Limiting Measures and Energy Growth for Sequences of Solutions to Taubes’s Seiberg–Witten Equations

Abstract

We consider sequences of solutions $(\psi _n,A_n)_{n=1}^\infty$ to Taubes’s modified Seiberg–Witten equations, associated with a fixed volume-preserving vector field X on a 3-manifold and corresponding to arbitrarily large values of the strength parameter $r_n\rightarrow \infty$. In Taubes’s work, the asymptotic behavior of these solutions is related to the dynamics of X. We consider the rather unexplored case of sequences of solutions whose energy is not uniformly bounded as $n\rightarrow \infty$. Our first main result shows that when the energy grows more slowly than $r_n^{1/2}$, the limiting nodal set of the solutions converges to an invariant set of the vector field X. The main tool we use is a novel maximum principle for the solutions with the key property that it remains valid in the unbounded energy case. As a byproduct, in the usual case of sequences of solutions with bounded energy, we obtain a new, more straightforward proof of Taubes’s result on the existence of periodic orbits that does not involve a local analysis or the vortex equations. Our second main result proves that, contrary to what happens in the bounded energy case, when the energy is unbounded there are no local restrictions to the limiting measures that may arise in the modified Seiberg–Witten equations. Furthermore, we obtain a connection between the dimension of the support of the limiting measure (as expressed through a d-Frostman property) and the energy growth of the sequence of local solutions we construct.

1 Introduction

Taubes’s celebrated proof of the Weinstein conjecture in dimension 3 hinges on the analysis of a modified version of the Seiberg–Witten equations [10], which were originally introduced to study supersymmetric gauge theories in four dimensions. To define Taubes’s equations, one starts off with a closed oriented 3-manifold M, endowed with a smooth volume form $\mu$, and an exact volume-preserving vector field X that does not vanish. We recall that X is said to be exact if $i_X\mu$ is an exact 2-form. The modified Seiberg–Witten equations are then a gauge-invariant semilinear elliptic system on M that depends on the vector field X and on a large parameter r. The gist of Taubes’s approach is to relate the dynamics of the vector field X with the concentration properties of a certain sequence of solutions as $r\rightarrow \infty$.

Let us record here the form of the system of PDEs considered by Taubes. For this, one starts by noting that one can take an adapted metric, that is, a Riemannian metric g on M such that $\mu$ is the corresponding volume form and X is a unit vector in this metric: $g(X,X)=1$. There is no loss of generality in assuming that $\mu$ is normalized so that $\int _M\mu =1$. If we now denote by

$$\begin{aligned} \lambda :=g(X,\cdot ) \end{aligned}$$

the 1-form dual to the vector field X, Taubes’s modified Seiberg–Witten equations is a system of equations defined using the metric g and depending on a real parameter $r\gg 1$. The unknowns are A, which is a connection on a complex line bundle, and $\psi$, which is a section of a related ${\mathbb {C}}^2$ bundle of spinors. The equations read as

$$\begin{aligned} \begin{aligned} *F_{A}&= r(\lambda -\psi ^\dagger \sigma \psi ) + \varpi ,\\ D_{A}\psi&= 0, \end{aligned} \end{aligned}$$

(1.1)

where $*F_{A}$ denotes the Hodge dual of the curvature 2-form of the connection A (which we take to be real valued), $D_{A}$ is the Dirac operator defined by this connection and the Riemannian metric and $\psi ^\dagger \sigma \psi$ is a 1-form, depending quadratically on the spinor $\psi$, which is defined using Clifford multiplication on the spinor bundle. The equations depend on an auxiliary 1-form $\varpi$ and on a reference connection $A_0$, which must be chosen carefully and are bounded in the $C^3$ norm by a constant independent of r. Precise definitions will be provided in Sect. 2.

Remark 1.1

For convenience, we follow the usual notation according to which a connection on a complex line bundle is locally written as $-iA$, so that A is a real 1-form. The curvature $F_A$ is the 2-form written locally as $F_A=dA$. In other words, if $A_{T}$ and $F_{A_T}$ denote the imaginary valued connection and its corresponding curvature (as in Taubes’s article [10]), we have $A_{T}=-i A$ and $F_{A_T}=-i F_{A}$.

A key quantity in Taubes’s analysis of the concentration properties of a sequence of solutions to the modified Seiberg–Witten equations is the so-called energy of the connection A,

$$\begin{aligned} {{\mathcal {E}}}(A):= \int _M \lambda \wedge F_{A}. \end{aligned}$$

Although it is not obvious a priori, one can show [10] that, for any solution of the Seiberg–Witten equations, the energy can be estimated as

$$\begin{aligned} -C<{{\mathcal {E}}}(A)< Cr. \end{aligned}$$

The first part of Taubes’s proof of the Weinstein conjecture in dimension 3 is to show, in a technical tour de force building upon the work of Kronheimer and Mrowka [4], that if X is the Reeb field of a contact form, then one can construct a sequence $(r_{n},\,\psi _n,\,A_n)_{n=1}^\infty$ of solutions of fixed degree to the modified Seiberg–Witten equations with $r_n\rightarrow \infty$ and bounded energy (i.e., ${{\mathcal {E}}}({{\mathcal {A}}}_n)<C$); see [10, Section 3] for a definition of the degree of a solution. The second part of the proof consists in analyzing the limiting measures defined by a sequence of solutions with fixed degree and bounded energy.

The state of the art concerning our knowledge of limiting measures for the Seiberg–Witten equations is summarized in the following theorem. The statement uses the helicity of the exact vector field X [1, 12], which can be written as

$$\begin{aligned} {{\mathcal {H}}}(X):= \int _M \gamma \wedge d\gamma = \int _M *(\gamma \wedge d\gamma )\, \mu \end{aligned}$$

in terms of the 1-form $\gamma$ defined by the equation $i_X\mu = d\gamma$ modulo a closed 1-form which does not contribute to the integral. Here $*$ denotes the Hodge star operator.

Remark 1.2

For ease of notation, in the statement of the theorem below and in what follows, we often identify 3-forms and their corresponding signed measures in the obvious way: if $\Omega$ is a 3-form then there is a signed measure (which we will denote by $\Omega$ or $d\Omega$ when no confusion may arise) defined as

$$\begin{aligned} \int _{M} f \, d \Omega :=\int _{M} f \, \Omega =\int _{M} f\, (*\Omega ) \,\mu \end{aligned}$$

for each $f\in C(M)$.

Theorem 1.3

(Taubes [10, 11]) Suppose that the helicity of the vector field X is positive. Then there exists a sequence $(r_{n},\,\psi _n,\,A_n)_{n=1}^\infty$ of solutions to the modified Seiberg–Witten equations (1.1) with $r_n\rightarrow \infty$ and fixed degree. Furthermore:

• (i) If the sequence of energies ${{\mathcal {E}}}_n:={{\mathcal {E}}}(A_n)$ is bounded (i.e., ${{\mathcal {E}}}_n< C$), then the vector field X possesses at least one periodic orbit.

• (ii) If the sequence of energies is not bounded, the signed measures

  $$\begin{aligned} \sigma _n:=\frac{\lambda \wedge F_{A_n}}{{{\mathcal {E}}}_n} \end{aligned}$$

  converge, possibly after passing to a subsequence, to an invariant probability measure $\sigma _{\infty }$ of X. This measure satisfies

  $$\begin{aligned} \int _M *(\gamma \wedge d\gamma )\, d\sigma _\infty \leqslant 0, \end{aligned}$$

  so it is not the volume.

Remark 1.4

In Theorem A.2 we will show that the last assertion can be refined to show that, in fact, one can take a subsequence so that

$$\begin{aligned} \int _M *(\gamma \wedge d\gamma )\, d\sigma _\infty = 0, \end{aligned}$$

provided that the energy growth is sublinear.

When X is the Reeb field of a contact form, then there exists a sequence of solutions of fixed degree with bounded energy [10]. For other kinds of exact volume preserving vector fields, however, all sequences of solutions could have unbounded energy.

One is thus naturally led to the goal of extracting more properties of the invariant measure $\sigma _{\infty }$ in the unbounded energy case. This is an interesting question on geometric analysis and could provide new techniques to study the dynamics of volume-preserving 3-dimensional vector fields. Despite its promise, there have not been any further developments in this direction, and any other properties of the invariant measures $\sigma _{\infty }$ remain a mystery.

Our objective in this paper is to analyze the limiting measures for sequences of solutions with unbounded energy. Specifically, we shall next present two theorems which illustrate, and under suitable hypotheses provide precise statements of, the following two rough guiding principles:

• (i) The support of the limiting measure is contained in the set where $|\psi _n|$ tends to 0 (Theorem 1.5 and Proposition 1.7).

• (ii) There are no local obstructions to the limiting measures when the energy is unbounded, so the problem is inherently global (Theorem 1.8).

Needless to say, we do not expect these principles to hold is all generality; however, the theorems we state below show that they do provide useful intuitions. We hope that these results will spark further developments on this subject.

The main difficulty in the unbounded energy case is that, over small scales, the solutions to the Seiberg–Witten equations can no longer be interpreted as approximate solutions to the 2-dimensional vortex equations with finite energy. This asymptotic small-scale behavior is a key ingredient in Taubes’s approach.

To overcome this difficulty we resort to a combination of various tools, the most important of which is a new maximum principle for the Seiberg–Witten equations (Theorem 3.1). The key feature of this maximum principle is that it applies no matter if the energy is bounded or not. Although we are mostly interested in the latter case, when the energy is bounded, this provides a substitute of Taubes’s local analysis based on the vortex equations. This enables us to provide a different, more straightforward proof of the corresponding results.

1.1 Limiting Measures Supported on the Set Where ${|\psi _n|\rightarrow 0}$

An important observation of Taubes [10] (which follows immediately from the bounds in Lemma 2.3) is that

$$\begin{aligned} 1-|\psi _n|^2 \geqslant -\frac{C}{r_n}, \end{aligned}$$

so for all large n and any $p\in M$, $|\psi _n(p)|^2$ is bounded by a constant as close to 1 as desired. This does not imply that $|\psi _n|^2$ converges to an indicator function because the smooth functions $|\psi _n|^2$ can oscillate wildly. However, the way the “zeros and ones” of $|\psi _n(p)|^2$ are distributed across the manifold has much to do with the dynamics of the vector field X, and the analysis of the sets where $|\psi _n|^2$ tends to 0 or 1 lies at the very heart of Taubes’s proof.

Our first main result shows that when the energy grows slower than $r^{\frac{1}{2}}$, the set of points of M where $|\psi _n|$ tends to 0 (that is, the limiting nodal set) is invariant under the flow of X. The tools we develop to prove this result provide, in the special case of sequences with bounded energy, a direct proof of Taubes’s celebrated periodic orbit theorem (item (i) in Theorem 1.3), see Sect. 4.3. Contrary to Taubes’s, this proof does not rely on the relationship between the small scale behavior of the Seiberg–Witten and the vortex equations. We want to emphasize that the following theorem is the natural generalization of Taubes’s periodic orbit theorem for solutions with unbounded energy.

Theorem 1.5

Suppose that X has positive helicity and consider a sequence of solutions $(r_{n},\psi _{n},A_{n})_{n=1}^\infty$ to the associated Seiberg–Witten equations with $r_n\rightarrow \infty$ and ${{\mathcal {E}}}_n= o (r_n^{\frac{1}{2}})$, i.e.,

$$\begin{aligned} \limsup _{n\rightarrow \infty } \frac{{{\mathcal {E}}}_n}{r_n^{1/2}}=0. \end{aligned}$$

Then:

• (i) For any fixed $\theta \in (0, 1)$, the set

  $$\begin{aligned} Z^{\theta }_n:=\left\{ p \in M: 1-|\psi _n|^2(p) \geqslant \theta \right\} \end{aligned}$$

  is non-empty for n large enough, and any convergent subsequence (in the Hausdorff metric) converges to a closed subset $Z^\theta _{\infty }$ which is invariant under the flow of X.

• (ii) The collection of limiting sets $Z^\theta _{\infty }$ is independent of $\theta$, in the sense that, for any converging subsequence $Z^\theta _n$, the corresponding subsequence $Z_n^{\theta '}$, for any $\theta '\ne \theta$, is also converging with the same limit, i.e., $Z^{\theta '}_\infty =Z^\theta _\infty$.

• (iii) There is a constant C, independent of n, such that any convergent subsequence of sets

  $$\begin{aligned} Z_n:=\left\{ p \in M: |\psi _n|^2(p) \leqslant C\max (r_n^{-\frac{1}{4}},{{\mathcal {E}}}_n r_n^{-\frac{1}{2}}) \right\} \end{aligned}$$

  also converges to an invariant set $Z_{\infty }$. The collection of such limiting sets coincides with the limiting sets $Z^\theta _\infty$ in the sense specified above.

In the case that the sequence of energies is uniformly bounded, the invariant set $Z_\infty$ consists of a finite collection of periodic orbits of X.

Remark 1.6

If instead of taking the Hausdorff limit of the sequences of sets in Theorem 1.5, we take the upper Kuratowski limit, this is always compact and unique (independent of the subsequence), so we can write $Z^\theta _\infty =Z_\infty$ for all $\theta \in (0,1)$.

This theorem is proved in Sect. 4, using the maximum principle presented in Sect. 3. It should be emphasized that, in general, the limiting invariant set $Z_{\infty }$ could be the whole manifold M. Indeed, because of the high oscillations of $|\psi _n|$ for large n, the fact that a point p is in $Z_{\infty }$ does not imply that $1-|\psi _n|^2(p) >0$ for all large enough n; it could very well happen that $|\psi _n|(p)=1$ for all n. We can only characterize $Z_\infty$ when the energy is uniformly bounded.

Concerning points where $|\psi _n|$ tends to 1, the next proposition establishes that if $|\psi _n|\rightarrow 1$ on an open set U, then the limiting measure does not charge this set. This result is proved in Sect. 4.4.

Proposition 1.7

Let $(r_n,\psi _n, A_n)_{n=1}^\infty$ be a sequence of solutions with unbounded energy. If $|\psi _n|\rightarrow 1$ pointwise on an open set $U\subset M$ as $n\rightarrow \infty$, then $\sigma _\infty (U)=0$.

1.2 Absence of Local Obstructions for the Limiting Measures

Our second main result proves that, locally, any invariant measure can arise as the limiting measure for some sequence of solutions to the Seiberg–Witten equations. Thus, contrary to what happens in the bounded energy case, when the energy is unbounded any attempt to derive some restrictions to the possible invariant sets of the vector field X from the PDE must involve global arguments.

To state the theorem, we start by fixing a flow box ${{\mathcal {C}}}\subset M$ of the vector field X. We choose local coordinates and identify ${{\mathcal {C}}}= (0,1)\times \mathbb {{D}}$, where $\mathbb {{D}}$ is the (open) unit 2-dimensional disk, and assume that $X=\partial _t$ with t being the coordinate on the interval (0, 1). Note that any X-invariant measure on ${{\mathcal {C}}}$ can then be written as

$$\begin{aligned} \sigma =\sigma _{\mathbb {{D}}}\otimes dt \, \end{aligned}$$

where $\sigma _{\mathbb {{D}}}$ is a measure supported on $\mathbb {{D}}$ and dt is the Lebesgue measure on the interval. Without loss of generality, we can normalize $\sigma _{\mathbb {{D}}}$ and assume that it is a probability measure.

The following theorem does not only show that there are no local obstructions for the limiting measure obtained from solutions to the Seiberg–Witten equations. Furthermore, it also suggests that there is a connection between the dimension of the support of the invariant measure and the energy sequence. Roughly speaking, the faster the growth of ${{\mathcal {E}}}_n$ that we allow, the larger the dimension of the support of the measure $\sigma _\infty$. A convenient way of articulating this connection is by recalling that a probability measure $\sigma$ on $\mathbb {{D}}$ is d-Frostman if the measure of any ball $B(x,\varepsilon )$ of radius $\varepsilon$ is bounded as

$$\begin{aligned} \sigma (B(x,\varepsilon ))\leqslant C\varepsilon ^d, \end{aligned}$$

for all $\varepsilon >0$ and $x\in \mathbb {{D}}$. It is standard [8, Exercise 1.15.20] that this property implies that the Hausdorff dimension of the support of $\sigma$ is at least d (i.e., $\dim _H \text {(supp } \sigma )\geqslant d$), but this property is strictly stronger in that it provides some quantitative control on the measure. It is worth mentioning that the dimension of the support of the metric is also connected with its regularity (i.e., very roughly speaking, the better the integrability properties of the weak derivatives of the measure, as estimated using Sobolev or Besov spaces, the higher the Hausdorff dimension of its support). For the benefit of the reader, we specify this connection in Proposition 5.4.

Theorem 1.8

Let $\sigma _{\mathbb {{D}}}$ be any probability measure on $\mathbb {{D}}$. There is an adapted metric g on ${{\mathcal {C}}}$ and a sequence of solutions $(r_n,\psi _n,A_n)$ to the Seiberg–Witten equations (1.1) with $\varpi =0$ on the flow box ${{\mathcal {C}}}$ such that

• (i) Setting ${{\mathcal {E}}}_n:= \int _{{{\mathcal {C}}}} \lambda \wedge F_{A_n}$, we have

  $$\begin{aligned} \frac{\lambda \wedge F_{A_n}}{{{\mathcal {E}}}_n}\rightarrow \sigma _{\mathbb {{D}}}\otimes \, dt \end{aligned}$$

  in the sense of weak convergence of measures.

• (ii) If $\sigma _{\mathbb {{D}}}$ is d-Frostman for some $d>0$, then we can choose the sequence of solutions such that

  $$\begin{aligned} \lim _{n\rightarrow \infty }{{\mathcal {E}}}_{n} r_{n}^{-\theta }= 0 \end{aligned}$$

  with $\theta :=\min \big \{\frac{1}{4}, \, \frac{d}{2(d+1)}\big \}$.

1.3 Structure of the Paper

In Sect. 2 we recall the definition of the various objects appearing in the modified Seiberg–Witten equations and some properties of the solutions. Some further auxiliary equations are derived too. In Sect. 3 we prove a new maximum principle for these equations that can be effectively applied to sequences of solutions with unbounded energy. This result turns out to be a fundamental tool to analyze the properties of the limiting invariant measures. The proofs of Theorem 1.5 and Proposition 1.7 are presented in Sect. 4. As an additional application of the new maximum principle, we also include an alternative proof of Taubes’s periodic orbit theorem. The proof of Theorem 1.8 on the absence of local obstructions for the limiting measure is given in Sect. 5. Finally, in Sect. 6 we show that the vector field X cannot be ergodic provided that the energy growth is linear. We also include Appendix A with an additional result that can be useful for future work on the subject. Concretely, we reinterpret the concentration properties of solutions to the Seiberg–Witten equations using Sullivan’s theory of currents, thus implying as a particular consequence the refinement stated in Remark 1.4.

2 Setting and Preliminary Results

In this section, following Taubes [10] (see also [7]), we include the precise formulation of the modified Seiberg–Witten equations, Taubes’s theorem on the existence of solutions (Theorem 2.2) and the basic a priori estimates (Lemmas 2.3 and 2.4). Our main contribution is Proposition 2.5, which shows that the function $|\alpha |^2$ defined below satisfies an explicit second order elliptic PDE on M. This statement is not included in Taubes’s works and is key to prove the new maximum principle we present in Sect. 3.

2.1 Definitions and Existence of Solutions

Let us recall the definition of the modified Seiberg–Witten equations. The reader can find further details in [3, 10, 11]. Throughout, g denotes a fixed Riemannian metric on the 3-manifold M adapted to the volume-preserving vector field X, and $\lambda := g(X,\cdot )$ stands for the dual 1-form. In what follows, we will assume that the helicity of the vector field X is positive.

We start by recalling that a spin-c structure on M is a pair $\mathfrak {s}= ({{\mathbb {S}}},\sigma )$, where ${{\mathbb {S}}}$ is a rank-2 Hermitian vector bundle on M, called the spinor bundle, and

$$\begin{aligned} \sigma : TM \longrightarrow {\text {End}}({{\mathbb {S}}}) \end{aligned}$$

is a bundle map, called Clifford multiplication, such that for each $p\in M$:

• (i) If $V,W\in T_pM$, then

  $$\begin{aligned} \sigma (V)\sigma (W) + \sigma (W)\sigma (V) = -2g(V,W). \end{aligned}$$

• (ii) If $e_1,e_2,e_3$ is an oriented orthonormal frame for $T_pM$, then

  $$\begin{aligned} \sigma (e_1)\sigma (e_2)\sigma (e_3)=1. \end{aligned}$$

Of course, taking a local trivialization, we can identify ${\mathbb {S}}$ with ${\mathbb {C}}^2$.

