# Ginzburg-Landau Equations on Non-compact Riemann Surfaces

Nicholas M. Ercolani  Department of Mathematics, University of Arizona,
Tucson, AZ 85721-0089, USA<EMAIL_ADDRESS>,  Israel Michael Sigal
Department of Mathematics, University of Toronto, Toronto, ON M5S 2E4, Canada
<EMAIL_ADDRESS>and  Jingxuan Zhang  Department of Mathematical Sciences,
University of Copenhagen, Copenhagen 2100, Denmark  Department of Mathematics,
University of Toronto, Toronto, ON M5S 2E4, Canada<EMAIL_ADDRESS>
(Date: April 17, 2023)

###### Abstract.

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.

###### Key words and phrases:

Ginzburg–Landau equations; Superconductivity; Bifurcation theory; Elliptic
equations on Riemann surfaces

###### 2020 Mathematics Subject Classification:

35Q56 (primary); 35J66 (secondary)

##  1\. Introduction

We consider the Ginzburg-Landau equations on a line bundle $E$ over a Riemann
surface $\Sigma$ with a Hermitian metric $h$:

(GL) $\displaystyle-\Delta_{a}\psi$
$\displaystyle=\kappa^{2}\left(1-\left\lvert\psi\right\rvert^{2}\right)\psi,$
$\displaystyle d^{*}da$
$\displaystyle=\operatorname{Im}\left(\bar{\psi}\nabla_{a}\psi\right).$

Here $\kappa>0$ is a fixed parameter. $\psi$ and $a$ are respectively a
section of and a connection $1$-form on the line bundle $E$. $\nabla_{a}$ is
the covariant derivative induced by the $1$-form $a$, and
$-\Delta_{a}=\nabla_{a}^{*}\nabla_{a}$ is the covariant Laplacian, both acting
on sections of $E$. $d$ denotes the exterior derivative on $\Sigma$. Note that
the adjoint $\nabla_{a}^{*}$ depends on the metric $h$. See Appendix A for
detailed definitions.

In the standard Ginzburg-Landau equations, $\kappa$ is the dimensionless
Ginzburg-Landau material parameter, $\psi$ the complex order parameter for the
electronic condensate on $\Sigma$, and $a$ the vector potential, with the
$2$-form $da$ giving the magnetic field, $\left\lvert\psi\right\rvert^{2}$ the
local density of superconducting electrons, and
$J(\psi,a):=\operatorname{Im}(\bar{\psi}\nabla_{a}\psi)$ the supercurrent
density.

(GL) are the Euler-Lagrange equations for the Ginzburg-Landau energy,

(1.1)
$\mathcal{E}(\psi,a,h)=\int_{\Sigma}\left(\frac{1}{2}\left\lvert\nabla_{a}\psi\right\rvert^{2}+\frac{1}{2}\left\lvert
da\right\rvert^{2}+\frac{\kappa^{2}}{4}\left(\left\lvert\psi\right\rvert^{2}-1\right)^{2}\right)\,\omega.$

The Hermitian metric $h$ enters (1.1) through the area $2$-form $\omega$
induced by $h$.

Physically, (1.1) corresponds to the Ginzburg-Landau Helmholz free energy. By
the Chern-Weil correspondence (see Section 2.3 below), $\mathcal{E}(\psi,a,h)$
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 [MR3381531].

One can think of solutions to (GL) as non-commutative versions of the
Abrikosov vortex lattices, with commutative lattice $\mathcal{L}$ acting on
$\mathbb{C}$ by translations replaced by a non-commutative one – a Fuchsian
group $\Gamma$ acting on the Poincaré half-plane $\mathbb{H}$. 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 [MR1426137]
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) $(\psi,a)\mapsto(g\psi,a+g^{-1}dg),$

where $g$ is a $U(1)$-valued isomorphism 111The fact that (1.2) is indeed a
symmetry for (GL) follows from the relations
$\nabla_{a}(g\psi)=g\nabla_{a}\psi+(dg)\psi=g(\nabla_{a}+g^{-1}dg)\psi=g\nabla_{a+g^{-1}dg}\psi$,
and
$\operatorname{Im}(\overline{g\psi}\nabla_{a}(g\psi))=\operatorname{Im}(\overline{g\psi}g\nabla_{a+g^{-1}dg}\psi)=\operatorname{Im}(\bar{\psi}g\nabla_{a+g^{-1}dg}\psi).$
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 $\Sigma$ of finite volume. Our existence theory holds on the
arithmetic surfaces

(1.3) $\Sigma\cong\mathbb{H}/\Gamma(N),\quad N=1,2,\ldots,$

where $\mathbb{H}$ is the Poincaré half-plane, equipped with a hyperbolic
metric with constant negative curvature, and $\Gamma(N)$ is the principal
congruence subgroup of level $N$, acting on $\mathbb{H}$ by Möbius transform.
(GL) on a non-compact Riemann surface $\Sigma$ of the form (1.3) could serve
as a toy model of a superconducting circuit with several narrow open ends
[Pnueli]. 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 $(\Sigma,h)$,
together with a unitary line bundle $E\to\Sigma$, and then seek a solution
pair $(\psi,a)$ consisting of a section of and a connection on $E$. The
parameter $\kappa>0$ in (GL) is a fixed number. Unless otherwise stated,
dependence of various quantities on $\kappa$ is not displayed but always
understood.

###  1.1. Main Results

To state our main result, we introduce some definitions. To fix the ideas, we
consider the family of hyperbolic metrics on $\Sigma$ given by

(1.4) $h_{r}=\frac{r}{(\operatorname{Im}z)^{2}}dz\otimes d\bar{z}\quad(r>0).$

The corresponding area $2$-form is given by

(1.5) $\omega_{r}=\frac{ri}{2(\operatorname{Im}z)^{2}}dz\wedge d\bar{z}.$

Each $h_{r}$ turns $\Sigma$ into a surface with constant curvature $-1/r$.
Denote $\left\lvert\Sigma\right\rvert$ the area of $\Sigma$ w.r.t. the
standard hyperbolic $2$-form
$\omega_{1}\equiv\tfrac{i}{2(\operatorname{Im}z)^{2}}dz\wedge d\bar{z}.$ Then
the surface $(\Sigma,h_{r})$ has area
$\left\lvert\Sigma\right\rvert_{r}=r\left\lvert\Sigma\right\rvert$.

For fixed $\Sigma$, $E$, and each hyperbolic $h_{r}$, (GL) has the following
constant curvature (or magnetic field) solutions:

(1.6) ${\psi\equiv 0,\quad a=a^{b_{r}}},$

where $\psi$ is the zero-section on the line bundle $E$, and $a^{b_{r}}$ is a
constant curvature connection satisfying

(1.7) $da^{b_{r}}={b_{r}}\omega_{r},$

and

(1.8) ${b_{r}}\equiv b(r,\Sigma,E):=\frac{2\pi\deg
E}{\left\lvert\Sigma\right\rvert_{r}}=\frac{b}{r},$

where $\deg E$ is the degree (or the first Chern number) of the bundle $E$
(see Section A.2 for definitions), and

(1.9) $b:=\frac{2\pi\deg E}{\left\lvert\Sigma\right\rvert}.$

The value of $b_{r}$ in (1.8) is determined by the Chern-Weil correspondence,
see (2.17) below. Once $\Sigma,\,E$ 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 $\Sigma$, $E$, let $b_{r}=b/r$ with $r>0$. The value $b_{r}$ turns
out to be the smallest eigenvalue of $-\Delta_{a^{b_{r}}}$ (see Section 3.2
for discussions). We denote by

(1.10) $\displaystyle K(r)=\operatorname{Null}(-\Delta_{a^{b_{r}}}-b_{r})$

the finite dimensional null space of $-\Delta_{a^{b_{r}}}-b_{r}$ acting on the
space of square integrable sections on $E\to\Sigma$. Now, we define the
function

(1.11)
${\beta(r):=\min\left\\{{\langle\left\lvert\xi\right\rvert^{4}\rangle}:\xi\in
K(r),\langle\left\lvert\xi\right\rvert^{2}\rangle=1\right\\}.}$

Here and below, $\langle
f\rangle:=\frac{1}{\left\lvert\Sigma\right\rvert_{r}}\int f$. Note that, by
Hölder’s inequality, $\beta(r)>1$. Eq. (1.11) contains information about the
energy of the solutions as a function of $r$ (see Corollary 1.6).

