Solutions of the Ginzburg-Landau equations with vorticity concentrating near a nondegenerate geodesic

It is well-known that under suitable hypotheses, for a sequence of solutions of the (simplified) Ginzburg-Landau equations $-\Delta u_\varepsilon +\varepsilon^{-2}(|u_\varepsilon|^2-1)u_\varepsilon = 0$, the energy and vorticity concentrate as $\varepsilon\to 0$ around a codimension $2$ stationary varifold -- a (measure theoretic) minimal surface. Much less is known about the question of whether, given a codimension $2$ minimal surface, there exists a sequence of solutions for which the given minimal surface is the limiting concentration set. The corresponding question is very well-understood for minimal hypersurfaces and the scalar Allen-Cahn equation, and for the Ginzburg-Landau equations when the minimal surface is locally area-minimizing, but otherwise quite open. We consider this question on a $3$-dimensional closed Riemannian manifold $(M,g)$, and we prove that any embedded nondegenerate closed geodesic can be realized as the asymptotic energy/vorticity concentration set of a sequence of solutions of the Ginzburg-Landau equations.

If M is simply connected, then given a sequence of solutions (u ε ) of (1.1) satisfying the energy bound E ε (u ε ) ≤ C, (1.2) the rescaled energy density |log ε| −1 e ε (u ε ) is known to concentrate as ε → 0, after possibly passing to a subsequence, around an (n − 2)-dimensional stationary varifold -a weak, measure-theoretic minimal surface. This is proved in an appendix in [27], following earlier results in simply-connected Euclidean domains, such as those in [4,15,17]. Similar but more complicated results hold when M is not simply connected; in this case, the limiting energy measure may have a diffuse part, but any concentrated part must again be an (n − 2)-dimensional stationary varifold. In this paper we address a sort of converse question: When can a given codimension 2 minimal surface be realized as the energy concentration set of a sequence of solutions of (1.1)?
A first answer is provided by Gamma-convergence results, see [10,1], that relate the Ginzburg-Landau functional and, roughly speaking, the (n − 2)-dimensional area (with multiplicity) of a limiting vorticity concentration set, where the vorticity associated to a wave function u, denoted Ju, is the 2-form defined by Ju := du 1 ∧ du 2 , where u = u 1 + iu 2 and u 1 , u 2 are real-valued. (1.3) (We will also sometimes refer to Ju as the Jacobian of u.) These results imply as a general principle that one should be able to find solutions u ε of (1.1) whose energy and vorticity concentrate around a locally area-minimizing minimal surface of codimension 2. In the Euclidean setting, specific instances of this general principle, for particular compatible choices of boundary conditions on the minimal surface and the solutions u ε of (1.1), have been established in [1,25,22]. However, arguments based on Gamma-convergence are of limited use for capturing the behaviour of non-minimizing critical points. The corresponding question is also very well-understood for minimal hypersurfaces and the Allen-Cahn equation, i.e. the scalar counterpart of (1.1), see for example [13,23,14,7] among many others. Many of these results are based on gluing techniques and elliptic PDE arguments, which can be used to construct a great variety of solutions and establish detailed descriptions of them. These techniques seem to be hard to implement for the Ginzburg-Landau equation in 3 or more dimensions.
A particularly basic case in which our question remains open concerns the Ginzburg-Landau equation (1.1) on a smooth bounded domain Ω ⊂ R 3 containing an unstable geodesic with respect to natural boundary conditions, i.e. a line segment in Ω meeting ∂Ω orthogonally at both ends, admitting perturbations that decrease the arclength quadratically, and satisfying a natural nondegeneracy condition.
In this situation one would like to prove the existence of a sequence (u ε ) of solutions of the Ginzburg-Landau equations, also with natural (Neumann) boundary conditions, whose energy and vorticity concentrate around the given line segment. Such solutions would satisfy lim ε→0 E ε (u ε ) = L =: the length of the geodesic.
(1.4) Partial progress toward this goal was achieved in [11], which develops a general framework for using Gamma-convergence to study convergence, not of critical points, but of critical values, then uses this framework to prove the existence of solutions of (1.1), in the situation described above, that satisfy (1.4), but without control over the limiting concentration set. An example in the same paper (Remark 4.5) shows that the general framework is too weak to characterize asymptotic behaviour of critical points -in this context, to determine where the energy and vorticity concentrate. For this, more detailed information about the sequence of solutions is needed. The results of [11] were extended to the Riemannian setting in the Ph.D. thesis of Jeffrey Mesaric in [20] which, starting with a nondegenerate unstable closed geodesic on a closed, oriented 3 dimensional Riemannian manifold (M, g), uses machinery from [11] to construct solutions to (1.1) satisfying (1.4). Again, this result does not establish whether the energy of the solutions concentrates along the geodesic.
In the main result of this paper, we fill in this gap in the Riemannian case. Our main result is the following theorem. Theorem 1.1. Let (M, g) be a closed oriented 3-dimensional Riemannian manifold, and let γ be a closed, embedded, nondegenerate geodesic of length L. Assume in addition that γ = ∂S in the sense of Stokes' Theorem for some 2dimensional submanifold S of M .
Then there exists ε 1 > 0 such that for every 0 < ε < ε 1 , there is a solution u ε of the Ginzburg-Landau equation (1.1) such that In fact we will prove a slightly stronger result; see Theorem 5.1 for the full statement.