A spin-c connection A on ${\mathbb {S}}$ is a connection that behaves naturally with respect to the Clifford multiplication: for any section $\psi : M \rightarrow {\mathbb {S}}$ and any vector fields $X, Y \in TM$, we have

$$\begin{aligned} \nabla ^{A}_X (\sigma (Y) \psi )=\sigma (\nabla ^{LC}_{X} Y) \psi +\sigma (Y)\nabla ^{A}_{X} \psi \end{aligned}$$

where $\nabla ^{LC}$ is the Levi-Civita connection on TM induced by the metric g.

Consider the oriented 2-plane field $K^{-1}:={\text {Ker}}\lambda$, which we regard as a Hermitian line bundle. It is standard that $K^{-1}$ determines a distinguished spin-c structure $\mathfrak {s}_\lambda =({\mathbb {S}}_\lambda ,\sigma _\lambda )$ on M, in which

$$\begin{aligned} {{\mathbb {S}}}_\lambda ={\underline{{\mathbb {C}}}}\oplus K^{-1}, \end{aligned}$$

(2.1)

where ${\underline{{\mathbb {C}}}}$ denotes the trivial complex line bundle over M. Any spin-c structure $\mathfrak {s} =({\mathbb {S}},\sigma )$ is obtained from this one by tensoring with a suitable Hermitian line bundle E, so that

$$\begin{aligned} {{\mathbb {S}}} = E \otimes {{\mathbb {S}}}_\lambda = E\oplus K^{-1}E\, \end{aligned}$$

(2.2)

and $\sigma =\sigma _{\lambda }\otimes \text {Id}_{E}$. In this decomposition, E is the $+i$ eigenspace of Clifford multiplication by the vector field dual to the 1-form $\lambda$, while $K^{-1}E$ is the $-i$ eigenspace.

In what follows, E stands for a fixed line bundle whose first Chern class is such that $c_1(K^{-1})+ 2 c_1(E)$ is torsion in $H^2(M;{\mathbb {Z}})$. This is equivalent to requiring that $\det ({\mathbb {S}})$ has torsion first Chern class, since any spin-c connection on ${\mathbb {S}}$ induces a connection on the determinant bundle $\det ({\mathbb {S}})$ that can be written as $A_0+2A$, where $A_0$ is a connection on $K^{-1}$ (inducing a natural spin-c connection on ${\mathbb {S}}_{\lambda }$) and A is a connection on E.

The unknowns in the Seiberg–Witten equations are a spinor $\psi$, which is a section of ${\mathbb {S}}$, and a connection A on E, whose curvature we denote by $F_A$. We also need an auxiliary connection $A_0$ on $K^{-1}$, which we pick (following Taubes) as the only connection such that

$$\begin{aligned} D_{A_0}\psi _0=0, \end{aligned}$$

(2.3)

where $\psi _0:=(1,0)$ is a section of the distinguished spin bundle ${\mathbb {S}}_\lambda$, and $D_{A_0}$ should be understood as the Dirac operator associated with the unique spin-c connection on ${\mathbb {S}}_\lambda$ defined by the connection $A_0$ on $K^{-1}$ and the trivial connection on ${\mathbb {C}}$. It is well known that the connections $A_0$ on $K^{-1}$ and A on E determine a unique spin-c connection on $\mathfrak {s}$, which we will denote by $\nabla _A$; the associated Dirac operator is then defined as $D_A:=\sigma (\nabla _A)$. Taubes’s modified Seiberg–Witten equations read

$$\begin{aligned} \begin{aligned} *F_{A}&= r(\lambda -\psi ^\dagger \sigma \psi ) + \varpi ,\\ D_{A}\psi&= 0, \end{aligned} \end{aligned}$$

(2.4)

where $r>1$ is a real parameter. Here $\varpi$ is a given 1-form whose significance will become clear in a moment, and $\psi ^\dagger \sigma \psi$ is the 1-form that acts on any vector field V as:

$$\begin{aligned} \psi ^\dagger \sigma \psi (V):=-i \psi ^\dagger \sigma (V)\psi . \end{aligned}$$

Notice that the properties of the Clifford map $\sigma$ ensure that the 1-form $\psi ^\dagger \sigma \psi$ is real valued.

Remark 2.1

In local coordinates on a ball $B\subset M$, if $\{e_1,e_2,e_3\}$ is a local orthonormal frame so that $\{e_2,e_3\}$ span ${\text {Ker}}\lambda$, then $\psi$ is a function from B to ${\mathbb {C}}^2$, A and $A_0$ are (real-valued) 1-forms,

$$\begin{aligned} \sigma (X)=i\Big [(X\cdot e_1)\sigma _1+(X\cdot e_2)\sigma _2+(X\cdot e_3)\sigma _3\Big ], \end{aligned}$$

where $\sigma _k$ are the Pauli matrices, and the covariant derivative $\nabla _A \psi$ can be understood as two complex-valued vector fields on B given by

$$\begin{aligned} \nabla _A \psi =\nabla \psi + \Lambda \psi - \frac{i}{2} (A_0^{\sharp }+2 A^{\sharp }) \psi . \end{aligned}$$

Here $A^\sharp$ is the vector field associated with the 1-form A and $\Lambda$ is the $2 \times 2$ matrix-valued vector field given by

$$\begin{aligned} \Lambda = \frac{1}{8} g(\nabla _{e_j} e_{m}, e_{n}) [\sigma (e_n), \sigma (e_m)] e_{j}. \end{aligned}$$

Summation over repeated indices is understood.

We are now ready to state the fundamental existence theorem due to Taubes [10] that we will need in this paper. A caveat is that we have not defined what one means by the degree of the solutions to the modified Seiberg–Witten equations whose existence is proved here. The notion of degree can be defined using the Seiberg–Witten–Floer homology but, since we will not need it in the following, we refer to [3, 10] for the precise definition. We stress that in the case of Reeb fields, the value of the energy (which is always finite) can be related to the degree [7].

Let us also record here that a solution $(A,\psi )$ is called irreducible if $\psi$ is not identically 0. Finally, we will denote by $\varpi _K$ the harmonic 1-form on M with the property that the Hodge dual $*\varpi _K$ represents the image in the cohomology group $H^2(M)$ of the first Chern class $c_1(K)$. Equivalently, one can set

$$\begin{aligned} \varpi _K:=-\frac{1}{2}* F_{A_0}+ * d \Gamma , \end{aligned}$$

(2.5)

where the 1-form $\Gamma$ satisfies

$$\begin{aligned} d* d \Gamma =\frac{1}{2}d * F_{A_0}. \end{aligned}$$

Theorem 2.2

(Taubes) Let X be a nonvanishing volume-preserving vector field with positive helicity. There is a real number $0<\theta <1$ and an infinite set of negative integers ${{\mathcal {K}}}$ such that, for each fixed $k \in {{\mathcal {K}}}$, we have:

• (i) There exists a smooth 1-form $\varpi '$, of arbitrarily small $C^3$ norm, such that the Seiberg–Witten Equations (2.4) with $\varpi :=\varpi _K+*d \varpi '$ has an irreducible solution $(\psi ,A)$ of degree k provided that the value of the parameter r belongs to a certain increasing sequence $(r_n)_{n=1}^\infty \subset (1,\infty )$ (depending on k) without any accumulation points.

• (ii) The aforementioned sequence of solutions $(\psi _n,A_n)$ of degree k corresponding to the value of the parameter $r_n$ satisfy the uniform bound

  $$\begin{aligned} \Vert 1-|\psi _n|^2\Vert _{L^\infty (M)}>\theta \,. \end{aligned}$$

2.2 A Priori Estimates and a Useful Equation

Let us henceforth employ the shorthand notation

$$\begin{aligned} \psi =:(\alpha ,\beta ) \end{aligned}$$

for the decomposition according to the splitting (2.2) of the spinor part of the solution $(A,\psi )$ to the modified Seiberg–Witten equations. In view of Remark 2.1, it is clear that both $\alpha$ and $\beta$ can be locally understood as complex-valued functions.

In what follows let $(r_n,\psi _n,A_n)$ be a sequence of solutions as in Theorem 2.2 (see also Theorem 1.3). For future reference we record here an identity connecting the signed measures $\sigma _n$ and the decomposition $\psi _n=(\alpha _n,\beta _n)$ that will be useful in the case when ${{\mathcal {E}}}_n\rightarrow \infty$:

$$\begin{aligned} \sigma _n= \frac{ r_n(1-|\alpha _n|^2)\, \mu }{{{\mathcal {E}}}_n}+ O({{\mathcal {E}}}_n^{-1})\,\mu . \end{aligned}$$

(2.6)

This follows easily from the second estimate in Lemma 2.3 and the fact that

$$\begin{aligned} (*\psi ^\dagger \sigma \psi ) \wedge \lambda =\psi ^\dagger \sigma \psi (X)\,\mu =(|\alpha |^2-|\beta |^2)\,\mu . \end{aligned}$$

Now we recall Taubes’s a priori estimates for solutions of the Seiberg–Witten equations [10, Lemmas 2.2 and 2.3]. With a slight abuse of notation, here and in what follows we will use $\nabla _{A}$ to denote both the covariant derivative defined by the connection A on E and the covariant derivative defined by A and $A_0$ on $E\otimes K^{-1}$. In other words, in local coordinates $\nabla _A\alpha =\nabla \alpha -iA^\sharp \alpha$ and $\nabla _A\beta =\nabla \beta -i(A^\sharp +A_0^\sharp )\beta$.

Lemma 2.3

There exists a constant C such that the solution $(\psi ,A)$ is bounded as

$$\begin{aligned} |\alpha |&\le 1 + \frac{C}{r}\,,\\ |\beta |^2&\le \frac{C}{r}\left| 1-|\alpha |^2\right| + \frac{C}{r^2}\,,\\ |\nabla _{A} \alpha |&\le C \sqrt{r}\,,\\ |\nabla _{A} \beta |&\le C\,, \\ |\nabla ^2_{A} \alpha |&\le C r\,,\\ |\nabla ^2_{A} \beta |&\le C \sqrt{r}\,. \end{aligned}$$

In particular, the negative part of $1-|\alpha |^2$ is bounded as

$$\begin{aligned} 1-|\alpha |^2\geqslant -\frac{C}{r}. \end{aligned}$$

A first refinement of these a priori estimates we need, which is implicit in Taubes’s work, is a set of anisotropic estimates that provide finer control of some geometric quantities. To emphasize this anisotropy, it is convenient to introduce some further notation. Given a scalar function f on M, we let

$$\begin{aligned} \nabla ^{\parallel }f&:=(X\cdot \nabla f) X \,,\\ \nabla ^{\perp }f&:= \nabla f- \nabla ^{\parallel }f \end{aligned}$$

denote the components of its gradient that are parallel and perpendicular to the field X, respectively. For sections of the vector bundles ${\mathbb {S}}$, E and $E \otimes K^{-1}$, $\nabla ^{\parallel }_{A}$ and $\nabla ^{\perp }_{A}$ are defined analogously.

Lemma 2.4

A solution $(\psi ,A)$ to the modified Seiberg–Witten equations satisfies the following anisotropic bounds:

$$\begin{aligned} |\nabla ^{\parallel }|\alpha |^2|&\leqslant C\,,\\ |\nabla ^{\perp }|\alpha |^2|&\leqslant C\sqrt{r}\,,\\ \big |\nabla \nabla ^{\parallel }|\alpha |^2\big |&\leqslant C\sqrt{r}\,,\\ |\nabla ^2 |\alpha |^2|&\leqslant Cr\,,\\ |X\cdot \nabla _A\alpha |&\leqslant C\,. \end{aligned}$$

Proof

Since A is a Hermitian connection on E, we have, for any vector field V,

$$\begin{aligned} V \cdot \nabla |\alpha |^2= 2{{\,\textrm{Re}\,}}( \bar{\alpha }V \cdot \nabla _{A} \alpha ), \end{aligned}$$

so we readily get

$$\begin{aligned} |\nabla |\alpha |^2| \leqslant 2|\alpha | |\nabla _A \alpha |. \end{aligned}$$

Similarly, since

$$\begin{aligned} W\cdot \nabla (V \cdot \nabla |\alpha |^2)= 2 {{\,\textrm{Re}\,}}(\overline{W \cdot \nabla _A \alpha } \, V\cdot \nabla _A \alpha )+ 2 {{\,\textrm{Re}\,}}[\bar{\alpha }W \cdot \nabla _A ( V\cdot \nabla _A \alpha ) ] \end{aligned}$$

for any vector fields V, W, one finds that

$$\begin{aligned} |\nabla ^2 |\alpha |^2| \leqslant 2|\nabla _A \alpha |^2+ 2|\alpha | |\nabla ^2_{A} \alpha |. \end{aligned}$$

These equations together with the bounds in Lemma 2.3 automatically imply the second and fourth estimates we aimed to prove.

To derive the other bounds, we first observe that the Dirac equation implies (see e.g. [11, Equation 3.7])

$$\begin{aligned} |X\cdot \nabla _A\alpha |\leqslant C (|\nabla _A \beta |+|\beta |+|\alpha |). \end{aligned}$$

Combining all the previous estimates together with the bounds on the derivatives of $\beta$ in Lemma 2.3, the remaining inequalities follow. $\square$

Finally, we are ready to state the main result of this section. In the following proposition we show that the function $|\alpha |^2$ satisfies an explicit second order elliptic PDE on M. This result will be instrumental in the proof of a new maximum principle for solutions of the Seiberg–Witten equations, cf. Theorem 3.1.

Proposition 2.5

The absolute value of $\alpha$ satisfies the equation

$$\begin{aligned} |\alpha |^{2}\Delta |\alpha |^{2}- {|\nabla |\alpha |^2|^2}+2r|\alpha |^{4}(1-|\alpha |^{2}-|\beta |^2) =H(\psi ,A)\, \end{aligned}$$

(2.7)

where the term $H(\psi ,A)$ is pointwise bounded as

$$\begin{aligned} |H(\psi ,A)|&\leqslant C\Big (1+|\nabla ^{\perp }|\alpha |^2|\Big )\ \end{aligned}$$

for an r-independent constant C.

Proof

As proved in [10, Section 6.1], $|\alpha |^2$ satisfies the equation

$$\begin{aligned} |\alpha |^2\Delta |\alpha |^2-2|\alpha |^2|\nabla _{A} \alpha |^2+2r|\alpha |^4(1-|\alpha |^2-|\beta |^2)=G, \end{aligned}$$

where G has the form

$$\begin{aligned} G:=-|\alpha |^2\Big (\tau (\alpha , \beta )+\tau (\alpha , \nabla _A \beta )+\tau (\alpha , \alpha )\Big ) \end{aligned}$$

(the notation $\tau (\cdot , \cdot )$ will henceforth represent bilinear maps that only depend on the metric, and that may change from line to line).

Observe that, by virtue of Lemma 2.3, we have the pointwise bound $|G|\leqslant C$. Thus, to prove the proposition it suffices to show that

$$\begin{aligned} 2|\alpha |^2|\nabla _A \alpha |^2= |\nabla |\alpha |^2|^2+G', \end{aligned}$$

where $G'$ is some function of $A, \psi$ and its derivatives that satisfies the bound

$$\begin{aligned} |G'| \leqslant C(1+|\nabla ^{\perp }|\alpha |^2|). \end{aligned}$$

First, we notice that

$$\begin{aligned} |\alpha |^2 |\nabla _A \alpha |^2=|{{\,\textrm{Re}\,}}( \bar{\alpha }\nabla _A \alpha )|^2+|{{\,\textrm{Im}\,}}(\bar{\alpha }\nabla _A \alpha ) |^2=\frac{1}{4} |\nabla |\alpha |^2|^2+|{{\,\textrm{Im}\,}}(\bar{\alpha }\nabla _A \alpha ) |^2. \end{aligned}$$

(2.8)

Next, for convenience we introduce some local notation. For each point $p\in M$, one can pick two vector fields $e_1^p,e_2^p$, defined on a neighborhood $V^p$ of p, such that $\{X(q),e_1^p(q),e_2^p(q)\}$ is an oriented orthonormal basis of the tangent space $T_qM$ for any $q\in V^p$. For ease of notation, we will henceforth omit the superscript p. The vectors $\{e_1(q), e_2(q)\}$ span the transverse distribution ${\text {Ker}}\lambda$ at each point $q \in V$. We also denote by J(q) the almost complex structure on this 2-plane field defined at q by

$$\begin{aligned} J(e_1(q)):=e_2(q),\qquad J(e_2(q)):=-e_1(q). \end{aligned}$$

This almost complex structure does not depend on the particular choice of orthonormal vector fields and is well defined globally on ${\text {Ker}}\lambda$.

Since the complex structure J on ${\text {Ker}}\lambda$ preserves the scalar product, Eq. (2.8) can be rewritten as

$$\begin{aligned} |\alpha |^2 |\nabla _A \alpha |^2=\frac{1}{4} |\nabla |\alpha |^2|^2+|{{\,\textrm{Im}\,}}\bar{\alpha }\nabla ^{\parallel }_A \alpha |^2+|{{\,\textrm{Im}\,}}(J \bar{\alpha }\nabla ^{\perp }_A \alpha ) |^2. \end{aligned}$$

(2.9)

Now the crucial observation is that one can infer from the Dirac equation $D_{A} \psi =0$ that

$$\begin{aligned} J i \nabla ^{\perp }_{A} \alpha =\nabla ^{\perp }_{A} \alpha +\Theta \nabla ^{\parallel }_{A} \beta +\Theta \beta . \end{aligned}$$

(2.10)

(Here and in what follows, we will use $\Theta$ to represent linear maps between the corresponding bundles that depend only on the metric.) Indeed, on the one hand, on the local frame $\{X, e_1, e_2\}$ we have

$$\begin{aligned} J i \nabla ^{\perp }_{A} \alpha =(i e_1 \cdot \nabla _A \alpha )e_2-(i e_2 \cdot \nabla _A \alpha ) e_1, \end{aligned}$$

and on the other hand, the Dirac equation implies the following relation between the derivatives of $\alpha$ and $\beta$ (see e.g. [11, Equation 3.7])

$$\begin{aligned} X \cdot \nabla _A\beta = -i e_1\cdot \nabla _A\alpha + e_2\cdot \nabla _A\alpha +\Theta \beta . \end{aligned}$$

(2.11)

Using this relation to write $e_1 \cdot \nabla _A \alpha$ in terms of $e_2 \cdot \nabla _A \alpha$, and viceversa, we readily get Eq. (2.10).

This understood, we can write:

$$\begin{aligned} {{\,\textrm{Im}\,}}(J \bar{\alpha }\nabla ^{\perp }_A \alpha )&=-\frac{1}{2} J \big (i\bar{\alpha }\nabla ^{\perp }_A \alpha -i (\overline{\nabla ^{\perp }_A \alpha }) \alpha \big )\\&=-\frac{1}{2}\big ( \bar{\alpha }(J i \nabla ^{\perp }_A \alpha )+(\overline{J i \nabla ^{\perp }_A \alpha }) \alpha \big )=\\&=-{{\,\textrm{Re}\,}}(\bar{\alpha }\nabla ^{\perp }_A \alpha )-\tau (\alpha , \nabla ^{\parallel }_A \beta )-\tau (\alpha , \beta )\,, \end{aligned}$$

where we have used Eq. (2.10) in the last equality.