We define another function of threshold Ginzburg-Landau parameter,
$\kappa_{c}(r)$, as

(1.12) ${\kappa_{c}(r):=\sqrt{\frac{1}{2}\left(1-\frac{1}{\beta(r)}\right)}.}$

Finally, let $\mathcal{H}^{k}$ and $\vec{\mathcal{H}}^{k}$ be the Sobolev
spaces of order $k$ of sections and weakly co-closed $1$-forms (i.e.,
$d^{*}\alpha=0$ in the distributional sense) on the line bundle $E\to\Sigma$,
$O_{\mathcal{H}^{k}}$ and $O_{\vec{\mathcal{H}}^{k}}$ stand for error terms in
the sense of the norms in $\mathcal{H}^{k}$ and $\vec{\mathcal{H}}^{k}$, and
let

(1.13) $\displaystyle X^{k}:=\mathcal{H}^{k}\times\vec{\mathcal{H}}^{k}.$

The definitions of these spaces are standard. See Appendix A for details.

Now, we are ready to formulate our main results:

######  Theorem 1.1 (Existence and uniqueness).

Let $(\Sigma,h_{r}),\,r>0$, be a non-compact Riemann surface equipped with the
hyperbolic metric (1.4) with finite area. Let $E\to\Sigma$ be a unitary line
bundle, with $b:=2\pi\deg E/\left\lvert\Sigma\right\rvert>0.$ Suppose

(1.14) ${b\neq 1/2},\quad$

and $r>0$ satisfies

(1.15) $\displaystyle{0<\left\lvert\kappa^{2}r-{b}\right\rvert\ll
1,\quad{(\kappa-\sqrt{b_{r}})(\kappa-\kappa_{c}(r))}>0},$ (1.16)
$\displaystyle{\dim_{\mathbb{C}}K(r)=1.}$

Then, for each $r$ as above, there exists a solution

(1.17) $(\psi(r),a(r))$

to (GL) in a neighbourhood of $U\subset X^{k}$, $k\geq 2$, around
$(0,a^{{b_{r}}})$.

Moreover, the solution (1.17) is unique in $U\subset X^{k}$ 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 $\left\lvert\kappa^{2}r-b\right\rvert\ll 1$. Then
there exists $\epsilon>0$ s.th. (GL) with metric (1.4) has a $C^{2}$ branch of
solutions $(\psi_{s},a_{s},r_{s})$, $s\in\mathbb{C}$, $\left\lvert
s\right\rvert\leq\epsilon$, satisfying

(1.18) $\displaystyle\psi_{s}=s\xi+O_{\mathcal{H}^{k}}(\left\lvert
s\right\rvert^{3}),$ (1.19) $\displaystyle a_{s}=a^{{b_{r_{s}}}}+\left\lvert
s\right\rvert^{2}\alpha+O_{\vec{\mathcal{H}}^{k}}(\left\lvert
s\right\rvert^{4}),$

where $\xi=O_{\mathcal{H}^{k}}(1)$ is gauge-equivalent to a holomorphic
section of $E$ corresponding to $a^{{b_{r_{s}}}}$,
$\alpha=O_{\vec{\mathcal{H}}^{k}}(1)$ is a co-closed $1$-form and satisfies,
with $*$ denoting the Hodge operator,

(1.20) $\displaystyle
d\alpha=\frac{1}{2}*\left(1-\left\lvert\xi\right\rvert^{2}\right).$

Moreover, if (1.15) and (1.16) hold, then we can take $s\in\mathbb{R}_{\geq
0}$, and the solution $(\psi_{s},a_{s},r_{s})$ is unique, and the equation
$r=r_{s}$ can be solved for $s$ to obtain

(1.21)
$\displaystyle{s}^{2}={\frac{\kappa^{2}-b_{r}}{(\kappa^{2}-\tfrac{1}{2})\beta(r)+\tfrac{1}{2}}+O((\kappa^{2}-b_{r})^{2})}.$

Furthermore, writing (1.21) as $s=s(r)$ gives, for any $r>0$ as above, the
solution

(1.22) $\displaystyle(\psi(r),a(r))=(\psi_{s(r)},a_{s(r)}),$

as in Theorem 1.1 (see (1.17)).

This theorem is proved in Sects. 4–6.

######  Remark 1.

Condition (1.15) guarantees the r.h.s. of (1.21) is positive, and was first
isolated as a criterion for the existence of the Abrikosov vortex lattice in
[MR2560758, MR2904275, MR3123370, MR3758428]. Through (1.8), condition (1.15)
gives rise to the critical magnetic field $b_{r}=\kappa^{2}$.

######  Remark 2.

The reason of condition (1.14) is explained in Section 4, where we also show
that an explicit bundle $E\to\Sigma$ satisfying condition (1.14) is

(1.23) $\displaystyle\Sigma=\mathbb{H}/\Gamma(6),\quad\deg E=12.$

######  Remark 3.

As we explain in Sections 2.2 and 2.3, the number $\kappa^{2}r$ 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 $\Sigma$ 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 [MR1086749].

######  Remark 4.

Similar results as ours have been obtained for compact Riemann surfaces in
[MR3790522, MR4049917, MR4213771, nagy2022nonminimal, MR1451329]. 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:

######  Proposition 1.3.

The constant curvature solution $(\psi\equiv 0,a=a^{b_{r}})$ (see (1.6) ) is a
minimizer of the energy $\mathcal{E}(\psi,a,h_{r})$ if and only if

(1.24) ${b_{r}>\kappa^{2}.}$

We make two conjectures concerning the energy of the solution constructed in
Theorem 1.2.

######  Conjecture 1.4.

Under the assumption of Theorem 1.2, solution (1.22) is a local minimizer of
the energy $\mathcal{E}(\psi,a,h_{r})$ if and only if

(1.25) ${b_{r}<\kappa^{2}.}$

We expect that conjecture above can be proven similarly to the corresponding
result for the original Ginzburg-Landau equations proven in [MR2904275, 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:

######  Conjecture 1.5.

The constant curvature solution $(\psi\equiv 0,a=a^{b_{r}})$ (see (1.6) ) is a
global minimizer of the energy $\mathcal{E}(\psi,a,h_{r})$ if (1.24) holds. If
(1.25) holds, then solution (1.22) is a global minimizer of the energy
$\mathcal{E}(\psi,a,h_{r})$.

The energy associated to the constant curvature solution (1.6) is

(1.26)
$\displaystyle\mathcal{E}(0,a^{b_{r}},h_{r})=\frac{1}{2}\left({b_{r}^{2}}+\frac{\kappa^{2}}{2}\right)\left\lvert\Sigma\right\rvert_{r}.$

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 $(\psi_{s(r)},a_{s(r)})$ constructed in (1.22), we have,

(1.27)
${{\mathcal{E}(\psi_{s(r)},a_{s(r)},h_{r})=\mathcal{E}(0,a^{b_{r}},h_{r})}{}}-{\frac{\left\lvert\Sigma\right\rvert_{r}}{4}\frac{\left\lvert\kappa^{2}-{b_{r}}\right\rvert^{2}}{(\kappa^{2}-\tfrac{1}{2})\beta(r)+\tfrac{1}{2}}+O\left(\left\lvert\kappa^{2}-{b_{r}}\right\rvert^{3}\right)}.$

This corollary is proved at the end of Section 6.

######  Remark 6.

Since
$(\kappa^{2}-\tfrac{1}{2})\beta(r)+\tfrac{1}{2}=\beta(r)(\kappa^{2}-\kappa_{c}^{2}(r))$,
expansion (1.27) ensures that
$\mathcal{E}(\psi_{s(r)},a_{s(r)},h_{r})<\mathcal{E}(0,a^{b_{r}},h_{r})$,
provided

(1.28) $\kappa>\kappa_{c}(r).$

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 $\kappa_{c}(r)=\kappa$. Thus, at
$\kappa=\kappa_{c}(r)$, the gauge-translational 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 $\beta(r)\equiv\beta(r,\Sigma,E)$ yields the asymptotics
of the GL energy of bundles over Riemann surfaces.

