Compactness in the adiabatic limit of disk vortices

Abstract

This paper is the first input towards an open analogue of the quantum Kirwan map. We consider the adiabatic limit of the symplectic vortex equation over the unit disk for a Hamiltonian G-manifold with Lagrangian boundary condition, by blowing up the metric on the disk. We define an appropriate notion of stable solutions in the limit, and prove that any sequence of disk vortices with energy uniformly bounded has a subsequence converging to such a stable object. We also proved several analytical properties of vortices over the upper half plane, which are new type of bubbles appearing in our compactification.

Appendices

Appendix 1: Analysis of vortices

In this appendix we establish several necessary estimates related to the problem and provide proofs of Theorem 14 and Theorem 16. The techniques used here are standard, and all results which don’t involve boundary conditions are essentially covered in the previous literature such as [8] and [17]. A new technical result here is the proof of the existence of admissible almost complex structures with respect to a G-Lagrangian (Lemma 49).

1.1 Preliminaries

1.1.1 A neighborhood of $\mu ^{-1}(0)$

We assumed that 0 is a regular value of $\mu$. Therefore we can fix two numbers $\updelta >0$, ${\varvec{m}} = {\varvec{m}}(\updelta )>0$ satisfying that for any $x \in X$ with $|\mu (u)|\le \updelta$, the map is injective and

(7.1)

We fix these two numbers throughout this appendix.

1.1.2 G-invariant metrics

Let h be a G-invariant Riemannian metric on X with respect to which J is isometric. We denote by $\nabla$ the Levi-Civita connection of h, and R the Riemannian curvature.

Let $\Xi \subset {\mathbb H}$ be an open subset with coordinates (s, t). For any smooth map $(u, \phi , \psi ): \Xi \rightarrow X \times {\mathfrak g} \times {\mathfrak g}$, we denote

Moreover, we have a natural covariant derivative on $u^* TX \oplus {\mathfrak g}$ associated with $(u, \phi , \psi )$, defined as follows. For any $V \in \varGamma (u^* TX)$, we define

(7.2)

for $\xi : \Xi \rightarrow {\mathfrak g}$, we define

$$\begin{aligned} \nabla _{A, s} \xi = \partial _s \xi + \left[ \phi , \xi \right] ,\ \nabla _{A, t} \xi = \partial _t \xi + \left[ \psi , \xi \right] . \end{aligned}$$

By the G-invariance of h one can check that this covariant derivative preserves the metric.

We can extend the covariant derivative to tensor fields along u, by Leibniz rule. We denote , . Then we have

Lemma 47

If T is a G-invariant tensor field on X and $\nabla$ is the Levi-Civita connection of a G-invariant metric, then $\nabla _{A, s} T = \nabla _{v_s} T$. In particular, if J is a G-invariant almost complex structure, then $\nabla _{A, s}J = \nabla _{v_s} J$.

It is straightforward to extend the above result to ${\mathfrak g}$-valued tensor fields. We denote

$$\begin{aligned} \rho (v_s, V) = \nabla _{A, s} \left( d\mu \cdot J V \right) - d\mu \cdot \left( J \nabla _{A, s} V \right) ,\ \rho (v_t, V) = \nabla _{A, t} \left( d\mu \cdot JV \right) - d\mu \cdot \left( J \nabla _{A, t} V \right) . \end{aligned}$$

To estimate the energy density, it is convenient to have a special type of metrics on X.

Definition 48

Let L be a G-Lagrangian of $(X, \omega , \mu )$. Let J be a G-invariant almost complex structure. A $(J, L, \mu )$-admissible Riemannian metric is a G-invariant Riemannian metric h on X satisfying

• 1. J is isometric;

• 2. J(TL) and TL are orthogonal with respect to h;

• 3. L is totally geodesic with respect to h;

• 4. $T\mu ^{-1}(0)$ is orthogonal to for all $\xi \in {\mathfrak g}$.

In the non-equivariant case Frauenfelder [12] proved the existence of a similar type of metric for a Lagrangian submanifold satisfying (1)–(3). Here we generalize this result.

Lemma 49

For any G-Lagrangian L, and G-invariant almost complex structure J, there exists a $(J, L, \mu )$-admissible Riemannian metric.

Proof

By [12, Lemma A.3], there exists a Riemannian metric $\bar{h}$ on the symplectic quotient $\bar{X}$ satisfying (1)–(3) for the Lagrangian $\bar{L}$. Now we construct a suitable lift of $\bar{h}$ to $\mu ^{-1}(0)$. Let $h_0$ be the metric on X induced by $\omega$ and J. Let $H\subset T\mu ^{-1}(0)$ be the orthogonal complement (with respect to $h_0$) of the distribution $T{\mathfrak g}$ generated by infinitesimal G-actions. Then it is easy to see that H is G-invariant, and we have an isomorphism $H \simeq \pi ^* T\bar{X}$, where $\pi : \mu ^{-1}(0) \rightarrow \bar{X}$ is the projection. Then we can pull-back $\bar{h}$ to a G-invariant metric on H, and choose a G-invariant metric on $T{\mathfrak g}$ such that . Let $h'$ be the direct sum of the two components, which is a Riemannian metric on $\mu ^{-1}(0)$.