We briefly sketch the main ideas, not in the order in which they appear in the body of the paper. Terminology such as "nondegenerate" and "stationary varifold" are defined in Section 2 below.
• In Section 4 we show that for any δ > 0, there exists ε 0 > 0 such that for 0 < ε < ε 0 and any τ > 0, one can find a solution u ε of the Ginzburg-Landau heat flow whose vorticity is initially concentrated near the geodesic Γ := γ([0, L)), and such that See Proposition 4.1. This relies heavily on tools developed in the earlier papers [11,20].
The main point of the proof of Theorem 1.1 is to strengthen this by showing that for such solutions, if ε and δ are small enough, the vorticity 1 π Ju ε (·, t) does not stray very far from Γ for any t ∈ [0, τ ].
• We carry this out in Section 5, using an argument by contradiction and passing to limits to obtain a stationary 1-dimensional varifold that is close, but not equal, to the varifold associated to Γ. This argument requires, among other ingredients, an extension to the Riemannian setting of an important theorem of Bethuel, Orlandi, and Smets [5]. The extension we need is stated in Theorem 2.3 and is proved in a companion paper, see [6]. The stationary varifold satisfies additional good properties, notably including lower density bounds.
• To obtain a contradiction, we prove that this stationary varifold cannot exist. This is the content of Proposition 3.1, which is a measure theoretic strengthening of the classical fact that a nondegenerate closed geodesic is isolated; it is the only closed geodesic in a tubular neighborhood of itself. The proof relies, among other ingredients, on results from [2] about the structure of stationary 1-dimensional varifolds on Riemannian manifolds.
We believe that something like Theorem 1.1 should be valid in much greater generality, including on higher-dimensional manifolds and on smooth, bounded subsets of R n , n ≥ 3, with natural boundary conditions both for the geodesic Γ (or codimension 2 minimal surface, for n ≥ 4) and the Ginzburg-Landau equation. Our proof does not adapt in a straightforward way to either of these settings.
• Our strategy requires a sufficiently good version of Theorem 2.3. On a bounded set Ω ⊂ R n , even for n = 3, such a result is not known. If Ω is convex, a result of this type for the scalar parabolic Allen-Cahn equation was proved several years ago in [21]. A similar strategy could probably be pursued for the Ginzburg-Landau heat flow, but global convexity is not a natural assumption for any analog of Theorem 1.1.
• Our reliance on results from [2] about stationary 1-dimensional varifolds would seriously complicate any effort to adapt our argument to dimensions n ≥ 4, where one would confront stationary varifolds of dimension n − 2 ≥ 2.
2 Background and notation 2.1 Geometric notions regarding a non-degenerate geodesic Throughout this document we use M or (M, g) to denote a closed oriented three dimensional Riemannian manifold where "closed" means compact and without boundary. We let T M be the bundle over M whose fiber T p M at p ∈ M is the tangent space to M at p. We use the notation (·, ·) g to denote the inner product on T M given by g. We also use |·| g to denote the corresponding norm, where we will omit mention of g when no confusion will arise. We write vol g to denote the Riemannian volume form associated to the metric g.
We will write r 0 > 0 to be a number, fixed throughout this paper, such that Throughout this paper, a central role will be played by a geodesic γ that we take to be parametrized by arclength. That is, we will assume the existence of an injective map γ : R/LZ → M whose range consists of a simple closed curve Γ := {γ(t) : t ∈ R/LZ} of length L such that We will insist that this curve Γ bounds an orientable smooth surface S Γ ⊂ M , i.e. Γ = ∂S Γ .
We introduce here the notation for a neighborhood of Γ. For t ∈ R/LZ, we then let A normal vector field along γ is a map ξ : R/LZ → T M such that ξ(t) ∈ N γ(t) Γ for every t. We also introduce the coordinates ψ : where Ξ 1 , Ξ 2 are fixed normal vector fields which are orthogonal for each t ∈ (0, L). We note for r < r 0 , this map is smoothly invertible. For future use, we will use the notation ψ −1 (x) = (y(x), τ (x)) ∈ B r (0) × (0, L), so that for x ∈ K r , ψ(y, t) = x ⇐⇒ y(x) = y and τ (x) = t. (2.6) We observe that the mapping τ simply assigns to an x ∈ K r the parameter value t corresponding to the closest point on Γ to x. Given two normal vector fields along γ, denoted by ξ,ξ, we can define the L 2 inner product in the natural way: We will write L 2 (N Γ) to denote the space of square integrable normal vector fields, a Hilbert space with the above inner product.
For ξ ∈ L 2 (N Γ), we will use the notation where exp denotes the exponential map. We next recall the Jacobi operator L J which acts on smooth normal vector fields ξ along γ, and is defined by where R denotes the curvature tensor. We say that a geodesic is nondegenerate if 0 is not an eigenvalue of L J . With this notion in hand, we add another crucial hypothesis on the geodesic by assuming henceforth that γ : R/LZ → M is a simple, closed, nondegenerate geodesic with |γ ′ | ≡ 1, (2.9) One says that γ has finite index if the total number (algebraic multiplicity) of negative eigenvalues of L J is finite. Since M is closed, this is always true, as a consequence of standard Sturm-Liouville theory. Our standing assumption (2.9) that γ is nondegenerate then imples there exists some ℓ ≥ 0 and a nondecreasing sequence of eigenvalues of L J , together with an associated orthonormal basis of L 2 (N Γ) consisting of (smooth) eigensections {ξ j } ∞ j=1 . We will always assume that ℓ > 0, since otherwise the results presented here admit much simpler proofs. We define We will say that ξ is Lipschitz, and we will write ξ ∈ Lip, if γ ξ is Lipschitz continuous. It is clear that The standard fact that the Jacobi operator, cf. (2.8), is the second variation of arclength, together with the definition (2.11) of H − , implies that there exist c 0 , r 0 > 0 such that for all ξ L ∞ ≤ r 0 one has (2.12)

