Topological defects in the abelian Higgs model

We give a rigorous description of the dynamics of the Nielsen-Olesen vortex line. In particular, given a worldsheet of a string, we construct initial data such that the corresponding solution of the abelian Higgs model will concentrate near the evolution of the string. Moreover, the constructed solution stays close to the Nielsen-Olesen vortex solution.


Introduction
In 1973 Nielsen and Olesen [20] conjectured a relationship between the abelian Higgs model in R 1+3 and the Nambu-Goto action. In this paper we show that their conjecture follows from a conjecture of Jaffe and Taubes about the 2-dimensional Euclidean abelian Higgs model. In particular, since the Jaffe-Taubes conjecture is known to hold for a range of values of a coupling parameter appearing in the abelian Higgs model, our results show that the Nielsen-Olesen scenario holds in these situations. The abelian Higgs model (see (1.1) below) arises in various branches of physics: in highenergy physics, as perhaps the simplest Yang-Mills-Higgs theory; in solid-state physics, in connection with superconductivity; and in cosmology where, for reasons stemming from its relevance to high-energy physics, it provides a basis for studies of the possible behavior of cosmic strings, should any such objects exist.
The Nambu-Goto action of a (1 + 1)-dimensional string in (1 + 3)-dimensional Minkowski space is (proportional to) the Minkowski area of its worldsheet, see (1.5) below for a precise formulation. The associated equations of motion are exactly the condition that the Minkowski mean curvature of the worldsheet vanishes. This is the simplest natural model for the relativistic dynamics of a string in Minkowski space. We refer to a solution of the equations of motion as a "timelike Minkowski minimal surface", by analogy with ordinary (Euclidean) minimal surfaces. The action is due to Nambu [18] and Goto [6], and it has its origins in the early days of string theory, as a description of the evolution of a closed (dual) string. See [5] for a nice historical perspective.
The relationship between these two models proposed in [20] is that solutions of the abelian Higgs model exhibit, for suitable initial data, features known as vortex lines that, Nielsen and Olesen argued, should sweep out worldsheets that are approximately governed by the Nambu-Goto action. This proposal has subsequently been investigated particularly intensively by cosmologists interested in possible cosmic strings, starting with work of Kibble [15]. Models for cosmic strings assume that some form of Yang-Mills-Higgs (YMH) equation, perhaps arising from some yet-unknown grand unified theory, is relevant to descriptions of the distribution of matter in the universe. One can associate to a YMH model an object called the "vacuum manifold", and it is believed that qualitative features of solutions known as topological defects are determined by the topology of the vacuum manifold. In particular, string-like defects are expected to form when the vacuum manifold has a nontrivial fundamental group. The abelian Higgs model, for which the vacuum manifold is given by S 1 , provides the simplest case of this scenario, and it is thus studied as a useful prototype for more general models whose vacuum manifold is not simply connected.
There is a large body of mathematics describing strings and other defects in solutions of elliptic and parabolic equations with vacuum manifolds that are either disconnected or non-simply-connected. References and a more detailed discussion may be found in [12].
On the other hand, there is not a great deal of rigorous mathematical work describing dynamics of topological defects in nonlinear hyperbolic equations, and most of it deals with defects that can be thought of as point particles or 0-dimensional defects, see for example [23,11,17,8]. Higher-dimensional defects are however treated in [3] and [12]. In particular, the latter work proves that topological defects in certain semilinear hyperbolic equations, including a non-gauged analog of the abelian Higgs model, do indeed approximately sweep out timelike minimal surfaces for suitable initial data. This covers the case of the domain wall, one of the basic examples of topological defects considered by cosmologists, associated to the real scalar equation u + −2 (u 2 − 1)u = 0. Cosmic strings, as in the abelian Higgs model, have a much richer mathematical structure, and are also considered more likely to be present in our universe than domain walls.
The basic scheme we use here draws on that developed in [12]. To show that this scheme works for a gauge theory such as the abelian Higgs model we must, among other things, formulate and establish suitable stability estimates, relating energy and vorticity for the 2-dimensional Euclidean abelian Higgs model, and a large part of our work is devoted to these tasks.
We next present some necessary background about the abelian Higgs model, the Nambu-Goto action, the 2d Euclidean abelian Higgs model and the Jaffe-Taubes conjecture, and normal coordinates around a string. With this done, we will finally state our main result.
1.1. The abelian Higgs model. We will write the Lagrangian for the abelian Higgs model in the form A a 1-form with components A α : R 3+1 → R, α ∈ 0, . . . , 3, D α denotes the covariant derivative D α ϕ := (∂ α − iA α )ϕ, and F := dA, so that One may regard A as a U (1) connection and F as the associated curvature. We write f, g to denote the real inner product f, g = Re(f g).
We will consider the scaling 0 < 1, which is relevant to models describing cosmic strings, where typically ∼ 10 −16 in the units we have (implicitly) chosen.
