Ginzburg-Landau Equations on Non-compact Riemann Surfaces

We study the Ginzburg-Landau equations on line bundles over non-compact Riemann surfaces with constant negative curvature. We prove existence of solutions with energy strictly less than that of the constant curvature (magnetic field) one. These solutions are the non-commutative generalizations of the Abrikosov vortex lattice of superconductivity. Conjecturally, they are (local) minimizers of the Ginzburg-Landau energy. We obtain precise asymptotic expansions of these solutions and their energies in terms of the curvature of the underlying Riemann surface. Among other things, our result shows the spontaneous breaking of the gauge-translational symmetry of the Ginzburg-Landau equations.

Here κ > 0 is a fixed parameter. ψ and a are respectively a section of and a connection 1-form on the line bundle E. ∇ a is the covariant derivative induced by the 1-form a, and −∆ a = ∇ * a ∇ a is the covariant Laplacian, both acting on sections of E. d denotes the exterior derivative on Σ. Note that the adjoint ∇ * a depends on the metric h. See Appendix A for detailed definitions.
In the standard Ginzburg-Landau equations, κ is the dimensionless Ginzburg-Landau material parameter, ψ the complex order parameter for the electronic condensate on Σ, and a the vector potential, with the 2-form da giving the magnetic field, |ψ| 2 the local density of superconducting electrons, and J(ψ, a) := Im(ψ∇ a ψ) the supercurrent density.
(GL) are the Euler-Lagrange equations for the Ginzburg-Landau energy, The Hermitian metric h enters (1.1) through the area 2-form ω induced by h. Physically, (1.1) corresponds to the Ginzburg-Landau Helmholz free energy. By the Chern-Weil correspondence (see Section 2.3 below), E(ψ, a, h) can be parametrized by the average magnetic field in the sample. It is related to the Ginzburg-Landau Gibbs free energy, depending on applied magnetic field through the Legendre transform. For more discussions, see [31].
One can think of solutions to (GL) as non-commutative versions of the Abrikosov vortex lattices, with commutative lattice L acting on C by translations replaced by a non-commutative one -a Fuchsian group Γ acting on the Poincaré half-plane H. See Section 2 for details.
One can also connect (GL) to the Ginzburg-Landau equations on a thin superconducting membrane, and we conjecture that the mathematical techniques developed in this paper can be applied to this latter model. See [24] for a review of the physics problem.
(GL) is the first and arguably the simplest gauge theory. Indeed, (GL) is invariant under local U (1)-gauge transforms (1.2) (ψ, a) → (gψ, a + g −1 dg), where g is a U (1)-valued isomorphism 1 and a is a gauge field related to the standard connection on the principal U (1)-bundle.
In this paper, we construct nontrivial solutions to the Ginzburg-Landau equations (GL) defined on a unitary line bundle E over a non-compact Riemann surface Σ of finite volume. Our existence theory holds on the arithmetic surfaces  [26]. The precise results are given in Thms. 1.1-1.2.
We also obtain asymptotics for the energy of the our solutions and we show that under condition (1.28), the energy of the our solutions is lower than that of the constant curvature solution (see (1.6) below). The precise energy estimate is given in Corollary 1.6.
Throughout the paper, we fix some non-compact Riemann surface (Σ, h), together with a unitary line bundle E → Σ, and then seek a solution pair (ψ, a) consisting of a section of and a connection on E. The parameter κ > 0 in (GL) is a fixed number. Unless otherwise stated, dependence of various quantities on κ is not displayed but always understood. 1 The fact that (1.2) is indeed a symmetry for (GL) follows from the relations ∇a(gψ) = g∇aψ + (dg)ψ = g(∇a + g −1 dg)ψ = g∇ a+g −1 dg ψ, and Im(gψ∇a(gψ)) = Im(gψg∇ a+g −1 dg ψ) = Im(ψg∇ a+g −1 dg ψ).

Main Results.
To state our main result, we introduce some definitions. To fix the ideas, we consider the family of hyperbolic metrics on Σ given by (1.4) h r = r (Im z) 2 dz ⊗ dz (r > 0).
For fixed Σ, E, and each hyperbolic h r , (GL) has the following constant curvature (or magnetic field) solutions: where ψ is the zero-section on the line bundle E, and a br is a constant curvature connection satisfying (1.7) da br = b r ω r , where deg E is the degree (or the first Chern number) of the bundle E (see Section A.2 for definitions), and (1.9) b := 2π deg E |Σ| .
The value of b r in (1.8) is determined by the Chern-Weil correspondence, see (2.17) below. Once Σ, E are fixed, the value b r can be computed explicitly using the Gauss-Bonnet formula (2.9) and the curvature parameter r in the background metric (1.4).
In the standard Ginzburg-Landau equations, solutions of the form (1.6) correspond to normal, non-superconducting states.
For fixed Σ, E, let b r = b/r with r > 0. The value b r turns out to be the smallest eigenvalue of −∆ a br (see Section 3.2 for discussions). We denote by K(r) = Null(−∆ a br − b r ) (1.10) the finite dimensional null space of −∆ a br − b r acting on the space of square integrable sections on E → Σ. Now, we define the function (1.11) β(r) := min |ξ| 4 : ξ ∈ K(r), |ξ| 2 = 1 .
Here and below, f := 1 |Σ| r f . Note that, by Hölder's inequality, β(r) > 1. Eq. (1.11) contains information about the energy of the solutions as a function of r (see Corollary 1.6).
Finally, let H k and H k be the Sobolev spaces of order k of sections and weakly co-closed 1-forms (i.e., d * α = 0 in the distributional sense) on the line bundle E → Σ, O H k and O H k stand for error terms in the sense of the norms in H k and H k , and let The definitions of these spaces are standard. See Appendix A for details. Now, we are ready to formulate our main results: Then, for each r as above, there exists a solution (1.17) (ψ(r), a(r)) to (GL) in a neighbourhood of U ⊂ X k , k ≥ 2, around (0, a br ). Moreover, the solution (1.17) is unique in U ⊂ X k up to a gauge symmetry (see (1.2)).
