{\mu }_{\varphi }(w),{({\tau }^{-1})}^{*}A⟩\right)\hfill \\ & \le & {\int }_{[0,+\infty )\times S^1}\left(v^{*}{\omega }_{\varphi }-d⟨{\mu }_{\varphi }(v),{({\tau }^{-1})}^{*}A⟩\right),\hfill \end{array}$$

where $A_k={\Phi }_{k}ds+{\Psi }_{k}dt$. From the condition (30),

$$E_{a,b}([0,+\infty )\times S^1)=\underset{\varphi \in {\mathcal{T}}_{a,b}}{sup}{\int }_{(0,+\infty )\times S^1}\left(v_{\varphi }^{*}{\omega }_{\varphi }-d⟨{\mu }_{\varphi }(v_{\varphi }),A_{\varphi }⟩\right)<\infty$$

and the assumption that moment map $\mu$ is proper, we have

$$\sum _k{\int }_{D_{{ϵ}_k{R}_k}(0)}\left(w_k^{*}{\omega }_{\varphi }-d⟨{\mu }_{\varphi }(w_k),A_k⟩\right)\le E_{a,b}([0,+\infty )\times S^1)<\infty .$$

So,

$${\int }_{\mathbb{C}}\left(w_{\infty }^{*}{\omega }_{\varphi }-d⟨{\mu }_{\varphi }(w_{\infty }),A_{\infty }⟩\right)=\underset{k\to \infty }{lim}{\int }_{D_{{ϵ}_k{R}_k}(0)}\left(w_k^{*}{\omega }_{\varphi }-d⟨{\mu }_{\varphi }(v),A⟩\right)=0,$$

where $A_{\infty }={lim}_{k\to \infty }{A}_k=\Phi (s^{*},t^{*})ds+\Psi (s^{*},t^{*})dt$ is a constant 1-form. So

$$d⟨{\mu }_{\varphi }(w_{\infty }),A_{\infty }⟩=0\mathrm{and}{\int }_{\mathbb{C}}{w}_{\infty }^{*}{\omega }_{\varphi }=0.$$

Hence, $w_{\infty }$ must be constant, and this is a contradiction with (34).

• Case 2: $w_k(0,0)=w(s_k,t_k)$ has a subsequence diverging to $\{+\infty \}\times SZ$.

Restricting to this subsequence, suppose $w_k(0,0)\in \left\{r_k\right\}\times SZ$, and thus $r_k\to +\infty$. Without loss of generality, we assume that $r_k>r_0$ for all k. Denote by ${\widehat{R}}_k\in (0,{ϵ}_k{R}_k]$ for each k the largest radius such that $w_k(D_{{\widehat{R}}_k})\subset (r_0,\infty )\times SZ$. We see that ${\widehat{R}}_k\to \infty$ since $|d{w}_k|\le 2$. Then, we use the $\mathbb{R}$-translation maps

$$T_c:\mathbb{R}\times SZ\to \mathbb{R}\times SZ:(r,x)\mapsto (r+c,x)$$

(35)

to define

$${\widehat{w}}_k:=T_{-r_k}\circ w_k{|}_{D_{{\widehat{R}}_k}}:D_{{\widehat{R}}_k}\to \mathbb{R}\times SZ.$$

Recall we are using a translation-invariant metric on $[r_0,\infty )\times SZ$, so $|d_A{\widehat{w}}_k|=|d_A{w}_k|$ is also bounded. Then, ${\widehat{w}}_k$ is a uniformly $C_{loc}^1$-bounded sequence of maps into $\mathbb{R}\times SZ$. Note that moment map ${\mu }_{\varphi }$ is invariant under the $\mathbb{R}$-translation since the G-action is trivial along the $\mathbb{R}$-direction. So $A_k$ is also invariant and the sequence of pairs $(A_k,{\widehat{w}}_k)$ always satisfies Equation (21). Note that $V_{\Phi }$, $V_{\Psi }$ and J are invariant under the $\mathbb{R}$-translation. From Equation (33) and the elliptic regularity, we see that, up to gauge transformations, there exists a subsequence convergent in $C_{loc}^1(\mathbb{C})$ to a J-holomorphic plane

$$w_{\infty }:\mathbb{C}\to \mathbb{R}\times SZ$$

with respect to an almost complex structure J extended over $\mathbb{R}\times SZ$. We will verify

$${\tilde{E}}_{a,b}(w_{\infty };\mathbb{C})=\underset{\varphi \in {\mathcal{T}}_{a,b}}{sup}{\int }_{\mathbb{C}}{w}_{\infty }^{*}{\omega }_{\varphi }<\infty$$