We remark that the Lagrangian (1.1) is invariant under action of the U (1) group, so for any sufficiently smooth function X : R 1+3 → R we have L(ϕ, A) = L(e iX ϕ, A + dX).
The Euler-Lagrange equations associated to the action functional R 1+3 L(ϕ, A) are Our main theorem describes the behavior of certain solutions of this system, for well-chosen initial data. H(y 0 , y 1 ) = (y 0 , h(y 0 , y 1 )) for some h : (−T, T ) × S 1 → R 3 .
Here and throughout this paper, S 1 denotes R/LZ for some L > 0, so that S 1 is a circle of arbitrary positive length L. We write Γ for the image of such a map H, and the induced metric on Γ is denoted by where we implicitly sum over repeated indices α, β = 0, . . . , 3. A surface Γ is said to be timelike if det(γ ab ) < 0 at every point in (−T, T ) × S 1 . The Nambu-Goto action is proportional to This is well-known in the physics literature, see for example [24,Section 6.2], and is proved in [2]. We will always assume H is a smooth timelike 1 embedding on (−T, T ) × S 1 and that (1.4), (1.6) hold. With a conformal parametrization, (1.6), h(y 0 , y 1 ) may be written in the form 1 2 (a(y 0 + y 1 ) + b(y 0 − y 1 )) for functions a, b : S 1 → R 3 such that |a | = |b | = 1, and conversely every map of this form parametrizes a minimal surface. In particular, if h 0 : S 1 → R 3 is an arclength parametrization of a smooth embedded curve Γ 0 = image(h 0 ), then the timelike minimal surface that agrees with Γ 0 at time t = 0 and with zero initial velocity can be written in the form (1.4), with This is the situation that we will always consider, although we will rarely need the explicit formula (1.7).

1.3.
The Euclidean abelian Higgs model in 2 dimensions. We will write the 2d abelian Higgs energy density in the form . Note that is just a scaling parameter in (1.9), and one can easily change variables to set = 1. That is, given a configuration However, we find it convenient to include the scaling parameter in the energy. We define the (2-dimensional) current j(U ) and vorticity ω(U ), given by As we will recall in slightly more detail in Section 2, if U is a finite-energy configuration then ω(U ) ∈ L 1 (R 2 ), and moreover for every finite energy U.
It follows that every finite-energy U belongs to exactly one of the sets (1.14) (These sets are called weak homotopy classes by Rivière [21], who establishes a slightly different description of them.) Note also that, while R 2 e ν ,λ (U ) certainly depends on λ, the condition R 2 e ν ,λ (U ) < ∞ is independent of λ, and hence the homotopy classes H n are also independent of and λ. We will use the notation Our main results describe solutions of the 1 + 3-dimensional abelian Higgs model in terms of solutions, when they exist, of the 2d minimization problem: For 0 < 1, the regime that interests us, we always assume that a minimizer U m ,λ is obtained by starting from a fixed minimizer of the = 1 problem and scaling as in (1.10), so that the energy and vorticity concentrate near the origin.
Remark 1.1. For every n ∈ Z and λ > 0, there exists an equivariant U (n) ∈ H n solving the Euler-Lagrange equations associated to the minimization problem (1.16), see [4]. Here "equivariant" implies for example that φ (n) can be written in the form f (r)e inθ . The equivariant solution is known to be linearly stable if |n| = 1 or λ ≤ 1, and linearly unstable (and hence not even a local energy minimizer) if λ > 1 and |n| ≥ 2, see Gustafson and Sigal [7].
Remark 1.2. The conjecture of Jaffe and Taubes [9, Chapter III.1, Conjectures 1 and 2] mentioned earlier holds that the equivariant solution solves problem (1.16) for all parameter values for which it is linearly stable, and that no minimizer exists whenever the equivariant solution is linearly unstable. This is known to be true in the case λ = 1, which has a special structure that will be recalled in Section 2, and for all sufficiently large λ, due to work of Rivière [21]. Otherwise it is open, as far as we know. Remark 1.3. In Theorem 4.1 we establish a general sufficient condition, involving the behavior of the map m → E λ m , for existence of solutions of problem (1.16). In particular we deduce from this that a minimizer exists for |n| = 1 and 1 5 < λ < 5.  [21], and we note in Lemma 2.1 that it is easily verified for 1 2 ≤ λ ≤ 2. It is expected that (1.17) holds for all λ > 0, and more generally that n → E λ n is increasing for n ∈ N. (It is easy to check that E λ −n = E λ n for all n.) A statement similar to (1.17) is proved for certain non-gauged generalized Ginzburg-Landau-type models by Almog et al in [1], but adapting their arguments to the gauged case seems not to be easy.
1.4. Normal coordinates. Next we describe a useful coordinate system, which we will refer to as normal coordinates, for a neighbourhood of a minimal surface Γ. A key point in our analysis (as in [12]) will be to obtain estimates in these coordinates.
We will arrange that y 0 is a timelike coordinate and y 1 , . . . , y 3 spacelike.