Recalling that ${{\,\textrm{Re}\,}}(\bar{\alpha }\nabla ^{\perp }_A \alpha )= \frac{1}{2} \nabla ^{\perp }|\alpha |^2$, we obtain

$$\begin{aligned} |{{\,\textrm{Im}\,}}(J \bar{\alpha }\nabla ^{\perp }_A \alpha ) |^2= & {} \frac{1}{4}|\nabla ^{\perp }|\alpha |^2|^2+\nabla ^{\perp }|\alpha |^2 \cdot \big ( \tau (\alpha , \nabla ^{\parallel }_A \beta )+\tau (\alpha , \beta ) \big )\\{} & {} +|\tau (\alpha , \nabla ^{\parallel }_A \beta )+\tau (\alpha , \beta )|^2. \end{aligned}$$

Plugging this identity into Eq. (2.9) we easily infer that

$$\begin{aligned} |\alpha |^2 |\nabla _A \alpha |^2&=\frac{1}{4} |\nabla |\alpha |^2|^2+\frac{1}{4}|\nabla ^{\perp }|\alpha |^2|^2+|{{\,\textrm{Im}\,}}\bar{\alpha }\nabla ^{\parallel }_A \alpha |^2\\&\quad +\nabla ^{\perp }|\alpha |^2 \cdot \big ( \tau (\alpha , \nabla ^{\parallel }_A \beta )+\tau (\alpha , \beta ) \big )\\&\quad +|\tau (\alpha , \nabla ^{\parallel }_A \beta )+\tau (\alpha , \beta )|^2\,, \end{aligned}$$

so substituting $|\nabla ^{\perp }|\alpha |^2|^2$ by $|\nabla |\alpha |^2|^2-|\nabla ^{\parallel }|\alpha |^2|^2$ we finally obtain

$$\begin{aligned} |\alpha |^2 |\nabla _A \alpha |^2&=\frac{1}{2}|\nabla |\alpha |^2|-\frac{1}{4}|\nabla ^{\parallel }|\alpha |^2|^2+|{{\,\textrm{Im}\,}}\bar{\alpha }\nabla ^{\parallel }_A \alpha |^2\\&\quad +\nabla ^{\perp }|\alpha |^2 \cdot \big ( \tau (\alpha , \nabla ^{\parallel }_A \beta )+\tau (\alpha , \beta ) \big )\\&\quad +|\tau (\alpha , \nabla ^{\parallel }_A \beta )+\tau (\alpha , \beta )|^2\,. \end{aligned}$$

The proposition then follows taking into account the bounds in Lemmas 2.3 and 2.4. $\square$

3 A Maximum Principle for Solutions with Unbounded Energy

In this section we prove a new maximum principle for solutions of the modified Seiberg–Witten equations. Specifically, we establish a dichotomy for the large r behavior of local minima of $|\alpha |^2$ on small disks: either they are close to 0 or close to 1 as $r\rightarrow \infty$. The main consequence of this result is Theorem 3.2 below, which is instrumental in the proof of Theorem 1.5. We stress that all the constants appearing in this section are independent of r. In what follows, $(r,\psi ,A)$ is a sequence of solutions as in Taubes’s Theorem 2.2, and we recall that $\psi =(\alpha ,\beta )$.

Theorem 3.1

Let $\rho :(1,\infty )\rightarrow (0,1)$ be any continuous function with

$$\begin{aligned} \lim _{r\rightarrow \infty }\rho (r)=0. \end{aligned}$$

Suppose that $\Sigma$ is a disk of radius $\rho (r)$, embedded in M, transverse to the vector field X and perpendicular to it at some point p. If a point $q \in \Sigma$ is a local minimum of the restriction of $|\alpha |^2$ to $\Sigma$, then either

$$\begin{aligned} |\alpha (q)|^2 \leqslant C\eta (r)^{1/2 } \end{aligned}$$

or

$$\begin{aligned} \big ||\alpha (q)|^2-1\big | \leqslant C\eta (r). \end{aligned}$$

Here

$$\begin{aligned} \eta (r):= r^{-1/2}+\rho (r)^2 \end{aligned}$$

is another continuous function that tends to 0 at infinity.

Proof

Let us start by noticing that the Laplacian (on $\Sigma )$ of the restriction of a scalar function f to the surface $\Sigma$, which we denote by $\Delta _\Sigma f$, and the restriction to $\Sigma$ of the Laplacian of f are related through the following formula:

$$\begin{aligned} \Delta f|_\Sigma&= \sum _{j=1}^2 \nabla ^{2} f (V_j, V_j) + \nabla ^{2} f(N, N)\\&= \Delta _\Sigma f + N \cdot \nabla ( N \cdot \nabla f)+ \sum _{j=1}^2 ({{\,\textrm{div}\,}}V_j -{{\,\textrm{div}\,}}_\Sigma V_j)\, V_j\cdot \nabla f + ({{\,\textrm{div}\,}}N) N \cdot \nabla f\,. \end{aligned}$$

Here $\{V_1,V_2, N\}$ is a local orthonormal basis of the tangent space of M chosen so that N is perpendicular to $\Sigma$ at every point. Further, ${{\,\textrm{div}\,}}_\Sigma V_j$ denotes the divergence of the vector field $V_j$ (which is tangent to $\Sigma )$ with respect to the induced area form on $\Sigma$, $i_{N} \mu$.

If the point q is a local minimum of the restriction of $|\alpha |^2$ to $\Sigma$, it follows that, at q,

$$\begin{aligned} \Delta _\Sigma |\alpha |^2\geqslant 0,\qquad \nabla _{\Sigma } |\alpha |^2=0, \end{aligned}$$

where the gradient on $\Sigma$ is

$$\begin{aligned} \nabla _{\Sigma } |\alpha |^2= \nabla |\alpha |^2-(N\cdot \nabla |\alpha |^2)\, N. \end{aligned}$$

Accordingly, the fact that $\nabla _\Sigma |\alpha |^2(q)=0$ implies

$$\begin{aligned} \nabla |\alpha |^2(q)= (N\cdot \nabla |\alpha |^2)\, N(q)=(N-X) \cdot \nabla |\alpha |^2(q)+X \cdot \nabla |\alpha |^2(q). \end{aligned}$$

In then follows from the a priori estimates in Lemma 2.4 and the obvious bound $\Vert X-N\Vert _{L^\infty (\Sigma )} < C\rho (r)$, that

$$\begin{aligned} |\nabla |\alpha |^2(q)|\leqslant C[1+\sqrt{r}\,\rho (r)]. \end{aligned}$$

In view of the formula for $\Delta |\alpha |^2|_\Sigma$, and using again that $\Vert X-N\Vert _{L^\infty (\Sigma )} < C\rho (r)$, we infer that, always at the point q,

$$\begin{aligned} \Delta |\alpha |^2 (q)&\geqslant N \cdot \nabla ( N \cdot \nabla |\alpha |^2)+ ({{\,\textrm{div}\,}}N) N \cdot \nabla |\alpha |^2\\&= (N-X) \cdot \nabla ((N-X)\cdot \nabla |\alpha |^2) + X \cdot \nabla ((N-X)\cdot \nabla |\alpha |^2) \\&\quad + (N-X) \cdot \nabla (X\cdot \nabla |\alpha |^2) + X \cdot \nabla (X\cdot \nabla |\alpha |^2)+ ({{\,\textrm{div}\,}}N) N \cdot \nabla |\alpha |^2\\&\geqslant -C\rho (r)^2|\nabla ^2 |\alpha |^2| - C\rho (r)|\nabla \nabla ^{\parallel }|\alpha |^2| -|\nabla ^{\parallel }\nabla ^{\parallel }|\alpha |^2|- C | \nabla |\alpha |^2|\\&\geqslant -C[1+r^{1/2}\rho (r) +r\, \rho (r)^2]\,. \end{aligned}$$

On the other hand, Proposition 2.5 allows us to write

$$\begin{aligned} r|\alpha |^{4}(1-|\alpha |^{2}-|\beta |^2)\leqslant - \frac{1}{2}|\alpha |^{2}\Delta |\alpha |^{2}+ \frac{1}{2} {|\nabla |\alpha |^2|^2} +C\big ( 1+|\nabla ^{\perp }|\alpha |^2| \big ). \end{aligned}$$

If we now evaluate at q the inequalities that we have derived and invoke the bounds obtained in Lemma 2.4, we infer that, at the point q

$$\begin{aligned} r|\alpha |^{4}(1-|\alpha |^{2}-|\beta |^2)\leqslant C [1+r^{1/2} \rho (r) + r\rho (r)^2]\,. \end{aligned}$$

Since $\beta \rightarrow 0$ as $r\rightarrow \infty$ by Lemma 2.3, we finally conclude that

$$\begin{aligned} |\alpha |^{4}(1-|\alpha |^{2})\leqslant C[r^{-1}+r^{-1/2}\rho (r)+\rho (r)^2] \leqslant C[r^{-1/2}+\rho (r)^2] \end{aligned}$$

at the point q. This is the bound stated in the theorem. $\square$

The main strength of the maximum principle stated in Theorem 3.1 is that it does not assume that the sequence of solutions $(r,\psi ,A)$ has uniformly bounded energy. The following result exploits this property to show that if the energy growth is smaller than $r^{1/2}$, there are points on M where $|\alpha |^2\rightarrow 0$. This turns out to be an effective alternative to Taubes’s local analysis of solutions with bounded energy using the vortex equation, and it will be crucially used in the proof of Theorem 1.5.

In the proof, it is convenient to use suitable flow boxes adapted to the vector field X. To define a flow box, let p be any point in M and $\{e_1,e_2,X\}$ an orthonormal basis at $T_pM$. Consider, for positive constants $\varepsilon$ and R, the map

$$\begin{aligned} \Phi _p: (0, \varepsilon ) \times {\mathbb {D}}_R\longrightarrow M, \end{aligned}$$

defined by

$$\begin{aligned} \Phi _p(t, x):=\phi ^{t}_{X} \big (\exp _{p} (x_1 e_1(p)+x_2 e_2 (p))\big ), \end{aligned}$$

where $\mathbb {{D}}_R:=\{x \in {\mathbb {R}}^2: |x| < R \}$ is the disk of radius R, $\phi _X^t$ is the time-t flow of X, and $\exp _p: T_p M \rightarrow M$ is the exponential map. With $\varepsilon$ and R small enough, $\Phi _p$ is a smooth diffeomorphism into its image, which we will denote by ${{\mathcal {C}}}_p(R, \varepsilon )$. From now on, we will refer to ${{\mathcal {C}}}_p(R, \varepsilon )$ as the flow box based at p of radius R and length $\varepsilon$.

Theorem 3.2

Let $(r_n,\psi _n,A_n)$ be a sequence of solutions to the Seiberg–Witten equations with $r_n\rightarrow \infty$ and such that ${{\mathcal {E}}}_n=o(r_n^{1/2})$, i.e.,

$$\begin{aligned} \limsup _{n\rightarrow \infty }{{\mathcal {E}}}_nr_n^{-1/2}=0. \end{aligned}$$

Let $\{p_n\}$ be a sequence of points in M for which there is a positive constant $\theta$ such that

$$\begin{aligned} 1-|\alpha _n|^2(p_n)\geqslant \theta . \end{aligned}$$

Then there is a constant L (independent of n) such that the following holds: if n is large enough, there are disks $\Sigma _n\subset M$ of radius $\rho _n:= L{{\mathcal {E}}}_n r_n^{-1/2}$, transverse to the vector field X and perpendicular to it at $p_n$, and points $q_n\in \Sigma _n$ such that

$$\begin{aligned} |\alpha _n|^2(q_n)\leqslant C(r_n^{-1/4}+\rho _n). \end{aligned}$$

Proof

The existence of the sequence of points $\{p_n\}$ is ensured by Theorem 2.2, so let us take disks $\Sigma _n$ centered at $p_n$ as in the statement. We claim that the bound on the energy growth ensures that there is a local minimum of $|\alpha _n|^2|_{\Sigma _n}$ in the interior of the disk $\Sigma _n$, provided that n is large enough. In order to prove this, we proceed by contradiction.

Consider the connected component ${{\mathcal {V}}}_n$ of the compact set

$$\begin{aligned} \{ q\in \overline{\Sigma _n}: 1-|\alpha _n|^2(q)\geqslant \theta \} \end{aligned}$$

that contains the point $p_n$. Let us assume that ${{\mathcal {V}}}_n\cap \partial \Sigma _n$ is nonempty. Then there exists a continuous curve $\Gamma _n:[0,1)\rightarrow \Sigma _n$ with $\Gamma _n(0)=p_n$ and $\overline{\Gamma _n([0,1))}\cap \partial \Sigma _n\ne \emptyset$ such that

$$\begin{aligned} 1-|\alpha _n|^2(\Gamma _n(s))\geqslant \theta \end{aligned}$$

for all $s\in [0,1)$.

Take a small enough constant $\Lambda >0$ that will be fixed later. Since the length of the curve $\Gamma _n([0,1))$ is at least $\rho _n$, one can take at least

$$\begin{aligned} K_n:=\frac{r_n^{1/2}\rho _n}{2 \Lambda \theta } \end{aligned}$$

pairwise disjoint flow-boxes

$$\begin{aligned} {{\mathcal {C}}}_{p_{n,k}}\bigg (\frac{\Lambda \theta }{\sqrt{r_n}}, \Lambda \theta \bigg )\, \end{aligned}$$

centered at different points $\{p_{n, k}\}_{k=1}^{K_n}$ lying on the image of the curve $\Gamma _{n}$, with $p_{n,1}:= p_n$. If $\Lambda <\Lambda _0$, Lemma 3.3 below and the definition of $\rho _n$ imply that the signed measures $\sigma _n$ (cf. Eq. 2.6) satisfy

$$\begin{aligned} \sigma _n(M)&\geqslant \bigcup _{k=1}^{K_n}\sigma _n\bigg ({{\mathcal {C}}}_{p_{n,k}}\bigg (\frac{\Lambda \theta }{\sqrt{r_n}}, \Lambda \theta \bigg )\bigg )-C{{\mathcal {E}}}_n^{-1} \\&\geqslant \frac{c_0}{2} \Lambda ^2 \theta ^3 r_n^{1/2}\rho _n {{\mathcal {E}}}_n^{-1}-C{{\mathcal {E}}}_n^{-1}\\&\geqslant \frac{c_0}{2} \Lambda ^2 \theta ^{3} L-C{{\mathcal {E}}}_n^{-1}\,. \end{aligned}$$

Here the constant $c_0$ comes from Lemma 3.3 and does not depend on n or L.

We then infer that picking a large enough constant L in the definition of $\rho _n$ yields a contradiction with the fact that $\sigma _{n}(M)=1$ (even in the case that ${{\mathcal {E}}}_n$ is uniformly bounded). Therefore, ${{\mathcal {V}}}_n \cap \partial \Sigma _n$ is empty and the compactness of ${{\mathcal {V}}}_n$ implies that, for large enough n, there is a global minimum $q_n$ of $|\alpha _n|^2|_{{{\mathcal {V}}}_n}$ on ${{\mathcal {V}}}_n$.

Since $|\alpha _n|^2(q_n)\leqslant 1-\theta$, the maximum principle stated in Theorem 3.1 allows us to write the bound

$$\begin{aligned} |\alpha _n|^2(q_n)\leqslant C(r_n^{-1/4}+\rho _n), \end{aligned}$$

which completes the proof of the theorem. $\square$

The following technical lemma is invoked in the proof of Theorem 3.2. We use the same notation as before.

Lemma 3.3

Let $(r,\psi ,A)$ be a sequence of solutions to the modified Seiberg–Witten equations. Assume that p is a point in M such that $1-|\alpha |^2(p)\geqslant \theta$ for some uniform $0<\theta <1$. Then there are positive constants $c_0$ and $\Lambda _0$, independent of $\theta$ and r, such that

$$\begin{aligned} \sigma _r \bigg ({{\mathcal {C}}}_{p}\bigg (\frac{\Lambda \theta }{\sqrt{r}}, \Lambda \theta \bigg )\bigg ) \geqslant \frac{ c_0\Lambda ^3\theta ^4}{{{\mathcal {E}}}_r} \end{aligned}$$

for all $\Lambda <\Lambda _0$.

Proof

First, Eq. (2.6) implies that, for any open set $U \subset M$

$$\begin{aligned} \sigma _r (U) {{\mathcal {E}}}_r \geqslant r \int _U (1-|\alpha |^2)\mu -C \mu (U) \end{aligned}$$

for some constant C independent of r.

Since $1 -|\alpha |^2\geqslant \theta$ at p, it follows from the a priori estimates for the derivatives of $|\alpha |^2$ in Lemma 2.4 that

$$\begin{aligned} 1 -|\alpha |^2\geqslant \frac{\theta }{2} \end{aligned}$$

in a flow box of the form ${{\mathcal {C}}}_{p}(\frac{\Lambda \theta }{\sqrt{r}}, \Lambda \theta )$, provided that the constant $\Lambda$ is smaller than some constant $\Lambda _0$ (independent of r and $\theta$). Therefore

$$\begin{aligned} {{\mathcal {E}}}_r \sigma _r \bigg ({{\mathcal {C}}}_{p}\bigg (\frac{\Lambda \theta }{\sqrt{r}}, \Lambda \theta \bigg )\bigg )&\geqslant r \int _{{{\mathcal {C}}}_{p}(\frac{\Lambda \theta }{\sqrt{r}}, \Lambda \theta )} (1 -|\alpha |^2)\mu -C\mu \bigg ({{\mathcal {C}}}_{p}\bigg (\frac{\Lambda \theta }{\sqrt{r}}, \Lambda \theta \bigg )\bigg ) \geqslant \\&\geqslant \mu \bigg ({{\mathcal {C}}}_{p}\bigg (\frac{\Lambda \theta }{\sqrt{r}}, \Lambda \theta \bigg )\bigg ) \bigg (\frac{\theta r}{2}-C \bigg )\\&\geqslant C\Lambda ^3\theta ^4+O(r^{-\frac{1}{2}}) \geqslant c_0\Lambda ^3\theta ^4\,, \end{aligned}$$

as claimed. $\square$

4 Nodal Sets and Limiting Measures: Proof of Theorem 1.5

In most of this section we are concerned with solutions of the modified Seiberg–Witten equations whose energy is bounded as

$$\begin{aligned} \limsup _{n\rightarrow \infty }\frac{{{\mathcal {E}}}_n}{r_n^{1/2}}=0. \end{aligned}$$

Specifically, in Sects. 4.1 and 4.2 we prove Theorem 1.5, which establishes a connection between some invariant sets of the vector field X and the set of points where $|\alpha _n|\rightarrow 0$. Our proof exploits the new maximum principle presented in Theorem 3.2. In particular, since it applies to solutions with uniformly bounded energy, this allows us to obtain an alternative proof of Taubes’s theorem on the existence of periodic orbits without using the vortex equations, as discussed in Sect. 4.3. Finally, in Sect. 4.4, we prove Proposition 1.7, which is a sort of converse to Theorem 1.5: the open sets of M where $|\alpha _n|\rightarrow 1$ do not charge the invariant measure $\sigma _\infty$. No constraint on the energy growth is assumed in this case.

4.1 Step 1: Construction of an Invariant Set

We first observe that we can define the sets $Z_n^\theta$ and $Z_n$ using $|\alpha _n|^2$ rather than $|\psi _n|^2$ (by the a priori estimates in Lemma 2.3). Let us pick any $\theta \in (0, 1)$. By Theorem 2.2, the compact set

$$\begin{aligned} Z^{\theta }_n:=\{p \in M,\, 1-|\alpha _n|^2(p)\geqslant \theta \} \end{aligned}$$

is non-empty for $\theta$ small enough, and in fact, by Theorem 3.1, it is non-empty for any $\theta \in (0, 1)$ and all large enough n.

Fix a subsequence which converges in the Hausdorff metric, which we still denote by $Z_n^\theta$, and let $Z^{\theta }_{\infty }$ be its limit. In this step, our aim is to show that $Z^{\theta }_{\infty }$ is invariant under the flow of X. Notice that by compactness of M, non-empty compact subsets of M with the Hausdorff metric form a compact metric space, so such a limiting set always exists and it is not the empty set.

Before proceeding, let us explain the main idea of the proof. For any point $p \in Z^{\theta }_{\infty }$, we will show that there is a set $S_p$ containing p, contained in $Z^{\theta }_{\infty }$, and invariant under the flow of X. This clearly implies that $Z^{\theta }_{\infty }$ is invariant. The set $S_p$ will turn out to be the closure of the orbit of X passing through p.

The Hausdorff convergence implies that, for any $p \in Z^{\theta }_{\infty }$, there is a sequence of points $p_n \in Z^{\theta }_n$ converging to p. By definition, the points $p_n$ satisfy

$$\begin{aligned} 1-|\alpha _n|^2(p_n)\geqslant \theta \end{aligned}$$

for all n, with $\theta >0$.

Since ${{\mathcal {E}}}_n=o(r_n^{-1/2})$, it follows from Theorem 3.2 that for each large enough n, there exists a point $q_n^1$ on a disk $\Sigma _n^1$ centered at $p_n$, of radius at most

$$\begin{aligned} \rho _n:=cr_n^{-1/2}{{\mathcal {E}}}_n \end{aligned}$$

and orthogonal to X at $p_n$, such that

$$\begin{aligned} |\alpha _n(q_n^1)|^2<\varepsilon _n, \end{aligned}$$

where $\varepsilon _n:=C(r_n^{-1/4}+ \rho _n)$. We then consider the cylinder

$$\begin{aligned} {{\mathcal {C}}}_{n,1}:={{\mathcal {C}}}_{q_n^1}(\rho _n,T) \end{aligned}$$

of radius $\rho _n$ centered at this point and length T, where T is a small positive constant independent of n.