We remark that our results in Section 7 enable us to establish the first, existence part of Theorem 1.1 without imposing the non-degeneracy condition (1.16). Theorem 1.1 follows from the following: Theorem 1.2 (Parametrization and asymptotics). Let (1.14) hold and assume κ 2 r − b ≪ 1. Then there exists ǫ > 0 s.th. (GL) with metric (1.4) has a C 2 branch of solutions (ψ s , a s , r s ), s ∈ C, |s| ≤ ǫ, satisfying (1.19) where ξ = O H k (1) is gauge-equivalent to a holomorphic section of E corresponding to a br s , α = O H k (1) is a co-closed 1-form and satisfies, with * denoting the Hodge operator, Moreover, if (1.15) and (1.16) hold, then we can take s ∈ R ≥0 , and the solution (ψ s , a s , r s ) is unique, and the equation r = r s can be solved for s to obtain Furthermore, writing (1.21) as s = s(r) gives, for any r > 0 as above, the solution (ψ(r), a(r)) = (ψ s(r) , a s(r) ), (1.22) as in Theorem 1.1 (see (1.17)).
Remark 2. The reason of condition (1.14) is explained in Section 4, where we also show that an explicit bundle E → Σ satisfying condition (1.14) is Remark 3. As we explain in Sections 2.2 and 2.3, the number κ 2 r corresponds to the average magnetic field in superconductors. Hence, condition (1.15) gives an estimate of the neighbourhood of the critical average field strength we work in, in terms of the topological degree deg E, and the hyperbolic area of Σ through the quantization relation (1.8). In this sense (1.15) can be compared to Bradlow's condition for the existence of magnetic vortex on compact Riemann surfaces [3].
Remark 4. Similar results as ours have been obtained for compact Riemann surfaces in [5,22,23,25,27]. It seems that ours is the first rigorous existence theory for (GL) on non-compact Riemann surfaces. Remark 5. It is possible to extend Thms. 1.1-1.2 by dropping the second condition in (1.14), as we explain in Section 7.
The proof of Theorem 1.2 implies also the following proposition, whose proof can be found at the end of Section 3: We make two conjectures concerning the energy of the solution constructed in Theorem 1.2.
We expect that conjecture above can be proven similarly to the corresponding result for the original Ginzburg-Landau equations proven in [32,Thm. 4].
The significance of such a result is that it would show that by decreasing the curvature b, one passes from the (dynamically) stable constant curvature (normal) solution (1.6) to the (dynamically) stable variable curvature solution (1.22).
In fact, we expect stronger statements to be true: As a corollary of Theorem 1.2, we obtain the following energy expansion: Corollary 1.6. Let conditions (1.14), (1.15), (1.16) hold as in Theorem 1.2. Then, for the solution (ψ s(r) , a s(r) ) constructed in (1.22), we have, This corollary is proved at the end of Section 6.
Therefore, if (1.28) holds, then the solutions constructed in Theorem 1.2 are energetically favourable compared to the constant curvature one. Moreover, Corollary 1.6 shows that there is an energy crossover between the trivial and nontrivial solutions at κ c (r) = κ. Thus, at κ = κ c (r), the gaugetranslational symmetry (ensuring that a has a constant curvature) is broken and a non-gauge-translational invariant "ground state" emerges at this point.
Finally, the function β(r) ≡ β(r, Σ, E) yields the asymptotics of the GL energy of bundles over Riemann surfaces.
Remark 7. As was already indicated above, in physics, ψ is called, depending on the area, either the order parameter or the Higgs field. In our context, it is represented by the following two equivalent objects: (1) (Geometry) Sections of the unitary line bundle E with n = deg E = 0 over the surface (Σ, h ≡ h r ) (as in this section); (2) (Number theory) Γ-automorphic functions with weight k = 4πn/ |Σ| and trivial multiplier system (see Appendix A.3 and [29]). Similar parallel can be drawn for the constant curvature connections on E, weighted Maass operators on Σ, and gauge potentials with constant magnetic fields with strength b. See Section 2.3 for details.
Remark 8. Conceptually, if one drops the second part in condition (1.14), it is useful to introduce also the extended Abrikosov function First, in Section 3, we study the linearized problem associated to (GL), which reduces to understanding the spectral properties of the Laplacian −∆ a b associated to a constant curvature connection a b , viewed as an operator acting on squareintegrable sections of the unitary line bundle E → Σ. We show that the essential spectrum of −∆ a b is given by the half-line [ 1 4 + b 2 , ∞), and the lowest eigenvalue of this operator equals to b whenever the space of cusp forms on Σ is non-trivial, in which case we give explicit description of Null(−∆ a b − b). For precise statements, see Theorem 3.2.
The operator −∆ a b is known in the physics literature as the magnetic Laplacian at constant field strength b, and is studied in e.g. [2,7,8,26]. For b = 0, −∆ a b reduces to the Laplace-Beltrami operator acting on the Poincaré half-plane, whose spectral properties are well studied in [21]. For b = 0, eigenfunctions of −∆ a b are precisely the weighted Maass forms in number theory. See e.g. [4,Sec. 2], [29], and the references therein.
Next, in Section 4, we use Lyapunov-Schmidt reduction to show that a nontrivial branch of solution of the form (1.18)-(1.19) bifurcates from the constant curvature solutions (1.6), provided the metric on Σ satisfies condition (1.15). In Section 5, we solve the key bifurcation equation (4.30), which, by results from the previous section, amounts to solving (GL). In Section 6, we derive precise asymptotics for the solutions constructed in Sects. 4-5. This proves Theorem 1.2 and Corollary 1.6.
Lastly, in Section 7, we explain how to drop the non-degenerate condition in (1.14) and extend the main results above to dim K > 1.

Preliminaries
In this section, we explain the geometric setting for the results and proofs in this paper.
In the remainder of this paper, the following geometric assumptions are always understood: (1) The underlying surface Σ is of the form (2.1), with finite area, g genus, m cusp, and no elliptic points (e.g. the principal congruence subgroup Σ = H/Γ(N ) with N ≥ 2, defined in (2.4) below); (2) b > 0 in (1.8) (which can always be achieved by changing orientation so that deg E = 1, 2, . . .).

2.1.