Forms and currents
We denote, for k ∈ N ∪ {0}, the space of smooth k-forms on M by where ∧ k M is an abbreviated notation for ∧ k T * M . We denote the dual space We refer to the elements of D k (M ) as k-currents. For a k-current T , we define We will be most interested in 1-currents. A basic class of examples consists of 1-currents we shall write as T λ whose action on φ ∈ D 1 (M ) takes the form where λ : (a, b) → M is a Lipschitz curve. (2.13) We will say a 1-current is integer multiplicity rectifiable if it can be written as a finite or countable sum of currents of the form (2.13). We will write For a 1-current J, we write J to denote the associated total variation measure, defined through its action on continuous, nonnegative functions f : For a k-current S, the boundary of S is the (k − 1)-current ∂S defined by We define and for T ∈ F ′ 1 (M ), we will write T F := inf{M(S) : T = ∂S}.
We also define . We note that the 1-current T γ associated with the geodesic γ via (2.13), in particular, bounds a finite mass 2-current; that is, (2.15) in light of the assumption (2.3). Lastly, we will at times wish to identify the Jacobian (i.e. vorticity) of a map u ∈ H 1 (M ; C) with an element of D 1 (M ), which we denote ⋆J(u), and which is defined through its action on 1-forms φ by where J(u) = du (1) ∧ du (2) for u = u (1) + iu (2) where u (1) , u (2) are real-valued.

Gamma-limit of the Ginzburg-Landau functional
Below we state the version we will need of standard Gamma-convegence results for the Ginzburg-Landau functional. We first fix the notation V = F ′ 1 (M ), with the flat norm v V := v F . We also define the functional Thus E V is an extension to V of the "arclength functional" in the sense that if λ : (a, b) → M is an injective Lipschitz continuous curve and T λ is the corresponding current, then E V (T λ ) = arclength of Image(λ).
The geodesic γ is a saddle point of the arclength with respect to smooth perturbations, as reflected in (2.12). For use in combination with Theorem (2.1), one needs to identify a sense in which the corresponding current T γ is a saddle point of E V . We defer a discussion of this and related issues to Section 4.

Varifolds
We briefly recall the definition of a rectifiable 1-varifold and introduce some notation that will be used later. After doing this we will introduce the definition of a general 1-varifold. We note that the general definition will only be used in the proof of Proposition 3.1. For general varifolds we will follow [2] with some terminology from [26].
For any 1-dimensional rectifiable set Σ, basic theory (see for example [26], Lemma 11.1) shows that there exists a countable family of and H 1 (N 0 ) = 0, and every point in Σ \ N 0 is contained in exactly one Λ j . We then define, for We will write τ Σ (x) to denote a unit vector in apT x Σ. First, we recall that if S is a countably 1-rectifiable, H 1 measurable subset of M and Θ : S → (0, ∞) is a locally H 1 -integrable function on S then we can use the pair (S, Θ) to form the measure H 1 Θ, where we have extended Θ to be zero outside of S. We refer to such a measure as a rectifiable 1-varifold. We also refer to the function Θ as the multiplicity function of this rectifiable 1-varifold and, at times, we will write Θ S to emphasize the association. We will also sometimes use the alternate notation ΘH 1 S for H 1 Θ S . If Θ happens to be integer-valued H 1 -almost everywhere then we will say this rectifiable varifold is of integer multiplicity. Finally, if there is a λ > 0 such that Θ ≥ λ at H 1 -almost every point then we say that the rectifiable varifold has density bounded below. A particular example of an integer multiplicity rectifiable 1-varifold that we will be interested in will be integration over a countable collection of geodesics.
Next, for a smooth Riemannian manifold, M , we let P M be the bundle whose fiber P a M at a ∈ M consists of the lines through the origin in T a M . If x ∈ M and ξ is a unit vector in T x M , we will sometimes abuse notation slightly and write (x, ξ) to denote the element of P M Thus (x, ξ) and (x, −ξ) correspond to the same element of P M . Suppose that η is a smooth function on M . When representing points in P M as described above, a mapping such as (x, ξ) ∈ P M → |∇ ξ η(x)| 2 is well-defined as a function P M → R, since it is independent of the choice of sign for the unit vector ξ. We let π : P M → M be the bundle projection. We refer to a measure V ∈ M(P M ) as a 1-varifold. Observe that to a rectifiable 1-varifold V = H 1 Θ S , we may associate a 1-varifold V defined by Roughly speaking, the difference between a rectifiable 1-varifold and the associated general 1-varifold is that the latter explicitly records information about the approximate tangent spaces to the set S on which the former lives.