We claim that L is totally geodesic in $\mu ^{-1}(0)$ with respect to $h'$. Let $\nabla '$ be the Levi-Civita connection of $h'$. It suffices to check that $(\nabla ')_X Y \in TL$ for any vector fields X, Y tangent to L. Since this is a pointwise condition, we assume that X, Y are both G-invariant. For any Z orthogonal to TL in $\mu ^{-1}(0)$, we have (the inner products in the following are the ones for $h'$)

$$\begin{aligned} \begin{aligned} \langle (\nabla ')_X Y, Z \rangle =&\ X \langle Y, Z \rangle - \langle Y, (\nabla ')_X Z \rangle \\ =&\ - \langle Y, [ X, Z] \rangle - \langle Y, (\nabla ')_Z X \rangle \\ =&\ \langle Y, [Z, X] \rangle - Z \langle Y, X \rangle + \langle (\nabla ')_Z Y, X \rangle \\ =&\ \langle Y, [Z, X] \rangle - Z \langle Y, X \rangle + \langle [Z, Y], X \rangle + \langle (\nabla ')_Y Z, X \rangle \\ =&\ \langle Y, [Z, X] \rangle - Z \langle Y, X \rangle + \langle [Z, Y], X \rangle - \langle Z, (\nabla ')_Y X \rangle \\ =&\ \langle Y, [Z, X] \rangle - Z\langle Y, X \rangle + \langle [Z, Y], X \rangle - \langle Z, (\nabla ')_X Y \rangle . \end{aligned} \end{aligned}$$

(7.3)

We choose Z to be pull-backed from $\bar{X}$ so that Z is G-invariant and $\pi _* Z$ is a smooth vector field on $\bar{X}$ and orthogonal to $T\bar{L}$.

If $X, Y \in \varGamma (H \cap TL) \simeq \varGamma (\pi ^* T\bar{L})$, then we choose X, Y to be pull-backed from $\bar{X}$ and such that $\langle X, Y \rangle$ is a constant. Then $\pi _* X, \pi _* Y$ are smooth vector fields on $\bar{X}$. Then by the definition of $h'$, we see that

$$\begin{aligned} \langle Y, [Z, X]\rangle = \langle \pi _* Y, [ \pi _* Y, \pi _* Z] \rangle _{\bar{h}},\ \langle [Z, Y], X\rangle = \langle [\pi _* Z, \pi _* Y], \pi _* X \rangle _{\bar{h}}. \end{aligned}$$

Then by the same calculation as (7.3), we know that the sum of the above two terms is equal to $2\langle \bar{\nabla }_{\pi _* X} (\pi _* Y), \pi _* Z \rangle _{\bar{h}}$, which vanishes by the totally geodesic assumption on $\bar{h}$.

On the other hand, if $X \in \varGamma ( T{\mathfrak g})$ and $Y\in \varGamma (H \cap TL)$, then $\langle Y, X\rangle \equiv 0$. We can choose Z being G-invariant and [Y, Z] vanishing at a point where we want to evaluate (7.3). Then (7.3) vanishes at that point. Finally, if $X, Y \in \varGamma (T{\mathfrak g})$, then we take , for $\xi , \eta$ constants. Then $\langle Y, X \rangle$ is a constant and $[X, Z] = [Y, Z]= 0$. So (7.3) vanishes.

Now we would like to extend $h'$ to a metric on X which satisfies (1)–(4).

We choose local coordinates $\theta _1, \ldots , \theta _k,x_1, \ldots , x_m$ of L where the first $k = \mathrm{dim} G$ coordinates are coordinates of G-orbits. Extend them to coordinates

$$\begin{aligned} \theta _1, \ldots , \theta _k, x_1, \ldots , x_m, \tau _1, \ldots , \tau _k, y_1, \ldots , y_m \end{aligned}$$

on X such that the first k coordinates are still coordinates of G-orbits, L is parametrized by $(\theta , x, 0, 0)$ and satisfying

$$\begin{aligned} J \frac{\partial }{\partial \theta _i} = \frac{\partial }{\partial \tau _i},\ i=1,\ldots , k,\ J \frac{\partial }{\partial x_j} = \frac{\partial }{\partial y_j},\ j=1, \ldots , m, \end{aligned}$$

on L. We remark that $\frac{\partial }{\partial y_j}$ may not be tangent to $\mu ^{-1}(0)$ but $\mu ^{-1}(0)$ can be parametrized as $( \theta , x, \tau (\theta , x, y), y)$ where $\tau$ satisfies $\tau (\theta , x, 0) = 0$. We write J as

$$\begin{aligned} J = \left( \begin{array}{cc} A(\theta , x, \tau , y) &{}\quad B(\theta , x, \tau , y) \\ C(\theta , x,\tau , y) &{}\quad D(\theta , x, \tau , y) \end{array} \right) \end{aligned}$$

where A, B, C, D are of size $(m+k) \times (m+k)$ and $A(\theta , x, 0, 0) = D(\theta , x, 0, 0) = 0$, $C(\theta , x, 0, 0) = - B(\theta , x, 0, 0)^T = I_{m+k}$. We consider a locally defined metric

$$\begin{aligned} \widetilde{h}(\theta , x, \tau , y) = \left( \begin{array}{cc} \widetilde{a}(\theta , x,\tau , y) &{}\quad \widetilde{b}^T(\theta , x, \tau , y) \\ \widetilde{b}(\theta , x,\tau , y) &{}\quad \widetilde{c}(\theta , x,\tau , y) \end{array}\right) . \end{aligned}$$

where the matrix decomposition is written with respect to the same coordinates. Then the value of $\widetilde{a}(\theta , x, \tau (\theta , x, y), y)$ and part of $\widetilde{b}(\theta , x, \tau (\theta , x, y), y)$ and $\widetilde{c}(\theta , x, \tau (\theta , x, y), y)$ have been determined by the choice of $h'$. Then we choose the undetermined part of $\widetilde{h}|_{\mu ^{-1}(0)}$ such that for any $\xi \in {\mathfrak g}$ with respect to $\widetilde{h}|_{\mu ^{-1}(0)}$ and such that . Moreover, we require that the 1-jet of $\widetilde{h}$ along L in the $\tau$-direction satisfy

$$\begin{aligned} \frac{\partial }{\partial \tau _i} \widetilde{a}(\theta , x, 0, 0) + \frac{\partial }{\partial \tau _i} \left( A^T \widetilde{a} A + C^T \widetilde{b} A + A^T \widetilde{b}^T C + C^T \widetilde{c} C \right) (\theta , x, 0, 0) = 0,\ i \ = \ 1, \ldots , k. \end{aligned}$$

(7.4)

Since $A|_L\equiv 0$, the left-hand-side of this equation is $\partial _{\tau _i} \widetilde{a}(\theta , x, 0, 0)$ plus terms which don’t contain derivatives of $\widetilde{a}$. Therefore (7.4) has a solution subordinate to all the constrains we have put on the 0-jet of $\widetilde{h}$. This gives us a metric $\widetilde{h}$ defined in the coordinate patch.