Writing B ν (ρ) := {y ν = (y ν1 , y ν2 ) : |y ν | < ρ}, we will restrict ψ to a set of the form (−T 1 , T 1 ) × S 1 × B ν (ρ 0 ) on which ψ is injective and satisfies other useful properties; see Section 5 for details. Since our argument will rely heavily on specific properties of the abelian Higgs model when written with respect to the new coordinates, we find it useful to distinguish between the Higgs field and connection when written in terms of the original, standard coordinates for Minkowski spacetime, which we will write (ϕ, A), and the same objects written in terms of the new coordinates, which we will denote (φ, A). These are related by . The components of the curvature in the two coordinate systems will be denoted F αβ and F αβ respectively. We will also write U to denote a pair (ϕ, A) and similarly U for a pair (φ, A), and we will write (1.23) U = ψ * U when U and U are related as in (1.22).
1.5. main theorem. Our main result, stated below, asserts the existence of a solution whose energy concentrates around a minimal surface Γ, and that in a neighborhood of Γ is close to a configuration that in the y coordinates takes the form U NO = (φ NO , A NO ), with ,λ is a ground state of the 2d minimization problem (1.16). Thus, in standard coordinates this configuration can be written Note that U NO is only defined in the domain of ψ −1 , which is a neighborhood of Γ.
Theorem 1.4. Let Γ be a codimension 2 timelike minimal surface, given as the image of a conformal parametrization H (so that H satisfies (1.4), (1.6)) that is a smooth embedding in (−T, T ) × S 1 . Assume also that the initial velocity of Γ at t = 0 is everywhere 0.
Let λ > 0 and m ∈ Z be such that the 2d minimization problem (1.16) has a solution, and in addition assume that E λ m ≤ E λ n whenever |m| ≤ |n|. Then, given T 0 < T , there exists an neighborhood N 1 ⊂ Image(ψ) of Γ in (−T 0 , T 0 ) × R 3 , and a constant C, both independent of , such that given ∈ (0, 1], there exists a solution U of the abelian Higgs model (1.2), (1.3) satisfying the following estimates. First, in a suitable gauge, for U NO defined in (1.25). Second, where d ν : N 1 → R is the distance in normal coordinates to Γ, so that d ν • ψ(y) := |y ν |. And finally, Remark 1.5. From the construction of the initial data and conservation of energy, we will have {t}×R 3 |Dϕ| 2 + 2 |F| 2 ≥ C > 0 for every t. Thus (1.28), (1.27) contain highly nontrivial information about energy concentration around Γ = {d ν = 0}.
Remark 1.6. In fact we prove a more general stability result, giving estimates for any solution that at t = 0 has a vortex filament near Γ 0 and satisfies certain smallness conditions. For details see Proposition 6.1, from which one can also extract further estimates satisfied by the particular solution U of Theorem 1.4.
• any λ > 0, and m minimizing n → E λ n among nonzero integers. This again follows from Theorem 4.1. They are believed to hold for all λ > 0 when |m| = 1, and for all m ∈ Z when λ ∈ (0, 1). (See the next remark and remarks 1.2 and 1.3) . Remark 1.9. The estimates obtained in [12,Theorem 2] for the non-gauged analog of the abelian Higgs model are similar to (1.27), (1.28) but much weaker. Indeed, they show that the total energy diverges like | ln |, whereas the weighted energy (as in (1.27), (1.28)) is bounded as → 0. Thus energy concentrates very weakly around the manifold Γ. In addition, no useful estimate along the lines of (1.26) is obtained in [12].
Remark 1.10. The assumption on Γ to have zero initial velocity is there to avoid some technicalities of [12]. We also use it in the construction of the initial data. In principle, one could repeat the steps and assume nonzero velocity.
1.6. global well-posedness for the abelian Higgs model. The abelian Higgs model is a U (1) version of the Yang-Mills-Higgs (YMH), where the gauge group is in general nonabelian. For the purposes of this article, we need the 1 + 3 dimensional abelian Higgs model to be well posed for H 1 loc × H 1 loc data (this is made precise in Section 1.7 below). In addition, since we are going to rescale, we need the well-posedness for large data and for all time (we would like the analysis to hold at least for the existence of the time-like minimal surface). Finally, we are interested in the topological behavior at infinity, |ϕ| → 1 instead of having |ϕ| → 0.
The strongest well-posedness result in the literature for YMH is due to Keel [14]. It shows global well-posedness of the 1 + 3 solution in the energy class for any size data in the temporal gauge. Moreover, the Higgs potential is taken to be energy critical, V (ϕ) = |ϕ| p with p = 6; the power six is the highest power that can be controlled using the Sobolev embedding by the kinetic part of the energy. A power p < 6 is called subcritical, and is significantly easier to handle. Since we have a quartic potential, the global well-posedness for the abelian Higgs model we need, in the temporal gauge, is implied by [14]. The only detail left is then addressing |φ| → 1 (see Section 1.7 below).