Consider the point $\widetilde{q}_n^1:=\phi _X^T(q_n^1)$ and take a disk $\Sigma _n^2$ centered at $\widetilde{q}_n^1$, of radius $\rho _n$ and orthogonal to X at $\widetilde{q}_n^1$. The bounds for the derivatives of $|\alpha |^2$ (Lemmas 2.3 and 2.4) ensure that

$$\begin{aligned} |\alpha _n(\widetilde{q}_n^1)|^2<\varepsilon _n+ CT. \end{aligned}$$

Hence if T is small, Theorem 3.2 ensures again that there exists a point $q_n^2\in \Sigma _n^2$ such that

$$\begin{aligned} |\alpha _n(q_n^2)|^2<\varepsilon _n. \end{aligned}$$

Let us now define the flow-box

$$\begin{aligned} {{\mathcal {C}}}_{n,2}:={{\mathcal {C}}}_{q_n^2}(\rho _n,T) \end{aligned}$$

and note that the volumes of ${{\mathcal {C}}}_{n,k}$ (with $k=1,2$) and of the intersection ${{\mathcal {C}}}_{n,1}\cap {{\mathcal {C}}}_{n,2}$ can be estimated as

$$\begin{aligned} \mu ({{\mathcal {C}}}_{n,k})&> C\rho _n^2\,,\\ \mu ({{\mathcal {C}}}_{n,1}\cap {{\mathcal {C}}}_{n,2})&< C\rho _n^3\,. \end{aligned}$$

Since $\rho _n\rightarrow 0$, this means that, for large enough n, the volume of the intersection ${{\mathcal {C}}}_{n,1}\cap {{\mathcal {C}}}_{n,2}$ is just a small fraction of that of either of the cylinders.

By repeating the argument (considering both the forward flow of X and the backward flow), one obtains a sequence of points $(q_n^k)_{k\in {\mathbb {Z}}}$, which give rise to flow boxes ${{\mathcal {C}}}_{n,k}:={{\mathcal {C}}}_{q_n^k}(\rho _n,T)$ satisfying

$$\begin{aligned} \mu ({{\mathcal {C}}}_{n,k})&> C\rho _n^2\,,\\ \mu ({{\mathcal {C}}}_{n,k}\cap {{\mathcal {C}}}_{n,k+1})&< C\rho _n^3\,,\\ |\alpha _n(q_n^k)|^2&<\varepsilon _n \end{aligned}$$

for all k and all large enough n.

Consider now, for some constant D independent of n, the thinner cylinders $\widetilde{{{\mathcal {C}}}_{n, k}}:={{\mathcal {C}}}_{q_n^k}(Dr_n^{-\frac{1}{2}},T)\subset {{\mathcal {C}}}_{n,k}$. If D and T are chosen small enough, the bounds for the derivatives of $|\alpha |^2$ in Lemmas 2.3 and 2.4 ensure that for any point $q' \in \widetilde{{{\mathcal {C}}}_{n, k}}$ we have

$$\begin{aligned} |\alpha _n(q')|^2<\varepsilon _n+C (D+T) < 1-\theta \end{aligned}$$

(4.1)

for all k and all large enough n. For each positive integer K, let us set

$$\begin{aligned} S_{n,K}:= \bigcup _{k=-K}^K \widetilde{{{\mathcal {C}}}_{n,k}}. \end{aligned}$$

By construction, $S_{n,K}$ is contained in a neighborhood of width $K\rho _n$ of the portion

$$\begin{aligned} S_{K}^n:= \{\phi _X^t p_n: |t|\leqslant KT\} \end{aligned}$$

of the integral curve of X passing through $p_n$. If the integral curve is periodic, this length may mean that this set winds around the integral curve more than once. In particular, setting

$$\begin{aligned} S_{K}:= \{\phi _X^t p: |t|\leqslant KT\}, \end{aligned}$$

it is clear that both $S_{n,K}$ and $S_{K}^n$ converge to $S_K$ as $n\rightarrow \infty$, albeit this convergence does not need to be uniform in K.

Finally, let us define the compact invariant set $S_p$ as the closure of the integral curve of X passing through p, which obviously arises as the Hausdorff limit of $S_K$ as $K\rightarrow \infty$. We claim that $S_p \subset Z^{\theta }_{\infty }$.

To see this, observe that Eq. (4.1) ensures that, for all K and any large enough n, $S_{n,K} \subset Z^{\theta }_n$. Consider an infinite sequence of integers $\cdots< K_{-1}< K_0< K_1 <\cdots$. It is clear that any $q \in S_p$ is the limit as $|i| \rightarrow \infty$ of some sequence of points $q_i \in S_{K_i}$, and the points $q_i$ are themselves the limits as $n\rightarrow \infty$ of some sequence of points $p^{n}_i \subset S_{n,K_i} \subset Z^{\theta }_n$. Upon choosing a diagonal sequence of $p^{n}_i$, we conclude that q is the limit as $(n, |i|) \rightarrow (\infty , \, \infty )$ of a sequence of points in $Z^{\theta }_n$. The uniqueness of the Hausdorff limit (recall that we have fixed a converging subsequence at the beginning) implies that $q \in Z^{\theta }_{\infty }$, as we wanted to prove.

4.2 Step 2: The Collection of Limiting Sets is Independent of $\theta$

Let us recall the definition of the sets $Z_n$:

$$\begin{aligned} Z_n:=\left\{ p \in M: |\alpha _n|^2(p) \leqslant C\text { max }(r_n^{-\frac{1}{4}},{{\mathcal {E}}}_n r_n^{-\frac{1}{2}}) \right\} . \end{aligned}$$

Notice that $Z_n \subset Z^{\theta }_n$ for all n large enough.

We claim that given any $\theta \in (0, 1)$, and any converging subsequence $Z^{\theta }_{n}$ (in the Hausdorff metric), there is a converging subsequence $Z_{n}$ such that the limits coincide: $Z^{\theta }_{\infty }=Z_{\infty }$. Reciprocally, given a convergent subsequence $Z_n$, there is a subsequence $Z^{\theta }_{n}$ with the same limit.

Recall that the Hausdorff distance between the sets $Z_{n}$ and $Z^{\theta }_{n}$ is defined as

$$\begin{aligned} {{\,\textrm{dist}\,}}_{H}(Z_{n}, Z^{\theta }_{n})=\text {max}\bigg (\sup _{x \in Z_n} {{\,\textrm{dist}\,}}(x, Z^{\theta }_{n}), \sup _{y \in Z^{\theta }_{n}} {{\,\textrm{dist}\,}}(y, Z_{n})\bigg ), \end{aligned}$$

for each n. Fix a converging subsequence $Z^{\theta }_{n}$, and consider the corresponding sequence of sets $Z_n$. We claim that ${{\,\textrm{dist}\,}}_{H}(Z_{n}, Z^{\theta }_{n}) \rightarrow 0$ as $n\rightarrow \infty$.

Indeed, by Theorem 3.2, for any sequence of points $p_n \in Z^{\theta }_n$ we can find another sequence $q_n$ such that

$$\begin{aligned}{} & {} |\alpha _n|^2(q_n)\leqslant C(r_n^{-1/4}+\rho _n),\\{} & {} {{\,\textrm{dist}\,}}(p_n, q_n) < \rho _n, \end{aligned}$$

with $\rho _n:=c{{\mathcal {E}}}_n r^{-\frac{1}{2}}_{n}$ going to zero as $n\rightarrow \infty$. From this we infer that $q_n \in Z_n$ for all n and we conclude that

$$\begin{aligned} \sup _{y \in Z^{\theta }_{n}} {{\,\textrm{dist}\,}}(y, Z_{n}) < \rho _n \rightarrow 0 \end{aligned}$$

as $n\rightarrow \infty$. Since $Z_n \subset Z^{\theta }_n$ for all large enough n, we can write

$$\begin{aligned} \sup _{x \in Z_n} {{\,\textrm{dist}\,}}(x, Z^{\theta }_{n})=0, \end{aligned}$$

thus implying that ${{\,\textrm{dist}\,}}_{H}(Z_{n}, Z^{\theta }_{n}) \rightarrow 0$ as claimed. In particular, $Z_n$ converges to a compact set $Z_\infty$ which is equal to the limiting set $Z^\theta _\infty$. The same argument shows that if a subsequence $Z_n$ converges to a set $Z_\infty$, the corresponding subsequence $Z_n^\theta$ converges to the same Hausdorff limit. Analogously, combining the previous argument with the fact that $Z_n^{\theta '}\subset Z_n^\theta$ if $\theta '\geqslant \theta$, it follows that any converging subsequence $Z_n^\theta$ yields a converging subsequence $Z_n^{\theta '}$ with the same limit, for any $\theta '\ne \theta$. This completes the proof of Theorem 1.5.

4.3 The Bounded Energy Case: Taubes’s Result Revisited

All the previous arguments, as well as the maximum principle proved in Sect. 3, apply to sequences of solutions with uniformly bounded energy. In fact, in this case, a simple boundedness argument allows us to prove that the invariant set $Z_{\infty }$ must consist of a finite collection of periodic orbits of X. Of course, this recovers Taubes’s periodic orbit theorem, but without making use of the local analysis that compares Seiberg–Witten with the vortex equations. To see this, in this short subsection we shall assume that ${{\mathcal {E}}}_n \leqslant C$.

We claim that for any $p \in Z^{\theta }_{\infty }$, the invariant set $S_p$ that we constructed in Sect. 4.1 is a periodic orbit of X. Indeed, suppose $S_{p}$ is not a periodic orbit. Then one has that, for any K as large as desired, the cylinders $\widetilde{{{\mathcal {C}}}_{n,k}}$ satisfy the small intersection condition

$$\begin{aligned} \mu ( \widetilde{{{\mathcal {C}}}_{n,j}}\cap \widetilde{{{\mathcal {C}}}_{n,k}})<C \overline{\rho _n}^3 \end{aligned}$$

for all $-K\leqslant j< k\leqslant K$ and all large enough n (depending on K). Here, we set $\overline{\rho _n}:=D r^{-\frac{1}{2}}$.

Let us now define the slightly cut out cylinders

$$\begin{aligned} {{\mathcal {C}}}_{n,k}':= \widetilde{{{\mathcal {C}}}_{n,k}}\backslash \overline{\widetilde{{{\mathcal {C}}}_{n,k-1}}}, \end{aligned}$$

which are pairwise disjoint by construction. In view of Eq. (4.1) and Lemma 3.3, we have

$$\begin{aligned} {{\mathcal {E}}}_n \sigma _n({{\mathcal {C}}}_{n,k}')>\delta \end{aligned}$$

for some constant $\delta >0$ depending on $\theta , T$ and D (which are taken sufficiently small), but not on n.

Observe that, by definition, ${{\mathcal {E}}}_n \sigma _n(M)={{\mathcal {E}}}_n$. Moreover, the sets ${{\mathcal {C}}}_{n,k}'$ are pairwise disjoint, so Eq. (2.6) and the bound for the negative part of $1-|\alpha |^2$ in Lemma 2.3 imply

$$\begin{aligned} {{\mathcal {E}}}_n\geqslant \sum _{k=-K}^K {{\mathcal {E}}}_n \sigma _n({{\mathcal {C}}}_{n,k}')-C\geqslant (2K+1) \delta -C, \end{aligned}$$

where all the constants are independent of n. Since K can be taken as large as desired and ${{\mathcal {E}}}_n$ is uniformly bounded by hypothesis, this yields a contradiction. So $S_p$ must be a periodic orbit.

The same argument shows that the number of periodic orbits in $Z^{\theta }_{\infty }$ must be finite; otherwise we could construct an unbounded number of disjoint cylinders with ${{\mathcal {E}}}_n\sigma _n$ bounded from below, contradicting the boundedness of the energy.

4.4 Additional Concentration Properties: Proof of Proposition 1.7

In this final section we prove Proposition 1.7. This concerns the set of points where $|\alpha _n|\rightarrow 1$, which is not considered in Theorem 1.5. We show that any open component of such a set has zero measure with respect to $\sigma _\infty$. The only hypothesis on the energy sequence is that it is unbounded. We stress that for sequences of bounded energy, an analogous result follows from the analysis in [10], which makes use of the convergence of the Seiberg–Witten equations towards the vortex equations at small scales.

Proof of Proposition 1.7

We first observe that the assumption $|\psi _n|\rightarrow 1$ on U is equivalent to $|\alpha _n|\rightarrow 1$ on U, by the a priori estimates (Lemma 2.3).

Let us define $v_n:=|\alpha _n|^2-1$. Equation (2.7) in Proposition 2.5 can be written as:

$$\begin{aligned} (1+v_n)\Delta v_n- |\nabla v_n|^2=2r(1+v_n)^2 v_n+H(v_n), \end{aligned}$$

(4.2)

where the term $H(v_n)$ satisfies the pointwise bound

$$\begin{aligned} |H(v_n)| \leqslant C\Big (1+|\nabla ^{\perp }v_n|\Big ). \end{aligned}$$

(4.3)

Here we have used that $|\beta |^2\leqslant \frac{C}{r}$, cf. Lemma 2.3.

Fix a sufficiently small constant $\rho >0$, and consider a geodesic ball $B_{\rho }\subset U$ of radius $\rho$. It is convenient to define a cut-off $C^\infty$ function $\chi : M\rightarrow [0,1]$ that vanishes on the complement of $B_{\rho }$, is positive on $B_\rho$ and is equal to 1 on $B_{\frac{\rho }{2}}\subset B_{\rho }$. It is easy to see that $\chi$ can be chosen to satisfy the pointwise bounds:

$$\begin{aligned} |\nabla \chi |^2 \leqslant \frac{C}{\rho ^2},\qquad |\Delta \chi | \leqslant \frac{C}{\rho ^2}. \end{aligned}$$

(4.4)

Now, multiply Eq. (4.2) by $\chi ^2 v_n$ to obtain:

$$\begin{aligned} \chi ^2 v_n(1+v_n)\Delta v_n- \chi ^2 v_n|\nabla v_n|^2=2r\chi ^2(1+v_n)^2 v^2_n+\chi ^2 v_n H(v_n). \end{aligned}$$

Taking into account that

$$\begin{aligned} \Delta (\chi ^2 v_n)=v_n \Delta \chi ^2+\chi ^2 \Delta v_n+4\chi \nabla \chi \cdot \nabla v_n, \end{aligned}$$

we can write

$$\begin{aligned}&v_n(1+v_n)\Delta (\chi ^2 v_n)-v^2_n (1+v_n) \Delta \chi ^2 - 4 v_n(1+v_n)\chi \nabla \chi \cdot \nabla v_n-\chi ^2 v_n |\nabla v_n|^2\\&\quad =2r\chi ^2(1+v_n)^2 v^2_n+\chi ^2 v_n H(v_n)\,. \end{aligned}$$

If we integrate this equation over the ball $B_{\rho }$, and integrate by parts the term $v_n(1+v_n)\Delta (\chi ^2 v_n)$, we get the following expression for the local $L^2$ norm of the derivatives of $v_n$:

$$\begin{aligned} -\int _{B_{\rho }} \chi ^2|\nabla v_n|^2&=3 \int _{B_{\rho }} v_n \chi ^2|\nabla v_n|^2+8 \int _{B_{\rho }}v^2_n\chi \nabla \chi \cdot \nabla v_n\\&\quad +6 \int _{B_{\rho }}v_n\chi \nabla \chi \cdot \nabla v_n+ \int _{B_{\rho }}v^2_n (1+v_n) \Delta \chi ^2\\&\quad +2r \int _{B_{\rho }} \chi ^2 v^2_n(1+v_n)^2+\int _{B_{\rho }} \chi ^2 v_n H(v_n)\,. \end{aligned}$$

Our objective now is to bound the integrals in the right hand side of this equation. We will use repeatedly the bounds in Eq. (4.4) and the fact that, for any $\delta >0$,

$$\begin{aligned} \int _{B_{\rho }} \chi |\nabla v_n| \leqslant \delta \int _{B_{\rho }}\chi ^2 |\nabla v_n|^2+\frac{C}{\delta }, \end{aligned}$$

with C a constant independent of $\delta$ and n. This follows from the elementary inequality

$$\begin{aligned} \chi f \leqslant \delta \chi ^2 f^2+\frac{1}{4 \delta }. \end{aligned}$$

Noting that the $L^\infty$ norm $\Vert v_n\Vert _{\infty }$ of $v_n$ on the ball $B_\rho$ is bounded by 1 (as a consequence of the a priori bound on $|\alpha _n|^2$), we have

$$\begin{aligned} \Big | \int _{B_{\rho }} v_n \chi ^2|\nabla v_n|^2 \Big |\leqslant & {} \Vert v_n\Vert _\infty \int _{B_{\rho }} \chi ^2|\nabla v_n|^2,\\ \Big |\int _{B_{\rho }}v^2_n\chi \nabla \chi \cdot \nabla v_n \Big |\leqslant & {} \frac{C\Vert v_n\Vert ^2_\infty }{\rho } \int _{B_{\rho }} \chi |\nabla v_n| \leqslant \frac{C\Vert v_n\Vert _\infty }{\rho } \Big ( \delta \int _{B_{\rho }}\chi ^2 |\nabla v_n|^2+\frac{C}{\delta }\Big ),\\ \Big | \int _{B_{\rho }} v_n\chi \nabla \chi \cdot \nabla v_n \Big |\leqslant & {} \frac{C\Vert v_n\Vert _\infty }{\rho } \int _{B_{\rho }}\chi |\nabla v_n| \leqslant \frac{C\Vert v_n\Vert _\infty }{\rho } \Big ( \delta \int _{B_{\rho }}\chi ^2 |\nabla v_n|^2+\frac{C}{\delta }\Big ),\\ \Big | \int _{B_{\rho }}v^2_n (1+v_n) \Delta \chi ^2 \Big |\leqslant & {} \frac{C\Vert v_n\Vert _\infty }{\rho ^2}. \end{aligned}$$

Furthermore, using the bounds for H in Eq. (4.3), we deduce

$$\begin{aligned} \Big | \int _{B_{\rho }} v_n \chi ^2 H \Big |\leqslant & {} C\Vert v_n\Vert _\infty \Big (\rho ^3+\int _{B_{\rho }} \chi |\nabla v_n|\Big ) \\\leqslant & {} C\Vert v_n\Vert _\infty \Big (\rho ^3+\delta \int _{B_{\rho }}\chi ^2 |\nabla v_n|^2+\frac{C}{\delta }\Big ). \end{aligned}$$

Finally, applying Eq. (2.6), we obtain

$$\begin{aligned} 2r \int _{B_{\rho }} \chi ^2 v^2_n(1+v_n)^2 \leqslant C\Vert v_n\Vert _\infty \Big ({{\mathcal {E}}}_n |\sigma _n(B_{\rho })|+1\Big ). \end{aligned}$$

Plugging all these estimates into the integral identity we obtained for $\chi ^2|\nabla v_n|^2$, and using that $\delta$ and $\rho$ are small, we conclude:

$$\begin{aligned} \Bigg (1-\Vert v_n\Vert _{\infty }\Big (3+\frac{C\delta }{\rho }\Big )\Bigg )\int _{B_{\rho }}\chi ^2 |\nabla v_n|^2 \leqslant C\Vert v_n\Vert _{\infty } \Big (\frac{1}{\rho ^2 \delta }+{{\mathcal {E}}}_n |\sigma _n(B_{\rho })|\Big ) \end{aligned}$$

Now we use that, by assumption, $\Vert v_n\Vert _{\infty }$ goes to zero as $n\rightarrow \infty$ (because $|\alpha _n|\rightarrow 1$ on $U\supset B_{\rho }$). Fixing a constant $\delta$, for large enough n we have

$$\begin{aligned} \int _{B_{\frac{\rho }{2}}} |\nabla v_n|^2\leqslant \int _{B_{\rho }}\chi ^2 |\nabla v_n|^2 \leqslant C\Vert v_n\Vert _{\infty } \Big (\frac{1}{\rho ^2 \delta }+{{\mathcal {E}}}_n |\sigma _n(B_{\rho })|\Big ). \end{aligned}$$

Therefore, as $n \rightarrow \infty$

$$\begin{aligned} \frac{1}{{{\mathcal {E}}}_n}\int _{B_{\frac{\rho }{2}}} |\nabla |\alpha _n|^2|^2=\frac{1}{{{\mathcal {E}}}_n}\int _{B_{\frac{\rho }{2}}} |\nabla v_n|^2\leqslant C\Vert v_n\Vert _{\infty } \Big (\frac{1}{\rho ^2 \delta {{\mathcal {E}}}_n}+|\sigma _n(B_{\rho })|\Big )\rightarrow 0, \end{aligned}$$

(4.5)

which holds even for solutions with uniformly bounded energy ${{\mathcal {E}}}_n$.