Non-compact Riemann Surfaces. Let Σ be a connected Riemann surface.
The Uniformization Theorem states that if Σ is non-compact and not flat, then where H is the Poincaré half-plane, and Γ is a Fuchsian group, i.e. a discrete subgroup of acting freely on H. Here, the action of SL(2, R) on H is by Möbius transform, By convention, we define γ∞ = a/c. where Γ(N ) is the principal congruence subgroup of level N , By definition, Γ(N ) is a normal subgroup of the modular group SL(2, Z) for each N .
2.2. Metric and Rescaling. Let r > 0. As in Section 1, we equip the space H with the following families of hyperbolic metrics and induced area 2-forms: These carry over to Riemann surfaces of the form Σ = H/Γ. The surface (Σ, h r ) has constant curvature −1/r and surface area |Σ| r = r |Σ|, where |Σ| denotes the area of Σ w.r.t. the standard hyperbolic ω ≡ ω r | r=1 .
Suppose (Σ, h ≡ h 1 ) has finite area, g genus, m cusps, and no elliptic points. Then Gauss-Bonnet formula gives For h = h r given by (2.7), the Ginzburg-Landau energy functional (1.1) reduces to defined on the base surface (Σ, h ≡ h 1 ). To see this, we distinguish the quantities related to the metric h r by tildes and consider Plugging (2.13) into (1.1) and using ω r = rω 1 (see (2.8)), we find which gives (2.11).
The Euler-Lagrange equations for E r (ψ, a) are the rescaled (GL), given by in the space X k , k ≥ 2 defined in (1.13 The proof of this theorem is given in Appendix C. Since deg E ∈ Z, equation (2.16) implies quantization of the average magnetic field (magnetic flux).
(2.17) relates the average magnetic field (curvature) on Σ to the geometry of Σ, and the topology of the line bundle E → Σ. Indeed, (2.17) shows that varying the metric on Σ as in (2.7)-(2.8) amounts to varying the constant curvature solution.

Linearized Ginzburg-Landau Equations
From now on, the central object of study will be the rescaled GL equations (2.15). As explained in Section 2.2, these equations are posed on the unscaled surface (Σ, h ≡ h 1 ). In this section, we consider the linear problem associated to (2.15).
As a consequence of the Chern-Weil correspondence (2.16), since a b is a constant curvature connection on (Σ, h ≡ h 1 ), the number b is given by (1.9). Linearizing (2.15) at the solution (3.1), we get a decoupled system So α must be closed; but by the definition of (1.13) it must also be co-closed and hence harmonic. This establishes the following.  Thus in what follows we fix the canonical choice a b as in (3.6).
In the remaining subsections, we study the spectral properties of the magnetic Laplacian −∆ a b in the l.h.s. of (3.2). Locally, in the rectangular coordinate z = x + iy, we have This follows from direct computation using the standard definitions in Section A.
We prove the following.
Theorem 3.2. Let Σ = H/Γ be a non-compact Riemann surface with m cusps and no elliptic points.  We also obtain a description of the solution space to the static Schrödinger equation (3.2). Before we state this result, we need some preliminary definitions.
Let Γ ⊂ P SL(2, R) be a Fuchsian group. Let c i be a cusp of Γ. Then the stabilizer of c i is an infinite cyclic group generated by some parabolic transform. In symbols, 3.3. Weitzenböck-type formula. In this subsection, let a be an arbitrary unitary connection on a smooth complex line bundle E → Σ (see Appendix A for the definitions).
We decompose the covariant derivative ∇ a into (1, 0) and (0, 1) parts as With the definitions above, we can rewrite (3.9) as Throughout this subsection, a c denotes the complexification of a real valued 1-form, and is not to be confused with the constant curvature connection a b .
In the reverse direction, we have a = 2 Re a c and In terms of a c , the curvature is given by F a = 2 Re∂a c . Now we prove the following relations: Proof. We compute in local coordinates. Using the relations , together with the expression (A.3) for the curvature F a , we compute (3.16) which gives (3.13).
To find the expression for ∆ a , we use the relation ∆ a = ∇ i ∇ i to compute which gives (3.14).
Since ∂ ′′ a * ∂ ′′ a ≥ 0, eq. (3.14) implies (3.15). Proposition 3.3 and the fact that * F a b = b imply the next two corresponding relations in the constant curvature case: Proof of Theorem 3.2, Parts (b), (c). Part (b) follows from (3.18). To complete the proof of Part (c), we now claim where Ω 0 (E) denotes the space of sections of the line bundle E, and H 0 (E) the space of holomorphic sections.
Indeed, suppose (3.20) holds. By the equivalence described at the end of Section 1.1, the space H 0 (E) of holomorphic sections of a unitary line bundle E are isomorphic (as a complex vector space) to the space M k (Σ) of modular forms on Σ, with weight k = 2b = 4π deg E/ |Σ|. Since Σ is non-compact, the intersection For higher dimensional manifolds, condition (3.21) provides the means to construct a canonical holomorphization of E. The precise result is the theorem below, which gives (3.20), whereby completing the proof of Theorem 3.2(c).
The following result was obtained in [5,11]. [5] develops a more constructive approach when M is a closed Riemann surface.
3.4. Essential Spectrum of the Magnetic Laplacian. In this subsection, we compute the essential spectrum of the linearized operator −∆ a b . Here we use a direct method of geometric decomposition, and we will use freely the standard definitions from Appendix A and results from Appendix B. First, we identify Σ with a fundamental domain F Σ ⊂ H of Γ, with the sides identified as in the proof of Theorem 2.2. Let Then γ i ∈ SL(2, R) and See Figure 3 below. Now, we decompose F Σ into a compact connected set, U 0 , and neighbourhoods U i of the cusps c i , in such a way that . . , m, the map ϕ i maps the domains U i isometrically onto the half-cylinder for some s i ≫ 1. So long as s i ≫ 1, such decomposition of F Σ is easy to construct, see Figure 2 below. Im z In the next two lemmas, we analyze the spectral property of −∆ a b in each domain U i separately.
Proof. 1. First, note that the compact domain U 0 and the cuts do not contribute to the essential spectrum. Next, let Figure 2).