Definitions: first variation, stationarity, Brakke flow
For a rectifiable 1-varifold ν given by ν = H 1 Θ Σ , where Σ is a 1-rectifiable set, the first variation of ν is a distribution, denoted δν, whose action on smooth vector fields X is defined by (Note that since τ Σ appears quadratically, the choice of unit vector in apT x Σ does not matter.) A 1-d varifold ν of the given form is stationary if We remark that in light of (2.2), of course it follows from an integration by parts that one can associate a multiplicity-one stationary varifold with the geodesic γ. Properties of stationary varifolds will be recalled later as needed.
For simplicity, we discuss Brakke flows and related notions from geometric measure theory only in the case of 1-dimensional varifolds in the 3-dimensional manifold (M, g). Let be a family of rectifiable 1-varifolds in M . To say that (ν t * ) t>0 is a Brakke flow means that for t > 0 there exists a ν t * -integrable vector field H(·, t) along Σ t ν (that is, H(x, t) ∈ T x M for ν t * almost every x ∈ Σ t ν ) such that the following hold. First, for all C 1 vector fields X. Second, for every t > 0 and every nonnegative Another simple fact we will need is the following.
Proof. Clearly, if (2.25) holds, then by taking χ = 1 in (2.23), we find that H = 0 a.e. in Σ t ν for every t ∈ (a, b). It follows that ν t * is stationary for such t. It is also easy to see that t → ν t * is constant for t ∈ (a, b). Indeed, given any nonnegative On the other hand, since H = 0, it follows from (2.23) that These together imply that t → M χ j dν t * is constant for j = 1, 2. Since this holds for all nonnegative χ 1 ∈ C 2 (M ), it easily follows that ν t * does not depend on t ∈ (a, b), proving (2.26).
We will make heavy use of results from a paper of Allard and Almgren [2] on stationary 1-dimensional varifolds with positive density in a Riemannian manifold. Among other results, they prove that a stationary 1-d varifold with density bounded away from 0 is supported on a finite or countable union of geodesic segments terminating at singular points. From these singular points multiple segments emanate, with a balance condition on the weighted sum, at each singular point, of the tangent vectors generating the geodesics that meet there. Other results from [2] will be cited as the need arises.

Asymptotic analysis of the Ginzburg-Landau heat flow
As a last preliminary, we state a recently established extension to the Riemannian setting of a theorem of Bethuel, Orlandi and Smets [5], who built on prior work of a number of authors, including [9,3,16].
The theorem quoted below is proved in [6].
For every t ≥ 0, let µ t ε be the measure on N defined by Then after passing to a subsequence (still denoted simply by ε), there exist mea- weakly as measures for every t > 0.
Moreover, there exists a smooth function Φ * solving the heat equation on N × (0, ∞) and a family of measures (ν t * ) t>0 on N , such that for every t > 0 where Σ t ν is an (n − 2)-dimensional rectifiable subset of N and Θ * is a bounded measurable function. In addition, there exists a function η : for H n−2 a.e. x ∈ Σ t ν , for a.e. t > 0. Finally, the family (ν t * ) t>0 is a Brakke flow.

A non-existence result for stationary 1-varifolds near a non-degenerate geodesic
The proof of our main result hinges crucially on showing there is no stationary varifold sitting over a 1-current that is nearby T γ , the 1-current associated with the non-degenerate geodesic γ. While the non-degeneracy assumption (2.9) easily precludes the existence of another nearby smooth geodesic, it is the need to rule out proximity in the weaker sense of (3.2), (3.3) below and within the larger class of varifolds that makes the result below much more challenging to establish.
Proposition 3.1. Let T γ be the 1-current in M corresponding to integration over the nondegenerate geodesic γ, and let η > 0 be given. Then there exists r 0 > 0 depending on M, γ, and η, such that for 0 < r < r 0 there is no stationary 1-dimensional rectifiable varifold V * and 1-current

2)
and The starting point of the proof is provided by the following lemma, established by Mesaric [20].
Lemma 3.2. For T γ as above, let J 1 ∈ R 1 ∩ F ′ 1 (M ) be a current satisfying (3.2), and such that Then provided r is taken sufficiently small, there is a 1-current J * 1 ∈ R 1 (M ) such that the support of J * 1 , denoted by Γ * , consists of a single Lipschitz curve with no boundary satisfying