On the other hand, the proof in [14] is more sophisticated than what we need, and not only because of the critical power of the potential. An intermediate step leading to the global result is changing to a Coulomb gauge. In the nonabelian case, the Coulomb gauge can be constructed only locally in space, and hence the nonabelian case is much more technical than the abelian one, where the global Coulomb gauge can be constructed. Therefore, due to having subcritical potential and the abelian problem, heuristically speaking, the global well-posedness we need can also follow from the work done on the Maxwell-Klein-Gordon problem [16,22]. 1.7. initial data. The solution U that we find in Theorem 1.4 will be obtained by invoking the results in [14,Theorem 1.2]. To do this we will impose the temporal gauge and we require initial data such that with the compatibility condition (stated in the temporal gauge) 3, which will be the case for us. The initial data U| t=0 is carefully constructed in the proof of Theorem 1.4 in Section 7. It has a rather explicit description near Γ ∩ {t = 0}, in terms of the minimizer U m from (1.16) and the diffeomorphism ψ from (1.21), and away from Γ ∩ {t = 0} it has the form for some smooth q. From these facts it follows that (1.30) holds, and from [14] we then obtain a global solution in the temporal gauge.
We note that in particular we will consider data such that (1.33) holds in R 3 \ B R for some R. By finite propagation speed and an easy explicit calculation, ϕ(t) = e iq , A(t) = dq is a solution on |x| > R + t. By uniqueness, it must agree with the solution obtained using [14].
1.8. some notation. As mentioned above, we implicitly sum over repeated upper and lower indices. We use the convention that greek indices α, β, µ, ν... run from 0 to 3, and latin indices i, j, k.... run from 1 to 3.
For the convenience of the reader, we include the following summary of the different solutions we work with Sections 5 and 6 consider the abelian Higgs model in 1 + 3-dimensional Minkowski space. A basic ingredient in our analysis, as in [12], is supplied by weighted energy estimates in the normal coordinate system, introduced in Section 1.4. These estimates are proved in Section 5, using results about the vorticity confinement functional from Section 3. Finally, section 6 is devoted to the proof of Theorem 1.4.

Energy and vorticity in 2 dimensions
In the next three sections, we focus on Euclidean abelian Higgs model in 2 dimensions. In this section we record some facts, mostly well-known, relating the energy e ν ,λ and the vorticity ω, defined in (1.9) and (1.12) respectively. We recall that the parameter is just a scaling parameter, see (1.10), so that all results in this section reduce to the case = 1. However, due to the role it plays elsewhere in this paper, it seems useful to formulate things here for general > 0.
Remark 2.2. With a little more work one can prove by similar arguments that (2.4) holds for a slightly larger range of λ, but these sorts of simple arguments have no hope of proving the natural conjecture, which is that it is valid for all λ > 0.
We also remark that it is known from [21] that if λ is sufficiently large then (2.4) is true for all m and n.
We conclude this section by proving the lemma mentioned above, which shows that the vorticity is approximately quantized on a set on which the boundary energy is not too large. For this we need Lemma 2.3. There exists constant C such that if S ⊂ R 2 is a bounded, connected, and simply connected set, and ∂S is Lipschitz with |∂S| ≥ , and if ρ is a smooth nonnegative function on a neighborhood of ∂S, then where ∇ τ denotes the tangential derivative along ∂S.
This is proved in [10, lemma 2.3] when S is a ball, and exactly the same argument applies here, since the proof only involves integrating along ∂S, which is isometric to a circle. Lemma 2.4. Assume that S ⊂ R 2 is connected and simply connected with Lipschitz boundary. Let λ > 0. There exists a constant C, depending on λ, such that if |∂S| ≥ then Proof. Case 1: If inf ∂S |φ| < 1 2 , then since |∇ τ |φ|| ≤ |∇ A φ|, we can apply Lemma 2.3 to ρ = |φ| to find that .
Since min n∈Z |a − πn| ≤ π 2 for every a ∈ R, this implies (2.5). Case 2: We assume that Note in particular that this occurs if ∂S e ν ,λ (U ) ≤ 1 C , due to Lemma 2.3. Because of (2.6) we can then write φ = ρe iη on ∂S, and in this notation, And the conclusion now follows by noting that and, recalling (2.6),

2d Vorticity confinement functional
Let m be a positive integer. For a configuration where of course C depends on R. We expect D ν m (U ; R) to be small (or negative) if (at least) m quanta of vorticity are concentrated near the center of the ball B ν (R).
The main results of this section are the following two propositions, both of which relate e ν ,λ and D ν m . They together yield stability properties that are used in a crucial way in our proof of Theorem 1.4.
Our first proposition will allow us to control changes in D ν m . In its statement and proof, we write points in (0, T ) × B ν (R) in the form (y 0 , y ν ).
We will in fact prove something stronger than (3.3), but here we have recorded only the conclusion that is needed for the proof of Theorem 1.4.
The other main result of the section shows that control over D ν m implies good lower energy bounds.