Now we define $h(v, w) = \frac{1}{2} \big ( \widetilde{h}(v, w) + \widetilde{h}(Jv, Jw) \big )$ and we claim that h satisfies Definition 48 inside the coordinate patch we are considering, except for the G-invariance. As in [12], we can use such locally constructed metrics and a partition of unity to construct a global metric h, satisfying (1)–(4).

Indeed, the first condition is automatic. For the fourth condition, for $Y \in T\mu ^{-1}(0)$ and , decompose $Y = Y_1 + Y_2$ where $Y_1 \in H$ and for some $\eta \in {\mathfrak g}$. Then by the condition required for $\widetilde{h}$. Therefore, J(TL) is orthogonal to TL.

For the totally geodesic condition, we see that (7.4) implies that for X, Y tangent to L, $\nabla _X Y \in T\mu ^{-1}(0)|_L$ where $\nabla$ is the Levi-Civita connection of h. Moreover, by the condition that along $\mu ^{-1}(0)$, we see that $h|_{\mu ^{-1}(0)} = \widetilde{h}|_{\mu ^{-1}(0)}$. Since L is totally geodesic with respect to $\widetilde{h}|_{\mu ^{-1}(0)} = h'$, this implies that $\nabla _X Y \in TL$ and L is totally geodesic in the local coordinate patch.

Now the metric h constructed above may not be G-invariant. We integrate h against the Haar measure of G, getting a G-invariant metric $\hat{h}$. The point-wise conditions (1), (2), (4) are clearly preserved. To see that L is totally geodesic with respect to $\hat{h}$, it suffices to show that for any vector fields X, Y tangent to L, $\hat{\nabla }_X Y$ is tangent to L, where $\hat{\nabla }$ is the Levi-Civita connection of h. Indeed, if we denote by $\nabla$ the Levi-Civita connection of h, then

$$\begin{aligned} \hat{\nabla }_X Y = \int _G \left( g_* \right) ^{-1}\nabla _{g_* X} (g_* Y) dg. \end{aligned}$$

Since L is G-invariant, $g_* X$, $g_* Y$ are both tangent to L. Therefore we see that $\hat{\nabla }_X Y$ is tangent to L. Therefore $\hat{h}$ is a $(J, L, \mu )$-admissible metric. $\square$

Now we fix a $(J, L, \mu )$-admissible metric h on X. We use $\langle \cdot , \cdot \rangle$ to denote the inner product of h in the remaining of this appendix. Then by (4) of Definition 48, there exists ${\varvec{n}}= {\varvec{n}}(\updelta )>0$ such that for any $x \in X$ with $|\mu (x)| \le \updelta$, and any $Y \in T_x X$,

Since a rescaling of h is still $(J, L, \mu )$-admissible, we may assume instead

(7.5)

Moreover, we may assume that h coincides with a small constant multiple of $\omega (\cdot , J \cdot )$ whenever $|\mu (x)| \ge \updelta$, so that (7.5) holds throughout X.

1.2 The isoperimetric inequality

We first recall Poźniak’s isoperimetric inequality ([31, Lemma 3.4.5]). Let $(Y, \omega )$ be a symplectic manifold and let $L_0, L_1\subset Y$ be two compact Lagrangian submanifolds which intersect cleanly in Y.

Lemma 50

[31, Lemma 3.4.5] There exist constants $\updelta ' = \updelta '(L_0, L_1) >0$ and ${\varvec{c}}' = {\varvec{c}}'(L_0, L_1)>0$ satisfying the following condition. Let $x: [0, \pi ] \rightarrow Y$ be a $C^1$-path with $x(0) \in L_0, x(1) \in L_1$ and $\int _0^\pi |x'(t)|^2 dt \le ( \updelta ')^2$. Then there exists a $C^1$-map $v: [0,1]\times [0, \pi ] \rightarrow X$ with

$$\begin{aligned} v(s, 0) \in L_-, v(s, \pi ) \in L_+, v(0, t) \in L_- \cap L_+, v(1, t) = x(t). \end{aligned}$$

Moreover, if we define the symplectic action of the path x as

$$\begin{aligned} {\varvec{a}}(x) = - \iint _{[0,1]\times [0, \pi ]} v^* \omega , \end{aligned}$$

(7.6)

then

$$\begin{aligned} |{\varvec{a}}(x)|\le {\varvec{c}}' \int _0^\pi |x'(t)|^2 dt. \end{aligned}$$

(7.7)

Now we consider two G-Lagrangian submanifolds $L_0, L_1$ of the Hamiltonian G-manifold $(X, \omega , \mu )$. Suppose they intersect cleanly in $\bar{X}$. Consider the path spaces

Then we define a “local action functional” analogous to that in [11] and [17] when a path is sufficiently close to . More precisely, for , we denote

Then we have

Lemma 51

There exist positive constants $\updelta = \updelta (L_0, L_1)$ and ${\varvec{c}} = {\varvec{c}}(L_0, L_1)$ such that for with

(7.8)

there exists such that

$$\begin{aligned} \sup _{t\in [0,\pi ]} \left| \eta (t) - \widetilde{\eta }(t) \right| \le {\varvec{c}} l(x, \eta ),\ d(x(t), \widetilde{x}(t)) \le {\varvec{c}} \left( \left| \mu (x(t)) \right| + l(x, \eta ) \right) . \end{aligned}$$

Proof

For small $\updelta >0$, there exist a unique path $(x_0, \eta _0): [0,\pi ]\rightarrow \mu ^{-1}(0) \times {\mathfrak g}$ such that