To show that $\sigma _{\infty }(U)=0$, we first observe that, for n large enough we have

$$\begin{aligned} \frac{1}{\sqrt{2}}\leqslant |\alpha _n|^2\leqslant 1+Cr^{-1}_{n} \end{aligned}$$

at any point on $B_\rho$, the upper bound coming from Lemma 2.3. Together with Eq. (2.6), this implies that

$$\begin{aligned} \frac{r_n}{{{\mathcal {E}}}_n}\int _{B_{\frac{\rho }{4}}} |\alpha _n|^4 (1-|\alpha _n|^2) \geqslant \frac{r_n}{2 {{\mathcal {E}}}_n} \int _{B_{\frac{\rho }{4}}} (1-|\alpha _n|^2)-\frac{C}{{{\mathcal {E}}}_{n}} \rho ^3 \geqslant \frac{1}{2} \sigma _n(B_{\frac{\rho }{4}})-\frac{C}{{{\mathcal {E}}}_{n}} \rho ^3\,. \end{aligned}$$

(4.6)

Now, let $\chi ':M\rightarrow [0,1]$ be a smooth cut-off function supported on the ball $B_{\frac{\rho }{2}}$, equal to one on $B_{\frac{\rho }{4}}$ and positive on $B_{\frac{\rho }{2}}$. We assume that it satisfies the same bounds as in (4.4). Multiplying Eq. (2.7) by $\chi '$ and integrating, we deduce

$$\begin{aligned} \frac{2r_n}{{{\mathcal {E}}}_n}\int _{B_{\frac{\rho }{4}}} |\alpha _n|^4 (1-|\alpha _n|^2)\leqslant & {} \frac{1}{{{\mathcal {E}}}_n}\Bigg ( \int _{B_{\frac{\rho }{2}}} \chi ' |\nabla |\alpha _n|^2|^2-\int _{B_{\frac{\rho }{2}}} \chi ' |\alpha _n|^2 \Delta |\alpha _n|^2\\{} & {} + \int _{B_{\frac{\rho }{2}}} \chi ' H +C\rho ^3\Bigg ), \end{aligned}$$

where we have used the a priori bounds for $|\beta _n|^2$ and that $1-|\alpha _n|^2+Cr_n^{-1}\geqslant 0$. Accordingly, from Eq. (4.6) we get

$$\begin{aligned} \sigma _n(B_{\frac{\rho }{4}}) \leqslant \frac{1}{{{\mathcal {E}}}_n}\Bigg ( \int _{B_{\frac{\rho }{2}}} \chi ' |\nabla |\alpha _n|^2|^2-\int _{B_{\frac{\rho }{2}}} \chi ' |\alpha _n|^2 \Delta |\alpha _n|^2+ \int _{B_{\frac{\rho }{2}}} \chi ' H +C\rho ^3\Bigg ). \end{aligned}$$

(4.7)

Next, let us estimate the second term on the right hand side of this equation. Integrating by parts we obtain

$$\begin{aligned} -\int _{B_{\frac{\rho }{2}}} \chi ' |\alpha _n|^2 \Delta |\alpha _n|^2=\int _{B_{\frac{\rho }{2}}} \chi ' |\nabla |\alpha _n|^2|+\int _{B_{\frac{\rho }{2}}} |\alpha _n|^2 \nabla \chi ' \cdot \nabla |\alpha _n|^2. \end{aligned}$$

By the elementary inequality $a \leqslant a^2+\frac{1}{4}$, the bound for $|\nabla \chi '|$ and the fact that $|\alpha _n|^2 \leqslant 1+Cr_n^{-1}$, we can write

$$\begin{aligned} \int _{B_{\frac{\rho }{2}}} |\alpha _n|^2 \nabla \chi ' \cdot \nabla |\alpha _n|^2 \leqslant \frac{C}{\rho } \int _{B_{\frac{\rho }{2}}} |\nabla |\alpha _n|^2|^2+C\rho ^2, \end{aligned}$$

with C independent of $\rho$ and n. Summing up, we obtain the bound

$$\begin{aligned} -\int _{B_{\frac{\rho }{2}}} \chi ' |\alpha _n|^2 \Delta |\alpha _n|^2 \leqslant \frac{C}{\rho }\int _{B_{\frac{\rho }{2}}} |\nabla |\alpha _n|^2|^2+C\rho ^2. \end{aligned}$$

Using this estimate in Eq. (4.7), it follows that

$$\begin{aligned} \sigma _n(B_{\frac{\rho }{4}})\leqslant & {} \frac{1}{{{\mathcal {E}}}_n}\bigg ( \frac{C}{\rho }\int _{B_{\frac{\rho }{2}}} |\nabla |\alpha _n|^2|^2+\int _{B_{\frac{\rho }{2}}} \chi ' H +C\rho ^2\bigg )\\\leqslant & {} \frac{C}{{{\mathcal {E}}}_n}\bigg ( \frac{1}{\rho }\int _{B_{\frac{\rho }{2}}} |\nabla |\alpha _n|^2|^2+\rho ^2\bigg ), \end{aligned}$$

where we have used that Eq. (4.3) implies the bound

$$\begin{aligned} \bigg |\int _{B_{\frac{\rho }{2}}}\chi ' H\bigg | \leqslant C\int _{B_{\frac{\rho }{2}}} |\nabla |\alpha _n|^2|^2+C\rho ^3. \end{aligned}$$

Then we infer from Eq. (4.5) and the assumption ${{\mathcal {E}}}_n \rightarrow \infty$, that

$$\begin{aligned} \sigma _{n}(B_{\frac{\rho }{4}}) \rightarrow 0 \end{aligned}$$

(4.8)

as $n\rightarrow \infty$. Since for any point $p\in U$ we can take a small enough neighborhood $N_p$ whose closure is contained in U, Eq. (4.8) implies that $\sigma _\infty (N_p)=0$, thus completing the proof of the proposition. $\square$

5 Absence of Local Obstructions for the Invariant Measures

In this section we prove Theorem 1.8. To this end, in Sect. 5.1 we show that, locally, the modified Seiberg–Witten equations can be reduced to a rescaled version of the vortex equations. This allows us to study the limiting invariant measures using the 2-dimensional vortex equations, cf. Sect. 5.2.

As defined before stating Theorem 3.2, we denote by ${{\mathcal {C}}}$ a flow box adapted to the vector field X. We recall that a flow box is the image of the cylinder $(0, 1) \times {\mathbb {D}}$ under an appropriate map

$$\begin{aligned} \Phi : (0, 1) \times {\mathbb {D}}\longrightarrow M, \end{aligned}$$

which is a diffeomorphism into its image and which satisfies

$$\begin{aligned} d \Phi (\partial _t)=X. \end{aligned}$$

Here t is the coordinate in the interval (0, 1). By the volume-preserving flow box theorem, we can choose the local diffeomorphism $\Phi$ so that the volume form $\mu$ on the flow box coordinates is given by

$$\begin{aligned} \mu =C dx \wedge dy \wedge dt \end{aligned}$$

for some small enough constant C, and coordinates $(x, y) \in \mathbb {{D}}$, $t \in (0, 1)$.

The standard Euclidean metric

$$\begin{aligned} g_{0}=dx^2+dy^2+dt^2 \end{aligned}$$

is then an adapted metric for the vector field X on ${{\mathcal {C}}}$. It is easy to see that we can construct a global metric g on M adapted to the vector field X so that $\Phi _{*} g_0=g|_{{{\mathcal {C}}}}$.

5.1 From Seiberg–Witten to the Rescaled Vortex Equations

In this section we use the notation and constructions introduced in Sect. 2.1. We always work in the flow box ${{\mathcal {C}}}$ using the aforementioned coordinates and adapted metric. The 1-form $\lambda =i_{X} g$ is dt, the Hermitian line bundle $K={\text {Ker}}\lambda$ is spanned by the vector fields $\{\partial _{x}, \partial _{y}\}$ and thus it is trivial, and the base connection $A_0$ defined by Eq. (2.3) is $A_0=0$, and then the 1-form $\varpi _{K}$ introduced in Eq. (2.5) is also 0.

We take the associated line bundle E to be the trivial bundle ${\mathbb {C}}\times {{\mathcal {C}}}$. We endow the rank-two complex bundle ${\mathbb {S}}=E\oplus K^{-1}E$ with the following spin structure, defined via the Clifford multiplication:

$$\begin{aligned} \sigma (\partial _t):=\begin{pmatrix} i &{} 0 \\ 0 &{} -i \end{pmatrix}, \qquad \sigma (\partial _y):=\begin{pmatrix} 0 &{} -1 \\ 1 &{} 0 \end{pmatrix}, \qquad \sigma (\partial _x):=\begin{pmatrix} 0 &{} i \\ i &{} 0 \end{pmatrix}. \end{aligned}$$

Since all the bundles are trivial, the spinor can be identified with a map $\psi =(\alpha , \beta ): {{\mathcal {C}}}\rightarrow {\mathbb {C}}^2$ and the connection with a 1-form $A=A_t dt+A_x dx+ A_{y} dy$ on ${{\mathcal {C}}}$. The modified Seiberg–Witten equations then read as

$$\begin{aligned} \partial _y A_{x}-\partial _x A_{y}&=r(1-|\alpha |^2+|\beta |^2)\,, \end{aligned}$$

(5.1)

$$\begin{aligned} \partial _x A_{t}-\partial _t A_x&=ir({\bar{\alpha }}\beta -{\bar{\beta }} \alpha )\,, \end{aligned}$$

(5.2)

$$\begin{aligned} \partial _t A_y-\partial _y A_t&=r({\bar{\alpha }}\beta +{\bar{\beta }}\alpha )\,, \end{aligned}$$

(5.3)

and the second equation (the Dirac equation) is

$$\begin{aligned}&-(\partial _t \beta -iA_t \beta )+(\partial _x-i\partial _y) \alpha -i(A_x-iA_y)\alpha =0\,, \end{aligned}$$

(5.4)

$$\begin{aligned}&(\partial _t \alpha +iA_t \alpha )+(\partial _x+i\partial _y) \beta +i(A_x+iA_y)\beta =0\,. \end{aligned}$$

(5.5)

These equations can be simplified if we look for t-independent solutions that satisfy

$$\begin{aligned} \beta = A_t= \partial _t A_x=\partial _t A_y=\partial _t \alpha =0, \end{aligned}$$

(5.6)

in which case the modified Seiberg–Witten equations reduce to the well-known rescaled vortex equations on $\mathbb {{D}}\subset {\mathbb {C}}$ with the complex variable $z:=x+i y$:

$$\begin{aligned}&*d a=r(1-|\phi |^{2})\,, \end{aligned}$$

(5.7)

$$\begin{aligned}&{\overline{\partial }}_{a}\phi :={\overline{\partial }}_z\phi -i(a_{x}-ia_{y})\phi =0\,. \end{aligned}$$

(5.8)

Here we have set $\phi :=\alpha$ and $a=a_x dx+a_y dy:=A_x dx+A_y dy$. Equations (5.7) and (5.8) are obtained from the standard vortex equations using the change of variables $z=\sqrt{r}z'$ (see e.g. [2]).

The finite-energy solutions to the vortex equations are well understood. In particular, the following result was proved by Taubes, see [2, 9]. It will be instrumental to prove Theorem 1.8, so we state it for future reference.

Theorem 5.1

(Taubes [2, 9]) Let ${{\mathcal {P}}}:=\{z_j\}_{j=1}^k$ be a finite set of distinct points $z_j\in {\mathbb {C}}$, and let $\{m_j\}_{j=1}^k$ be an associated set of positive integers. There is a smooth solution $(a, \phi )$ to the vortex equations (5.7) and (5.8) with $r=1$ such that $\phi ^{-1}(0)={{\mathcal {P}}}$, and such that the zero $z_j$ of $\phi$ has multiplicity $m_j$. Furthermore, the solution satisfies the additional properties:

• (i) $|\phi | < 1$ on ${\mathbb {C}}$ and $|\phi |\rightarrow 1$ as $|z|\rightarrow \infty$.

• (ii) The energy of the solution is given by

  $$\begin{aligned} {{\mathcal {E}}}:=\int _{{\mathbb {C}}} da=\int _{{\mathbb {C}}} (1-|\phi |^2)=2\pi \sum _j m_j. \end{aligned}$$

• (iii) There is a universal constant C, not depending on the particular configuration of points ${{\mathcal {P}}}$ nor on their multiplicities, such that

  $$\begin{aligned} |\nabla |\phi |^{2}(z)|\leqslant |\nabla _a \phi (z)| \leqslant C. \end{aligned}$$

• (iv) Let $\Omega ^{-}(\phi )$ denote the set of points in ${\mathbb {C}}$ where $|\phi |^{2}\leqslant \frac{1}{2}$. There is a universal constant $c\in (0, 1)$, not depending on the particular configuration of points nor their multiplicities, such that, for any $z\in {\mathbb {C}}$ with ${{\,\textrm{dist}\,}}(z, \Omega ^{-}(\phi ))\geqslant c^{-1}$,

  $$\begin{aligned} |1-|\phi (z)|^{2}|\leqslant & {} e^{-c{{\,\textrm{dist}\,}}(z, \Omega ^{-}(\phi ))}, \end{aligned}$$

  (5.9)
  $$\begin{aligned} |\nabla |\phi |^{2}(z)|\leqslant & {} |\nabla _a \phi (z)| \leqslant c^{-1}e^{-c{{\,\textrm{dist}\,}}(z, \Omega ^{-}(\phi ))}. \end{aligned}$$

  (5.10)

From this theorem we deduce the following important corollary, which we will use in the next section. It follows from the trivial observation that if $(a(z), \phi (z))$ is a solution to the vortex equations with $r=1$ and zeros at $\{z_j\}^{k}_{j=1}$, then

$$\begin{aligned} (a_r(z), \phi _r(z)):=(\sqrt{r}a(\sqrt{r}z), \phi (\sqrt{r}z)) \end{aligned}$$

(5.11)

is a solution to the rescaled vortex equations (5.7)–(5.8) with zeros at $\{\frac{z_j}{\sqrt{r}}\}^{k}_{j=1}$. All the items in Corollary 5.2 then follow from Theorem 5.1 by rescaling according to Eq. (5.11).

Corollary 5.2

Let ${{\mathcal {P}}}:=\{z_j\}_{j=1}^k$ be a finite set of points $z_j\in \mathbb {{D}}$, and let $\{m_j\}_{j=1}^k$ be an associated set of positive integers. For each $r>0$, there is a solution $(a_r, \phi _r)$ to the rescaled vortex equations (5.7) and (5.8) on ${\mathbb {C}}$ with $|\phi _r|^{-1}(0)={{\mathcal {P}}}$ and with each zero $z_j$ having multiplicity $m_j$. Furthermore, the solution $(a_r, \phi _r)$ is bounded as $|\phi _r| < 1$ and has the following properties:

• (i) $|\nabla |\phi _r|^{2}(z)|\leqslant |\nabla _{a_r} \phi _r (z)| \leqslant C \sqrt{r}$.

• (ii) ${{\mathcal {E}}}_r:=r \int _{\mathbb {C}}(1-|\phi _r|^2)=2\pi \sum _j m_j$.

• (iii) If we define

  $$\begin{aligned} \Omega ^{-}_{r}:=\Big \{ z\in {\mathbb {C}}\text { such that } |\phi _r|^{2}(z)\leqslant \frac{1}{2}\Big \}, \end{aligned}$$

  there is a constant c such that, if ${{\,\textrm{dist}\,}}(z, \Omega ^{-}_{r}) \geqslant \frac{1}{c\sqrt{r}}$, we have

  $$\begin{aligned} |1-|\phi _{r}|^2(z)|\leqslant e^{-c \sqrt{r}{{\,\textrm{dist}\,}}(z, \Omega ^{-}_{r}) } \end{aligned}$$

  and

  $$\begin{aligned} |\nabla |\phi _{r}|^2(z)|\leqslant c^{-1} \sqrt{r}e^{-c \sqrt{r}{{\,\textrm{dist}\,}}(z, \Omega ^{-}_{r})} \end{aligned}$$

5.2 Proof of Theorem 1.8

The theorem follows from the following key proposition, whose proof is relegated to Sect. 5.3:

Proposition 5.3

Let $\sigma _{\mathbb {{D}}}$ be a probability measure on the disk. There is an increasing sequence of constants $r_n$ that tends to $\infty$, a sequence ${{\mathcal {P}}}_{n}:=\{z_{jn}\}_{j=1}^{k_{n}} \subset \mathbb {{D}}$ of finite sets of points, with $\{m_{jn}\}_{j=1}^{k_{n}}$ an associated collection of positive integers, and a sequence of solutions $(a_{r_n},\phi _{r_n})$ to the Eqs. (5.7) and (5.8) such that:

• (i) $\phi _{r_n}^{-1}(0)={{\mathcal {P}}}_{n}$, with multiplicities $\{m_{jn}\}$.

• (ii) The sequence of measures

  $$\begin{aligned} \sigma _{n}:=\frac{r_n(1-|\phi _{r_n}|^{2}) dx \wedge dy}{\int _{\mathbb {{D}}} da_{r_n}} \end{aligned}$$

  converges weakly to $\sigma _{\mathbb {{D}}}$.

• (iii) As $n\rightarrow \infty$, we have

  $$\begin{aligned} \frac{\int _{\mathbb {{D}}} d a_{r_n}}{ 2 \pi N_n} \rightarrow 1, \end{aligned}$$

  where $N_{n}:=\sum _{j=1}^{k_n} m_{jn}$.

• (iv) If $\sigma _\mathbb {{D}}$ is d-Frostman for some $d>0$, then $N_n$ is bounded as

  $$\begin{aligned} \lim _{n\rightarrow \infty }N_n r_{n}^{-\theta }=0, \end{aligned}$$

  with $\theta :=\min \big \{\frac{1}{4}, \, \frac{d}{2(d+1)}\big \}$.

Let us first show how item (i) in Theorem 1.8 follows from Proposition 5.3. Given the sequence of solutions $(a_{r_n},\phi _{r_n})$, the discussion in Sect. 5.1 shows that $\psi _r:=(\phi _{r_n},0)$ and $A_r:=a_{r_n}$ is a sequence of solutions of the modified Seiberg–Witten equations on ${{\mathcal {C}}}$. Obviously $F_{A_r}=da_{r_n}$, $\lambda \wedge F_{A_r}=r_n(1-|\phi _{r_n}|^2)dx\wedge dy\wedge dt$ and the energy of the solutions is ${{\mathcal {E}}}_r=\int _\mathbb {{D}}da_{r_n}$. The sequence of measures of the Seiberg–Witten equations is then

$$\begin{aligned} \sigma _n\otimes dt, \end{aligned}$$

and therefore item (ii) above implies that it converges weakly to $\sigma _\mathbb {{D}}\otimes dt$, as claimed. The energy ${{\mathcal {E}}}_r$ is bounded as $N_{n}$, when $r\rightarrow \infty$, by item (iii). Assuming that the measure $\sigma _\mathbb {{D}}$ is d-Frostman, item (iv) provides an estimate for $N_{n}$, which immediately implies item (ii) in Theorem 1.8, which completes the proof.

In the following proposition, we show the connection between the regularity of a measure and its Frostman properties alluded to in the Introduction. Recall that the Sobolev space $W^{-1,p}_{\mathbb {{D}}}$ is defined as

$$\begin{aligned} W^{-1,p}_{\mathbb {{D}}}:=\left\{ \phi \in W^{-1,p}({\mathbb {R}}^2): {\text {supp}}(\phi )\subset {\overline{\mathbb {{D}}}}\right\} . \end{aligned}$$

It easily follows from a duality argument and the Sobolev embedding theorem that any measure $\sigma _{\mathbb {{D}}}$ is in $W^{-1, p}_{\mathbb {{D}}}$ for all $p<2$. This result is sharp, as evidenced by the Dirac measure $\delta _{p_0}$ supported at a point $p_0\in \mathbb {{D}}$. When the measure is slightly more regular, we infer that it is d-Frostman for some $d>0$:

Proposition 5.4

Assume that the probability measure $\sigma _{\mathbb {{D}}}$ is in the Sobolev space $W^{-1,p}_{\mathbb {{D}}}$ for some $p\in (2,\infty ]$. Then $\sigma _\mathbb {{D}}$ is d-Frostman with $d:=1-\frac{2}{p}$.