Then for every R > 0, there exist constants κ 1 and C, depending on R, λ, m, such that if We remark that, although one could extract from our arguments estimates of how various constants depend on λ, we have not made any effort to optimize this dependence, and indeed we appeal several times to (2.2), which is far from sharp when 0 < λ 1 or λ 1.
3.1. Proof of Proposition 3.1. We will use the following lemma Lemma 3.3. Assume the hypotheses of Proposition 3.1. Then for every r ∈ (0, R) and t ∈ (0, T ) where H 2 is 2-dimensional Hausdorff measure and e 3d ,λ (U ) denotes the 3-dimensional energy density Note that although y 0 is naturally identified with time, we think of and write e 3d ,λ as 3-dimensional rather than 1 + 2 dimensional, since 1 2 2 a=0 |D a φ| 2 is a sort of Euclidean (rather than Minkowski) norm-squared of the covariant derivative.
Proof. For this proof only, we will write ω to denote the 3-dimensional vorticity, which we identify with the 2-form d Breaking ∂((0, t) × B ν (r)) into pieces, we deduce that is understood to have the standard orientation for s = 0, t (rather than the orientation inherited as part of ∂((0, t) × ∂B ν (r)).) The left-hand side of this identity is just the left-hand side of (3.6) in slightly different notation, so it suffices to estimate the right-hand side, which can be written more explicitly as where τ 0 (y), τ 1 (y) is a properly oriented orthonormal basis for T y ((0, t) × ∂B ν (r)), and ω(U )(τ 0 , τ 1 ) at a point y denotes the number obtained by the two-form ω(U (y)) acting on the vectors τ 0 (y), τ 1 (y). Thus it suffices in fact to show that for any orthonormal vectors τ 0 , τ 1 .
To prove this, note that ω(τ 0 , τ 1 ) at a point y is just 3 the two-dimensional vorticity of the restriction of U to the (suitably oriented) plane through y spanned by τ 0 , τ 1 , and so (2.2) implies that |ω(U )(τ 0 , τ 1 )| is bounded by max{1, λ −1 } times the 2-dimensional energy e ν ,λ of the restriction of U to the same plane, and this clearly implies (3.7).
Using the above lemma, we complete the 3 This is easily verified by fixing an orthonormal basis τ0, τ1, τ2 for R 3 , and then noting that ω(τ0, τ1) = and this immediately implies (3.3).

Lower Energy Bounds.
A large part of the proof of Proposition 3.2 is contained in the following lemma.
then n(s) := deg( φ |φ| ; ∂B ν (s)) ∈ Z is well-defined, and We defer the proof of this until the end of this section. We note however that when λ = 1, a weaker version of the conclusion (with an error term of order rather than 2 ) follows immediately from Lemma 2.4 and the fact that E 1 n = π|n|. Proof of of proposition 3.2. 1. We may assume that as otherwise the conclusion of the proposition is immediate. Also, standard density arguments allow us to assume that U is smooth. Consider balls B ν (s) = B s , ≤ s ≤ R. We say that s is good if it satisfies the hypotheses of Lemma 3.4 for some C 1 > 2 R to be chosen below, so that < s < R − 1 C 1 and If s is not good, then it is said to be bad. Lemma 3.4 implies that there exists some C 2 (depending on C 1 ) such that if s is good, then Because n → E λ n is increasing for n ≥ 0 by hypothesis, see (3.4), the proposition follows if (3.14) there exists some good s such that n(s) ≥ m.
We therefore assume that (3.14) does not hold, which in view of Lemma 2.4 and the definition of good s implies that (3.15) Bs ω ≤ π(m − 1) + C for all good s, (for C depending on C 1 ). We will show that this implies the desired lower bound (3.5). Toward this end, first note that, owing to (3.12), As a result, if 0 < < 0 and 0 ≤ 1 C 1 , then 2. It follows from (3.9) and the definition ( Bs ω dy ν ds. Then we deduce from (3.15) that Bs ω dy ν ds + bad s
Rearranging and using (3.16), we find that .
Also, it follows from (2.2) that Then (3.5) follows from (3.17) for all sufficiently small , if we choose κ 1 = π 2 and C 1 ≥ R 2 such that C 1 ≥ max{1, λ −1 } g ∞ E λ m (2 + E λ m ) for example. We now prove the lemma that was used above to guarantee a nearly sharp lower energy bound for a ball bounded by a "good radius".
Proof of Lemma 3.4. We have assumed that U satisfies It follows from this and Lemma 2.3 that for small enough, 1. We first claim that there is a configurationŨ on R 2 such thatŨ = U in B ν (s), (3.20)Ũ ∈ H n , and Although our definition of H n requires thatÃ ∈ H 1 loc , it suffices to constructŨ such thatÃ ∈ L 1 loc and the distributional exterior derivative satisfies dÃ = F 12 dy 1 ∧ dy 2 , with F 12 ∈ L 2 , since any suchŨ can be approximated arbitrarily well by smooth (hence H 1 loc ) functions, via a standard mollification procedure for example.