######  Remark 7.

As was already indicated above, in physics, $\psi$ 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. (1)

(Geometry) Sections of the unitary line bundle $E$ with $n=\deg E\neq 0$ over
the surface $(\Sigma,h\equiv h_{r})$ (as in this section);

  2. (2)

(Number theory) $\Gamma$-automorphic functions with weight $k=4\pi
n/\left\lvert\Sigma\right\rvert$ and trivial multiplier system (see Appendix
A.3 and [MR750670]).

Similar parallel can be drawn for the constant curvature connections on $E$,
weighted Maass operators on $\Sigma$, 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

$\begin{array}[]{ccrcl}{\beta}&\colon&{\operatorname{Null}(-\Delta_{a^{b_{r}}}-b_{r})\times\mathbb{R}_{>0}\times\left\\{\text{Fuchsian
groups}\right\\}}&\longrightarrow&{\mathbb{R}_{\geq 0}}\\\
\mbox{}&\mbox{}&{(\xi,r,\Gamma)}&\longmapsto&{\left\lVert\xi\right\rVert_{\mathcal{L}^{4}}^{4}/\left\lVert\xi\right\rVert_{\mathcal{L}^{2}}^{2}}\end{array}.$

Then we have
$\beta(r,\Gamma)=\inf\left\\{\beta(\xi,r,\Gamma):\xi\in\operatorname{Null}(-\Delta_{a^{b_{r}}}-b_{r})\right\\}$,
c.f. (1.11).

###  1.2. Organization of the paper

In Section 2, after giving some preliminary definitions, we show that (GL) on
$(\Sigma,h_{r})$ with $h_{r}$ as in (1.4) is equivalent to the rescaled
Ginzburg-Landau equations, (2.15), posed on $(\Sigma,h\equiv h_{1})$. Then the
proof of Theorem 1.2 consists of two parts of analysis on the rescaled
equation (2.15).

First, in Section 3, we study the linearized problem associated to (GL), which
reduces to understanding the spectral properties of the Laplacian
$-\Delta_{a^{b}}$ associated to a constant curvature connection $a^{b}$,
viewed as an operator acting on square-integrable sections of the unitary line
bundle $E\to\Sigma$. We show that the essential spectrum of $-\Delta_{a^{b}}$
is given by the half-line $[\tfrac{1}{4}+b^{2},\infty)$, and the lowest
eigenvalue of this operator equals to $b$ whenever the space of cusp forms on
$\Sigma$ is non-trivial, in which case we give explicit description of
$\operatorname{Null}(-\Delta_{a^{b}}-b)$. For precise statements, see Theorem
3.2.

The operator $-\Delta_{a^{b}}$ is known in the physics literature as the
magnetic Laplacian at constant field strength $b$, and is studied in e.g.
[MR776146, MR870891, MR1069475, Pnueli]. For $b=0$, $-\Delta_{a^{b}}$ reduces
to the Laplace-Beltrami operator acting on the Poincaré half-plane, whose
spectral properties are well studied in [MR1942691]. For $b\neq 0$,
eigenfunctions of $-\Delta_{a^{b}}$ are precisely the weighted Maass forms in
number theory. See e.g. [MR1431508, Sec. 2], [MR750670], and the references
therein.

Next, in Section 4, we use Lyapunov-Schmidt reduction to show that a non-
trivial branch of solution of the form (1.18)–(1.19) bifurcates from the
constant curvature solutions (1.6), provided the metric on $\Sigma$ 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 $\dim K>1$.

##  2\. 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. (1)

The underlying surface $\Sigma$ is of the form (2.1), with finite area, $g$
genus, $m$ cusp, and no elliptic points (e.g. the principal congruence
subgroup $\Sigma=\mathbb{H}/\Gamma(N)$ with $N\geq 2$, defined in (2.4)
below);

  2. (2)

$b>0$ in (1.8) (which can always be achieved by changing orientation so that
$\deg E=1,2,\ldots$).

###  2.1. Non-compact Riemann Surfaces

Let $\Sigma$ be a connected Riemann surface. The Uniformization Theorem states
that if $\Sigma$ is non-compact and not flat, then

(2.1) $\Sigma\cong\mathbb{H}/\Gamma,$

where $\mathbb{H}$ is the Poincaré half-plane,

$\mathbb{H}:=\left\\{z:z\in\mathbb{C},\,\operatorname{Im}z>0\right\\},$

and $\Gamma$ is a Fuchsian group, i.e. a discrete subgroup of

$PSL(2,\mathbb{R}):=SL(2,\mathbb{R})/\left\\{\pm\mathbf{1}\right\\}$

acting freely on $\mathbb{H}$. Here, the action of $SL(2,\mathbb{R})$ on
$\mathbb{H}$ is by Möbius transform,

(2.2) $\gamma z=\frac{az+b}{cz+d}\quad\left(\gamma=\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\in SL(2,\mathbb{R})\right).$

By convention, we define $\gamma\infty=a/c$.

An important class of examples are the Riemann surfaces

(2.3) $\Sigma:=\mathbb{H}/\Gamma(N),\quad N=1,2,\ldots,$

where $\Gamma(N)$ is the principal congruence subgroup of level $N$,

(2.4) $\Gamma(N):=\left\\{\gamma=\begin{pmatrix}a&b\\\ c&d\end{pmatrix}\in
SL(2,\mathbb{Z}):a\equiv d\equiv 1,b\equiv c\equiv 0\mod N\right\\}.$

By definition, $\Gamma(N)$ is a normal subgroup of the modular group
$SL(2,\mathbb{Z})$ for each $N$.

######  Definition 2.1 (cusp).

Let $\Gamma$ be a Fuchsian group. A point
$c\in\mathbb{R}\cup\left\\{\infty\right\\}$ is called a cusp of $\Gamma$ if
and only if there is an element $\gamma\in\Gamma$ that is conjugate-equivalent
to some horizontal translation $z\mapsto z+h,\,h\in\mathbb{R}$, s.th. $\gamma
c=c$.

$\operatorname{Im}z=0$$z=0$$z=-1$$z=1$ Figure 1. A fundamental domain of
$\Gamma(2)$ in $\mathbb{H}$ with three cusps $c_{1}=1,c_{2}=0,c_{3}=\infty$.

For example, if $\Gamma=SL(2,\mathbb{Z})$, then the only cusp of $\Gamma$ is
$c=\infty$, as every integral translation $z\mapsto z+n,\,n\in\mathbb{Z}$
fixes $c$. If $\Gamma=\Gamma(2)$, then there are three cusps with
$c_{1}=0,c_{2}=1,c_{3}=\infty$. See Figure 1. (Note that $-1$ is equivalent to
$c_{2}$ through translation $z\mapsto z+2$.)

Denote by $m$ the number of distinct cusps and $g$ the number of genera of
$\Gamma$. For the principal congruence subgroup $\Gamma=\Gamma(N)$ defined in
(2.4), the classical results from [MR0314766, Sec. 1.6] show that

(2.5) $\displaystyle m_{N}=$ $\displaystyle\left\\{\begin{aligned}
&3\quad(N=2),\\\ &\frac{1}{2}N^{2}\prod_{p\mid
N}\left(1-\frac{1}{p^{2}}\right)\quad(N\geq 3),\end{aligned}\right.$ (2.6)
$\displaystyle g_{N}=$ $\displaystyle 1+\frac{N-6}{12}m_{N}.$

###  2.2. Metric and Rescaling

Let $r>0$. As in Section 1, we equip the space $\mathbb{H}$ with the following
families of hyperbolic metrics and induced area $2$-forms:

(2.7) $\displaystyle h_{r}$
$\displaystyle=\frac{r}{(\operatorname{Im}z)^{2}}dz\otimes d\bar{z},$ (2.8)
$\displaystyle\omega_{r}$
$\displaystyle=\frac{ri}{2\left(\operatorname{Im}z\right)^{2}}dz\wedge
d\bar{z}.$

These carry over to Riemann surfaces of the form $\Sigma=\mathbb{H}/\Gamma$.
The surface $(\Sigma,h_{r})$ has constant curvature $-1/r$ and surface area
$\left\lvert\Sigma\right\rvert_{r}=r\left\lvert\Sigma\right\rvert$, where
$\left\lvert\Sigma\right\rvert$ denotes the area of $\Sigma$ w.r.t. the
standard hyperbolic $\omega\equiv\omega_{r}|_{r=1}$.