Proof

Take a smooth bump function $\chi :{\mathbb {R}}^2\rightarrow [0,1]$ such that $\chi (x)=1$ if $|x|\leqslant 1$ and $\chi (x)=0$ if $|x|\geqslant 2$. Defining $\chi _\varepsilon (x):=\chi \big (\frac{x-x_0}{\varepsilon }\big )$ for any point $x_0\in {\mathbb {R}}^2$ and any $\varepsilon >0$, we obviously have

$$\begin{aligned} \sigma _{\mathbb {{D}}}(B(x_0,\varepsilon ))\leqslant \int _{{\mathbb {R}}^2} \chi _\varepsilon (x)\, d\sigma _{\mathbb {{D}}}(x). \end{aligned}$$

(5.12)

Let $p':=p/(p-1)\in [0,2)$ be the dual exponent to p. By scaling, the $W^{1,p'}$ norm of $\chi _\varepsilon$ satisfies

$$\begin{aligned} \Vert \chi _\varepsilon \Vert _{W^{1,p'}}= \Vert \chi _\varepsilon \Vert _{L^{p'}}+ \Vert \nabla \chi _\varepsilon \Vert _{L^{p'}} \leqslant C\varepsilon ^{2-\frac{2}{p}}+ C\varepsilon ^{1-\frac{2}{p}}\leqslant C\varepsilon ^{1-\frac{2}{p}}\,. \end{aligned}$$

The generalized Hölder inequality then allows us to estimate (5.12) as

$$\begin{aligned} \sigma _{\mathbb {{D}}}(B(x_0,\varepsilon ))\leqslant \Vert \sigma _{\mathbb {{D}}}\Vert _{W^{-1,p}} \Vert \chi _\varepsilon \Vert _{W^{1,p'}}\leqslant C\Vert \sigma _{\mathbb {{D}}}\Vert _{W^{-1,p}}\varepsilon ^{1-\frac{2}{p}}. \end{aligned}$$

The lemma then follows. $\square$

5.3 Proof of Proposition 5.3

The proof is divided in five steps. Items (i) and (iii) are established in Steps 2 and 4, respectively, while items (ii) and (iv) are proved in Step 5. The proof of some intermediate lemmas is postponed to Sect. 5.4.

5.3.1 Step 1: Choice of a Sequence of Points

We claim that we can choose a sequence of finite sets of points ${{\mathcal {P}}}_n=\{z_{jn}\}_{j=1}^{k_n} \subset \mathbb {{D}}$ with multiplicities $\{m_{jn}\}_{j=1}^{k_n}$, $n \in {\mathbb {N}}$, and $N_{n}=\sum _j m_{jn}$, such that the Dirac measures

$$\begin{aligned} \delta _{{{\mathcal {P}}}_n}:=\frac{1}{N_n} \sum _{j=1}^{k_n} m_{jn}\delta (z-z_{jn}) \end{aligned}$$

converge weakly to $\sigma _{\mathbb {{D}}}$ as $n \rightarrow \infty$, and, moreover, if we define

$$\begin{aligned} \epsilon _{n}:=\frac{1}{2} \min \bigg (\min _{j, k} |z_{jn}-z_{kn}|, \, \min _{j} {{\,\textrm{dist}\,}}(z_{jn}, \partial \mathbb {{D}})\bigg ). \end{aligned}$$

(5.13)

we have:

• (i) There is a decreasing continuous function $F: (0, \infty ) \rightarrow (0, \infty )$ with

  $$\begin{aligned} \lim _{x\rightarrow \infty } F(x)=0 \end{aligned}$$

  and so that

  $$\begin{aligned} \epsilon _n \geqslant F(N_{n}). \end{aligned}$$

• (i) If $\sigma _{\mathbb {{D}}}$ is d-Frostman with $d>0$, this function can be taken $F(x):=Cx^{-\frac{1}{d}}$ for some constant $C>0$, so that

  $$\begin{aligned} \epsilon _{n} \geqslant \frac{C}{ N_{n}^{\frac{1}{d}}}. \end{aligned}$$

Indeed, in the case that $\sigma _\mathbb {{D}}$ is a point measure there is $n_0\geqslant 1$ such that ${{\mathcal {P}}}_n={{\mathcal {P}}}_{n_0}$ for all $n\geqslant n_0$, and the same with $N_n$ and $\epsilon _n$; the claim is then obvious because the function F can be chosen so that $F(N_n)\leqslant \epsilon _{n_0}$ for all $n\geqslant n_0$. Otherwise, it is standard that we can always approximate the measure $\sigma _{\mathbb {{D}}}$, in the sense of weak convergence, by a sequence of Dirac probability measures of the form

$$\begin{aligned} \Sigma _n:=\frac{1}{M_n}\sum _{j=1}^{k_n} \sigma _{\mathbb {{D}}}(B_{\epsilon '_n}(z_{jn})) \delta (z-z_{jn}) \end{aligned}$$

for some sequence of points $\{z_{jn}\}_{j=1}^{k_n}\subset \mathbb {{D}}$. Here, $\{B_{\epsilon '_n}(z_{jn})\}$ is a disjoint collection of balls of radius $\epsilon '_n$ inside the disk, which cover it when $n\rightarrow \infty$ (and hence $\epsilon '_n\rightarrow 0$), and $M_n:=\sum _{j=1}^{k_n} \sigma _{\mathbb {{D}}}(B_{\epsilon '_n}(z_{jn}))$. By density, we can safely assume that each $\sigma _{\mathbb {{D}}}(B_{\epsilon '_n}(z_{jn}))M_n^{-1}$ is a rational number of the form

$$\begin{aligned} \frac{\sigma _{\mathbb {{D}}}(B_{\epsilon '_n}(z_{jn}))}{M_n}=\frac{m_{jn}}{N_n} \end{aligned}$$

for some positive integers $m_{jn}$ and $N_n$. Obviously, $\sum _j m_{jn}=N_n$, $k_n\leqslant N_n$ and $M_n\rightarrow 1$ as $n\rightarrow \infty$. The measure $\Sigma _n$ is then of the form $\delta _{{{\mathcal {P}}}_n}$ stated above. Moreover, the numbers $\epsilon _n$ defined in Eq. (5.13) are bounded from below as $\epsilon _n\geqslant \epsilon '_n$.

If $\sigma _{\mathbb {{D}}}$ is d-Frostman, then $\sigma _{\mathbb {{D}}}(B_{\epsilon }(x)) \leqslant C \epsilon ^{d}$, so taking n large enough so that $M_n\geqslant \frac{1}{2}$, we deduce the relation

$$\begin{aligned} N_n\epsilon _n^d\geqslant N_n (\epsilon '_n)^d\geqslant \frac{C}{2}, \end{aligned}$$

which proves the item (ii) above. If the measure $\sigma _\mathbb {{D}}$ is not d-Frostman, there is no explicit relation between $\epsilon _n$ and $N_n$. However, using that the sequence $\epsilon _n$ can be chosen to be decreasing, we can always define $F(N_n):= \epsilon _n$, and find a decreasing positive function F interpolating those values such that $\lim _{x\rightarrow \infty }F(x)=0$, so that item (i) is trivially true.

In what follows we fix a sequence ${{\mathcal {P}}}_n$ and associated multiplicities $\{m_{jn}\}_{j=1}^{k_n}$ with the properties stated above. In particular, if the measure is d-Frostman, we assume that F is given as in item (ii).

5.3.2 Step 2. Choice of a Sequence of Rescaled Vortex Solutions

By Corollary 5.2, for any sequence of positive real numbers $\{r_{n}\}_{n=0}^{\infty }$ we can find a sequence of solutions $(a_{r_n}, \phi _{r_n})$ to the $r_{n}$-rescaled vortex equations on ${\mathbb {C}}$, with $\phi _{r_n}^{-1}(0)={{\mathcal {P}}}_{n}$, and with associated multiplicities $\{m_{jn}\}$. This already proves item (i) of the proposition.

For the rest of the proof, it is convenient to fix a sequence $\{r_{n}\}$ so that the following conditions are satisfied as $n\rightarrow \infty$:

$$\begin{aligned} 0=\lim _{n\rightarrow \infty }\frac{N_n}{r^{\frac{1}{4}}_{n}} =\lim _{n\rightarrow \infty } \frac{N_n}{F(N_n) \sqrt{r_n}}=\lim _{n\rightarrow \infty } \frac{\log r_{n}}{F(N_n) \sqrt{r_n}}, \end{aligned}$$

(5.14)

where F is the function defined in Step 1. Observe that such a sequence always exists because it suffices to take $r_n$ large enough for each n.

It is easy to see that in the case that $\sigma _{\mathbb {{D}}}$ is d-Frostman for some $d>0$, and so $F(N_n)=C N_n^{-\frac{1}{d}}$, Eq. (5.14) is satisfied if we choose a sequence $r_n$ verifying

$$\begin{aligned} \lim _{n\rightarrow \infty }N_nr^{-\theta }_{n}= 0 \end{aligned}$$

(5.15)

with

$$\begin{aligned} \theta :=\min \bigg \{\frac{1}{4},\, \frac{d}{2(d+1)}\bigg \}. \end{aligned}$$

5.3.3 Step 3: Some Key Auxiliary Lemmas

It is convenient to define

$$\begin{aligned} \Omega _{n}^{+}&:=\mathbb {{D}}\setminus \bigcup _{j} B(z_{jn}, F(N_n))\,,\\ \Omega _{n}^{-}&:=\left\{ z \in \mathbb {{D}}: |\phi _{r_n}(z)|^{2} < \frac{1}{2}\right\} \end{aligned}$$

and to let $\Omega _{n}^{-}(z_{jn})$ denote the connected component of $\Omega _{n}^{-}$ which contains $z_{jn}$. Note that

$$\begin{aligned} \Omega _n^- = \bigcup _j \Omega _n^-(z_{jn}) \end{aligned}$$

because there is a zero of $\phi$ in each connected component of $\Omega _n^-$ (simply because, by the form of the vortex equations, $\phi$ must vanish at each minimum of $|\phi |^2$).

Our goal is to show that, as $n\rightarrow \infty$, the function $|\phi _{r_n}|$ is exponentially close to 1 in the set $\Omega _{n}^{+}$, while in the set $\Omega _{n}^{-}$ the solution goes to zero with a polynomial bound. Before stating the lemmas that establish these properties, we notice that Eq. (5.14) implies, for any fixed constant C independent of n, that

$$\begin{aligned} B\left( z_{jn}, \frac{C N_n}{\sqrt{r_n}}\right) \subset B(z_{jn}, F(N_n)) \end{aligned}$$

(5.16)

provided that n is large enough. The proofs of these lemmas will be presented in Sect. 5.4.

Lemma 5.5

$\Omega ^{-}_{n}(z_{jn}) \subset B(z_{jn}, \frac{CN_n}{\sqrt{r_n}})$ for some constant C independent of n, for all j and all large enough n.

Lemma 5.6

For any $z \in \Omega _{n}^{+}$ and all $n>0$, the solution to the vortex equations is bounded as

$$\begin{aligned} 1-|\phi _{r_n}(z)|^2\leqslant & {} e^{-c F(N_n) \sqrt{r_n}},\\ |\nabla |\phi _{r_n}|^2|\leqslant & {} c^{-1} \sqrt{r_n} e^{-c F(N_n) \sqrt{r_n}}. \end{aligned}$$

Moreover, for any $z \in {\mathbb {C}}\setminus \mathbb {{D}}$ the estimate is

$$\begin{aligned} 1-|\phi _{r_n}(z)|^2\leqslant & {} e^{-c\sqrt{r_n}(||z|-1|+F(N_n))},\\ |\nabla |\phi _{r_n}|^2 |\leqslant & {} c^{-1} \sqrt{r_n} e^{-c\sqrt{r_n} (||z|-1|+F(N_n))}. \end{aligned}$$

Here c is a constant that does not depend on n.

Observe that the third condition in Eq. (5.14) implies that $F(N_n) \sqrt{r_n} \rightarrow \infty$ as $n\rightarrow \infty$ faster than $\log r_n$ (even in the case where $N_n$ is constant for all $n\geqslant n_0$), so all the upper bounds in Lemma 5.6 go to 0 as $n \rightarrow \infty$.

Lemma 5.7

For some constant $C>0$ (independent of n), on each ball $B(z_{jn}, \frac{C N_n}{\sqrt{r_n}})$ we can write

$$\begin{aligned} |\phi _{r_n}(z)|^2=r^{m_{jn}}_{n} h_{jn}(z) |z-z_{jn}|^{2m_{jn}}, \end{aligned}$$

where $h_{jn}(z)$ is a smooth function satisfying $h_{jn}(z)>0$ and

$$\begin{aligned} \bigg |\frac{1}{N_n} \sum _{j} \int _{B(z_{jn}, \frac{CN_n}{\sqrt{r_n}})}\log (h_{jn}) \bigg | \rightarrow 0 \end{aligned}$$

as $n \rightarrow \infty$.

5.3.4 Step 4: Proof of Item (iii)

Taking into account item (ii) in Corollary 5.2, it is enough to show that

$$\begin{aligned} \lim _{n\rightarrow \infty } r_{n}\int _{{\mathbb {C}}\setminus \mathbb {{D}}}(1-|\phi _{r_n}|^2) = 0. \end{aligned}$$

By Lemma 5.6, we have the estimate

$$\begin{aligned} \int _{{\mathbb {C}}\setminus \mathbb {{D}}} r_n(1-|\phi _{r_n}|^{2}) \leqslant r_n e^{-c \sqrt{r_n} F(N_n)} \int _{{\mathbb {C}}\setminus \mathbb {{D}}} e^{-c \sqrt{r_n} |z-1|}. \end{aligned}$$

Accordingly, since $F(N_n) \sqrt{r_n} \rightarrow \infty$ faster than $\log r_n$, the claim follows.

5.3.5 Step 5: Proof of Items (ii) and (iv)

Once item (iii) has been established, to prove item (ii) it is enough to show that, for any function $f\in C^\infty ({{\overline{\mathbb {{D}}}}})$,

$$\begin{aligned} \int _{\mathbb {{D}}} r_{n}(1-|\phi _{r_n}|^2) f =2 \pi \sum _{z_{jn} \in {{\mathcal {P}}}_n} m_{jn}f(z_{jn})+e(r_n), \end{aligned}$$

with an error satisfying $\lim _{n\rightarrow \infty }{e(r_n)}/{N_n} = 0$.

It will be convenient to work with the function $u_{r_n}$ defined as

$$\begin{aligned} u_{r_n}:=\log |\phi _{r_n}|^2. \end{aligned}$$

Since $|\phi _{r_n}|< 1$ (cf. Corollary 5.2), the function $u_{r_n}$ is negative. It is not hard to check that the function $u_{r_n}$ satisfies, as a distribution, the PDE (see e.g. [2, Chapter 3.3])

$$\begin{aligned} \Delta u_{r_n} +2 r_n (1-e^{u_{r_n}})=4\pi \underset{z_{j_n} \in {{\mathcal {P}}}_n}{\sum } m_{jn} \delta (z-z_{jn}). \end{aligned}$$

(5.17)

In terms of $u_{r_n}$, the measure $\sigma _{n}$ reads as

$$\begin{aligned} \sigma _{n}=\frac{r_n(1-e^{u_{r_n}}) dx \wedge dy}{\int _{\mathbb {{D}}} da_{r_n}}. \end{aligned}$$

Noticing that Eq. (5.17) implies that, for any $f \in C^{\infty }({{\overline{\mathbb {{D}}}}})$,

$$\begin{aligned} r_n \int _{\mathbb {{D}}} (1-e^{u_{r_n}}) f =-\frac{1}{2}\int _{\mathbb {{D}}} f \Delta u_{r_n} + 2\pi \underset{ {{\mathcal {P}}}_n}{\sum }\ m_{jn}f(z_{jn}), \end{aligned}$$

we infer that item (ii) follows if we prove that

$$\begin{aligned} \lim _{n\rightarrow \infty }\frac{\bigg |\int _{\mathbb {{D}}} f \Delta u_{r_n} \bigg |}{N_n} =0, \end{aligned}$$

for all $f\in C^\infty ({{\overline{\mathbb {{D}}}}})$.

To this end, we first integrate by parts to obtain

$$\begin{aligned} \int _{\mathbb {{D}}} f \Delta u_{r_n}= -\int _{\mathbb {{D}}} \nabla u_{r_n} \cdot \nabla f+\int _{\partial \mathbb {{D}}} f \nabla u_{r_n} \cdot \nu \, d\theta , \end{aligned}$$

(5.18)

where $\nu$ is the outward pointing unit normal vector at the boundary of the disk. Now, by Lemma 5.6, for any point $z\in \partial \mathbb {{D}}$ and all n large enough, we have the estimate

$$\begin{aligned} |\nabla u_{r_n}|(z)=\frac{1}{|\phi _{r_n}(z)|^{2}}|\nabla |\phi _{r_n}(z)|^{2}| \leqslant C \sqrt{r_n} e^{-c F(N_n) \sqrt{r_n}}. \end{aligned}$$

Therefore,

$$\begin{aligned} \left| \int _{\partial \mathbb {{D}}} f \nabla u_{r_n}\cdot \nu \right| \leqslant C\Vert f\Vert _{L^1(\partial \mathbb {{D}})} \sqrt{r_n} e^{-c F(N_n) \sqrt{r_n}}, \end{aligned}$$

which goes to zero as $n\rightarrow \infty$ because $F(N_n) \sqrt{r_n}$ tends to infinity faster than $\log (r_n)$ (cf. Eq. (5.14)).

As for the first summand in Eq. (5.18), a second integration by parts yields

$$\begin{aligned} - \int _{\mathbb {{D}}} \nabla u_{r_n} \cdot \nabla f= \int _{\mathbb {{D}}} u_{r_n} \Delta f-\int _{\partial \mathbb {{D}}} u_{r_n} \nabla f \cdot \nu d\theta . \end{aligned}$$

Again, using Lemma 5.6, the rightmost term is bounded as

$$\begin{aligned} \bigg |\int _{\partial \mathbb {{D}}} u_{r_n} \nabla f \cdot \nu d\theta \bigg | \leqslant C\Vert f\Vert _{W^{1,1}(\partial \mathbb {{D}})} e^{-c F(N_n) \sqrt{r_n}}, \end{aligned}$$

which again goes to zero as $n\rightarrow \infty$. Finally, using that $u_{r_n}<0$, it is clear that

$$\begin{aligned} -\int _{\mathbb {{D}}} u_{r_n} \Delta f \leqslant -\Vert f\Vert _{C^{2}(\mathbb {{D}})} \int _{\mathbb {{D}}} u_{r_n}, \end{aligned}$$

so our main claim follows if we show that

$$\begin{aligned} \lim _{n\rightarrow \infty }\frac{1}{N_n}\int _{\mathbb {{D}}} u_{r_n} =0. \end{aligned}$$

(5.19)

To prove this, we divide the integral into two parts, the disks $B(z_{jn}, F(N_n))$, and the set $\Omega ^{+}_{n}$:

$$\begin{aligned} -\int _{\mathbb {{D}}} u_{r_n}=-\int _{\Omega _{n}^{+}} u_{r_n}-\sum _{{{\mathcal {P}}}_n} \int _{B(z_{jn}, F(N_n))} u_{r_n}. \end{aligned}$$

By Lemma 5.6, for any $z \in \Omega ^{+}_{n}$ we can write the bound

$$\begin{aligned} |1-|\phi _{r_n}(z)|^{2}| \leqslant e^{-cF(N_n)\sqrt{r_n}} \ll 1 \end{aligned}$$

provided that n is large enough. Thus, taking the Taylor expansion of

$$\begin{aligned} u_{r_n}={\log }(1-(1-|\phi _{r_n}|)^2) \end{aligned}$$

we obtain

$$\begin{aligned} -\int _{\Omega _{n}^{+}} u_{r_n}= & {} -\int _{\Omega _{n}^{+}} \log |\phi _{r_n}|^2 \leqslant \int _{\Omega _{n}^{+}} (1-|\phi _{r_n}|)^2+C \int _{\Omega _{n}^{+}} (1-|\phi _{r_n}|^2)^2 \\\leqslant & {} C e^{-c F(N_n) \sqrt{r_n}}, \end{aligned}$$

which tends to 0 as $n\rightarrow \infty$.