Let χ i be a partition of unity on Σ adapted to Then the IMS formula (see [9]) gives we expect that −∆ a b and T i have the same essential spectrum. To show this, we will repeatedly use the fact that compact operators form a two-sided ideal among the bounded operators. By The operator B is bounded on L 2 . Since χ 0 has its support bounded F Σ , the operator K is a compact operator. It follows that ∆ a b χ 0 (−∆ a b + 1) −2 = BK is compact. This proves the claim.
3. We will use a modified version of Weyl's Theorem for relative compact perturbations: . Let H and W be two self-adjoint operators s.th. W (H + 1) −n is compact for some n ≥ 1. Then We use this proposition with W = R from (3.29) and H = −∆ a b . Then it follows from the claim proved in Step 2 and Proposition 3.7 that Proof of the claim: By construction, Hence, one can show using Weyl's criterion that Next, we show that for each i = 1, . . . , m, Indeed, for each i = 1, . . . , m, we write where K i 's are defined by this relation. Explicitly, , and let {u n } be a Weyl sequence for T i and λ, i.e.
Since supp u n ⊂ U ′ i , each element in the sequence {u n } satisfies the Dirichlet boundary condition on U ′ i . By the choice of χ i , see (3.27), the factor 1 − χ i vanishes away from the bounded set U ′ i \ U i . Hence, it follows from (3.35)-(3.36) that K i u n → 0 strongly in L 2 as n → ∞ for every i = 1 . . . , m.
Proof. By construction, the map ϕ i in (3.23) maps U i isometrically onto the halfcylinder Z i in (3.24). This transformation maps −∆ a b Ui unitarily to another operator, say h i , acting on L 2 (Z i ) with the Dirichlet boundary conditions on It remains to compute the spectra of h i .
First, we note that by the invariance of the magnetic Laplacian, see (A.18), the spectral properties of −∆ a b are not affected under any isometric transform γ ∈ SL(2, R) acting on H. Hence, up to an initial gauge transform, we can assume h i is of the form (3.7), now acting on the half-cylinder Z i .
Next, we pass from h i to another unitary equivalent operator Now, we apply the Fourier transform in x for the operator p to obtain the decom- For k = 0, (3.41) can be written as and therefore we have the estimate Furthermore, by Hardy's inequality we have the lower bound This shows that σ b ≥ 1/4 + b 2 , and therefore by (3.43), it follows that σ(p) ⊂ . This, together with relations (3.39)-(3.40), proves the lemma.
Proof of Theorem 3.2, Part (d). Lemmas 3.6 and 3.8 together imply Here we note that as far as the bifurcation argument in Section 4 is concerned, a lower bound for the essential spectrum such as (3.44) suffices for our purposes.
To prove the inclusion Proof of Proposition 1.3. Consider the Ginzburg-Landau energy (1.1). The Hessian of E(·, h r ) at (0, a br ) is given by which can be computed as in (4.13). By Theorem 3.2(b) and rescaling, −∆ a br ≥ b r . Thus,

Bifurcation Analysis
In the previous sections, we have found that, for the constant curvature connection a b on the unitary line bundle E over a Riemann surface Σ, the ground state energy of the magnetic Laplacian −∆ a b equals to b. Moreover, the parameter b is determined by the degree of E and the signature of Σ.
Let n = deg E. Recall that |Σ| is the area of Σ w.r.t. the standard hyperbolic metric, and b = 2πn/ |Σ| is the critical value of the average field strength. In this section, we construct solutions to the rescaled GL equation, (2.15), for scaling parameter r close to b/κ 2 . This emerging solution corresponds to a second order phase transition as the applied field strength is lowered past the critical value b = 2πn/ |Σ|.
We follow the general approach of [5]. Here we note two additional difficulties specific to our situation: These lead to subtle technical issues due to bifurcations with higher multiplicity and bifurcation from essential spectrum. We sidestep those issues here by assuming that dim(−∆ a b − b) = 1 and b = 1/2 (see (1.14)), and explain how to overcome the second one in Section 7.
The main result of this section is the following: The solution u s is unique, up to a gauge symmetry, in a small neighbhourhood U ⊂ X k around 0.
Furthermore, r s has the following expansion, (4.11) and similarly for derivatives.
The remainder of this section is devoted to the proof of Proposition 4.1. First, we summarize the key properties of the map F (u, r) from (4.4) in the following proposition. Below, we identify H s with a real Banach space through ψ ↔ (Re ψ, Im ψ) and view F : X s × R → X s−2 as a map between real Banach spaces. From now till the end of this paper, we assume s ≥ 2 in (4.4).

Lemma 4.2. We have
(1) The map F is C 2 from the real Banach spaces X s × R to X s−2 , s ≥ 2; (2) F has gauge symmetry in the sense that for every θ ∈ R, we have Proof. The first claim follows from the fact that the map F is a polynomial in ψ,ψ, α and their derivatives up to the second order, together with the Sobolev inequalities and properties of fractional derivatives. The second claim follows directly from definition (4.4). The last claim is a rephrasing of (4.8).
Next, consider the linearized operator of F at the trivial branch (u, r) = (0, r), given explicitly by (c.f. For fixed r, this map is well-defined as the partial Fréchet derivative of F at u = 0. Moreover, d u F (0, r) is continuous from X s → X s−2 for s ≥ 2.
where Ω is given in (3.5) and K in (3.19).
The goal now is to show that a non-trivial branch of solution to (4.7) bifurcate from the space N if 0 < κ 2 r − b ≪ 1.

4.2.
Lyapunov-Schmidt Reduction. In this subsection, we use the Lyapunov-Schmidt reduction to reduce the infinite-dimensional problem (4.7) to a finitedimensional one.
Define a linear operator Q : X s → X s by Here, the operator Q ′ : H s → H s is the orthogonal projection onto Ω defined in (3.5), which can be identified with the space of equivariant harmonic 1-forms. R(z) is the resolvant of −∆ a b − b at 0, which is well-defined if b = 1/2. By construction, Q given in (4.16) is an orthogonal projection onto the space N ≡ N b/κ 2 from (4.15).