Moreover, there exists $c_2>0$ such that

On the other hand, since $\bar{L}_0$ and $\bar{L}_1$ intersect cleanly, there exists $\bar{q} \in \bar{L}_0 \cap \bar{L}_1$ such that

$$\begin{aligned} d( \bar{q}, \bar{x}_0(0)) \le c_3 \int _0^\pi |\bar{x}_0'(t)| dt. \end{aligned}$$

Then we can choose a lift $q \in L_0 \cap L_1$ of $\bar{q}$, such that

$$\begin{aligned} d \left( q, x_0(0) \right) \le c_4 l(x_0, \eta ) \le c_2 c_4 l(x, \eta ). \end{aligned}$$

On the other hand, choose $g: [0,\pi ]\rightarrow G$ such that $g(0) = \mathrm{Id}$, $g'(t) g(t)^{-1} = -\eta (t)$. Define

Then we see there exists $c_5>0$ such that

$$\begin{aligned} \begin{aligned} d(x(t), \widetilde{x}(t)) \le&\ d(x(t), x_0(t)) + d(x_0(t), \widetilde{x}(t)) \\ \le&\ c_5 |\mu (x(t))| + d ( g(t)^{-1}x_0 (t), q) \\ \le&\ c_5 |\mu (x(t))| + c_5 d(x_0(0), q) + c_5 \int _0^\pi \left| \frac{d}{dt} g(t)^{-1} x_0(t) \right| dt\\ \le&\ c_6 \left( |\mu (x(t))| + l(x, \eta ) \right) . \end{aligned} \end{aligned}$$

$\square$

For $\updelta >0$, denote by the subset of pairs $(x, \eta )$ that satisfy (7.8). Then we define the local action functional for all by

Lemma 52

There exist positive constants $\updelta = \updelta (L_0, L_1)$ and ${\varvec{c}} = {\varvec{c}}(L_0, L_1)$ such that for , we have

Proof

The same as in Step 3 of the proof of [11, Lemma 3.17]. $\square$

1.3 A priori estimates

Lemma 53

Let $u: B_{R+r} \rightarrow {\mathbb R}_+\cup \{0\}$ be a $C^2$-function and $a>0$ such that

$$\begin{aligned} \varDelta u \ge - a(u +u^2). \end{aligned}$$

Then for any $x \in B_R$,

$$\begin{aligned} \int _{B_r(x)} u \le \frac{\pi }{8a} \Longrightarrow u(x) \le \max \{ \frac{\pi }{8}, \frac{4a r^2 }{\pi } \} \frac{1}{r^2} \int _{B_r(x)} u. \end{aligned}$$

Proof

We first prove for the case that $r = 1$. Using the Heinz trick (cf. [25, Page 82]), define the function $f: [0, 1]\rightarrow {\mathbb R}$ by

$$\begin{aligned} f(\rho ) = (1- \rho )^2 \sup _{B_\rho (x)} u. \end{aligned}$$

Let $\rho ^* \in [0,1)$ be some number at which f attains its maximum. Choose $w^* \in B_{\rho ^*}(x)$ such that $u(z^*) = \sup _{B_{\rho ^*}(x)} u$ and denote $c^* = u(z^*)$. Denote $\delta = \frac{1 - \rho ^* }{2}< 1$. Then for $w \in B_\delta (w^*)$,

$$\begin{aligned} u(w) \le \sup _{B_{\rho ^* + \delta }(x)} u \le \frac{ ( 1- \rho ^*)^2 }{ ( 1- \rho ^* - \rho )^2 } \sup _{B_{\rho ^*}(x)} u = 4 c^*. \end{aligned}$$

Therefore on $B_\delta (w^*)$, we have

$$\begin{aligned} \varDelta u \ge - a (u +u^2) \ge - \left( 4ac^* + 16 a(c^*)^2 \right) =: -4 m c^*, \end{aligned}$$

which implies that the function $\widetilde{u}(w) = u(w) + m c^* |w- w^*|^2$ is subharmonic on $B_\delta (w^*)$. Therefore, for any $\rho \in (0, \delta ]$, we have

$$\begin{aligned} c^* = u(w^*) \le \frac{1 }{ \pi \rho ^2} \int _{B_\rho (w^*)} \left( u + mc^* |w- w^*|^2 \right) = \frac{1}{\pi \rho ^2}\int _{B_\rho (w^*)} u + \frac{1}{2} m c^* \rho ^2. \end{aligned}$$

Now if $m \delta ^2 \le 1$, then we take $\rho = \delta$, which implies that

$$\begin{aligned} \frac{1}{2} c^* \le c^* - \frac{1}{2} mc^* \delta ^2 \le \frac{1}{\pi \delta ^2} \int _{B_\delta (w^*)} u. \end{aligned}$$

Then

$$\begin{aligned} u(x) = f(0) \le f(\rho ^*) = 4 \delta ^2 c^* \le \frac{8}{\pi } \int _{B_\delta (w^*)} u \le \frac{8}{\pi } \int _{B_1(x)} u. \end{aligned}$$

On the other hand, if $m \delta ^2 > 1$, then take $\rho = \sqrt{ \frac{ 1 }{m } } < \delta$, we see

$$\begin{aligned} \frac{c^*}{2} \le \frac{ m}{\pi } \int _{B_\rho (w^*)} u \le \frac{ m}{\pi } \int _{B_1(x)} u. \end{aligned}$$

Therefore

$$\begin{aligned} \frac{\pi }{8a } = \epsilon \ge \int _{B_1(x)} u \ge \frac{ \pi c^* }{2 m } = \frac{ \pi c^* }{a + 4 a c^*}. \end{aligned}$$

It implies that $c^* \le \frac{1}{4}$. Therefore

$$\begin{aligned} u(x) \le u(w^*) = c^* \le \frac{2 ( a + 4 ac^*) }{\pi } \int _{B_1(x)} u \le \frac{4 a }{\pi } \int _{B_1(x)} u. \end{aligned}$$

In summary, we see that

$$\begin{aligned} \int _{B_1(x)} u \le \frac{\pi }{8 a} \Longrightarrow u(x) \le \max \{ \frac{8}{\pi }, \frac{4 a }{\pi } \} \int _{B_1(x)} u. \end{aligned}$$

For general $r>0$, the estimate follows by applying the above argument to $v(x) = u(rx)$ and a replaced by $ar^2$. $\square$

1.4 Mean-value estimate

Let h be a $(J, L, \mu )$-admissible metric on X satisfying (7.5). Let $\nabla$ be the Levi-Civita connection of h and we have the covariant derivatives defined by (7.2).