To bound the integral

$$\begin{aligned} -\int _{B(z_{jn}, F(N_n))} u_{r_n}, \end{aligned}$$

we write it for large enough n as

$$\begin{aligned} -\int _{B(z_{jn}, F(N_n))} u_{r_n}=-\int _{B(z_{jn}, \frac{CN_n}{\sqrt{r_n}})} u_{r_n}-\int _{B(z_{jn}, F(N_n))\setminus B(z_{jn}, \frac{CN_n}{\sqrt{r_n}}) } u_{r_n}, \end{aligned}$$

where C is the constant in Lemma 5.5 and we have used Eq. (5.16). By Lemma 5.5, we know that $|\phi _{r_n}|^2 \geqslant \frac{1}{2}$ on the set $B(z_{jn}, F(N_n)){\setminus } B(z_{jn}, \frac{CN_n}{\sqrt{r_n}})$, so on this set

$$\begin{aligned} 0\leqslant -u_{r_n} \leqslant \log 2, \end{aligned}$$

which allows us to write the bound

$$\begin{aligned} -\int _{B(z_{jn}, F(N_n))\setminus B(z_{jn}, \frac{CN_n}{\sqrt{r_n}}) } u_{r_n} \leqslant C F(N_n)^2. \end{aligned}$$

(5.20)

On the other hand, by Lemma 5.7, we can express $u_{r_n}$ in the disk $B(z_{jn}, \frac{C N_n}{\sqrt{r_n}})$ as

$$\begin{aligned} u_{r_n}(z)= \log (h_{jn})+m_{jn} \log r_n+2 m_{jn} \log (|z-z_{jn}|), \end{aligned}$$

so we deduce

$$\begin{aligned} -\int _{B(z_{jn}, \frac{CN_n}{\sqrt{r_n}})} u_{r_n}\leqslant & {} -\int _{B\big (z_{jn}, \frac{CN_n}{\sqrt{r_n}}\big )} \bigg [\log (h_{jn})-\pi m_{jn}\bigg (\frac{CN_n}{\sqrt{r_n}}\bigg )^2 \log r_n \\{} & {} -2\pi m_{jn} \bigg (\frac{CN_n}{\sqrt{r_n}}\bigg )^2 \log \bigg (\frac{CN_n}{\sqrt{r_n}}\bigg )+\pi m_{jn}\bigg (\frac{CN_n}{\sqrt{r_n}}\bigg )^2\bigg ]. \end{aligned}$$

The first term after the inequality divided by $N_n$ goes to zero because of Lemma 5.7, so putting the other terms together with the one coming from Eqs. (5.20), (5.19) follows if we show that the quantity

$$\begin{aligned}&\frac{1}{N_n} \sum _{{{\mathcal {P}}}_n} \bigg [C F(N_n)^2-\pi m_{jn} \bigg (\frac{CN_n}{\sqrt{r_n}}\bigg )^2 \log r_n \\&\qquad -2\pi m_{jn} \bigg (\frac{CN_n}{\sqrt{r_n}}\bigg )^2 \log \bigg (\frac{CN_n}{\sqrt{r_n}}\bigg ) +\pi m_{jn}\bigg (\frac{CN_n}{\sqrt{r_n}}\bigg )^2 \bigg ]\\&\quad \leqslant C\bigg ( F(N_n)^2+ \bigg (\frac{N_n}{\sqrt{r_n}}\bigg )^2 \log r_n + \bigg (\frac{N_n}{\sqrt{r_n}}\bigg )^2 \log N_n\bigg ) \end{aligned}$$

goes to zero as $n \rightarrow \infty$. But this is evident, because, by construction, if $\sigma _{\mathbb {{D}}}$ is not a point measure,

$$\begin{aligned} \lim _{n\rightarrow \infty }F(N_n)=0, \end{aligned}$$

and the other terms also tend to 0 as $n\rightarrow \infty$ by the conditions in Eq. (5.14). If $\sigma _{\mathbb {{D}}}$ is a point measure, and thus $N_{n}$ stays constant for all $n\geqslant n_0$, we reach the same conclusion by substituting in the argument above the sequence $F(N_n)$ by a sequence $F_{n}$ of positive numbers, smaller than $\epsilon _{n_0}$, and going to zero as $n \rightarrow \infty$.

This completes the proof of item (ii). Concerning item (iv), we simply recall that the condition (5.14) is verified by any d-Frostman measure upon choosing a sequence of $r_n$ satisfying Eq. (5.15). Proposition 5.3 then follows.

5.4 Proof of the Auxiliary Lemmas

In this section we prove Lemmas 5.5, 5.6 and 5.7, which are instrumental in the previous section. We follow the same notation and assumptions as before without further mention.

5.4.1 Proof of Lemma 5.5

The claim obviously follows if we show that for any points $p_n$ and $q_n$ in the same connected component of $\Omega ^{-}_{n}$ there is a constant C (independent of n) such that

$$\begin{aligned} {{\,\textrm{dist}\,}}(p_n, q_n)\leqslant \frac{CN_n}{\sqrt{r_n}}. \end{aligned}$$

Indeed, let $\gamma _{n}$ be a smooth embedded curve inside $\Omega ^{-}_{n}$, joining the points $p_n$ and $q_n$ (which exists because $\Omega ^{-}_{n}$ is an open set). By definition, any point $z \in \gamma _{n}$ satisfies that $|\phi _{r_n}|^{2}(z)< \frac{1}{2}$. Using that

$$\begin{aligned} |\nabla |\phi _{r_n}|^{2}(z)|\leqslant C \sqrt{r_n} \end{aligned}$$

for all $z\in {\mathbb {C}}$ by Corollary 5.2, we infer that there is a constant C (independent of n) such that, for any $\delta >0$ as small as desired, all the points within a distance $\frac{\delta }{C\sqrt{r}_n}$ of $\gamma _{n}$ satisfy $|\phi _{r_n}|^{2}\leqslant \frac{1}{2}+\delta$.

Fixing a small constant $\delta >0$, let us denote by $U_n$ the aforementioned set of points $z\in {\mathbb {C}}$ at a distance smaller or equal than $\frac{\delta }{C\sqrt{r}_n}$ from $\gamma _{n}$. The area of $U_n$ is bounded from below by

$$\begin{aligned} \int _{U_n} dx\wedge dy \geqslant |\gamma _{n}| \frac{\delta }{C\sqrt{r}_n}, \end{aligned}$$

where by $|\gamma _n|$ we denote the length of the curve $\gamma _n$. Therefore,

$$\begin{aligned} r_n\int _{U_n} (1-|\phi _{r_n}|^{2}) \geqslant r_n |\gamma _n| \Big (\frac{1}{2}-\delta \Big ) \frac{\delta }{C\sqrt{r}_n} \geqslant C\sqrt{r}_n |\gamma _n|, \end{aligned}$$

for some constant C that depends on $\delta$ but not on n. On the other hand, notice that, by Corollary 5.2,

$$\begin{aligned} r_n\int _{U_n} (1-|\phi _{r_n}|^{2}) \leqslant r_n \int _{{\mathbb {C}}} (1-|\phi _{r_n}|^{2}) =2\pi N_n. \end{aligned}$$

Therefore, combining both inequalities we can bound the length of $\gamma _n$ as

$$\begin{aligned} |\gamma _{n}| \leqslant C\frac{N_n}{\sqrt{r}_n}\, \end{aligned}$$

for some n-independent constant C. The claim follows because the length of $\gamma _n$ is always greater or equal than the distance between $p_n$ and $q_n$.

5.4.2 Proof of Lemma 5.6

We recall Eq. (5.16), i.e.,

$$\begin{aligned} B\left( z_j, \frac{CN_n}{\sqrt{r}_n}\right) \subset B(z_{jn}, F(N_n)). \end{aligned}$$

Then Lemma 5.5 implies that

$$\begin{aligned} \Omega _{n}^{-}(z_{jn}) \subset B(z_{jn}, F(N_n)). \end{aligned}$$

In particular, since $\epsilon _n\geqslant F(N_n)$ by definition, it follows that

$$\begin{aligned} \Omega _{n}^{-}(z_{jn}) \cap \Omega _{n}^{-}(z_{kn})=\emptyset \end{aligned}$$

for any $j \ne k$. Let us estimate the infimum of ${{\,\textrm{dist}\,}}(z, \Omega _{n}^{-})$ for $z \in \Omega _{n}^{+}$. It is clear that we can take z on the boundary $\partial B(z_{jn}, F(N_n))$ for some j, in which case Lemma 5.5 implies that, for some $C>0$,

$$\begin{aligned} {{\,\textrm{dist}\,}}(z, \Omega _{n}^{-}) \geqslant F(N_n)-\frac{C N_n}{\sqrt{r_n}}=F(N_n) \bigg (1-\frac{C N_n}{F(N_n)\sqrt{r_n}}\bigg ). \end{aligned}$$

Then, we deduce from Eq. (5.14) that for any given $\delta >0$ and any large enough n we have

$$\begin{aligned} {{\,\textrm{dist}\,}}(z, \Omega _{n}^{-}) \geqslant (1-\delta ) F(N_n) \gg \frac{1}{\sqrt{r_n}}, \end{aligned}$$

where we have used that $\lim _{n\rightarrow \infty }F(N_n)\sqrt{r_n}=\infty$. The first two statements in Lemma 5.6 then follow by applying item (iii) of Corollary 5.2.

The two other statements concerning points $z \in {\mathbb {C}}{\setminus } \mathbb {{D}}$ also follow from item (iii) in Corollary 5.2 upon noticing that

$$\begin{aligned} {{\,\textrm{dist}\,}}(z, \Omega _{n}^{-}) \geqslant {{\,\textrm{dist}\,}}(z, \partial \mathbb {{D}})+{{\,\textrm{dist}\,}}(\partial \mathbb {{D}}, \Omega _{n}^{-}) \geqslant ||z|-1|+ (1-\delta ) F(N_n). \end{aligned}$$

This completes the proof of the lemma.

5.4.3 Proof of Lemma 5.7

It is well known, cf. [2, Proposition 5.1], that any solution $\phi$ for the $r=1$ vortex equations can be written as

$$\begin{aligned} \phi (z)=(h_k (z))^{1/2} (z-z_k)^{m_k} \end{aligned}$$

on a disk that contains just one zero $z_{k}$. Here $h_k(z)$ is a smooth non-vanishing function on the disk and $m_k$ is the multiplicity of the zero. By rescaling, we get the first statement in Lemma 5.7, that is, we can represent $\phi _{r_n}$ as

$$\begin{aligned} |\phi _{r_n}(z)|^2=h_{jn}(z) r^{m_{jn}}_n |z-z_{jn}|^{2m_{jn}}. \end{aligned}$$

(5.21)

Notice that this representation holds on the disk $B(z_{jn},\frac{CN_n}{\sqrt{r_n}})$ for some constant $C>0$ because, by construction, $\epsilon _n\geqslant F(N_n)\geqslant \frac{CN_n}{\sqrt{r_n}}$, and hence $z_{jn}$ is the only zero of $\phi _{r_n}$ in such a disk.

For notational simplicity, we define the smooth function $v_{r_n}: B(z_{jn}, \frac{CN_n}{\sqrt{r_n}})\rightarrow {\mathbb {R}}$ as

$$\begin{aligned} v_{r_n}(z):=\log h_{jn}(z), \end{aligned}$$

and we set $B_{n}:= B(z_{jn},\frac{CN_n}{\sqrt{r_n}})$.

To prove the estimate for $h_{jn}$ in Lemma 5.7, we first notice that

$$\begin{aligned} \bigg |\int _{B_n} v_{r_n}\bigg | \leqslant \pi ^{\frac{1}{2}} \frac{C N_n}{\sqrt{r_n}} \Vert v_{r_n}\Vert _{L^2(B_n)}. \end{aligned}$$

(5.22)

Our goal is to bound the $L^2$ norm $\Vert v_{r_n}\Vert _{L^2(B_n)}$. To this end, we first observe that, if $w_{r_n}$ is the unique harmonic function on the disk $B_n$ that coincides with $v_{r_n}$ at the boundary, we have the inequality

$$\begin{aligned} \Vert v_{r_n}\Vert _{L^2(B_n)} \leqslant \Vert w_{r_n}\Vert _{L^{2}(B_n)}+\frac{1}{\lambda _{1}(B_n)} \Vert \Delta v_{r_n}\Vert _{L^2(B_n)}\, \end{aligned}$$

(5.23)

where $\lambda _{1}(B_{n})=\frac{c_0r_n}{N_n^2}$ is the first eigenvalue of the Dirichlet Laplacian on the disk $B_n$ (for some constant $c_0$). This estimate follows easily from the min–max characterization of Dirichlet eigenvalues.

Now, the maximum principle for harmonic functions allows us to write

$$\begin{aligned} \sup _{B_n} |w_{r_n}|=\sup _{\partial B_n} |w_{r_n}|=\sup _{\partial B_n} |v_{r_n}|, \end{aligned}$$

and therefore,

$$\begin{aligned} \Vert w_{r_n}\Vert _{L^2(B_n)} \leqslant \frac{CN_n}{\sqrt{r_n}}\,\sup _{\partial B_n} |v_{r_n}|. \end{aligned}$$

To obtain a bound of $\sup _{\partial B_n} |v_{r_n}|$, we recall that $\Omega _{n}^{-}(z_{jn}) \subset B_n$ by Lemma 5.5, so $|\phi _{r_n}|^2\geqslant \frac{1}{2}$ on $\partial B_n$, which implies by Eq. (5.21)

$$\begin{aligned} h_{jn}|_{\partial B_n} \geqslant \frac{1}{2} \frac{1}{(C N_n)^{2 m_{jn}}}, \end{aligned}$$

and hence

$$\begin{aligned} v_{r_n}|_{\partial B_n} \geqslant -\log 2-2m_{jn}\log (C N_n). \end{aligned}$$

On the other hand, $|\phi _{r_n}|^2 \leqslant 1$, so applying again Eq. (5.21) and taking the logarithm, we get the upper bound

$$\begin{aligned} v_{r_n}|_{\partial B_n} \leqslant - 2m_{jn}\log (C N_n). \end{aligned}$$

We then conclude that

$$\begin{aligned} \sup _{\partial B_n} |v_{r_n}| \leqslant |2m_{jn}\log (C N_n)+1| \leqslant 2m_{jn}|\log (C N_n)+1|, \end{aligned}$$

and therefore

$$\begin{aligned} \Vert w_{r_n}\Vert _{L^2(B_n)} \leqslant \frac{C N_n m_{jn}}{\sqrt{r_n}} |\log (C N_n)+1|. \end{aligned}$$

(5.24)

Finally, to obtain a bound for the $L^2$ norm of $\Delta v_{r_n}$, we use Eq. (5.17) and the fact that

$$\begin{aligned} \Delta \log |z-z_{jn}|^{2m_{jn}}=4 \pi m_{jn} \delta (z-z_{jn}), \end{aligned}$$

to infer that the smooth function $v_{r_n}$ satisfies the PDE

$$\begin{aligned} \Delta v_{r_n}=2 r_{n}(e^{u_{r_n}}-1). \end{aligned}$$

(5.25)

Accordingly,

$$\begin{aligned} \Vert \Delta v_{r_n}\Vert _{L^2(B_n)}=2 r_{n} \bigg (\int _{B_n} |e^{u_{r_n}}-1|^2 \bigg )^{\frac{1}{2}}, \end{aligned}$$

and since $|\phi _{r_n}|^2=e^{u_{r_n}} < 1$, we obtain the estimate

$$\begin{aligned} \Vert \Delta v_{r_n}\Vert _{L^2(B_n)} \leqslant C N_n \sqrt{r_n}. \end{aligned}$$

(5.26)

Putting together Eqs. (5.22), (5.23), (5.24) and (5.26) we get the bound

$$\begin{aligned} \bigg |\int _{B_n} v_{r_n}\bigg | \leqslant \frac{Cm_{jn}N^{2}_n}{r_n} |\log (C N_n)+1|+\frac{CN^{4}_n}{r_n} \,, \end{aligned}$$

(5.27)

and finally, using that $N_n=\sum _{{{\mathcal {P}}}_n} m_{jn}$ and $k_n \leqslant N_n$, we obtain

$$\begin{aligned} \frac{1}{N_n} \sum _{{{\mathcal {P}}}_n} \bigg |\int _{B_n} v_{r_n}\bigg |&\leqslant \frac{1}{N_n} \sum _{{{\mathcal {P}}}_n} m_{jn} \bigg (\frac{CN^{2}_n}{r_n} |\log (C N_n)+1|\bigg )+\frac{1}{N_n} \sum _{{{\mathcal {P}}}_n} \frac{CN^{4}_n}{2 r_n} \\&\leqslant \frac{CN^{2}_n}{r_n} |\log (C N_n)+1|+\frac{C N^{4}_n}{r_n} \end{aligned}$$

which goes to zero as $n\rightarrow \infty$ by the way the sequence of $r_n$ was constructed, cf. Equation (5.14). This completes the proof of the lemma.

6 Energy Growth and Ergodicity

In this final section we include a simple observation on the limiting invariant measures that one obtains when the energy growth of the sequence of solutions to the modified Seiberg–Witten equations is linear. By this we mean that there exists a positive constant $C>0$, independent of n, such that

$$\begin{aligned} C^{-1} r_n\leqslant {{\mathcal {E}}}_n\leqslant C r_n. \end{aligned}$$

Theorem 6.1

Let $(r_{n},\,\psi _{n},\,A_{n})_{n=1}^\infty$ be a sequence of solutions to the modified Seiberg–Witten equations as in Theorem 1.3. If the energy sequence ${{\mathcal {E}}}_n$ has linear growth, then the vector field X cannot be ergodic (with respect to the Lebesgue measure).

Proof

By Eq. (2.6), the signed measures $\sigma _n$ can be written as

$$\begin{aligned} \sigma _{n}(U)=\frac{r_n\int _{U}(1-|\alpha _n|^2) \mu }{{{\mathcal {E}}}_n}+O({{\mathcal {E}}}^{-1}_n) \end{aligned}$$

for any domain $U \subset M$. Accordingly, if the energy growth is linear, we obtain

$$\begin{aligned} \sigma _{n}(U) \leqslant C \int _{U}|1-|\alpha _n|^2| \mu + O(r_{n}^{-1}) \leqslant C\mu (U), \end{aligned}$$

where we have used that $|\alpha _n|$ is uniformly bounded. Taking the limit $n\rightarrow \infty$, this implies that $\sigma _{\infty }(U)=0$ whenever $\mu (U)=0$. In other words, $\sigma _{\infty }$ is absolutely continuous with respect to $\mu$.

Then, it is well known that we can write $\sigma _{\infty }=f \mu$, where $f \in L^1(M)$ is the Radon-Nikodym derivative of $\sigma _{\infty }$ with respect to $\mu$. Since both $\sigma _{\infty }$ and $\mu$ are invariant measures, f can be understood as an $L^1$ function that is invariant under the flow of X. Therefore, if X is ergodic, the ergodic theorem implies that f is constant, i.e., $f=\int _Mf\mu =1$, at almost every point of M.

However, the main observation is that f cannot be a.e. constant, because item (ii) in Theorem 1.3 ensures that, for any 1-form $\gamma$ such that $d\gamma =i_{X} \mu$:

$$\begin{aligned} \int _M *(\gamma \wedge d \gamma )f\mu =\sigma _{\infty }(*(\gamma \wedge d \gamma )) \leqslant 0, \end{aligned}$$

while $\int _M*(\gamma \wedge d \gamma )\mu ={{\mathcal {H}}}(X)>0$ by hypothesis. This contradiction shows that X cannot be ergodic. $\square$

Appendix A: The Seiberg–Witten Invariant Measures via Foliation Cycles

In this appendix, which is of independent interest, we revisit Taubes’s Theorem 1.3 from the viewpoint of Sullivan’s theory of foliation cycles, and refine it showing the property stated in Remark 1.4. Let us first recall some concepts.

1.1 A.1: Preliminaries

We denote by ${\mathcal {Z}}^{p}$ the space of p-currents on M, i.e., the continuous dual of the space of smooth p-forms. Notice that any 2-form $\Theta$ on M can be identified with a 1-current (denoted in the same way) as follows: for any 1-form $\theta$, the action of $\Theta$ on $\theta$ is given by $\int _M \Theta \wedge \theta$. Let ${\mathcal {Z}}_{X}$ and ${\mathcal {C}}_X$ be the set of foliation currents and of foliation cycles of the vector field X, respectively. We recall (see e.g. [5]) that a foliation current of a vector field X is a 1-current that can be approximated arbitrarily well (in the weak topology) by 1-currents supported on segments of orbits of the vector field. Equivalently, a foliation current can be approximated by 1-currents of the form

$$\begin{aligned} \sum _{i=1}^N c _i\, \delta ^{p_i}_X, \end{aligned}$$

with $N \in {\mathbb {N}}$, $c_i \in [0, \infty )$ and $p_i \in M$, and where for any $p\in M$ the 1-current $\delta ^{p}_X$ is defined as

$$\begin{aligned} \delta ^{p}_X(\theta )=\theta _{p}(X) \hbox { for any 1-form}\ \theta . \end{aligned}$$

A foliation cycle is a closed foliation current, i.e., a foliation current whose kernel contains the linear subspace of exact 1-forms. It is straightforward to see that foliation cycles are in one to one correspondence with invariant measures of the vector field X.