Suppose $(\Sigma,h\equiv h_{1})$ has finite area, $g$ genus, $m$ cusps, and no
elliptic points. Then Gauss-Bonnet formula gives

(2.9) $\left\lvert\Sigma\right\rvert=2\pi(2g-2+m),$

See e.g. [MR1942691, p.43, Eqn. (2.7)]. For example, if
$\Sigma_{N}=\mathbb{H}/\Gamma(N)$ with $N\geq 2$, then $\Sigma_{N}$ has no
elliptic point. Hence, by (2.5)–(2.6) and Gauss-Bonnet formula, we have

(2.10) $\left\lvert\Sigma_{N}\right\rvert=\frac{\pi Nm_{N}}{3}.$

For $h=h_{r}$ given by (2.7), the Ginzburg-Landau energy functional (1.1)
reduces to

(2.11)
$\mathcal{E}_{r}(\psi,a)=\int_{\Sigma}\left(\frac{1}{2}\left\lvert\nabla_{a}\psi\right\rvert^{2}+\frac{1}{2}\left\lvert
da\right\rvert^{2}+\frac{\kappa^{2}}{4}\left(\left\lvert\psi\right\rvert^{2}-r\right)^{2}\right)\,\omega_{1},$

defined on the base surface $(\Sigma,h\equiv h_{1})$. To see this, we
distinguish the quantities related to the metric $h_{r}$ by tildes and
consider

(2.12) $\displaystyle\tilde{\psi}=r^{-1/2}\psi,\quad\tilde{a}=r^{-1/2}a,$

Using the relations $\tilde{\nabla}=r^{-1/2}\nabla$ and $\tilde{d}=r^{-1/2}d$
(see e.g. [MR4049917, Sect. B.1]), we have

(2.13)
$\displaystyle{\nabla_{\tilde{a}}\tilde{\psi}}=\frac{1}{r}\nabla_{a}\psi,\quad\tilde{d}\tilde{a}=\frac{1}{r}da,\quad\left\lvert\tilde{\psi}\right\rvert^{2}=\frac{1}{r}\left\lvert\psi\right\rvert^{2}.$

Plugging (2.13) into (1.1) and using $\omega_{r}=r\omega_{1}$ (see (2.8)), we
find

$\displaystyle r\mathcal{E}(\tilde{\psi},\tilde{a},h_{r})=$
$\displaystyle\int_{\Sigma}\left(\frac{1}{2}\left\lvert\nabla_{\tilde{a}}\tilde{\psi}\right\rvert^{2}+\frac{1}{2}\left\lvert\tilde{d}\tilde{a}\right\rvert^{2}+\frac{\kappa^{2}}{4}\left(\left\lvert\tilde{\psi}\right\rvert^{2}-1\right)^{2}\right)r^{2}\omega_{1}$
(2.14) $\displaystyle=$
$\displaystyle\int_{\Sigma}\left(\frac{1}{2}\left\lvert\nabla_{a}\psi\right\rvert^{2}+\frac{1}{2}\left\lvert
da\right\rvert^{2}+\frac{\kappa^{2}}{4}\left(\left\lvert\psi\right\rvert^{2}-r\right)^{2}\right)\omega_{1}=\mathcal{E}_{r}(\tilde{\psi},\tilde{a}),$

which gives (2.11).

The Euler-Lagrange equations for $\mathcal{E}_{r}(\psi,a)$ are the rescaled
(GL), given by

(2.15) $\displaystyle-\Delta_{a}\psi$
$\displaystyle=\kappa^{2}\left(r-\left\lvert\psi\right\rvert^{2}\right)\psi,$
$\displaystyle d^{*}da$
$\displaystyle=\operatorname{Im}(\bar{\psi}\nabla_{a}\psi),$

in the space $X^{k},\,k\geq 2$ defined in (1.13). For suitable values of
$r>0$, we seek solution pair $(\psi,a)$ to (2.15) on the unscaled surface
$(\Sigma,h)$. In Section 2.3, we show that the parameter $r>0$ corresponds to
the average field (magnetic flux) strength. In Type II superconductors, the
variation of the latter triggers second-order phase transition. Hence, in what
follows we will use $r$ as the bifurcation parameter.

The rescaled Ginzburg-Landau equations (2.15) are the central objects of the
subsequent sections.

###  2.3. Quantization of Magnetic Flux

In this subsection, we state the extension of the Chern-Weil correspondence
(known in the physics literature as magnetic flux quantization) to non-compact
Riemann surfaces.

Recall that $\deg E$ denotes the topological degree, or the first Chern
number, of the unitary line bundle $E$, see Appendix A.

######  Theorem 2.2 (Chern-Weil correspondence for non-compact Riemann
surfaces).

Let $\Sigma$ be a non-compact Riemann surface, and $E\to\Sigma$ be a unitary
line bundle. For every connection $a$ on $E\to\Sigma$ with
$\left\lvert\int_{\Sigma}da\right\rvert<\infty$, there holds

(2.16) $\frac{1}{2\pi}\int_{\Sigma}da=\deg E.$

The proof of this theorem is given in Appendix C.

Since $\deg E\in\mathbb{Z}$, equation (2.16) implies quantization of the
average magnetic field (magnetic flux).

The Chern-Weil correspondence imposes a direct constrain on constant curvature
connections. Indeed, let $a^{\beta}$ be a constant curvature connection on
$(\Sigma,h_{r})$ satisfying $da^{\beta}=\beta\omega_{r}$ for some
$\beta\in\mathbb{R}$ (e.g. (1.7)). Then, by relation (2.16), we have

(2.17)
$\displaystyle\beta=\frac{1}{|\Sigma|_{r}}\int_{\Sigma}da=\frac{2\pi\deg
E}{|\Sigma|_{r}},$

which, together with the relation
$\left\lvert\Sigma\right\rvert_{r}=r\left\lvert\Sigma\right\rvert$, gives
$\beta=b_{r}$ as in (1.8).

(2.17) relates the average magnetic field (curvature) on $\Sigma$ to the
geometry of $\Sigma$, and the topology of the line bundle $E\to\Sigma$.
Indeed, (2.17) shows that varying the metric on $\Sigma$ as in (2.7)–(2.8)
amounts to varying the constant curvature solution.

##  3\. 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 $(\Sigma,h\equiv h_{1})$. In this section, we consider the linear
problem associated to (2.15).

The Ginzburg-Landau equations (2.15) have constant curvature solutions

(3.1) $(\psi,a)=(0,a^{b})$, where $a^{b}$ is any constant curvature connection
with $da^{b}={b}\omega$.

As a consequence of the Chern-Weil correspondence (2.16), since $a^{b}$ is a
constant curvature connection on $(\Sigma,h\equiv h_{1})$, the number $b$ is
given by (1.9).

Linearizing (2.15) at the solution (3.1), we get a decoupled system

(3.2) $\displaystyle(-\Delta_{a^{b}}-\kappa^{2}r)\phi$ $\displaystyle=0,$
(3.3) $\displaystyle d^{*}d\alpha$ $\displaystyle=0.$

These are the main objects of this section. At this point, the unknown in
(3.2)–(3.3) here is a (section, connection)-pair on a unitary line bundle
$E\to(\Sigma,h_{1})$. The curvature parameter $r$ from the unscaled GL
equations now enters only through the second term in the l.h.s. of (3.2).

Our goal now is to obtain an explicit description of the solutions to
(3.2)–(3.3) in the Sobolev space $X^{s}$ defined in (1.13).