Now we consider the vortex equation on $\Xi$. An area form can be written as $\sigma ds dt$ for a smooth function $\sigma : \Xi \rightarrow (0, +\infty )$. We assume that there exist $c^{(l)} >0$, $l = 1, \ldots$ such that

$$\begin{aligned} \left| \nabla ^l \sigma \right| \le c^{(l)} \sigma . \end{aligned}$$

(7.9)

The vortex equation is written as

$$\begin{aligned} v_s + J v_t = 0,\ \kappa + \sigma \mu (u) = 0. \end{aligned}$$

(7.10)

Using a G-invariant metric h, we define the energy density for a solution $(u, \phi , \psi )$ by

$$\begin{aligned} e(s, t) = | v_s(s, t) |_h^2 + \sigma (s, t) | \mu (u(s, t)) |^2, \end{aligned}$$

where second norm is the G-invariant metric on ${\mathfrak g}$ we used to define the vortex equation.

Lemma 54

Let $\Xi \subset {\mathbb C}$ be an open subset. For any compact subset $K\subset X$, there exists ${\varvec{c}} = {\varvec{c}}(K)>0$ depending on K (and also on the constants $c^{(l)}$ of (7.9)) such that for any solution $(u, \phi , \psi )$ of (7.10) satisfying $u(\Xi ) \subset K$, its energy density function e satisfies

$$\begin{aligned} \varDelta e\ge -{\varvec{c}} \left( e + e^2 \right) . \end{aligned}$$

(7.11)

Here $\varDelta$ is the standard Laplacian in coordinates (s, t). If $\sigma$ is constant, we have

$$\begin{aligned} \varDelta e \ge - {\varvec{c}} e^2. \end{aligned}$$

(7.12)

Proof

Since the covariant derivative respects the metric, we have

Denote $\bar{\rho }(v_s, v_t) = \rho (v_s, v_t) - \rho (v_t, v_s)$. Then there exist $c_K>0$ and for any $\epsilon >0$, $c_{K, \epsilon }>0$, depending on the metric h and the compact subset K such that

(7.13)

To estimate $\varDelta |v_s|^2$, we have the following standard calculations.

(7.14)

(7.15)

(7.16)

On the other hand, by the G-invariance of $\nabla J$ and Lemma 47, there exist tensors $L_1, L_2, L_3$ such that

$$\begin{aligned} \nabla _{A, s} \left( \left( \nabla _{v_s} J \right) v_t \right) =&\ L_1 (v_s, v_s, v_t) + L_2 \left( \nabla _{A, s} v_s, v_t \right) + L_3\left( v_s, \nabla _{A, s} v_t \right) ;\\ \nabla _{A, s} \left( \left( \nabla _{v_t}J \right) v_s \right) =&\ L_1 (v_s, v_t, v_s) + L_2 \left( \nabla _{A, s} v_t, v_s \right) + L_3 \left( v_t, \nabla _{A, s} v_s \right) . \end{aligned}$$

Therefore, we have

Therefore, since $u(\Xi )\subset K$, with abusive use of (small) $\epsilon$ and (big) $c_{K, \epsilon }$, we have

(7.17)

Therefore, by (7.9), (7.13) and (7.17), we have

(7.18)

Here the second inequality follows from (7.5). Moreover, if $|\mu | \le \updelta$, then for $\epsilon \le {\varvec{m}}$, the first term of the last line is nonnegative; if $|\mu |> \updelta$, then the first term is greater than or equal to $- \epsilon \updelta ^{-2} \sigma ^2 |\mu |^4$. In either case, there exist ${\varvec{c}}', {\varvec{c}}>0$ depending on $c_{K, \epsilon }$, ${\varvec{m}}$, ${\varvec{n}}$ and $\updelta$ such that

(7.19)

This implies (7.11). (7.12) follows by setting $c^{(1)}= c^{(2)} = 0$. $\square$

1.5 Removal of singularity at punctures

We prove the first part of Lemma 8, which we restate as follows.

Proposition 55

Let $L_-, L_+$ be two G-Lagrangians of X which intersect cleanly in $\bar{X}$. Let $(\varSigma , \partial \varSigma ) = \left( {\mathbb D}^* \cap {\mathbb H}, {\mathbb D}^* \cap {\mathbb R} \right)$ and $\partial _\pm \varSigma = {\mathbb D}^* \cap {\mathbb R}_\pm$. Suppose $\mathbf{v}$ is a bounded solution to (2.2) on $\left( \varSigma , \partial \varSigma \right)$ with respect to a smooth area form $\nu \in \varOmega ^2 ( \varSigma )$. Then there exists a smooth gauge transformation $g: \varSigma \rightarrow G$ such that $g^* u$ extends continuously to $\{0\}$.

Proof

Identify $\varSigma$ with $[0, +\infty ) \times [0, \pi ]$ via $\varSigma \ni z \mapsto w = s + {\varvec{i}} t= - \log z$ and view the strip as a subset of ${\mathbb H}$. Suppose $\nu = \sigma ds dt$, it is easy to check that (7.9) is satisfied. Let $\widetilde{e}: [0, +\infty ) \times [0,\pi ] \rightarrow {\mathbb R}_+ \cup \{0\}$ be the energy density function. By Lemma 54, we have

$$\begin{aligned} \varDelta \widetilde{e} \ge - {\varvec{c}} \left( 1 + \widetilde{e}^2 \right) . \end{aligned}$$

Here ${\varvec{c}}$ depends on a choice of G-invariant metric on X.