Define v = Qu, w = Q ⊥ u, where Q ⊥ = 1 − Q is the projection onto N ⊥ ⊂ X s . Then the key equation (4.7) is equivalent to the following two equations, eq. (4.18) has a unique solution w = w(v, r) ∈ ran Q ⊥ ⊂ X s which satisfies , and similarly for the derivatives of w.
Proof. 1. We first prove the existence and uniqueness of solution to (4.18). By Lemma 4.2 Parts (1), (3), and the Implicit Function Theorem, eq. (4.18) has a unique solution w = w(v, r) ∈ N in a small neighbourhood of (v, r) We first consider the ψ-component of this operator. By assumption, the ground state energy b = 1/2. Then by Theorem 3.2, Parts (c)-(d), b is an isolated eigenvalue of −∆ a b (see Remark 10). In this case, by elementary spectral theory (e.g. [19, Thm. 6.7]), the operator −∆ a b −b is invertible on K ⊥ .
It remains to consider the α-component of the diagonal operator (4.20), namely d * d. On the space of co-closed 1-forms, d * d equals to the Hodge Laplacian. By the lower bound on the essential spectrum of the Hodge Laplacian acting on 1forms, proved in e.g. [ whereL v,r := Q ⊥ d u F (v, r)Q ⊥ , and N v,r (w) is the nonlinearity defined through this relation. The explicit expansion forL 0,r follows from (4.13). By Proposition 3.1 and Theorem 3.2, the operatorL 0,b/κ 2 ≥ δQ ⊥ for some δ > 0 and is therefore invertible. By this fact and elementary perturbation theory, we find thatL v,r is invertible for v X s + κ 2 r − b ≪ 1, Direct computation using the definition (4.4) of F and the fact v ∈ N (see (4.14)-(4.15)) shows that Here and below, for a vector u ∈ X s we write u = ([u] ψ , [u] α ). Now we rewrite (4.21) as the fixed point equation ). Applying (4.24)-(4.26) to (4.27) and using triangle inequality, we find provided v X s + κ 2 r − b ≪ 1. Estimates (4.28)-(4.29) give (4.19). To obtain estimates on the derivatives of w, we differentiate (4.21) and then proceed with the resulting equation as above using estimates on N v,r (w) and its derivatives.  Recall here Q is the orthogonal projection onto N := Null d u F (0, b/κ 2 ) defined in (4.16). Solving (4.30) in v and r amounts to solving (4.17)-(4.18), and therefore gives a solution u = v + w(v, r) to (4.7).
By Lemma 4.2, Part (1), we can expand By the gauge symmetry (4.12) of F , it is not hard to check that any solution to the equation F (s, t, r) = 0 (4. 35) in V gives rise to a circle of solutions to the equation F (u, r) = 0, with u = v e iθ s,t + w e iθ s,t,r and any θ ∈ R. Thus, our goal now is to solve (4.35) in V .
To begin with, we derive an explicit expression of the map F (s, t, r). Estimate with suitable estimates on the error terms.
Next, since J(ψ, α) = Im(ψ∇ a b +α ψ) (see (4.5)) is quadratic in ψ, we find, with ψ str as in (4.36),  with corresponding estimates for derivatives in s, t, r on V . Moreover, in addition to using (4.36)-(4.37), we can also eliminate the even order terms in expansion (4.39) and odd order terms in (4.42), counting |s| and |t| as of the orders 1 and 2, respectively, by the gauge invariance (4.12).
4. Next, let G(s, t, r) = QF (u str , r), where u str is defined in (4.33) and write G = ([G] ψ , [G] α ). Our goal now is to derive a more explicit formula for G in terms of the parametrization (4.32).
Hence, using these facts, we can rewrite (4.39) and (4.42) as Clearly, any solution to (4.45)-(4.46) is also a solution to the bifurcation equation (4.30). In view of the parametrization (4.31), taking inner product of (4.45)-(4.46) w.r.t. ξ and η k , we can write these equations as κ 2 r − b+R ψ (s, t, r) = 0, (4.47) dim Ω l=1 B kl t l +R α,k (s, t, r) = 0, k = 1, . . . , dim Ω. (4.48) Here, the remainders R ψ and R α,k are C 2 functions from V to C and R dim Ω , respectively, since G is a C 2 map from V to N ⊂ X s−2 , s ≥ 2. The remainders satisfy the estimates  To obtain (4.48), we use self-adjointness of the orthogonal projection Q ′ : H s → H s , and the fact that η k ∈ Ω = ran Q ′ for every k.
Moreover, this solution satisfies and similarly for their derivatives.
This proposition is proved in Section 5. This completes the proof of Proposition 4.1.

Solvability of the Bifurcation Equation
In this section, we solve the bifurcation equations (4.47)-(4.48) by proving Lemma 4.4.
We differentiate w.r.t. t and use and the remainder estimate (4.50) to find that the Jacobian matrix of the l.h.s. of (4.48) at this zero is given by where B kl is as in (4.51) and 1 ≤ k, l ≤ dim Ω. Let t = (t 1 , . . . t dim Ω ) be an arbitrary non-zero real vector. We compute, for any Thus the matrix B in (5.1) is positive-definite and therefore invertible.
By construction, (4.48) is a real system of C 2 equations posed on a real vector space, R × R dim Ω × R. By the gauge symmetry (4.12), the functions R α,k depend only on |s| for all k. Hence, by the invertibility of B and the Implicit Function Theorem, eq. (4.48) has a unique C 2 solution t = (t 1 , . . . , t dim Ω ) in a small neighbourhood around (s, r) and similarly for its derivatives.
We note that, by the second relation in (4.12), the solution t depends only on |s| and r.

Precise Asymptotics of the Non-trivial branch
In this section we finish our proof of Theorem 1.2.
Here φ and γ are as in (4.32) and satisfy Moreover, we can take s ∈ R ≥0 and if, in addition, r satisfies (c.f. (1.15)) where κ c = κ c (1) is given by (1.12) , then, the equation r = r(s) can be solved for s to obtain s = s(r) with where β = β(1) is the Abrikosov function given by (1.11).