4)
and In addition, there exists a constant C 1 > 0 such that

7)
and This is demonstrated in Lemma 4.4 and comments following Lemma 4.6 of [20]. The proof is an adaptation to the Riemannian setting of arguments from [11], Lemma 5.5. The idea is that (3.2) and the definition of the flat norm imply that a large set of transverse slices to Γ must intersect J 1 at exactly one point, and this point must be close to Γ. Behavior of J 1 on other slices is constrained by the assumption that ∂J 1 = 0 and the mass bound. The proof also uses Federer's decomposition of integral 1-currents, [8], 4.2.25.
Proof of Proposition 3.1 Step 1: First we show that the rectifiable varifold V * does not have any mass outside of K 4 √ r . The idea is that if this fails, then the monotonicity formula and (3.7) would contradict the assumption M(V * ) ≤ L. This argument relies crucially on the uniform lower density bound for V * .
We recall that the Hessian Comparison Theorem, see Theorem 6.6.1 of [12], gives us that if µ > 0 is an upper bound on the absolute value of the scalar curvature over M and r > 0 is chosen so that then for each p ∈ M and all x ∈ B r (p) and v ∈ T x M we have In view of the lower density bound Θ * ≥ η, V * almost everywhere, it follows from a Riemannian version of the Monotonicity Formula (established with different notation in [2] in item (5) of Theorem 1 on pg. 87) that there exists r con (M ) > 0 such that for every 0 < s < r con , (v, v) does not depend on which unit tangent is chosen.) It follows from (3.9) that 2sη ≤ (1 + µs 2 )V * (B s (p)) for all 0 < s < 1 2 min r 0 , r con , π 2 √ µ and V * -almost every p ∈ M . We conclude that for all 0 < s < 1 2 min r 0 , r con , π 2 √ µ , 1 we have We now use this to prove that if r is chosen sufficiently small, then To verify (3.12), suppose to the contrary; then V * M \ K 4 √ r > 0 and so there is a point p of spt(V * ) in M \ K 4 √ r for which (3.11) holds. By (3.11) we have that By shrinking r 0 if necessary, we may assume that B 1 Hence, appealing to (3.7), we find that r.
Choosing r smaller if necessary, depending on C 1 , η, µ, this yields a contradiction. We conclude that (3.12) holds. Since J 1 ≤ V * , we remark that J 1 is also supported in K 4 √ r .
Step 2: Next we demonstrate that where v ∈ T x M is a unit vector, x ∈ K 4 √ r , and τ is the mapping defined in (2.6).
We prove only the statement about the Hessian, as the gradient estimate follows by similar arguments. In coordinates introduced by ψ : as defined in (2.5), we can write τ as These are what are called Fermi coordinates, and a basic fact, proved for example in Section V of [18], is that the vectors Ξ 1 , Ξ 2 in (2.5) can be chosen so that all Christoffel symbols vanish along the central geodesic, that is, when y = 0: Γ k ij (0, t) = 0 for i, j, k = 1, . . . , 3 where x k = y k for k = 1, 2, and x 3 = t. In general, the expression for the Hessian in coordinates is see for example [12], Definition 4.3.5. By combining these, we readily deduce that Hess g (τ )(ψ(0, t)) = 0 and thus that Hess g (τ )(ψ(y, t)) ij = O t (|y|) for 1 ≤ i, j ≤ 3. The Hessian estimate in (3.13) follows directly.
Step 3: Let V * be the 1-varifold associated as in (2.19) to the rectifiable 1-varifold V * . We next demonstrate that for each δ > 0 there is r 1 > 0 such that if 0 < r < r 1 in (3.2) and hence in (3.12), then where we recall our convention that a generic element of P M -that is, a line in T x M for some x ∈ M -is represented by a pair (x, ξ), where ξ is a unit vector in T x M spanning the given line, see (2.18). This will establish that most tangent vectors to the support of V * are, according to the measure V * , nearly parallel to ∇τ .
We suppose toward a contradiction that there is a δ > 0, a sequence (r k ) k∈N tending to 0 from the right, and a sequence of stationary rectifiable varifolds (V k ) k∈N on M satisfying the hypotheses of Proposition 3.1 with r replaced by r k in (3.2), and such that the associated 1-varifolds V k satisfy for all k ∈ N. In particular we have Since (V k ) k∈N is a sequence of stationary rectifiable varifolds we may combine (3.16) with the compactness result of Theorem 42.7, pg. 247 of [26] to conclude that there is a subsequence V kj j∈N and a rectifiable varifold V with associated multiplicity Θ V such that It follows from (3) and the fact that each V kj is stationary that V is also stationary. Then from (1) and the fact that spt(V kj ) ⊂ K 4 √ r k j we conclude that spt(V ) ⊂ Γ. Next, we observe that, due to (3.5), each V kj has support that contains a closed curve that meets every level set of τ . Hence, spt(V ) = Γ as a result of (1). Observe that since V is a stationary varifold with density bounded below and spt(V ) = Γ, then by the Theorem on page 89 of [2] we have that V is simply a constant multiplicity multiple of the stationary varifold V Γ associated with the geodesic γ. Applying the weak convergence (1) to (3.15), however, we see that an impossibility given that all tangent vectors ξ along Γ coincide with ±∇τ .
Step 4: Next we introduce three sets corresponding to slices normal to the central geodesic that are in some sense bad. We will argue that two of them correspond to sets of t-values of measure zero while the third is of small measure. We introduce the first such set, B 1 , through the function h : (0, L) → R given by Since h is non-decreasing, it is differentiable L 1 -almost everywhere and consequently L 1 (B 1 ) = 0. Now we recall that the singular set S V * , as defined in [2], is the set of points of M near which Θ V * , restricted to spt(V * ), is not constant. Then we introduce the set B 2 as the set of slices meeting the singular set: We claim that L 1 (B 2 ) = 0 as well.
To this end, we note that in the remark following the theorem on page 89 of [2], it is stated that V * (S V * ) = 0, (3.18) and so by (2.28) and (3.18) we have that We conclude that Since B 2 is the image of a subset of S V * by the Lipschitz map τ : K 4 √ r → (0, L), it follows that L 1 (B 2 ) = 0 as claimed.
The final 'bad' set of slices is B 3 defined by Replacing the role of the equality (3.18) by the inequality (3.14) in the argument above, the same line of reasoning goes to show that there is a constant C 2 > 0 such that where r is chosen sufficiently small.
Step 5: We now use the results obtained in Steps 1 and 2 to show that, for r and δ chosen sufficiently small, and a, b ∈ (0, L) \ B 1 we have where h is as defined in Step 4.
For s small and positive, we define the function and we let X be a smooth vector field on M such that X(x) = H a,b;s (τ (x))∇τ (x) for x ∈ K r0/2 . The fact that V * is stationary implies that δV * (X) = 0, cf. (2.20), and this means that We have used Step 1 to obtain that V * is concentrated on K 4 √ r . Next, we use Step 2, V * (M ) ≤ L, and the fact that H a,b;s L ∞ (R) = 1 to conclude that Then we observe that, by the definition of H a,b;s , we have