Now to derive pointwise decay of $\widetilde{e}$ as $s \rightarrow +\infty$, we have to extend $\widetilde{e}$ a bit beyond the boundary of $[0,+\infty ) \times [0,\pi ]$. For example, we use reflection to define

$$\begin{aligned} \widetilde{e}(s, t) = \widetilde{e}(s, -t),\ t \in (-a, 0) \end{aligned}$$

for $a > 0$ a small constant. Then $\widetilde{e}$ is extended to $[0, +\infty ) \times (-a, 1]$. To see that the extension still satisfies (7.11), it suffices to show that $\partial _t \widetilde{e}(s, 0) = 0$. This is the place where we need the properties of metrics defined by Definition 48. Choose a $(J, L_-, \mu )$-admissible metric $h_-$.

First, by the boundary condition, $\partial _t | \mu (u) |^2 = 0$. On the other hand,

Here the third equality follows from (7.14) and the last follows from $\nabla _{A, s} J = \nabla _{v_s} J$. Then evaluating at $t = 0$, we see that in the last row, the first term vanishes because $\nabla J$ is skew-adjoint; the second term vanishes because $L_-$ is totally geodesic and $JTL_-$ is orthogonal to $TL_-$; the third term vanishes by the boundary condition.

Therefore (7.11) holds on $[0, +\infty ) \times (-a, 1]$, for the constant ${\varvec{c}}$ associated with the metric $h_-$. By the mean value estimate ([32, Page 12]) for any $B_{\frac{a}{2}}(w) \subset [0, +\infty ) \times (-a, \pi ]$,

$$\begin{aligned} \lim _{s\rightarrow +\infty } \widetilde{e}(s, t) = 0,\ \forall t \in \left[ 0, \pi - \frac{a}{2} \right) . \end{aligned}$$

To achieve the estimate near the other boundary component, simply take a $(J, L_+, \mu )$-admissible metric $h_+$ and do the reflection along the other boundary. Since all metrics are equivalent, we see that $\widetilde{e}(s, t)$ converges to zero uniformly as $s \rightarrow +\infty$.

On the other hand, it is easy to see that there exists a gauge transformation $g: [0, +\infty ) \times [0,1] \rightarrow G$ which transforms (A, u) into temporal gauge, i.e.,

$$\begin{aligned} g^* (A, u) = (\psi dt, u). \end{aligned}$$

Since $\sigma$ decays exponentially as $s \rightarrow +\infty$, by the equation $\partial _s \psi + \sigma \mu (u) = 0$ and the uniform boundedness of $|\mu (u)|$, we see that

Therefore $\left| \partial _t u (s, t) \right| \rightarrow 0$ as $s \rightarrow +\infty$. Let $\gamma _s(t) = u(s, t)$. Since $L_-$ and $L_+$ intersect cleanly in X, by the Poźniak’s isoperimetric inequality Lemma 50, we can prove that there exists $x\in L_- \cap L_+$ such that $\lim _{s \rightarrow \infty } u(s, t) = x$. $\square$

1.6 Energy quantization of ${\mathbb H}$-vortices

Proposition 56

Let L be a G-Lagrangian of X and $K \subset X$ be a compact subset. Then there exists a constant $\epsilon _{K, L} > 0$ such that the following holds. Suppose $z_0 \in {\mathbb H}$ and $r>0$. Suppose . Then if $u( B_{2r}(z_0) \cap {\mathbb H})\subset K$ and $E(\mathbf{v}) < \epsilon _{K, L}$, then

$$\begin{aligned} \sup _{B_{r}(z_0) \cap {\mathbb H}} e(\mathbf{v}) \le \frac{8}{\pi r^{2}} E( \mathbf{v}, B_{2r}(z_0) \cap {\mathbb H}). \end{aligned}$$

Here $e(\mathbf{v}): B_{2r}(z_0) \cap {\mathbb H} \rightarrow {\mathbb R}_+ \cup \{0\}$ is the energy density function of $\mathbf{v}$ with respect to the standard metric on ${\mathbb H}$ and a $(J, L, \mu )$-admissible metric h.

Proof

For the same reason as in the proof of Proposition 55, e can be extended to $B_{2r}(z_0) \subset {\mathbb C}$ by reflection along the boundary of $B_{2r}(z_0) \cap {\mathbb H}$. This extension still satisfies

$$\begin{aligned} \varDelta e \ge -{\varvec{c}} e^2. \end{aligned}$$

Then by [25, Lemma 4.3.2], the estimate holds. $\square$

Proof (Proof of Theorem 16)

Choose $\epsilon _{X, L} = \epsilon _{K, L}$ for $K = X_{c_0}$ where $X_{c_0}$ is the one in (2.3). Suppose $\mathbf{v}$ is an ${\mathbb H}$-vortex and $E(\mathbf{v}) < \epsilon _{X, L}$. Then by the mean value estimate in Proposition 56, we see that for any $z\in {\mathbb H}$ and $r>0$,

$$\begin{aligned} e(z)\le \frac{8}{\pi r^{2}} E ( \mathbf{v}; B_r(z) ). \end{aligned}$$

Let $r\rightarrow \infty$, we have $e(z)=0$. Thus $E(\mathbf{v})=0$. $\square$

1.7 Annulus lemma for vortices on strips

Ziltener proved ([43, Proposition 45]) that for any annulus $A(r, R) = \{z\in {\mathbb C}\ |\ r\le |z|\le R\}$ and any small $\epsilon >0$, there exists a constant $E(r, \epsilon )$ such that for any vortex $\mathbf{v} = (A, u)$ on A(r, R) with respect to the standard area form, if $E(\mathbf{v}) \le E(r, \epsilon )$, then

$$\begin{aligned} \begin{aligned} E ( \mathbf{v}; A(ar, a^{-1} R) ) \le&\ c a^{ - 2 + \epsilon } E(\mathbf{v}),\\ \mathrm{diam}_G ( u(A(ar, a^{-r}R)) ) \le&\ c a^{-1 + \epsilon } \sqrt{ E(\mathbf{v})} \end{aligned} \end{aligned}$$

for some constant $c>0$ and for any $a \in \big [ 2, \sqrt{ R/r} \big ]$. Now we prove an analogue of this result on strips. Via the map $w = s + {\varvec{i}}t = \log z$, we identify the strip $A^+(r, R) = A(r, R) \cap {\mathbb H}$ with

$$\begin{aligned}{}[p, q]\times [0, \pi ] := [\log r, \log R] \times [0, \pi ]. \end{aligned}$$

Let $\nu = \sigma (s, t) ds dt$ be an area form satisfying (7.9).