###  3.1. Solving the Maxwell Equation

First, we solve the homogeneous equation (3.3) (the free Maxwell equation) for
the connection $a$.

By [MR4049917, Lem. 3.2], $\alpha$ solves (3.3) if and only if $\alpha$ is a
constant curvature connection. Thus $a^{b}$ is a solution. By the Chern-Weil
correspondence, the linearization, (3.3), around $(\psi,a)=(0,a^{b})$ must be
satisfied by a $1$-form $\alpha$ of degree 0. So again, by [MR4049917, Lem.
3.2], $\alpha$ must be a flat connection: In the distributional sense,

(3.4) $d^{*}d\alpha=0\iff\alpha\text{ is flat }\iff d\alpha=0.$

So $\alpha$ must be closed; but by the definition of (1.13) it must also be
co-closed and hence harmonic. This establishes the following.

######  Proposition 3.1.

$d^{*}d\geq 0$ and the solution space to (3.3) in $\vec{\mathcal{H}}^{2}$ is

(3.5)
$\Omega:=\operatorname{Null}d^{*}d|_{\vec{\mathcal{H}}^{2}}=\left\\{\text{harmonic
1-forms on}\,\,\,{\Sigma}\right\\}.$

###  3.2. The Magnetic Laplacian

Recall that $b=2\pi\deg E/\left\lvert\Sigma\right\rvert$, and we choose the
orientation of $E$ that makes $b>0$. By the classification of constant
curvature connections, Theorem B.2, an one-form $\beta$ on $\Sigma$ satisfies
$d\beta=b\omega$ and the equivariant condition if and only if $\beta$ is
gauge-equivalent to

(3.6) $a^{b}:=by^{-1}dx.$

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 $-\Delta_{a^{b}}$ in the l.h.s. of (3.2). Locally, in the
rectangular coordinate $z=x+iy$, we have

(3.7)
$\displaystyle-\Delta_{a^{b}}=-y^{2}(\partial_{xx}+\partial_{yy})+2iby\partial_{x}+b^{2}\text{
acting on }{F_{\Sigma}\subset\mathbb{H}}.$

This follows from direct computation using the standard definitions in Section
A.

We prove the following.

######  Theorem 3.2.

Let $\Sigma=\mathbb{H}/\Gamma$ be a non-compact Riemann surface with $m$ cusps
and no elliptic points.

  1. (a)

$-\Delta_{a^{b}}$ is self-adjoint;

  2. (b)

$-\Delta_{a^{b}}\geq b$;

  3. (c)

Let $\mathcal{S}(\Sigma)\equiv\mathcal{S}_{k}(\Sigma)$ denote the space of
cusp forms on $\Sigma$ with weight $k=2b=4\pi
n/\left\lvert\Sigma\right\rvert$. Then $b$ is an eigenvalue of
$-\Delta_{a^{b}}$ if and only if $\mathcal{S}(\Sigma)\neq\emptyset$, and the
multiplicity of $b$ equals to $\dim\mathcal{S}(\Sigma)$;

  4. (d)

The essential spectrum of $-\Delta_{a^{b}}$ consists of $m$ branches each of
which filling in the semi-axis $[1/4+b^{2},\infty)$. Hence,

(3.8) $\sigma_{\rm ess}(-\Delta_{a^{b}})=[1/4+b^{2},\infty).$

######  Remark 9.

If the field strength $b=0$, then $E$ is the trivial line bundle, and
$-\Delta_{a^{b}}$ reduces to the hyperbolic Laplacian $-\Delta$ on $\Sigma$.
The spectral properties of this operator have important bearings for number
theory. See e.g. [MR1942691] and the references therein.

For $b>0$, some of the results above have been obtained in [MR776146,
MR870891, MR1069475, Pnueli] for specific choices of $\Sigma$ in relevant
physics context. For the connection to weighted Maass forms in number theory,
see [MR1431508, Sec. 2].

######  Remark 10.

For $b=1/2$, the eigenvalue $b$ is embedded at the bottom of the essential
spectrum of $-\Delta_{a^{b}}$.

###### Proof of Theorem 3.2.

The proof of Part (a) is standard and can be found in [MR1431508, Sec. 2.1].
Parts (b)-(c) follow from the Weitzenböck formula given in Section 3.3. Part
(d) follows from the results in Section 3.4. ∎

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 $\Gamma\subset PSL(2,\mathbb{R})$ be a Fuchsian group. Let $c_{i}$ be a
cusp of $\Gamma$. Then the stabilizer of $c_{i}$ is an infinite cyclic group
generated by some parabolic transform. In symbols,

$Stab(c_{i},\Gamma)\equiv\left\\{\gamma\in\Gamma:\gamma
c_{i}=c_{i}\right\\}=\left\langle\begin{pmatrix}1&r_{i}\\\
0&1\end{pmatrix}\right\rangle\text{ for some }r_{i}\in\mathbb{R}.$

We call $\gamma_{i}\in SL(2,\mathbb{R})$ a scaling matrix of $c_{i}$ if
$\gamma_{i}c_{i}=\infty$ and $\gamma_{i}\left(\begin{smallmatrix}1&r_{i}\\\
0&1\end{smallmatrix}\right)\gamma_{i}^{-1}=\left(\begin{smallmatrix}1&1\\\
0&1\end{smallmatrix}\right).$ Such $\gamma_{i}$’s exist, and are unique up to
translation. See [MR1942691, Chapt. 2.2] for details.

###  3.3. Weitzenböck-type formula

In this subsection, let $a$ be an arbitrary unitary connection on a smooth
complex line bundle $E\to\Sigma$ (see Appendix A for the definitions).

We decompose the covariant derivative $\nabla_{a}$ into $(1,0)$ and $(0,1)$
parts as $\nabla_{a}=\partial_{a}^{\prime}+{\partial_{a}^{\prime\prime}}$,
where

(3.9) $\displaystyle\partial_{a}^{\prime}:=\partial+a_{c},\
\partial_{a}^{\prime\prime}:=\overline{\partial}+\bar{a}_{c}.$

Here $\partial:=\frac{\partial}{\partial z}\otimes dz$ and
$\overline{\partial}:=\overline{\frac{\partial}{\partial z}}\otimes
d\overline{z}$, where, as usual, $\frac{\partial}{\partial
z}\equiv\partial_{z}:=(\partial_{x_{1}}-i\partial_{x_{2}})/2$,
$\overline{\frac{\partial}{\partial
z}}\equiv\partial_{\bar{z}}:=(\partial_{x_{1}}+i\partial_{x_{2}})/2$, and

(3.10) $\displaystyle a_{c}:=\frac{1}{2}(a_{1}-ia_{2})\otimes dz,\
\bar{a}_{c}:=\frac{1}{2}(a_{1}+ia_{2})\otimes d\bar{z}.$

With the definitions above, we can rewrite (3.9) as

(3.11)
$\displaystyle\partial_{a}^{\prime\prime}=\tilde{\partial}_{a}^{\prime\prime}\otimes
d\overline{z},\
\quad\tilde{\partial}_{a}^{\prime\prime}:=\bar{\partial}_{z}+\bar{a}^{c},\
\quad a^{c}:=a_{1}-ia_{2}.$

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 $a=2\operatorname{Re}a_{c}$ and

(3.12)
$\tilde{\partial}_{a}^{\prime\prime}:=\frac{1}{2}(\nabla_{1}+i\nabla_{2}).$

In terms of $a_{c}$, the curvature is given by
$F_{a}=2\operatorname{Re}\bar{\partial}a_{c}$. Now we prove the following
relations:

######  Proposition 3.3.

We have

(3.13) $\displaystyle*F_{a}$
$\displaystyle=\frac{1}{2}[\tilde{\partial}_{a}^{\prime\prime},\tilde{\partial}_{a}^{\prime\prime*}],$
(3.14) $\displaystyle-\Delta_{a}$
$\displaystyle={\partial^{\prime\prime}_{a}}^{*}{\partial^{\prime\prime}_{a}}+*F_{a},$
(3.15) $\displaystyle-\Delta_{a}$ $\displaystyle\geq*F_{a}.$

###### Proof.

We compute in local coordinates. Using the relations
$\nabla_{1}=\frac{1}{2}(-\tilde{\partial}_{a}^{\prime\prime}+\tilde{\partial}_{a}^{\prime\prime*})$,
$\nabla_{2}=\frac{1}{2i}(\tilde{\partial}_{a}^{\prime\prime}+\tilde{\partial}_{a}^{\prime\prime*})$,
together with the expression (A.3) for the curvature $F_{a}$, we compute

(3.16) $\displaystyle
F_{a}=-\frac{1}{4}[\tilde{\partial}_{a}^{\prime\prime}-\tilde{\partial}_{a}^{\prime\prime*},\tilde{\partial}_{a}^{\prime\prime}+\tilde{\partial}_{a}^{\prime\prime*}]=\frac{1}{2}[\tilde{\partial}_{a}^{\prime\prime},\tilde{\partial}_{a}^{\prime\prime*}],$

which gives (3.13).