Proof. The solution branch u s from (4.53) is given by where v s = (φ(s), γ(t(s))) with φ and γ given in (4.32), t(s) = t(s, r(s)) and r s = r(s) given in (5.3)  It remains to prove (6.6). Observe that by gauge symmetry (4.12),R ψ (s, r) in (5.4) depends on s through |s|. Thus, we can assume s ∈ R. Again, by the reflection symmetry, which follows from gauge symmetry (4.12), the expansion of r in s contains only even powers. Thus, we have for some R ∈ R, and with corresponding estimates on derivatives in the remainder. Take ξ ∈ K as in (4.31). Using the expansion (6.1)-(6.2), together with the (6.8), we find Here, η := s −2 γ(t(s)), and we move to the last line using the fact that −∆ a b is self-adjoint, and the equation (6.3) satisfied by ξ.
This completes the proof of Proposition 6.1.
7. Extension to the case dim K > 1 In this section, we illustrate how to extend the main existence results in Section 1 to the degenerate case, with D = dim K > 1. We also show how to extend energy estimates to this more general setting (in Section 7.2). 7.1. Existence theory. We now make some remarks on the possibility of solving the bifurcation equations (4.7) in the case when D > 1 where D is the complex dimension of K. The Lyapunov-Schmidt reduction detailed in section 4.2 carries over mutatis mutandis for D > 1. The essential change comes in section 4.3, as we will now choose a hermitian orthonormal basis ξ j 1 ≤ j ≤ D for K. Define v = v st = (φ(s), γ(t)), where s = (s 1 , . . . , s D ), t = (t 1 , . . . , t dim Ω ), φ(s) = D j=1 s j ξ j , and γ(t) = dim Ω k=1 t k η k as in (4.32).
The solution of (7.2) is based on the following classical result of Krasnoselski that reduces, under appropriate conditions, a vector bifurcation problem to a scalar one.
We defer the elegant proof of Theorem 7.1 to an Appendix D, and turn to applying it to our problem.
The first step is to consider solutions of (7.1). We find that Lemma 7.2. Eq. (7.2) has a unique solution t = t(r, s) in a small neighbourhood of (b/κ 2 , 0), such that st(s, r) is C 2 .
We momentarily defer the proof of this lemma in order to apply it to solving the full bifurcation problem. As a consequence of the lemma, insertion of t(s, r) into the higher order terms (7.3) of (7.1) results in a biifurcation equation that is C 2 which will enable us to satisfy regularity conditions of Theorem 7.1. We now want to rewrite (7.1) in terms of real variables by setting s j = x j + iy j . We also apply a constant gauge transformation to reduce s 1 to s 1 = x 1 . With this gauge fix in place, we take s to denote the coordinates, s = (x 1 , x 2 , y 2 , . . . x D , y D ), which coordinatizes R 2D−1 . So in our application n = 2D − 1 which is odd so that condition of the theorem will be satisfied. With this gauge fix in place, (7.1) is precisely in the normal form (7.5). Indeed, setting ℓ = κ 2 r − b and substituting the solution t = t(r, s) found in Lemma 7.2 to (7.1), the equation takes the vector formG (ℓ, s) = ℓs + H(s, r) = 0 (7 .6) where H(s, r) = O(|s| 2 ). SinceG(ℓ, s) is C 2 , all conditions of Theorem 7.1 are satisfied. This establishes the existence of a supercritical zero. Given that, an application of the vector implicit function theorem yields the existence of a smooth branch of zeroes extending from the bifurcation point to the supercritical zero. Hence, Proposition 4.1 extends to be true for D > 1.
As a brief illustration we note how Theorem 7.1, for the case D = 1 aligns with our earlier proof of Lemma 4.4. From the proof of Theorem 7.1 given in Appendix D, when n = 1, the constraint simply reduces to sÊ(s) = 0. So for each s = 0, E(s) = 0. It follows that the unique C 2 curve ℓ(s) found in the proof of Theorem 7.1 determines the branch of supercritical solutions to the bifurcation equation. Applying this to (7.6), this is the branch κ 2 r(s) − b, determining a unique r(s) that satisfies We now turn to the proof of the Lemma 7.2.
Proof of Lemma 7.2. We consider the quadratic form associated to the leading terms of (7.2), t k t ℓ B kℓ (s) (7.8) = ω, |φ(s)| 2 ω L 2 (7.9) It is manifest from this that Φ is real valued and, away from s = 0, defined, differentiable and positive definite as a quadratic form in t.
To understand Φ in a neighborhood of s = 0 we do a (real) algebro-geometric blow-up of s-space at the origin. This construction replaces the origin by a compact hypersurface diffeomorphic to the projective space of all local directions through the origin. More precisely, in our situation, the blow-up is defined as follows. Let n = 2D − 1 and consider the product space R n × P n−1 with R n coordinatized by the affine coordinates w = ( x, y) and P n−1 coordinatized by the homogeneous coordinates z = [z 1 , . . . z n ]. The blow-up of R n at the origin, which we'll denote by M , is the closed subset of R n × P n−1 defined by the equations {w i z j − w j z i = 0 | i, j = 1, . . . , n}. One has a natural map π : M → R n induced by projection onto the first factor. It is fairly straightforward to check that away from w = 0, π is a diffeomorphism. However, π −1 (0) ≃ P n−1 ; it consists of all points of the form 0 × [z 1 , . . . z n ]. Now to see that points of π −1 (0) are in 1:1 correspondence with the set of lines through the origin in R n , note that a line L through the origin is parametrically given by w i = c i σ where the c i are not all zero. Then consider the lift of this line toL = π −1 (L − 0) in M − π −1 (0) whose parametrization is w i = c i σ, z i = c i σ. However, since z i are homogeneous coordinates, one may as well take this parametrization to be w i = c i σ, z i = c i . These equations are welldefined for σ = 0, defining the closure ofL in M which meets π −1 (0) in the point [c 1 , . . . , c n ] ∈ P n−1 . This defines the mapping, L → [c 1 , . . . , c n ] that gives the 1:1 correspondence between lines through the origin and points of π −1 (0). For further explanation and details we refer the reader to [18] on which the above is based.