Proposition 57

There exists ${\varvec{a}} = {\varvec{a}}(L_-, L_+) > 0$, $\upepsilon = \upepsilon (L_-, L_+)>0$ satisfying the following condition. Given a smooth solution $\mathbf{v} = (u, \phi , \psi )$ to (7.10) with boundary condition $u(A(r, R)\cap {\mathbb R}_\pm ) \subset L_\pm$. Suppose $\sigma$ is bounded from below by a constant $\underline{\sigma }$. If $E(\mathbf{v}) \le \upepsilon$, then for any $s \in \left[ \log 2, \frac{1}{2} ( q-p) \right]$, we have

$$\begin{aligned} E( \mathbf{v}; A^+ ( e^s r, e^{-s} R) )\le & {} {\varvec{a}} \exp \Big ( - \frac{s \min \{1, \underline{\sigma }\} }{{\varvec{c}} } \Big ) E ( \mathbf{v}; A^+ (r, R) ); \end{aligned}$$

(7.20)

$$\begin{aligned} \mathrm{diam}_G ( u( A^+( e^s r, e^{-s} R) ))\le & {} {\varvec{a}} \exp \Big ( -\frac{ s \min \{ 1, \underline{\sigma }\} }{2 {\varvec{c}} }\Big ) \sqrt{E ( \mathbf{v}; A^+(r, R) ) }. \end{aligned}$$

(7.21)

Here ${\varvec{c}} = {\varvec{c}} (L_-, L_+) >0$ is the ${\varvec{c}}(L_0, L_1)$ in Lemma 52 for $L_0 = L_+$, $L_1 = L_-$.

Proof

Let $\epsilon (s, t)$ be the energy density with respect to the cylindrical coordinates, so that

$$\begin{aligned} E (\mathbf{v}; A^+(r, R) ) = \int _p^q \int _0^\pi \epsilon (s, t) ds dt. \end{aligned}$$

By the estimate of Lemma 54 on the strip and Lemma 53, we know that there exists $\upepsilon >0$ such that if $E (\mathbf{v}; A^+(r, R) ) \le \upepsilon$, then for any $(s, t) \in [ p + \log 2, q - \log 2] \times [0, \pi ]$, we have $\epsilon (s, t)\le \updelta (L_-, L_+)$. Here $\updelta (L_-, L_+)$ is the one from Lemma 52. (Notice that when applying Lemma 53, we have to use $(J, L_\pm , \mu )$-admissible metrics to extend the energy density function beyond the boundary of the strip, and notice that the r of Lemma 53 is uniformly bounded). We write $A = d + \phi (s, t) ds + \psi (s, t) dt$. Then for $s \in [p + 2, q - 2]$, we have . So we can define the local equivariant action

For $s \in [ \log 2, ( q - p)/2 ]$, we denote $E(s) = E ( \mathbf{v}; A^+ ( e^s r, e^{-s} R) )$. Then by the isoperimetric inequality (Lemma 52), for ${\varvec{c}} = {\varvec{c}}(L_-, L_+)$, we have

Here $e(s, t) = |v_t(s, t)|^2 + \sigma (s, t)|\mu (u(s, t))|^2$. Abbreviate $\widetilde{\varvec{c}} = {\varvec{c}}/ \min \{1, \underline{\sigma }\}$. Then we have

$$\begin{aligned} E(s) \le E(2) \exp \Big ( - \frac{ s-2}{\widetilde{\varvec{c}}}\Big ) \le E ( \mathbf{v}; A^+(r, R) ) e^{2 /{\varvec{c}}} \exp \Big ( - \frac{s}{\widetilde{\varvec{c}}}\Big ). \end{aligned}$$

Therefore (7.20) holds.

To prove the estimate for the radius, apply Lemma 53 to $\epsilon$ again, for a choice of r uniformly bounded from below. Then e decays in a similar way as

$$\begin{aligned} \sup _{[p + s, q - s]\times [0, \pi ]} \epsilon = O ( \exp ( - s/\widetilde{\varvec{c}}) ). \end{aligned}$$

(7.22)

Integrating over $[ p + s, q - s]$ gives the upper bound on the equivariant diameter. $\square$

Now we prove the following asymptotic property of ${\mathbb H}$-vortices.

Proof (Proof of Theorem 14)

Let $q \rightarrow +\infty$ in (7.22), we obtain

$$\begin{aligned} \epsilon (s, t) \le {\varvec{a}} \exp ( -s/ {\varvec{c}}). \end{aligned}$$

Since $\epsilon (s, t) = e^{2s} e(s, t)$, we obtain (2.7). On the other hand, we could transform (A, u) into temporal gauge, i.e., $A = d + \psi d t$ such that

$$\begin{aligned} \lim _{s \rightarrow +\infty } \psi ( s, t) = 0. \end{aligned}$$

Then the decay of $\epsilon$ implies that in this gauge, u converges to a limit in $L_- \cap L_+$. $\square$

Appendix 2: Trees

In this appendix we fix notions and notations of trees.

In our convention, a tree consists of a finite set of vertices , a finite set of (finite) edges , and a finite set of semi-infinite edges . The semi-infinite edges are attached to vertices, by a map . A rooted tree is a tree with a distinguished vertex, which is usually denoted by $v_\infty$.