Following [6], associated to the vector field X, we also define the subset ${{\mathcal {F}}}_X$ of the space of 1-currents, consisting of the boundaries of zero-flux surfaces, i.e.,

$$\begin{aligned} {{\mathcal {F}}}_X =\Big \{\partial S:\, S \, \text { is a surface with } \int _{S} i_X \mu =0\Big \}. \end{aligned}$$

Notice that a 1-current c being in $\overline{{{\mathcal {F}}}_{X}}$ (where the closure is taken with respect to the weak topology in the space of 1-currents) is equivalent to the fact that, for any $\gamma$ such that $d \gamma =i_{X} \mu$

$$\begin{aligned} c(\gamma )=0. \end{aligned}$$

In particular, if the 1-current c is a smooth 2-form, this means that there is a 1-form b with $d b=c$ and

$$\begin{aligned} \int _M b \wedge i_{X} \mu =0. \end{aligned}$$

Before stating the main theorem of this appendix, following the same notation as in [11, Section 2] we introduce some functionals depending on solutions $(r, \psi _r, A_r)$ to the modified Seiberg–Witten equations. We first define the Chern–Simons functional

$$\begin{aligned} {\mathfrak {c}}{\mathfrak {s}}:=\int _{M} a_r \wedge da_r, \end{aligned}$$

where $a_r:=A_r-A_1$ is a 1-form and $A_1$ is a fixed connection on the bundle E satisfying

$$\begin{aligned} 2 F_{A_1}+F_{A_0}=0. \end{aligned}$$

Recall that the determinant bundle of $E \oplus K^{-1}E$ has torsion first Chern class, so $[2F_{A_1}+F_{A_0}]=0$, and such an $A_1$ always exists. We also define

$$\begin{aligned} \mathfrak {e}:=\int _{M} \gamma \wedge da_r, \end{aligned}$$

where $\gamma$ is a 1-form so that $d\gamma =i_X \mu$, and

$$\begin{aligned} \mathfrak {e}_{\nu }:=\int _{M} (\Gamma +\varpi ') \wedge da_r, \end{aligned}$$

where $\Gamma$ and $\varpi '$ are the perturbing 1-forms introduced in Eq. (2.5) and Theorem 2.2, respectively. Finally, we define the action functional

$$\begin{aligned} \mathfrak {a}:=\frac{1}{2} {\mathfrak {c}}{\mathfrak {s}}-r\mathfrak {e}-\mathfrak {e}_{\nu }+r \int _{M} \psi ^{\dagger }_{r} D_{A_r} \psi _r. \end{aligned}$$

One can check that for each fixed r, the solutions $(\psi _r,A_r)$ to the Seiberg–Witten equations are critical points of the functional $\mathfrak {a}$.

Finally, we state a more detailed version of Taubes’s Existence Theorem 2.2 (see [11, Proposition 4.1]), which ensures that, in fact, the sequence of solutions $(r_n, \psi _n, A_n)$ comes from a piecewise smooth 1-parameter family of solutions $(r, \psi _r, A_r)$:

Proposition A.1

(Taubes [11]) Let $(r_n, \psi _n,A_n)_{n=0}^{\infty }$ be a sequence of solutions to the Seiberg–Witten equations provided by Theorem 2.2. There is an increasing sequence $\{\rho _k\}_{k=1}^{\infty } \subset [1, \infty )$ with no accumulation points so that the following holds:

• (i) For each k, there is a smooth family of solutions $(r, {\tilde{\psi }}_r,{\tilde{A}}_r)$ to the Seiberg–Witten equations parametrized by $r \in (\rho _k, \rho _{k+1})$.

• (ii) The associated functions $\mathfrak {a}(r)$, ${\mathfrak {c}}{\mathfrak {s}}(r)$, $\mathfrak {e}(r)$, $\mathfrak {e}_{\nu }(r)$ and ${{\mathcal {E}}}_r$ defined by this family are smooth on the intervals $(\rho _k, \rho _{k+1})$. Moreover, there is a continuous function $\mathfrak {a}_{0}: [1, \infty ) \rightarrow {\mathbb {R}}$ such that for any $r \in [1, \infty ){\setminus } \{\rho _k\}_{k=1}^{\infty }$, $\mathfrak {a}_{0}(r)=\mathfrak {a}(r)$.

• (iii) The sequence $\{r_n\}_{n=0}^{\infty }$ is contained in the set $[1, \infty )\setminus \{\rho _k\}_{k=1}^{\infty }$, and moreover $(\psi _n, A_n)=({\tilde{\psi }}_{r_n}, {\tilde{A}}_{r_n})$ with energy ${{\mathcal {E}}}_n\equiv {{\mathcal {E}}}_{r_n}$.

1.2 A.2: Main Theorem

Theorem A.2

Suppose that the helicity of the vector field X is positive. Let $(r_{n},\,\psi _n,\,A_n)_{n=1}^\infty$ be a sequence of solutions to the modified Seiberg–Witten equations as in Proposition A.1. Assume that the sequence of energies ${{\mathcal {E}}}_n$ is not bounded (i.e., $\liminf _{n\rightarrow \infty } {{\mathcal {E}}}_n=\infty$). Then the sequence of 2-forms

$$\begin{aligned} \Sigma _n:=\frac{F_{A_n}}{{{\mathcal {E}}}_n} \end{aligned}$$

converges, possibly after passing to a subsequence, to a foliation cycle $\Sigma _{\infty }$ of X. Moreover, if the one-parameter family of energies ${{\mathcal {E}}}_r$ satisfies the estimate

$$\begin{aligned} C_1r^\theta \leqslant {{\mathcal {E}}}_r\leqslant C_2r^\theta , \end{aligned}$$

(A.1)

for some $\theta \in (0,1)$, some positive constants $C_1,C_2$ and all r large enough, we have

$$\begin{aligned} \Sigma _{\infty } \in \overline{{{\mathcal {F}}}_{X}} \cap {{\mathcal {C}}}_{X}. \end{aligned}$$

We remark that the 1-current $\Sigma _{\infty }$ obtained in this theorem is related to the invariant measure $\sigma _{\infty }$ of Taubes’s Theorem 1.3 in the following way: for any 1-form $\theta$ on M, we have

$$\begin{aligned} \Sigma _{\infty }(\theta )=\sigma _{\infty }(\theta (X)). \end{aligned}$$

However, the proof we give below is different from Taubes’s proof of the existence of the invariant measure $\sigma _\infty$. Additionally, we can interpret the property that $\Sigma _{\infty } \in \overline{{{\mathcal {F}}}_{X}}$ as follows: for any 1-form $\gamma$ satisfying $d \gamma =i_X \mu$, define the function $h_{\gamma }:=*(\gamma \wedge \, i_{X} \mu )=\gamma (X)$, which is precisely the density of the helicity functional. Then, as argued in the previous section,

$$\begin{aligned} 0=\Sigma _\infty (\gamma )=\sigma _{\infty }(h_{\gamma }), \end{aligned}$$

which is the refinement stated in Remark 1.4.

1.3 A.3: Proof of Theorem A.2

We divide the proof in three steps. First, we show that $\Sigma _{n}$ has a subsequence that converges to some non-trivial closed 1-current $\Sigma _{\infty }$ (this is straightforward). Then, we prove that $\Sigma _{\infty }$ is a foliation cycle of X. Finally, we establish that $\Sigma _{\infty }(\gamma )=0$ for $d \gamma =i_{X} \mu$, so that $\Sigma _{\infty }\in \overline{{{\mathcal {F}}}_{X}}\cap {{\mathcal {C}}}_X$.

1.3.1 Step 1: $\Sigma _{n} \rightarrow \Sigma _{\infty }$ and $\partial \Sigma _{\infty }=0$

For any 1-form $\theta$ it is straightforward to see that

$$\begin{aligned} |\Sigma _{n}(\theta )|=|\sigma _n(\theta (X))| \leqslant C \Vert \theta \Vert _{L^{\infty }(M)}, \end{aligned}$$

(A.2)

which implies that the sequence $\Sigma _{n}$ is bounded in the weak topology. Since in the space of 1-currents (with the weak topology) bounded subsets are precompact, there is a convergent subsequence. We denote by $\Sigma _{\infty }$ the limiting 1-current. It is obvious that $\Sigma _{\infty }$ is not trivial (the zero current) because $\Sigma _{\infty }(\lambda )=1$ (recall that $\lambda$ is the 1-form dual to X, so $\lambda (X)=1$).

We observe that the boundary operator $\partial : {\mathcal {Z}}^{1}(M) \rightarrow {\mathcal {Z}}^{0}(M)$ in the space of 1-currents is defined by duality as

$$\begin{aligned} \partial c(f):=c(df), \end{aligned}$$

and is continuous in the weak topology. Since the curvatures $F_{A_n}$ are closed 2-forms, we infer that for any $n \in {\mathbb {N}}$ and any smooth function f

$$\begin{aligned} \Sigma _{n}(df)=0, \end{aligned}$$

thus implying that $\Sigma _{\infty }(df)=0$, i.e., $\Sigma _{\infty }$ is a closed 1-current.

1.3.2 Step 2: $\Sigma _{\infty } \in {{\mathcal {C}}}_{X}$

We proceed by contradiction. As is well known, the space of foliation currents ${\mathcal {Z}}_{X}$ is a closed convex cone with compact convex base inside the space of 1-currents. Suppose that $\Sigma _{\infty } \notin {\mathcal {Z}}_{X}$. Then, by a standard application of the Hahn–Banach theorem, there is a hyperplane separating $\Sigma _{\infty }$ and ${\mathcal {Z}}_{X}$; in other words, there is a continuous linear functional ${{\mathcal {L}}}: {\mathcal {Z}}^{1} \rightarrow {\mathbb {R}}$ satisfying ${{\mathcal {L}}}(c) \geqslant 0$ for any $c \in {\mathcal {Z}}_{X}$ and ${{\mathcal {L}}}(\Sigma _{\infty }) <0$.

Since the space of 1-currents and the space of smooth 1-forms are continuous duals of each other, we can identify the functional ${{\mathcal {L}}}$ with a 1-form $\theta$ satisfying

$$\begin{aligned} \theta _p(X) \geqslant 0 \text { at any point } p \in M, \end{aligned}$$

and

$$\begin{aligned} \Sigma _{\infty } (\theta ) <0. \end{aligned}$$

Thus, to prove that $\Sigma _{\infty }$ is a foliation current it suffices to check that for any $\theta$ with $\theta _p(X)\geqslant 0$, we must have $\Sigma _{\infty }(\theta ) \geqslant 0$. Indeed, let $\theta$ be any such 1-form, then

$$\begin{aligned} \Sigma _n (\theta )= & {} \frac{\int _M F_{A_n} \wedge \theta }{{{\mathcal {E}}}_n}\\= & {} \frac{r_n \int _M (1-|\alpha _n|^2+|\beta _n|^2) \theta (X) \mu }{{{\mathcal {E}}}_n}\\{} & {} + \frac{\int _M (r_n\psi _n^{\dagger } \sigma ^{\perp } \psi _n + \varpi ) \wedge \theta }{{{\mathcal {E}}}_n}. \end{aligned}$$

Here we have used the notation $\psi ^{\dagger } \sigma ^{\perp } \psi := *(\psi ^{\dagger } \sigma \psi )- (\psi ^{\dagger } \sigma (X) \psi ) i_{X} \mu$. Now, taking an upper bound of the last term in the above equation, we can write

$$\begin{aligned} \Sigma _n (\theta ) \geqslant \frac{r_n \int _M (1-|\alpha _n|^2+|\beta _n|^2) \theta (X) \mu }{{{\mathcal {E}}}_n}- \Vert \theta \Vert _{L^{\infty }(M)} \frac{\int _M (r_n|\alpha _n| |\beta _n|+C)\mu }{{{\mathcal {E}}}_n}. \end{aligned}$$

Using the assumption $\theta (X)\geqslant 0$ on M and the fact that $r_n(1-|\alpha _n|^2) \geqslant -C$ by Lemma 2.3, we obtain

$$\begin{aligned} \Sigma _n (\theta )\geqslant -\frac{\Vert \theta \Vert _{L^\infty (M)}}{{{\mathcal {E}}}_n}\big (C + r_n \int _M |\alpha _n| |\beta _n|\mu \big ). \end{aligned}$$

Now, applying Lemma 2.3 again and using Eq. (2.6), we can bound the second summand in the above inequality as

$$\begin{aligned} r_n \int _{M} |\alpha _n| |\beta _n| \mu \leqslant C \sqrt{r_n} \int _M |1- |\alpha _n|^2|^{\frac{1}{2}} \mu \leqslant C\Big (r_n \int _M |1- |\alpha _n|^2|\mu \Big )^{\frac{1}{2}}\leqslant C{{\mathcal {E}}}_n^{\frac{1}{2}}. \end{aligned}$$

We then conclude that

$$\begin{aligned} \Sigma _n(\theta )\geqslant -\frac{\Vert \theta \Vert _{L^\infty (M)}}{{{\mathcal {E}}}_n}\big (C + {{\mathcal {E}}}_n^{\frac{1}{2}}\big )\rightarrow 0 \end{aligned}$$

as $n\rightarrow \infty$ because ${{\mathcal {E}}}_n$ is assumed to be an unbounded sequence, and therefore $\Sigma _\infty (\theta )\geqslant 0$. As we argued before, this implies that $\Sigma _\infty$ is a foliation current and being closed (by Step 1) we deduce that it is a foliation cycle, as we wanted to show.

1.3.3 Step 3: $\Sigma _{\infty } \in \overline{{{\mathcal {F}}}_X}$

Let $(r, \tilde{\psi }_r,{{\tilde{A}}}_r)$ be the 1-parameter family of solutions, whose existence is ensured by Proposition A.1, which coincides with $(r_n, \psi _n,A_n)$ at $r=r_n$. Observe that, since ${{\mathcal {E}}}_n$ is unbounded, we can write

$$\begin{aligned} \Sigma _{\infty }(\gamma )= \lim _{n \rightarrow \infty } \frac{\mathfrak {e}(r_n)}{{{\mathcal {E}}}_n}, \end{aligned}$$

(A.3)

so $\Sigma _{\infty }\in \overline{{{\mathcal {F}}}_{X}}$ if and only if

$$\begin{aligned} \lim _{n \rightarrow \infty } \frac{\mathfrak {e}(r_n)}{{{\mathcal {E}}}_n}=0. \end{aligned}$$

Since $\Sigma _{\infty }(\gamma ) \leqslant 0$ by Taubes’s Theorem 1.3, it is enough to prove that there is no constant $C>0$ such that

$$\begin{aligned} -\mathfrak {e}(r_n) \geqslant C {{\mathcal {E}}}_n. \end{aligned}$$

(A.4)

The following lemma is key in what follows.

Lemma A.3

For r large enough, there is a constant C independent of r such that

$$\begin{aligned} |{\mathfrak {c}}{\mathfrak {s}}(r)|\leqslant & {} C r^{\frac{2}{3}} {{\mathcal {E}}}_{r}^{\frac{4}{3}}\\ |\mathfrak {e}_{\nu }(r)|\leqslant & {} C {{\mathcal {E}}}_{r} \end{aligned}$$

Proof

The first bound on the ${\mathfrak {c}}{\mathfrak {s}}$ functional follows from the proof of [11, Lemma 4.3] and the assumption that ${{\mathcal {E}}}_r$ is unbounded. As for the second bound, we notice that

$$\begin{aligned} \frac{|\mathfrak {e}_{\nu }(r)|}{{{\mathcal {E}}}_r}=|\Sigma _{r}(\Gamma +\varpi ')|+O({{\mathcal {E}}}^{-1}_r) \end{aligned}$$

and clearly $|\Sigma _{r}(\Gamma +\varpi ')|\leqslant C$ for some constant C. $\square$

To show that there is no constant $C>0$ for which Eq. (A.4) holds, let us assume the contrary. Set $\mathfrak {b}(r):={-2\mathfrak {a}_{0}(r)}/{r}$. By Proposition A.1, the functions $\mathfrak {b}(r)$ and $\mathfrak {a}_{0}(r)$ are differentiable on the intervals $I_k:=(\rho _k, \rho _{k+1})$. Moreover, it is straightforward to check that they verify the relation (cf. [11, Section 4]):

$$\begin{aligned} \frac{d\mathfrak {b}}{dr} =\frac{{\mathfrak {c}}{\mathfrak {s}}}{r^2}-\frac{2 \mathfrak {e}_{\nu }}{r^2}, \end{aligned}$$

so, in view of Lemma A.3, we obtain the bound

$$\begin{aligned} \Big |\frac{d\mathfrak {b}}{dr}\Big | \leqslant C \Big (\frac{{{\mathcal {E}}}_{r}}{r}\Big )^{\frac{4}{3}} \end{aligned}$$

on $I_k$, from which we deduce that

$$\begin{aligned} |\mathfrak {b}(r)| \leqslant |\mathfrak {b}(r_0)|+\int _{r_0}^{r} \Big (\frac{{{\mathcal {E}}}_{\rho }}{\rho }\Big )^{\frac{4}{3}} d \rho \end{aligned}$$

and hence

$$\begin{aligned} |\mathfrak {a}_{0}(r)| \leqslant \frac{1}{2}r|\mathfrak {b}(r_0)|+\frac{1}{2}r\int _{r_0}^{r} \Big (\frac{{{\mathcal {E}}}_{\rho }}{\rho }\Big )^{\frac{4}{3}} d \rho . \end{aligned}$$

(A.5)

Since $\mathfrak {a}_0$ is a continuous function, the inequality (A.5) holds for all $r\geqslant r_0$ with $r,r_0\in [1,\infty )$.

Next, the definition of the functional $\mathfrak {a}(r)$ (notice that $D_{A_r}\psi _r=0$ for solutions of the Seiberg–Witten equations), Lemma A.3, and our assumption that $-\mathfrak {e}(r_n) \geqslant C {{\mathcal {E}}}_n$, allow us to write the bound

$$\begin{aligned} |\mathfrak {a}_{0}(r_n)| \geqslant C \bigg (r_n {{\mathcal {E}}}_n-{{\mathcal {E}}}_n-r_n^{\frac{2}{3}} {{\mathcal {E}}}_{n}^{\frac{4}{3}}\bigg ). \end{aligned}$$

Since we are assuming that the energy growth is bounded as ${{\mathcal {E}}}_n\leqslant Cr_n^\theta$ for some $\theta <1$, we easily infer that $\limsup _{n\rightarrow \infty } r_{n}^{\frac{2}{3}} {{\mathcal {E}}}^{\frac{4}{3}}_{n}(r_n {{\mathcal {E}}}_n)^{-1}=0$, and therefore, for large enough $r_n$, the previous bound implies

$$\begin{aligned} |\mathfrak {a}_{0}(r_n)| \geqslant C r_n {{\mathcal {E}}}_n. \end{aligned}$$

(A.6)

Accordingly, Eqs. (A.5) and (A.6) hold simultaneously, if

$$\begin{aligned} {{\mathcal {E}}}_n \leqslant C \int _{r_0}^{r_n} \Big (\frac{{{\mathcal {E}}}_{\rho }}{\rho }\Big )^{\frac{4}{3}} d \rho +C \end{aligned}$$

(A.7)

as $n \rightarrow \infty$. Finally, combining this estimate with the Assumption (A.1), we derive that

$$\begin{aligned} r_n^\theta \leqslant C(1+r_n^{\frac{4\theta -1}{3}}) \end{aligned}$$

provided that $\theta \ne \frac{1}{4}$, and

$$\begin{aligned} r_n^{\frac{1}{4}}\leqslant C\ln r_n \end{aligned}$$

when $\theta =\frac{1}{4}$, for all n large enough. This yields a contradiction with the fact that $\theta \in (0,1)$. We then conclude that there is no constant $C>0$ for which Eq. (A.4) holds, and hence $\lim _{n\rightarrow \infty } \frac{\mathfrak {e}(r_n)}{{{\mathcal {E}}}_n}=0$, which completes the proof of the theorem.