To find the expression for $\Delta_{a}$, we use the relation
$\Delta_{a}=\nabla_{i}\nabla_{i}$ to compute

(3.17) $\displaystyle\Delta_{a}$
$\displaystyle=\frac{1}{4}(\tilde{\partial}_{a}^{\prime\prime}-\tilde{\partial}_{a}^{\prime\prime*})^{2}-\frac{1}{4}(\tilde{\partial}_{a}^{\prime\prime}+\tilde{\partial}_{a}^{\prime\prime*})^{2}$
$\displaystyle=-\frac{1}{2}(\tilde{\partial}_{a}^{\prime\prime}\tilde{\partial}_{a}^{\prime\prime*}+\tilde{\partial}_{a}^{\prime\prime*}\tilde{\partial}_{a}^{\prime\prime})$
$\displaystyle=-\tilde{\partial}_{a}^{\prime\prime*}\tilde{\partial}_{a}^{\prime\prime}-\frac{1}{2}[\tilde{\partial}_{a}^{\prime\prime},\tilde{\partial}_{a}^{\prime\prime*}],$

which gives (3.14).

Since ${\partial^{\prime\prime}_{a}}^{*}{\partial^{\prime\prime}_{a}}\geq 0$,
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:

######  Corollary 3.4.

Let $a^{b}$ be a constant curvature connection s.th. $*F_{a}=b$ Then

(3.18) $\displaystyle-\Delta_{a^{b}}$ $\displaystyle\geq b,$ (3.19)
$\displaystyle K:=\operatorname{Null}(-\Delta_{a^{b}}-b)$
$\displaystyle=\operatorname{Null}{\partial^{\prime\prime}_{a^{b}}}.$

Hence, $b$ is an eigenvalue of $-\Delta_{a^{b}}$ if and only if
$\operatorname{Null}{\partial^{\prime\prime}_{a^{b}}}\neq\\{0\\}$.

###### Proof of Theorem 3.2, Parts (b), (c).

Part (b) follows from (3.18). To complete the proof of Part (c), we now claim

(3.20)
$\displaystyle\operatorname{Null}{\partial^{\prime\prime}_{a^{b}}}|_{\Omega^{0}(E)}=H^{0}(E),$

where $\Omega^{0}(E)$ 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
$M_{k}(\Sigma)$ of modular forms on $\Sigma$, with weight $k=2b=4\pi\deg
E/\left\lvert\Sigma\right\rvert$. Since $\Sigma$ is non-compact, the
intersection $\mathcal{L}^{2}(\Sigma)\cap M_{k}(\Sigma)$ equals to the space
of cusp forms $\mathcal{S}_{k}(\Sigma)$ with weight $k$. This fact follows
from the results in [MR1942691, Sect. 3] and [MR2112196, Sects. 3-4]. Since we
seek $H^{s}$ solution with $s\geq 0$, the desired result follows from here.

It remains to prove (3.20). On bundles over Riemann surfaces, we have that

(3.21) $\displaystyle
F^{0,2}_{a}=\partial^{\prime\prime}_{a}\circ\partial^{\prime\prime}_{a}=0.$

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 [MR1079726, MR4049917]. [MR4049917]
develops a more constructive approach when $M$ is a closed Riemann surface.

######  Theorem 3.5.

Given a $C^{\infty}$ complex vector bundle $E$ over $M$ with connection
$\nabla_{a}$ such that
$\partial^{\prime\prime}_{a}\circ\partial^{\prime\prime}_{a}=0$, there is a
unique holomorphic vector bundle structure on $E$ such that
$\partial^{\prime\prime}_{a}=\partial^{\prime\prime}$.

###  3.4. Essential Spectrum of the Magnetic Laplacian

In this subsection, we compute the essential spectrum of the linearized
operator $-\Delta_{a^{b}}$. 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 $\Sigma$ with a fundamental domain
$F_{\Sigma}\subset\mathbb{H}$ of $\Gamma$, with the sides identified as in the
proof of Theorem 2.2. Let

(3.22) $\gamma_{i}:=\left\\{\begin{aligned} &\begin{pmatrix}0&-1\\\
1&-c_{i}\end{pmatrix}\quad(c_{i}\neq\infty),\\\ &\begin{pmatrix}1&0\\\
0&1\end{pmatrix}\quad(c_{i}=\infty).\end{aligned}\right.$

Then $\gamma_{i}\in SL(2,\mathbb{R})$ and $\gamma_{i}c_{i}=\infty$, where the
action of $\gamma_{i}$ on $F_{\Sigma}\subset\mathbb{H}$ is the Möbius
transform (2.2). Denote by $\varphi_{i}$ the action of $\gamma_{i}$ on
$F_{\Sigma}\subset\mathbb{H}$. Then

(3.23) $\varphi_{i}:\left\\{\begin{aligned}
&z\mapsto-\frac{1}{z-c_{i}}\quad(c_{i}\neq\infty),\\\ &z\mapsto
z\quad(c_{i}=\infty).\end{aligned}\right.$

See Figure 3 below.

Now, we decompose $F_{\Sigma}$ into a compact connected set, $U_{0}$, and
neighbourhoods $U_{i}$ of the cusps $c_{i}$, in such a way that

  1. (1)

$U_{i}\cap U_{j}=\emptyset$ for $1\leq i\neq j\leq m$, and
$U_{0}:=F_{\Sigma}\setminus\bigcup_{i=1}^{m}U_{i}$ is compact;

  2. (2)

For $i=1,\ldots,m$, the map $\varphi_{i}$ maps the domains $U_{i}$
isometrically onto the half-cylinder

(3.24) $\displaystyle
Z_{i}:=\\{z\in\mathbb{C}:\operatorname{Im}z>s_{i}\\}/\mathbb{Z},$

for some $s_{i}\gg 1$.

So long as $s_{i}\gg 1$, such decomposition of $F_{\Sigma}$ is easy to
construct, see Figure 2 below.

$\operatorname{Im}z=0$$z=0$$z=-1$$z=1$$U_{1}$$U_{3}$$U_{2}$$U_{0}$ Figure 2.
Schematic diagram for the decomposition of a fundamental domain of $\Gamma(2)$
in $\mathbb{H}$ with three cusps $c_{1}=1,c_{2}=0,c_{3}=\infty$.
$\operatorname{Im}z$$Z_{2}$$s_{2}$$\varphi_{2}$$U_{2}$ Figure 3. Schematic
diagram illustrating the map $\varphi_{2}$ from (3.23) associated to cusps
$c_{2}=0$.

In the next two lemmas, we analyze the spectral property of $-\Delta_{a^{b}}$
in each domain $U_{i}$ separately.