(36)

for sufficiently small a, b, and

$${\tilde{E}}_Z(w_{\infty };\mathbb{C})={\int }_{\mathbb{C}}{w}_{\infty }^{*}{\pi }^{*}{\omega }_Z=0.$$

(37)

Let ${\varphi }_k(r):=\varphi (r-r_k)$. Then, for any $R>0$,

$${\int }_{D_R}{w}_{\infty }^{*}{\omega }_{\varphi }=\underset{k\to \infty }{lim}{\int }_{D_R}{\widehat{w}}_k^{*}{\omega }_{\varphi }=\underset{k\to \infty }{lim}{\int }_{D_R}{w}_k^{*}{T}_{-r_k}^{*}{\omega }_{\varphi }=\underset{k\to \infty }{lim}{\int }_{D_R}{w}_k^{*}{\omega }_{{\varphi }_k}.$$

Notice that the dependence of the last integral on ${\varphi }_k$ is limited to the interval $(r_0,\infty )$ since $w_k(D_R)\subset (r_0,\infty )\times SZ$. Note that for the symplectic form ${\omega }_{{\varphi }_k}$, ${\mu }_{{\varphi }_k}$ is invariant under translation. So $(A_k,w_k)$ is still a symplectic vortex with respect to ${\omega }_{{\varphi }_k}$. Then, by condition (8), we have for sufficiently large k

$$\begin{array}{ccc}\hfill {\int }_{D_R}\left(w_k^{*}{\omega }_{{\varphi }_k}-d⟨{\mu }_{{\varphi }_k}(w_k),A_k⟩\right)& \le & {\int }_{D_{{ϵ}_k{R}_k}}\left(w_k^{*}{\omega }_{{\varphi }_k}-d⟨{\mu }_{{\varphi }_k}(w_k),A_k⟩\right)\hfill \\ & =& {\int }_{D_{{ϵ}_k}(z_k)}\left(w^{*}{\omega }_{{\varphi }_k}-d⟨{\mu }_{{\varphi }_k}(w),A⟩\right)\hfill \\ & \le & {\int }_{(0,+\infty )\times S^1}\left(v^{*}{\omega }_{{\varphi }_k}-d⟨{\mu }_{{\varphi }_k}(v),A⟩\right)\hfill \\ & \le & E_{a,b}([0,+\infty )\times S^1)<\infty .\hfill \end{array}$$

Note that $d⟨\mu (w_{\infty }),A_{\infty }⟩=0$, so

$${\int }_{D_R}{w}_{\infty }^{*}{\omega }_{\varphi }=\underset{k\to \infty }{lim}{\int }_{D_R}\left(w_k^{*}{\omega }_{{\varphi }_k}-d⟨{\mu }_{{\varphi }_k}(w_k),A_k⟩\right)\le E_{a,b}([0,+\infty )\times S^1)<\infty .$$

This holds for every $R>0$ and every $\varphi \in {\mathcal{T}}_{a,b}$. Hence (36) is proved.

To verify (37), fix any $R>0$ and pick any $\varphi \in {\mathcal{T}}_{a,b}$. Since we can assume for a subsequence the disks $D_{{ϵ}_k}(z_k)$ are all disjoint, then we have