Applying this to Φ, we set A jm = FΣ ξ mξj ω ∧ * ω = µ jm + iν jm . Then one has a representation of Φ as Under the blow-up change of coordinates to ( c, σ, r), the higher order terms scale similarly but involve powers of σ greater than 2. Consequently, the terms in (1/σ 2 ) times the quadratic form associated to (7.2) are polynomial and so these equations are clearly C 2 in the blow-up coordinates. They have a leading term that is independent of σ. Hence the positivity seen in (7.9) is inherited by (1/σ 2 )Φ(s, t) from (7.11). But the latter is defined just in terms of c which are coordinates on a compact projective space. So by continuity, (1/σ 2 )Φ(s, t) realizes its infimum at some point in P n−1 . But this infimum is bounded away from zero. So B(s) is globally invertible and, by the implicit function theorem, t is smoothly defined in the blow-up coordinates and t is O(σ 2 ) in these coordinates and so vanishes in a continuous fashion as σ → 0. Hence t on the blowup M can be pushed forward under π to t(s, r) in the original s-space. Similarly for the first derivatives of t(s, r) and the second derivatives of st(s, r).

Declarations
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Appendix A. Standard definitions and notation We sketch very briefly some notions and definitions relevant for us . To fix ideas, in what follows, E is a line bundle over a Riemann surface Σ, with fibers isomorphic to C and the gauge group G = U (1).
A.1. Connections. A connection (gauge field), a, is a real-valued one-form with certain transformation properties. For a connection a, defines the covariant derivative, ∇ a , on sections of E, which can be written locally as ∇ a ψ = ∇ψ − iaψ. Here ∇ is a fixed connection (say, the Levi-Civita one).
A connection a defines the curvature two-form F a := da. A connection a on E is said to be a constant curvature connection if its curvature is of the form da = bω, (A.1) for some b ∈ R, where ω is the standard hyperbolic 2-form on Σ.
By [5,Lem. 3.2], a constant curvature connection, a, solves the (static) Maxwell equation Let x i , i = 1, 2 be a local coordinate on Σ. With the summation convention understood, we can write a connection (gauge field), a, and the covariant derivative, ∇ a , as a = a i dx i and ∇ a = ∇ i dx i , where ∇ j := ∂ i − ia j . Let d be the exterior derivative on Σ. Then, for a section ψ and 1-form and the curvature, F a = da, of a is given by A.2. Automorphy factor. Let Σ = H/Γ be a non-compact Riemann surface of the form (2.1). Let π 1 (Σ) be the first fundamental group of Σ, which we identify with Γ. A map ρ(γ, z) : is called an automorphy factor if it satisfies the important co-cycle condition, For fixed γ ∈ Γ, we often write ρ γ (z) ≡ ρ(γ, z) as a function from H to U (1). For every b ∈ R, we fix a canonical choice of automorphy factor with weight b as For results in this paper, we understood that b ∈ Z, but (A.5) can also be defined In fact, (A.6) defines an one-to-one correspondence, ρ ↔ E ρ , between (equivalence classes of) automorphy maps for Σ and line U (1)-bundles over Σ. See [16] for details. Through the correspondence (A.6), one can define a topological degree for a unitary line bundle over Σ, which by the Chern-Weil correspondence is equal to the Chern number, or the degree of E, as follows. Suppose E = E ρ as in (A.6) for an automorphy factor ρ satisfying (A.4). Suppose the Fuchsian group Γ has genus g and m cusps, with no elliptic points. Then Γ is generated by 2g hyperbolic transforms, A 1 , B 1 , . . . , A g , B g , and m parabolic transforms S 1 , . . . , S m , satisfying the relation Now, for every γ ∈ Γ, define a map σ = σ γ : H → R by the relation e iσγ (z) := ρ(γ, z). The first Chern number c 1 (σ) is defined by [17,Theorem 2A], c 1 (σ) is independent of z 0 , and takes values in Z.
Let E and E ′ be the line bundles corresponding to two automorphy factors e iσ , e iσ ′ respectively. Then we say that E and E ′ are equivalent if and only if c 1 (σ) = c 1 (σ ′ ).
Hence, by the one-to-one correspondence (A.4), c 1 (σ) is a topological invariant for a unitary line bundle E = E ρ over Σ = H/Γ. It turns out (see [15]) that c 1 (σ) is equal to the topological degree of E: where γ * is the pull-back by the Möbius transform (2.2) associated to γ ∈ Γ, and ρ(γ, z) is an automorphy factor, satisfying the co-cycle condition (A.4) Remark 14. For a standard lattice L ⊂ C ∼ = R 2 , equivariant solutions to (GL) are known as the Abrikosov lattices, predicted by A.A. Abrikosov in 1957 [1]. (This discovery was recognized by a Nobel prize.) These are ground state solutions to (GL) on the flat torus (A.12) T = C/L, which is a compact Riemann surface of genus 1. Existence and stability theory of solutions to (GL) on T are studied in [32][33][34][35]. Physically, these solutions correspond to regular arrays of vortices as seen in Type II superconductors. In (A.12), the lattice L acts on the complex plane by translation, and the action is commutative. In comparison, with the background geometry (2.1), the action of Γ on the Poincaré half plane H by Möbius transforms is in general non-commutative. In this sense, equivariant solutions to (GL) on (2.1) are noncommutative generalizations of the Abrikosov lattice. The equivariant states picture is useful for explicit computations. Below, we obtain an explicit description of the null space K from (3.19) in terms of equivariant functions in L 2 (H/Γ) and prove inclusion (3.45), using similar symmetrization argument as in [21], which studies the same problem with b = 0. This method was introduced by Selberg in 1950s, and is well-known to the number theorists. where each Γ i := Stab(c i , Γ) denotes the stabilizer of a distinct cusp c i of Γ, and γ i ∈ SL(2, R) is a scaling matrix of c i .
Moreover, there holds the asymptotics , which implies that ξ i (z) decays exponentially fast as z approaches the cusp c i .
Then for every cusp c i , we can form the Poincaré series This series (3.1) converges absolutely if φ satisfies certain growth condition in y.