Combing this with (3.20) and (3.21) yields
Letting s tend to zero, we find that Step 6: We next introduce a set of 'good' slices via G := a ∈ (0, L) : H 0 ({τ = a} ∩ spt(V * )) = 1 , and in this step we will demonstrate that From this and Step 4 it will follow that provided that r and δ are chosen sufficiently small.
Suppose by way of contradiction that there exists a value a such that Then in light of (3.5) we have that . We first argue that for such an a and every c, δ ∈ (0, 1) it must hold that provided r is chosen sufficiently small, depending on c. Fix 0 < c, δ < 1, and choose 1 < α < 1 δ . Note that since a ∈ B 2 then by item (3) of the theorem on page 89 of [2] we have that {τ = a} ∩ spt(V * ) consists only of interior points of the constituent geodesics (or "intervals" as they are referred to in [2]) making up V * . Moreover, the endpoints of these geodesics cannot accumulate at a. We conclude by compactness of spt(V * ) that {τ = a} ∩ spt(V * ) can intersect only finitely many of these constituents of spt(V * ). Also since a ∈ B 3 , it follows that this slice must intersect spt(V * ) transversally, so that {τ = a} ∩ spt(V * ) consists of finitely many points, say x 1 , . . . , x K , where K ≥ 2 by the choice of a.
It follows from the gradient estimate |∇τ | = 1 + O( 4 √ r) in K 4 √ r , established in (3.13), that for 0 < α < β < L, the geodesic distance between the sets Hence if r is small enough, then (Here and below, we tacitly assume that 0 < a − s < a + s < L and s < L/2.) Next we again use that a / ∈ B 3 to choose s 0 > 0 small enough so that and (y, ξ) ∈ spt(V * ). Combining these facts, for each 0 < s ≤ s 0 we estimate We now apply item (5) of the theorem on page 87 of [2] and let s → 0 + , using the differentiability of h at a guaranteed by the assumption a / ∈ B 1 , to find Here we have used Lemma 3.2 to assert that Γ * intersects each level set of τ with Θ * (x) ≥ 1 for x ∈ Γ * by (3.3), and that Θ * ≥ η in general by (5.12). Since α > 1 was arbitrary we may let α → 1 + to obtain (3.26). In light of (3.22), it then follows from (3.26) that for any b / ∈ B 1 we obtain Thus, choosing c, r, and δ sufficiently small and recalling that L 1 (B 1 ) = 0, we deduce that Thus, if there were a value a ∈ (0, L) satisfying (3.25), then Here we use (3.13) in the second inequality and the constant C 3 depends only on M and Γ. If we choose r sufficiently small, the contradiction is reached, establishing (3.23) and (3.24).
Step 7: In this step we show that It will immediately follow that the Lipschitz curve Γ * guaranteed by Lemma 3.2 represents the entire rectifiable varifold and is in fact a closed geodesic. Crucial use in this step will be made of the following general property of stationary 1-varifolds (cf. [2], pg. 88.): Every point p ∈ M is contained in an open set U p such that if V is any stationary varifold on M with support in U p , and if the support of δV consists of exactly two points, then V is the varifold corresponding to a constant multiple of the geodesic joining these two points.
This result is proved in [2] for possibly noncompact manifolds. Since M is compact, we may invoke the Lebesgue Number Lemma to conclude that there exists λ > 0 such that for any p ∈ M , the geodesic ball B λ (p) has the stated property.
For any A ⊂ (0, L), we will write By extending ψ to be periodic with respect to the t variable in the natural way, we can define K 4 √ r (A) for any A ⊂ R. By shrinking r and δ, we may arrange that if I is any interval of length at most 2C 2 δ/η, where C 2 is the constant appearing in (3.24), then K 4 √ r (I) is contained in a ball of radius λ.
We will prove the claim by showing that Indeed, for any t we can appeal to (3.24) to find some s 1 , s 2 ∈ G such that We now apply the result stated at the outset of this step to the varifold , in an open ball B λ (p) that contains K 4 √ r ([s 1 , s 1 ]). The definition of G implies that V * intersects {τ = s j } in exactly one point, say x j , and that δṼ is supported in {x 1 , x 2 }. Hence, this restriction of V * consists of a multiple of the geodesic joining these two points. This immediately implies the claim.
Since S V * = ∅, V * must simply be the rectifiable varifold associated with a single, closed, smooth geodesic.
By perhaps shrinking r one more time and applying the Morse-Palais Lemma, cf. [24], pg. 307, we may conclude that the central geodesic Γ, being a nondegenerate critical point of length, is isolated and so necessarily spt(V * ) = Γ. However, this contradicts (5.13) since r > 0, and the proof of Proposition 3.1 complete.