######  Lemma 3.6.

Let $-\Delta_{a^{b}}\big{|}_{U_{i}}$ be the restriction of $-\Delta_{a^{b}}$
on $L^{2}(U_{i})$ with the Dirichlet boundary conditions on $\partial U_{i}$.
Then

(3.25) $\displaystyle\sigma_{\rm
ess}(-\Delta_{a^{b}})=\bigcup_{i=1}^{m}\sigma_{\rm
ess}(-\Delta_{a^{b}}\big{|}_{U_{i}}).$

###### Proof.

1\. First, note that the compact domain $U_{0}$ and the cuts do not contribute
to the essential spectrum. Next, let
$\left\\{U_{0}^{\prime},\ldots,U_{m}^{\prime}\right\\}$ be an open covering
$\Sigma$, such that

  1. (1)

$U_{i}^{\prime}\supset U_{i}$ for every $i=0,\ldots,m$;

  2. (2)

$U_{i}^{\prime}\setminus U_{i}$ is bounded in $F_{\Sigma}$ for every
$i=0,\ldots,m$;

  3. (3)

$U_{i}^{\prime}\cap U_{j}^{\prime}=\emptyset$ for $1\leq i\neq j\leq m$.

For Item 3 above, we note that for sufficiently small $U_{i},\,i=1,\ldots m$,
we can always choose a slightly larger $U_{i}^{\prime}\supset U_{i}$ s.th.
$U_{i}^{\prime}$ are still mutually disjoint for $1\leq i\neq j\leq m$ (see
Figure 2).

Let $\chi_{i}$ be a partition of unity on $\Sigma$ adapted to
$U_{i}^{\prime}$, such that

(3.26) $\displaystyle\sum_{i=0}^{m}\chi_{i}^{2}$ $\displaystyle={\bf{1}},$
(3.27) $\displaystyle{\chi_{i}|_{U_{i}}\equiv
1},\quad\operatorname{supp}\chi_{i}$
$\displaystyle\subset{U_{i}^{\prime}}\quad(i=0,\ldots,m).$

Then the IMS formula (see [MR883643]) gives

(3.28) $\displaystyle-\Delta_{a^{b}}$ $\displaystyle=\sum_{i=1}^{m}T_{i}+R,\
\quad T_{i}:=\chi_{i}(-\Delta_{a^{b}})\chi_{i},$ (3.29) $\displaystyle R:$
$\displaystyle=\chi_{0}(-\Delta_{a^{b}})\chi_{0}-\sum_{i=0}^{m}|\nabla\chi_{i}|^{2}.$

2\. Since $R$ is localized in a compact domain

$\operatorname{supp}\chi_{0}\cup\operatorname{supp}\nabla\chi_{1}\cup\cdots\cup\operatorname{supp}\nabla\chi_{m},$

we expect that $-\Delta_{a^{b}}$ and $\sum T_{i}$ 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 Theorem 3.2, Part (b), the spectrum
$\sigma(-\Delta_{a^{b}})\subset\mathbb{R}_{\geq 0}$. Hence
$(-\Delta_{a^{b}}+1)^{-1}$ is well-defined and bounded on $\mathcal{L}^{2}$.

Claim: $R(-\Delta_{a^{b}}+1)^{-2}$ is a compact operator.

Proof of the claim: Since for each $i=1,\ldots,m$, the set
$\operatorname{supp}\left\lvert\nabla\chi_{i}\right\rvert$ is bounded in
$F_{\Sigma}$, it follows that the operators
$|\nabla\chi_{i}|^{2}(-\Delta_{a^{b}}+1)^{-2}$ are all compact. Now, write

$\displaystyle\Delta_{a^{b}}\chi_{0}(-\Delta_{a^{b}}+1)^{-2}=BK,$
$\displaystyle B:=\Delta_{a^{b}}(-\Delta_{a^{b}}+1)^{-1},$ $\displaystyle
K:=\chi_{0}(-\Delta_{a^{b}}+1)^{-1}-[\chi_{0},\Delta_{a^{b}}](-\Delta_{a^{b}}+1)^{-2}.$

The operator $B$ is bounded on $\mathcal{L}^{2}$. Since $\chi_{0}$ has its
support bounded $F_{\Sigma}$, the operator $K$ is a compact operator. It
follows that $\Delta_{a^{b}}\chi_{0}(-\Delta_{a^{b}}+1)^{-2}=BK$ is compact.
This proves the claim.

3\. We will use a modified version of Weyl’s Theorem for relative compact
perturbations:

######  Proposition 3.7 ([MR0493421] Chapt. XIII.4, Cor. 3).

Let $H$ and $W$ be two self-adjoint operators s.th. $W(H+1)^{-n}$ is compact
for some $n\geq 1$. Then

$\sigma_{\rm ess}(H+W)=\sigma_{\rm ess}(H).$

We use this proposition with $W=R$ from (3.29) and $H=-\Delta_{a^{b}}$. Then
it follows from the claim proved in Step 2 and Proposition 3.7 that

(3.30) $\displaystyle\sigma_{\rm ess}(-\Delta_{a^{b}})=\sigma_{\rm
ess}\left(\sum_{i=1}^{m}T_{i}\right).$

4\. Next, for $i=1,\ldots,m$, we denote by
$-\Delta_{a^{b}}\big{|}_{U_{i}^{\prime}}$ the restriction on $U_{i}^{\prime}$
with Dirichlet boundary condition on $\partial U_{i}^{\prime}$.

Claim:

(3.31) $\sigma_{\rm
ess}\left(\sum_{i=1}^{m}T_{i}\right)\subset\bigcup_{i=1}^{m}\sigma_{\rm
ess}(-\Delta_{a^{b}}\big{|}_{U_{i}}).$

Proof of the claim: By construction, $U_{i}^{\prime}\cap
U_{j}^{\prime}=\emptyset$ for $1\leq i\neq j\leq m$. Hence, one can show using
Weyl’s criterion that

(3.32) $\sigma_{\rm
ess}\left(\sum_{i=1}^{m}T_{i}\right)=\bigcup_{i=1}^{m}\sigma_{\rm
ess}\left(T_{i}\right).$

Next, we show that for each $i=1,\ldots,m$,

(3.33) $\sigma_{\rm ess}\left(T_{i}\right)=\sigma_{\rm
ess}\left(-\Delta_{a^{b}}\big{|}_{U_{i}}\right).$

Indeed, for each $i=1,\ldots,m$, we write

(3.34)
$T_{i}=-\Delta_{a^{b}}\big{|}_{U_{i}}-\Delta_{a^{b}}\big{|}_{U_{i}^{\prime}\setminus
U_{i}}+K_{i}\quad\text{ defined on the domain
}D(\Delta_{a^{b}}\big{|}_{U_{i}^{\prime}}),$

where $K_{i}$’s are defined by this relation. Explicitly,

(3.35)
$K_{i}=(-1+\chi_{i})\Delta_{a^{b}}\chi_{i}-\Delta_{a^{b}}(1-\chi_{i})\quad\text{acting
on }U_{i}^{\prime}.$

Take $\lambda\in\sigma_{\rm ess}(T_{i})$, and let $\\{u_{n}\\}$ be a Weyl
sequence for $T_{i}$ and $\lambda$, i.e.

(3.36) $\|u_{n}\|_{L^{2}}=1,\quad u_{n}\rightarrow 0\ \text{ weakly in
$L^{2}$, }\,\|(T_{i}-\lambda)u_{n}\|_{L^{2}}\rightarrow 0.$

Since $\operatorname{supp}u_{n}\subset U_{i}^{\prime}$, each element in the
sequence $\\{u_{n}\\}$ satisfies the Dirichlet boundary condition on
$U_{i}^{\prime}$.

By the choice of $\chi_{i}$, see (3.27), the factor $1-\chi_{i}$ vanishes away
from the bounded set $U_{i}^{\prime}\setminus U_{i}$. Hence, it follows from
(3.35)–(3.36) that

$K_{i}u_{n}\rightarrow 0\quad\text{strongly in $L^{2}$ as $n\to\infty$ for
every }i=1\ldots,m.$

Similarly, we have

$-\Delta_{a^{b}}\big{|}_{U_{i}^{\prime}\setminus U_{i}}u_{n}\rightarrow
0\quad\text{strongly in $L^{2}$ as $n\to\infty$ for every }i=1\ldots,m.$

Hence, we conclude from expansion (3.34) and assumption (3.36) that

(3.37) $\displaystyle(-\Delta_{a^{b}}\big{|}_{U_{i}}-\lambda)u_{n}$
$\displaystyle=$
$\displaystyle(T_{i}-\lambda)u_{n}-(K_{i}-\Delta_{a^{b}}\big{|}_{U_{i}^{\prime}\setminus
U_{i}})u_{n}\rightarrow 0\text{ strongly for each $i=1,\ldots,m$}.$

By Weyl’s criterion, this implies $\lambda\in\sigma_{\rm
ess}(-\Delta_{a^{b}}\big{|}_{U_{i}})$. Thus (3.33) holds.