Energy ofŨ . Since as noted above e ν ,λ (Ũ ) = 0 outside B s+ , to complete the proof of (3.20) it suffices to estimate the energy ofŨ in the annulus s < r < s + . So we henceforth restrict our attention to this set.
To close this section, we record for future reference the fact that Lemma 3.4 holds on domains more general than balls; this will be used in the proof of Theorem 4.1. Although we state the result for a square, which is what we need, it is clear that the proof remains valid for any domain that is bi-Lipschitz homeomorphic to a ball, with a constant depending on the domain. For simplicity, we prove the lemma with error terms of order rather than 2 , as this suffices for our later application. Proof. If Ψ : B → Q is a Lipschitz map between subsets of R 2 , and U is a configuration on Q, we will write Ψ * U to denote the configuration (φ • Ψ, Ψ * A) on B. Note that Define Ψ(0) := 0, and Ψ(y ν ) := |y ν |y ν max{|y 1 |,|y 2 |} , |y ν | = 0, where |y ν | is the standard Euclidean norm. Then Ψ(B ν (s)) = Q s , and DΨ ∞ ≤ 1 ≤ D(Ψ −1 ) ∞ ≤ √ 2. It follows from (3.24) and (3.26) and a change of variables that Thus deg( φ |φ| ; ∂Q s ) = deg( φ |φ| • Ψ, ∂B ν (s)) =: n(s) ∈ Z exists, and the proof of Lemma 3.4 shows that there exists a configurationŨ on R 2 such thatŨ = Ψ * U on B ν (s),

Minimizers of the 2d Euclidean energy
The main result of this section gives a criterion for existence of solutions of the minimization problem (1.16):  E λ N < min{E λ n 1 + · · · + E λ n i : n 1 + · · · + n i = N, at least two n j are nonzero}. Then there exists U ∈ H N such that R 2 e ν ,λ (U ) = E λ N . In particular, there exists U ∈ H ±1 minimizing R 2 e ν ,λ (·) if 1 5 < λ < 5.
It is proved in [21, Theorem I.2] that R 2 e ν ,λ (·) attains its minimum in H n for |n| = 1 and for all λ sufficiently large. Our argument is close in spirit to that of [21], and we omit some details that are either standard or can be found in [21].
Proof. We will show that there exists 0 > 0, to be specified below, such that for all ∈ (0, 0 ), there exists U ∈ H N such that R 2 e ν ,λ (U ) = E λ N . In view of scale invariance, see (1.10), this will establish the theorem.
1. We first remark that it follows from (2.3) and (2.2) that E λ m ≥ min{1, λ}π|m|, and hence that the minimum (4.1) is indeed attained, and in fact there exists δ λ N > 0 such that (4.2) if n 1 + · · · + n i = N and at least two n j are nonzero, then We may also assume that δ λ N ≤ 1. Assume that < 0 , and let (U k ) ⊂ H N be a minimizing sequence, so that R 2 e ν ,λ (U k ) → E λ N as k → ∞. We further assume, without loss of generality, that U k is smooth and that R 2 e ν ,λ (U k ) < (1 + 1 2 δ λ N )E λ N for every k. Let L := (R × Z) ∪ (Z × R), and for y ∈ R 2 , let τ y U k (x) := U k (x − y). Then it follows from Fubini's Theorem that for every k. Thus, replacing U k by a suitable translation τ y k U k for every k, we may arrange that 2. Now for every p = (p 1 , p 2 ) ∈ Z 2 , let Q p := p + (0, 1) 2 = {(x + p 1 , y + p 2 ) : (x, y) ∈ (0, 1) 2 }. It follows from (4.3) that ∂Q p e ν ,λ (U k ) ≤ 3E λ N for every k ∈ N and p ∈ Z 2 .
We can thus apply Lemma 3.5 on every square Q p to find that if is small enough, then n(p, k) := deg( φ k |φ k | ; ∂Q p ) is well-defined, and for all nonzero m and all < 0 . Then Also, as noted in Section 3, ω(U k ) is integrable, so by Lemma 2.4, the definition of H N , and the additivity of degree, Thus the definition of δ λ N implies that if at least two n(p, k) are nonzero, then for all k, since we assumed that δ λ N ≤ 1. But this is not the case for any k, by our choice of the sequence U k . We conclude for every k, (4.4) there exists p 0 = p 0 (k) ∈ Z 2 such that n(p 0 , k) = N , n(p, k) = 0 for p = p 0 .
Replacing (again) U k by a suitable translation, we may assume that p 0 = (0, 0) for every k.
3. The remainder of the existence proof is now standard, and very similar points are treated in detail in Rivière [21], so we summarize the arguments only briefly. First, if we impose the Coulomb gauge condition ∇·A k = 0 for every k, then the uniform energy bounds imply that the sequence U k is weakly precompact in ( We can thus extract a subsequence converging weakly to a limit U = (φ, A), and standard lower semicontinuity arguments imply that Next, using (4.4) and (4.3) we can check that n(p) = deg( φ |φ| , ∂Q p ) is well-defined and that n(p 0 ) = N and that n(p) = 0 if p = p 0 = (0, 0). It follows that U ∈ H N and hence that U is an energy-minimizer in H N .