In this paper we mainly consider rooted trees. There are obvious notions of morphisms between rooted trees, and rooted subtrees. We index vertices by letters $\alpha , \beta , \gamma$, etc.. For a rooted tree , there is a canonical partial order $\le$ in with $v_\infty$ the unique minimal element. Moreover, for notational purpose, we only consider edges with the correct orientation, i.e., for , we write $\alpha \succ \beta$ if and only if $v_\alpha$, $v_\beta$ are adjacent and $\beta \le \alpha$. For any , we denote by the unique vertex such that . For , denote by the end point of $e'$ which is closer to the root.

We regard a tree as a 1-dimensional simplicial complex. Note that a semi-infinite edge only has one end combinatorially, but the point at infinity on the semi-infinite edge is regarded as a point of the simplicial complex.

Definition 58

([13, Definition 1.1])

• 1. A rooted ribbon tree with $\underline{k}$ semi-infinite edges consists of a rooted tree with a topological embedding such that consists of the $\underline{k}$ infinities of these semi-infinite edges. As a convention we always order the semi-infinite edges by $e_1, \ldots , e_{\underline{k}}$ which respects the cyclic ordering induced by the embedding.

• 2. An isomorphism between two rooted ribbon trees and consists of a rooted tree isomorphism together with an isotopy as embedding of pairs between ${\mathfrak i}'\circ \rho$ and ${\mathfrak i}$. Two ribbon trees with $\underline{k}$ semi-infinite edges are equivalent if there is an isomorphism between them. An isotopy class of embeddings ${\mathfrak i}$ for a rooted tree is called a ribbon structure on .

• 3. A based rooted tree consists of a rooted tree and a rooted subtree where the latter is equipped with a ribbon structure.

• 4. A colored rooted tree is a rooted tree together with an order-reversing map (called the coloring) such that within every path of , ${\mathfrak s}^{-1}(1)$ consists of at most one vertex.

• 5. A vertex $v_\alpha$ of a based colored rooted tree is stable if one of the followings is true.

  • $v_\alpha = v_\infty$;

  • and ;

  • and ;

  • .

Definition 59

Let be a based colored rooted tree. A growth is another based colored rooted tree with a morphism , which is the composition of finitely many elementary growths of the following types.

In this case with . $\rho$ contracts the edge . corresponds to sphere bubbling downstairs.

In this case with . $\rho$ contracts the edge . corresponds to disk bubbling downstairs.

In this case with . $\rho$ contracts the edge . corresponds to the ${\mathbb C}$-vortex bubbling.

In this case with . $\rho$ contracts the edge $e_{\underline{\alpha }\underline{\alpha }'}$. corresponds to ${\mathbb H}$-vortex bubbling.

In this case with . $\rho$ contracts the edge $e_{\alpha \alpha '}$. corresponds to sphere bubbling upstairs.

In this case with . $\rho$ contracts the edge $e_{\underline{\alpha }\underline{\alpha }'}$. corresponds to disk bubbling upstairs.

In this case and $\rho$ contracts the path $\underline{\alpha }\succ \underline{\alpha }_1 \succ \cdots \succ \underline{\alpha }_s \succ \underline{\alpha }'$ in to the edge . corresponds to the appearance of connecting disk bubbles, either upstairs or downstairs.

1.1 Combinatorial types of stable holomorphic spheres and disks

In this subsection we set up some convention of expressing holomorphic spheres or disks.

1.1.1 Holomorphic spheres

Stable holomorphic spheres are modelled on ordinary trees. We will consider stable holomorphic spheres with a single marked point, which specifies a root of the tree. Therefore, rooted trees are what such objects are modelled on. Let be a rooted tree. A stable holomorphic sphere in an almost Kähler manifold $(X, \omega , J)$ modelled on is a collection of objects

where

• 1. For each , $u_\alpha : {\mathbb C} \rightarrow X$ is a holomorhpic map with finite energy (therefore extends to a holomorphic sphere with an evaluation $u_\alpha (\infty )$), such that $E(u_\alpha ) = 0$ implies that or $v_\alpha = v_\infty$ and .

• 2. For each , $z_{\alpha \beta } \in {\mathbb C}$ such that $u_\alpha (\infty ) = u_\beta ( z_{\alpha \beta })$ and for each , the collection of points $Z_\beta := ( z_{\alpha \beta } )_{\beta = \alpha '}$ are distinct.

In this situation, we call the rooted tree a branch.

1.1.2 Holomorphic disks

Stable holomorphic disks are modelled on based trees. We will consider stable holomorphic disks with a single boundary marked point, which specifies a root of the base. A stable J-holomorphic disk in (X, L) modelled on a based rooted tree is a collection

where

• 1. For each , $u_{\underline{\alpha }}: ({\mathbb H}, {\mathbb R}) \rightarrow (X, L)$ is a holomorphic map with finite energy (therefore extends to a holomorphic disk in (X, L) with an evaluation $u_{\underline{\alpha }}(\infty )$); for each , $u_\alpha : {\mathbb C} \rightarrow X$ is a holomorphic map with finite energy (therefore has an evaluation $u_\alpha (\infty )$); they should satisfy the stability condition: for and $E(u_\alpha ) = 0$, ; if and $E(u_\alpha ) = 0$, ; if $E(u_\infty ) = 0$ then .

• 2. (Denote $\varSigma _\alpha = {\mathbb H}$ if and $\varSigma _\alpha = {\mathbb C}$ otherwise.) For each , $z_{\alpha \beta }\in \partial \varSigma _\beta$; for each , $z_{\alpha \beta }\in \mathrm{Int} \varSigma _\beta$. They satisfy $u_\alpha (\infty ) = u_\beta ( z_{\alpha \beta })$ and for each , the collection of points $Z_\beta := (z_{\alpha \beta })_{\alpha ' = \beta }$ are distinct.

In this situation, we call the based rooted tree a based branch.