The Poincaré series has two important properties: (a) By construction, E ci,φ satisfies the equivariance condition (A.10); (b) If L is an invariant operator acting on H/Γ in the sense that for every γ ∈ Γ, φ ∈ H s , and Lφ = λφ for some λ ∈ C, then LE ci,φ = λE ci,φ . In view of these properties, to solve the eigenvalue problem (3.2) in L 2 (H/Γ), we first solve the following problem for some φ ∈ C 2 (H, C): This is calculated in e.g. [7,8]. Forming Poincaré series (A.17) w.r.t. each of the cusps of Σ gives (A.14). By construction, (A.14) are equivariant solution to (3.2).
By the classical results for the Fourier analysis of Poincaré series, e.g. [29,Sect. 2], there holds the Fourier expansion Here W β,µ (y) = O(e −y/2 ) is the Whittaker function, a decaying solution to the ODE This W β,µ (y) can be expressed in terms of the modified Bessel function of the second kind [14]. Expansion (A.21) implies the asymptotics (A.15).
Remark 15. Estimates for the coefficients A k in (A.21) are of significant interest in number theory, and have been obtained in [13,20].
Proof of (3.45). We use the same symmetrization method as in the proof of Proposition A.2. First, we seek C 2 (H, C)-solutions to the eigenvalue problem These φ k 's are the generalized eigenfunctions, which correspond to the spectral points (A.25) λ k := k 2 + 1 4 + b 2 (k ≥ 0).

Finally, letũ
Thenũ n ∈ L 2 (H/Γ) forms a Weyl sequence for −∆ a b and λ k , c.f. (3.36). This shows that λ k ∈ σ ess (−∆ a b ). Varying k, we find Appendix B. Classification of U (1)-Automorphy Factors and Constant Curvature U (1)-Connections In this section, we reproduce with minor modifications results of [5] on classification of automorphy factors and constant curvature connections.
Recall that a character of Γ is a homomorphism χ : Γ → U (1). For other standard definitions, see Appendix A.
Proof. Let s = a b c d , t = e f g h ∈ Γ. Using (B.1), we compute for any β, ρ β (s · t, z) = (ce + dg)z + (cf + dh) (ce + dg)z + (cf + dh) Using the formula for the Chern class, c 1 (ρ), of a co-cycle ρ (see [17], Theorem 2a), we compute c 1 (ρ β ) = n, provided β = 2πn |Σ| , where |Σ| is the area of Σ w.r.t. to the standard hyperbolic metric. Remark 16. The description in the above theorem does not depend at all on the complex structure of the underlying Riemann surface. Hence, if E n,χ is the unitary line bundle corresponding to the automorphy factors (B.2), then the projection of A b to E n,χ gives the distinguished connection a b,χ on E n,χ .
Proof of Theorem B.2. We begin with some preliminary constructions. We consider the trivial bundle,Ẽ := H × C with the standard complex structure on H associated to the standard hyperbolic metrich = (Im z) −2 |dz| 2 .
Since we work here on a global product space, it is natural to take the fiber metric to be induced from the metric on the base. So we take the metric on the fiber C over the point z ∈ H to be k z = (Im z) −2 |dw| 2 , where w is the coordinate on the fiber C z .
Let the connection A be given by A := A 1 dx 1 + A 2 dx 2 . We decompose the covariant derivative ∇ A into (1, 0) and (0, 1) parts as ∇ A = ∂ ′ A + ∂ ′′ A , where ∂ ′ A and ∂ ′′ A are defined in (3.9)-(3.10). Recall from Section 3.3 that, in terms of A c , the curvature is given by F A = 2 Re∂A c . Moreover, if A c satisfies the equivariance relation s * Ā c =Ā c − i∂f s , then A satisfies s * A = A + df s , with f s satisfying df s := 2 Im ∂f s . According to (3.10), the complexification of the connection A b given in the theorem is dz.
In the remaining of this proof, we omit the superindex b in A b and A b c . Proof of constant curvature. Using that ω = i 2 Im(z) −2 dz ∧ dz, we find ω.
Remark 18. A b is R -linear while A c is naturally C -linear. It is natural to ask what the action of i on A n c does when we map back to A n . A simple calculation shows that iA b c = i b 2 1 Im(z) dz is mapped to by −1 dy, which is flat. It turns out that the complex action of i induces a rotation into the space of flat connections. as well as the decomposition Here the plus sign denotes disjoint union.
Let A be an equivariant 1-form on F Σ that corresponds to the connection a on E → Σ, as in Sections A.2-A.3. By definition (A.9), the theorem is proved once we establish 1 2π ∂FΣ A = c 1 (σ).
Since the 1-form A is gauge-equivariant, we have (C.2) γ * A = A − iρ −1 γ dρ γ , ∀γ ∈ Γ, z ∈ ∂F Σ . This, together with the definition ρ(γ, z) = e iσγ (z) , implies where v i , v i+1 are the vertices of F Σ spanned by the side α i . Similarly we proceed with βi+β ′ i A and δj +δ ′ j A. In the last case, since the gauge exponent σ Si (z) is independent of z, we have δj +δ ′ j A = 0, so these term do not contribute.
Summing up these contributions and using (C.1), we arrive at where v i , v i+1 and w i , w i+1 are the vertices of F σ spanned by α i and β i , respectively. Note that the vertices v i , v i+1 , w i , w i+1 and u i , u i+1 are related as By construction, Manifestly, φ(0, 0) = 0 and ∂φ(0, 0)/∂ℓ = 1. It then follows from the implicit function theorem that there exists a unique function ℓ(s), for small s, such that φ(ℓ(s), s) = 0. SetÊ(s) = E(ℓ(s), s), where, recall, E(ℓ, s) = ℓs + H(ℓ, s), and let S ǫ := {s ∈ R n : |s| = ǫ} be the ǫ-sphere in R n . ThenÊ(s), regarded as a vector field on R n , is everywhere tangent to S ǫ . To see this, simply note that 0 = φ(ℓ(s), s) = 1 |s| 2 s, ℓ(s)s + H(ℓ(s), s) = 1 |s| 2 s,Ê(s) , which precisely states that the vector fieldÊ(s) is everywhere tangent to the sphere. Since n is odd, the sphere is even dimensional. But every vector field on an even dimensional sphere has as zero vector, i.e.Ê(s) = 0 has a solution on S ǫ . The theorem follows.