Finding good trajectories
The critical points of Ginzburg-Landau that we seek will be obtained as limits of certain carefully chosen trajectories of the Ginzburg-Landau heat flow. In this section we identify these trajectories.
In the remainder of this section we expand on this, aiming to provide enough detail to convey the main ideas, to explain where we depart from [11,20], and to make it possible, in principle, to check the terse proof given above.
We remark that the main difference between [11,20] and our present treatment is that those earlier works use a pseudo-gradient flow for the energy E ε , whereas we employ a small modification of the Ginzburg-Landau heat flow (4.1) for similar purposes. The use of (4.1) is necessary for our approach, due to our reliance in Section 5 below on Theorem 2.3.

Saddle point property of E V
Our assumptions about γ imply, roughly speaking, that the "arclength functional" has a local minmax geometry near γ, as reflected in (2.12), with respect to smooth perturbations. In particular, there is an ℓ-parameter family of arclength-decreasing perturbations of γ, and arclength increases for sufficiently transverse smooth perturbations. Here and below, ℓ is the index of γ, see (2.10).
The result below states that the "generalized arclength functional" E V defined in (2.17) has a a saddle point, in a suitable weak sense, at the current T γ ∈ V corresponding to γ. The relevant notion of saddle point was first introduced in [11].  [20]). For the geodesic γ satisfying (2.9) and (2.15), the associated current T γ is a saddle point of E V in the sense that there exist R, δ 0 > 0 and continuous functions such that P W V (T γ ) = 0, and the following conditions are satisfied: We sketch the proof from [20], although we note that this will not play any role in what follows, except that the notation γ w for the curve defined in (4.9) and T γw for the associated 1-current, cf. (2.13), will be used below.
To start, for w ∈ W we define ξ(w) = w 1 ξ 1 + . . . + w ℓ ξ ℓ (4.8) where ξ j denote eigenfunctions of the Jacobi operator, see in particular (2.10). We then define the curve γ w via cf. (2.7). We will always assume that R is small enough that With this in hand, we define Then (4.5) is immediate, and (4.7) follows directly from (2.12).
The construction of P W V is carried out in [20], Lemma 4.3, by designing an R ℓ -valued 1-form Φ such that for ξ ∈ L 2 (N Γ)∩Lip, as long as ξ L ∞ ≤ r 0 , a condition that can be guaranteed by a suitable choice of R. Here we recall that γ ξ is the curve given by (2.7) and T γ ξ is its associated 1-current. We then simply define P W V (T ) = T (Φ). With this choice, (4.6) follows directly from (4.8), (4.9), and (4.10). The hard part of the proof of Lemma 4.2 is the verification of (4.4). This is carried out in Proposition 4.1 of [20], to which we refer for the details. We remark that the main ideas in this proof, including the construction of Φ, are similar to elements in the proof of Theorem 5 in [28].
Note that we may shrink at will the parameter R in the definition (4.3) of W , and the conclusions of the lemma remain valid.

An ℓ-parameter family of solutions of (4.1)
To prove Proposition 4.1, we will define an ℓ-parameter family of solutions of (4.1) for every sufficiently small ε > 0. In the final step, given ε and τ (where ultimately we will take ε < ε 0 (δ)), we will choose from this family one solution The initial data for this family of solutions is provided by the following result.
Lemma 4.3. There exist R, ε 1 > 0 such that for every ε ∈ (0, ε 1 ) and w ∈ B ℓ R , there exists a function U ε,0 w ∈ H 2 (M ) satisfying the conditions: For fixed w, in view of (2.12) and the construction of γ w , conclusions (3) and (4) hold if U ε,0 w is a recovery sequence for the current T γw the Gamma-limit in Theorem 2.1. Such constructions are rather standard. It is easy to arrange that U ε,0 w L ∞ ≤ 1. The only points requiring attention are that the construction has to be carried out so that it depends continuously on w, in the H 1 norm, and with some control over the H 2 norm. The former point is carried out in [20], and the latter can be achieved by a small modification of the construction of [20]. We defer a more detailed discussion to Appendix A.
The H 2 estimate facilitates the proof of Lemma 4.4 below, whose need arises because we require the Ginzburg-Landau heat flow rather than the pseudogradient flows employed in [11,20].
Having constructed appropriate initial data for the Ginzburg-Landau flow, we are now ready to define the flow that we will use in our arguments below.
Lemma 4.4. For ε ∈ (0, ε 1 ) and w ∈ W , let U ε,1 w (x, t) solve the Ginzburg-Landau heat flow with initial data U ε,0 w . Then See Appendix A for the proof, which involves rather standard parabolic estimates.