$$\begin{array}{ccc}\hfill 0& =& \underset{k\to \infty }{lim}{\int }_{D_{{ϵ}_k}(z_k)}\left(w^{*}{\omega }_{\varphi }-d⟨{\mu }_{\varphi }(w),A⟩\right)\hfill \\ & =& \underset{k\to \infty }{lim}{\int }_{D_{{ϵ}_k{R}_k}}\left(w_k^{*}{\omega }_{\varphi }-d⟨{\mu }_{\varphi }(w_k),A_k⟩\right)\hfill \\ & =& \underset{k\to \infty }{lim}{\int }_{D_{{ϵ}_k{R}_k}}\left({\widehat{w}}_k^{*}{T}_{r_k}^{*}{\omega }_{\varphi }-d⟨{\mu }_{\varphi }(T_{r_k}\circ {\widehat{w}}_k),A_k⟩\right)\hfill \\ & \ge & \underset{k\to \infty }{lim}{\int }_{D_R}\left({\widehat{w}}_k^{*}{T}_{r_k}^{*}{\omega }_{\varphi }-d⟨{\mu }_{\varphi }(T_{r_k}\circ {\widehat{w}}_k),A_k⟩\right)\hfill \\ & =& \underset{k\to \infty }{lim}{\int }_{D_R}\left({\widehat{w}}_k^{*}{\omega }_{{\overline{\varphi }}_k}-d⟨{\mu }_{{\overline{\varphi }}_k}({\widehat{w}}_k),A_k⟩\right)\hfill \\ & =& \underset{k\to \infty }{lim}{\int }_{D_R}{\widehat{w}}_k^{*}{\omega }_{{\overline{\varphi }}_k}\hfill \end{array}$$

since $d⟨\mu (w_{\infty }),A_{\infty }⟩=0$, where ${\overline{\varphi }}_k(r):=\varphi (r+r_k)=\varphi \circ (T_{r_k}{|}_{\mathbb{R}})$ and

$${\omega }_{{\overline{\varphi }}_k}={\pi }^{*}{\omega }_Z+d({\overline{\varphi }}_k\alpha )={\pi }^{*}{\omega }_Z+{\overline{\varphi }}_k(r)d\alpha +{\overline{\varphi }}_k^{\prime }(r)dr\wedge \alpha .$$

We can choose special $\varphi$ such that ${\varphi }^{\prime }(r)={\varphi }^{\prime }(r+r_k)={\overline{\varphi }}_k^{\prime }(r)\to 0$ as $k\to \infty$, then the third term does not contribute to the limit of the integral. Let

$$\overline{\varphi }:=\underset{k\to \infty }{lim}{\overline{\varphi }}_k(r)=\underset{r\to +\infty }{lim}\overline{\varphi }(r),{\mu }_{\overline{\varphi }}:={\mu }_Z\circ \pi +{\mu }_{\overline{\varphi }\alpha }={\mu }_Z\circ \pi +\overline{\varphi }{\mu }_{\alpha }.$$

Then, the calculation above becomes

$$0\ge {\int }_{D_R}{w}_{\infty }^{*}({\pi }^{*}{\omega }_Z+\overline{\varphi }d\alpha ).$$

Recall that J is tamed by every ${\omega }_{\varphi }$, so the integral is non-negative. Since we assume $|a|$ and $|b|$ are sufficiently small, ${\pi }^{*}{\omega }_Z+\overline{\varphi }d\alpha$ is nondegenerate on ${\pi }^{*}TZ$ and also annihilates ${\partial }_r$ and the characteristic direction $\xi$. So the vanishing of this integral implies that $w_{\infty }$ is everywhere tangent to ${\partial }_r$ and $\xi$ over $D_R$. Since R is arbitrary, this is true on the whole plane, which is equivalent to

$${\int }_{\mathbb{C}}{w}_{\infty }^{*}{\pi }^{*}{\omega }_Z=0.$$

So (37) holds. Then, using Lemma 1 (1), (36) and (37) imply that $w_{\infty }$ is constant, contradicting the fact that $|d{w}_{\infty }(0,0)|=1$.

• Case 3: $w_k(0,0)=w(s_k,t_k)$ has a subsequence diverging to $\{-\infty \}\times SZ$.

The argument to obtain a contradiction is similar to Case 2 with the restriction of J to $(-\infty ,-r_0]\times SZ$. □

5. Conclusions

In the discussion above, for some simpler case, which is an open cylinder model for symplectic vortices near a codimension-2 Hamiltonian G-submanifold, we established the boundedness of differential $d_{A}u$ of a $\varphi$-symplectic vortex $(A,u)$ satisfying Equation (5) under the condition of the total energy $E_{a,b}([0,+\infty )\times S^1)<\infty$. The result is supposed to be useful for the future study of the relative version of Hamiltonian Gromov–Witten invariants. For instance, for a possible application, one can use the boundedness of differential $d_{A}u$ to study the asymptotic properties of symplectic vortices in an open cylinder model. That will be crucial for understanding the analytic behavior of symplectic vortices near a codimension-2 Hamiltonian G-submanifold.