abelian Higgs model: energy estimates in normal coordinates
In this section we consider the abelian Higgs model in the coordinate system introduced in Section 1.4.
In particular, recall the map ψ defined in (1.21), built around a timelike minimal surface parametrized by an embedding H : (−T, T ) × S 1 → R 1+3 as described in Section 1.4. Given T 0 < T , we henceforth restrict the domain of ψ to a set of the form (−T 1 , T 1 ) × S 1 × B ν (ρ 0 ). We do this in such a way that where ψ 0 denotes the 0th component of ψ, corresponding to the t variable. Given T 0 < T , and having fixed T 1 ∈ (T 0 , T ) and ρ 0 as above, we will write We will write (g αβ ) to denote the metric tensor written in the normal coordinate system 1, 1, 1) and we also use the notation (5.4) (g αβ ) := (g αβ ) −1 , g := det(g αβ ).
Thus, in the normal coordinate system the abelian Higgs model takes the form Here α, β run from 0 to 3, and we raise and lower indices with (g αβ ) and (g αβ ) respectively. We find it useful to write the above system as In [12,Prop.1] it is further shown that (5.10) . and that the vector field b defined in (5.7) satisfies All the above estimates are uniform in (−T 1 , T 1 ) × S 1 × B ν (ρ 0 ). We remark that the estimate |b ν | ≤ C|y ν | follows from the condition that Γ is a minimal surface, and it is the only place in our argument that we directly invoke this assumption. 5.2. energy. The natural energy for (5.5)-(5.6) is obtained from the stress-energy tensor We will state estimates in terms of the energy density e ,λ (U ) := 2T 0 0 . Explicitly, We define (a αβ ) so that a αβ ξ α ξ β = g αβ ξ a ξ b − g 0β ξ 0 ξ b , which leads to (5.12) a 00 = −g 00 , a i0 = a 0j = 0, a ij = g ij , i, j = 1, . . . , 3.
With this notation, We also remark that (5.14) where we remind the reader that repeated latin indices are summed from 1 to 3. We will need the following Lemma 5.1. There exist constants 0 < c ≤ C such that for every U = (φ, A), Proof. The pointwise inequalities (5.15) follow from (5.9), (1.6), (1.8) by routine computations.
Then there exists C > 0 such that As is well-known, the tensor T α β satisfies an exact conservation law ∂ α (T α β √ −g) = 0 for α = 0, . . . , 3. However, (5.19) is more useful for our purposes. Surprisingly, it does not seem to be easy to derive (5.19) directly from the exact conservation law.
Proof. We take the inner product of (5.5) with D 0 φ to find . Also, using the commutation relation Since −g 0β ξ β ξ 0 + 1 2 g αβ ξ a ξ β = 1 2 a αβ ξ α ξ β , by collecting terms we conclude that (5.20) We now rewrite the last term on the right-hand side. First, using the equation (5.6), We will leave the second term as it is. As for the first term, note that We combine (5.20)-(5.22) and collect all terms involving ∂ 0 on the left -hand side, to find that Note that the left-hand side is just ∂ 0 e ,λ (U ), and the first term on the right-hand side is −2 ∂ i T i 0 (φ, A). So we just need to estimate the other terms on the right-hand side. First, by (5.10) and (5.11), which we recall is essentially the assumption that Γ is minimal, Next, using (5.9)-(5.10). Finally, from (5.9) and (5.11) we similarly estimate 5.4. weighted energy estimate. In this subsection we establish an estimate that plays a key role in the proof of Theorem 1.4. We first introduce some notation.
Given a configuration U on S 1 × B ν (R) for some R > 0, if U is a configuration on S 1 × B ν (R) for some R > 0, then we will use the notation for m ∈ Z, where here U (y 1 ) denotes the function y ν → U (y 1 , y ν ). We recall that D ν m is defined in (3.1).
The proof follows that of Proposition 10 in [12].
2. Next, recalling the definition (5.26), (5.24) of ζ 2 , so it follows immediately from Proposition 3.1 that 3. We next show that Since (5.30) implies that it suffices to show that The s variable plays no role in this argument, so we regard it as fixed and do not display it. We will say that y 1 ∈ S 1 is good if |D ν m (U (y 1 ))| ≤ κ 1 , where κ 1 was fixed in Proposition 3.2, and y 1 is bad otherwise. As usual we estimate the size of the bad set by Chebyshev's inequality: So {y 1 ∈ S 1 : y 1 is good} ≥ |S 1 |−Cζ 2 (s), and we obtain the estimate we seek by applying the lower energy bounds from Proposition 3.2 in the normal variables for every good y 1 . Indeed, Rearranging gives (5.35). 4. We gather the previous steps to conclude In this section we complete the proof of Theorem 1.4. This involves, among other things, combining the weighted energy estimates of Proposition 5.3, expressed in the normal coordinate system and effective near Γ, with energy estimates in the standard coordinate system, effective away from Γ.