Choosing a good trajectory
We finally make use of the asymptotic saddle point geometry of E ε , inherited from E V via the Gamma-convergence Theorem 2.1, to complete the proof of Proposition 4.1.
We will use the notation Lemma 4.5. There exist δ 0 > 0 and R 0 > 0 such that for every R ∈ (0, R 0 ), there is some δ = δ(R) > 0 such that if we define as ε → 0, where L is the length of the geodesic γ.
The assertions about a ε and d ε follow directly from (2.12) and Lemma 4.3, and Step 3 of the proof of Theorem 4.4 of [11] shows exactly that c ε → L. The proof uses only ingredients that we have collected in Theorem 2.1 and Lemma 4.2 below.
Below we will not refer explicitly to the assertion about lim ε→0 a ε , but it plays a role in the proof of Lemma 4.6, and together with the lower bound for lim inf c ε , it reflects the asymptotic minmax geometry of E ε . Proposition 4.1 will essentially follow from the next fact.
Lemma 4.6. For each r > 0 there exists ε 0 > 0 and R > 0 such that for every 0 < ε < ε 0 and every τ > 0, there exists w = w(ε, τ ) such that As a result, w = w(ε, τ ) satisfies Proof. One may prove (4.14) by simply repeating the arguments from Steps 5-8 of the proof of Theorem 4.4 in [11], for r > 0 such that 0 < r < δ 0 , where δ 0 is the constant from item (4.4) of Lemma 4.2. Some comments are in order: First, the argument in [11] is stated for a pseudo-gradient flow (see Lemma 4.8, [11]) with certain properties that our flow (t, w) → U ε (t, w) does not possess. These are in fact not needed for the proof of (4.14), and some of them may appear in [11] only because there Lemma 4.8 is quoted directly from a standard text, which provides more than is actually needed. These are the only properties of the flow that are required for the proof of (4.14): • t → E ε (U ε (t, w)) is nonincreasing.
• continuity properties of the flow, as summarized in Lemma 4.4.
All of these are available here.
Without going into detail, we remark that the basic strategy of the proof is to apply degree theory arguments to the maps w → P W U (U ε (t, w)) : W → W as t varies from 0 to τ (where τ = 1 in [11], a harmless normalization).

Proof of the main result
The main result of this paper, stated more informally in the introduction as Theorem 1.1, can now be phrased precisely as Theorem 5.1. Let (M, g) be a closed oriented 3-dimensional Riemannian manifold, and let γ be a closed, embedded, nondegenerate geodesic of length L. Assume in addition that γ = ∂S (in the sense of Stokes' Theorem) for some 2-dimensional submanifold S of M .
Then for every r > 0, there exists ε 1 (r) > 0 such that if 0 < ε < ε 1 (r), then there is a solution u ε of the Ginzburg-Landau equations As a result, there exists a sequence (u ε ) ε>0 ⊂ H 1 (M ; C) of solutions of the Ginzburg-Landau equations such that We remark that standard Gamma-convergence results (see Theorem 2.1) imply that the sequence of solutions in (5.2) satisfies which is the last conclusion of Theorem 1.1. Indeed, since E ε (u ε ) is uniformly bounded, there exists some measure µ such that e ε (u ε ) π|log ε| → µ weakly as measures, after perhaps passing to a subsequence, and µ(M ) = lim ε→0 E ε (u ε ) = L. Standard Gamma-convergence results and (5.2) imply µ ≥ T γ , and since T γ (M ) = L = µ(M ), it follows that µ = T γ , proving (5.3).
Proof. The proof relies on an improvement on the properties of the flow defined in the previous section. The assertion is that the trajectory solving the Ginzburg-Landau flow identified in Proposition 4.1 remains close to T γ in the flat norm. More precisely, we will show:
Step 1: Under the assumption (5.5), we will argue that necessarily t k → ∞ as k → ∞. (In fact, below we only need to know that t k is bounded away from 0.) Assume toward a contradiction that lim inf k t k < K, for some K > 0. By passing to subsequences, relabelling, and invoking standard compactness, continuity, and Gamma-convergence results (i.e. Theorem 2.1), we may assume that the following hold.
Step 2: We will now reach a contradiction to (5.5), and so obtain Claim 5.4.
However, through an appeal to Proposition 3.1, we see that no such varifold can exist. Claim 5.4 is established.
Finally, we may insist that δ < r, and then it follows from Proposition 4.1 that |E ε (u ε ) − L| < r.
To specify the norm, we fix an open cover {U j } j∈J of M , with local coordinates ϕ j : U j → V j ⊂ R 3 on each patch, and a finite partition of unity {η j } subordinate to {U j }. We then define