We follow the proof of [12,Theorem 22]. We start by presenting all the details, to illustrate the basic argument, and we refer to [12] for the final part of the proof.
The proof will use some standard energy estimates for U and U respectively, which we recall for the reader's convenience.
Proof. Recall the energy identity for solutions of (1.2)-(1.3). This is standard and also can be deduced from (5.23) (replacing (g αβ ) by (η αβ )). We integrate by parts and use the fact that D i ϕ, D 0 ϕ + 2 F iν F 0ν ≤ e ,λ (U, η) and routine estimates to deduce (6.3). If χ has unbounded support, then one can approximate it by functions with compact support and use the fact that U has finite energy to pass to limits and obtain (6.3).
Proof. We fix ρ 1 as in the proof of Proposition 5.3 so that (5.29) holds. Then the proof of (6.5) is exactly like the proof of (6.3) in Lemma 6.2 above, except that we use for example (5.23) and (5.29) in place of their counterparts in standard coordinates.
We will also often use the fact that there exists some C > 0 such that . This is a straightforward consequence of the definition of ψ. A similar estimate holds for the restriction of ψ to {y : y 0 = 0}, which we will call ψ 0 .
2. Next we consider the standard coordinate system, and we show that the energy of U is small away from Γ for all t ∈ [0, t 1 ], for some t 1 > 0. The idea is to apply Lemma 6.2 with a = 0 and b = t ∈ (0, t 1 ] and a suitable cutoff function χ, and to use (6.9) to estimate the terms appearing on the right-hand side of (6.3).
To carry this out, let χ : [−T 0 , T 0 ] × R 3 → R be a smooth function such that ) . Next, the construction of ψ implies that ∂ 0 ψ > 0 everywhere and that ψ maps the set {y : y 0 = 0} into the set {(t, x) : t = 0}, and these facts imply that there exists some t 1 > 0 such that Now we apply Lemma 6.2 to find that for t ∈ (0, t 1 ], Moreover, (6.10) and (6.9) imply that And using properties of the support of χ with (6.9) and definition of ζ 0 , e ,λ (U, η) ≤ Cζ 0 .
To complete the proof of the theorem, then, it suffices only to show that after finitely many iterations of this argument, one can extend the bounds onζ i to all 0 ≤ s ≤ T 1 (for i = 1, 2, 3) and 0 ≤ t ≤ T 0 for i = 4. (The same conclusions for −T 1 ≤ s < 0 and −T 0 ≤ t < 0 then follow by time reversal symmetry.) A proof of this may be found in [12,proof of Theorem 22] for somewhat different equations, but exactly the same proof is valid here. The point is that the proof only involves piecing together estimates in the standard and normal coordinate systems, and the algorithm for doing so applies equally to any Lorenz-invariant equation.
(In fact the argument in [12] relies on a slightly different and more complicated iteration scheme than the one suggested above, but it remains true that the arguments there can be used in this setting with essentially no change.) We finally prove our main result.
The existence of the solution U follows from the discussion in Sections 1.6-1.7.
The quantitiesζ 1 ,ζ 2 are expressed in terms of the U = (φ, A) = ψ * U. So we must translate our assumptions about U at t = 0 into information about U for y 0 = 0.
In particular, (6.31) implies (1.28). Next, by (6.1), (6.6), and the definition of d ν , Then from (6.27) we see that (6.32) holds if and only if v µ A µ = 0 in N 1 . To arrange this, let f satisfy the linear transport equation (The condition g 00 < 0 implies thatv and hence v are timelike; thus v 0 never vanishes, so the above initial value problem is solvable.) Then after the gauge transform (ϕ, A) → (e if ϕ, A + df ), the equation that defines f states exactly that the new connection 1-form satisfies v µ A µ = 0. Thus we have achieved (6.32). Also, sinceζ i , i = 1, . . . 4 are gauge invariant, (6.30)-(6.31) still hold. Recall the form of U NO = (φ NO , A NO ): φ NO (y τ , y ν ) = φ m (y ν ), A NO (y τ , y ν ) := A m 1 (y ν )dy ν1 + A m 2 (y ν )dy ν2 , where U m = (φ m , A m ) = U m ,λ is a minimizer. We stipulate that U m ,λ is exactly the same minimizer out of which (φ m ,λ ,Ã m ,λ ) is constructed in Step 1. Then by a change of variables and a Poincaré inequality we have |∂ y 0 (A α − A NO α )| 2 (6.33) In the first integral on the right-hand side we use the explicit form of U NO and the gauge A 0 = 0 to write Inserting this into (6.33) and using (6.30), we conclude the first integral is bounded by C 2 .
To bound the second integral, using fundamental theorem of calculus we observe The second integral is bounded again using A 0 = 0 and (6.30), whereas C 2 bounds for the first integral follow from the construction of the data. The boundary term y 0 = −T 1 is treated exactly the same. This gives (1.26).