Eqs. (3.32)–(3.33) imply (3.31). This proves the claim.

5\. Running the argument from Step 4 backwards gives the other inclusion, i.e.

(3.38) $\sigma_{\rm ess}\left(\sum T_{i}\right)=\sigma_{\rm
ess}\left(\sum-\Delta_{a^{b}}\big{|}_{U_{i}}\right).$

Relations (3.30) and (3.38) together imply (3.25). ∎

######  Lemma 3.8.

For every $i=1,\ldots,m$, we have

$\sigma_{\rm ess}(-\Delta_{a^{b}}\big{|}_{U_{i}})\subset[1/4+b^{2},\infty).$

###### Proof.

By construction, the map $\varphi_{i}$ in (3.23) maps $U_{i}$ isometrically
onto the half-cylinder $Z_{i}$ in (3.24). This transformation maps
$-\Delta_{a_{b}}\big{|}_{U_{i}}$ unitarily to another operator, say $h_{i}$,
acting on $L^{2}(Z_{i})$ with the Dirichlet boundary conditions on $\partial
Z_{i}=\\{z\in\mathbb{C}:\operatorname{Im}z=s_{i}\\}/\mathbb{Z}$. Thus, we have

(3.39) $\displaystyle\sigma_{\rm
ess}(-\Delta_{a^{b}}\big{|}_{U_{i}})\subset\sigma_{\rm ess}(h_{i}).$

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 $-\Delta_{a^{b}}$ are not affected under any
isometric transform $\gamma\in SL(2,\mathbb{R})$ acting on $\mathbb{H}$.
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

$p:=y^{-1}h_{i}y=-y(\partial_{x}^{2}+\partial_{y}^{2})y+2iby\partial_{x}+b^{2}$

acting on $L^{2}(Z_{i},dxdy)$, with the Dirichlet boundary conditions on
$\partial Z_{i}$. Then

(3.40) $\displaystyle\sigma_{\rm ess}(h_{i})=\sigma_{\rm ess}(p).$

Now, we apply the Fourier transform in $x$ for the operator $p$ to obtain the
decomposition

$p=\bigoplus_{k\in\mathbb{Z}}p_{k}^{b},$

where $p_{k}^{b}$ is the operator

(3.41) $p_{k}^{b}:=y(-\partial_{y}^{2}+k^{2})y-2byk+b^{2}$

acting on $L^{2}(\mathbb{R}_{s})$, with
$\mathbb{R}_{s}:=\\{y\in\mathbb{R}:y>s\\}$.

For $k\neq 0$, (3.41) can be written as

$p_{k}^{b}:=-y\partial_{y}^{2}y+k^{2}y^{2}-2byk+b^{2}=-y\partial_{y}^{2}y+k^{2}(y-b/k)^{2},$

and therefore we have the estimate

(3.42) $p_{k}^{b}\geq p_{0}^{b}\quad(k\neq 0).$

By (3.42), we have

$\sigma(p_{k}^{b})\subset[\sigma^{b},\infty),\quad\sigma^{b}:=\inf\sigma(p_{0}^{b}|_{\mathbb{R}_{s}}),$

and therefore

(3.43)
$\sigma(p)=\bigcup_{k\in\mathbb{Z}}\sigma(p_{k}^{b}\big{|}_{\mathbb{R}_{s}})\subset[\sigma^{b},\infty).$

Furthermore, by Hardy’s inequality

$-\partial_{y}^{2}\big{|}_{\mathbb{R}_{s}}-\frac{1}{4}\frac{1}{y^{2}}\geq 0,$

we have the lower bound

$p_{0}^{b}:=-y\partial_{y}^{2}y+b^{2}\geq\frac{1}{4}+b^{2}.$

This shows that $\sigma^{b}\geq 1/4+b^{2}$, and therefore by (3.43), it
follows that $\sigma(p)\subset[1/4+b^{2},\infty)$. 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

(3.44) $\sigma_{\rm ess}(-\Delta_{a^{b}})\subset[1/4+b^{2},\infty).$

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

(3.45) $\displaystyle\sigma_{\rm
ess}(-\Delta_{a^{b}})\supset[1/4+b^{2},\infty),$

we compute the generalized eigenfunctions of $-\Delta_{a^{b}}$ in the space of
square-integrable equivariant functions. The details are delegated to the end
of Appendix A.3.

Combining (3.44)–(3.45) proves Part (d) of Theorem 3.2. ∎

###### Proof of Proposition 1.3.

Consider the Ginzburg-Landau energy (1.1). The Hessian of
$\mathcal{E}(\cdot,h_{r})$ at $(0,a^{b_{r}})$ is given by

(3.46) $L:=\operatorname{diag}(-\Delta_{a^{b_{r}}}-\kappa^{2},d^{*}d):X^{s}\to
X^{s-2},$

which can be computed as in (4.13). By Theorem 3.2(b) and rescaling,
$-\Delta_{a^{b_{r}}}\geq b_{r}$. Thus,

(3.47) $\displaystyle-\Delta_{a^{b_{r}}}-\kappa^{2}\geq 0\iff
b_{r}-\kappa^{2}\geq 0\iff b_{r}\geq\kappa^{2}.$

Next, by Proposition 3.1, $d^{*}d\geq 0$ on $\vec{\mathcal{H}}^{s}$, $s\geq
2$. This, together with (3.47) and formula (3.46), implies

(3.48) $\displaystyle L\geq 0\ \text{ if and only if }\ b_{r}\geq\kappa^{2}.$

This implies Proposition 1.3. ∎

##  4\. 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
$\Sigma$, the ground state energy of the magnetic Laplacian $-\Delta_{a^{b}}$
equals to $b$. Moreover, the parameter $b$ is determined by the degree of $E$
and the signature of $\Sigma$.

Let $n=\deg E$. Recall that $\left\lvert\Sigma\right\rvert$ is the area of
$\Sigma$ w.r.t. the standard hyperbolic metric, and $b=2\pi
n/\left\lvert\Sigma\right\rvert$ 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 $b/\kappa^{2}$. This emerging
solution corresponds to a second order phase transition as the applied field
strength is lowered past the critical value $b=2\pi
n/\left\lvert\Sigma\right\rvert$.

We follow the general approach of [MR4049917]. Here we note two additional
difficulties specific to our situation:

  1. (a)

If $b=1/2$, e.g. when $\deg E=1$ and $\Sigma=\mathbb{H}/\Gamma(3)$ (in which
case $\left\lvert\Sigma\right\rvert=4\pi$ by the Gauss-Bonnet formula (2.10)),
then ground state energy $b$ is embedded in the essential spectrum of
$-\Delta_{a^{b}}$.

  2. (b)

In general, by Theorem 3.2, Part (c), the lowest eigenvalue of
$-\Delta_{a^{b}}$ is not simple.

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 $\dim(-\Delta_{a^{b}}-b)=1$ and $b\neq 1/2$ (see
(1.14)), and explain how to overcome the second one in Section 7.

An explicit example satisfying (1.16) is given in (1.23). To see this, we note
that by Theorem 3.2, condition (1.16) holds if $b\neq 1/2$ and the complex
vector space $\mathcal{S}_{k}(\Sigma)$ of cusp forms on $\Sigma$ with weight
$k=2b=4\pi\deg E/\left\lvert\Sigma\right\rvert$ is one-dimensional. Let
$\Sigma=\mathbb{H}/\Gamma(N),\,N\geq 2$ be a non-compact Riemann surface,
where $\Gamma(N)$ is the principal congruence subgroup of level $N$ (see
Section 2.1 for definitions). Using classical dimension formulae found in
[MR0314766]*Secs. 1.6, 2.6, we find that if $k$ is even, i.e.
$b\in\mathbb{Z}$, then

(4.1) $\dim(\mathcal{S}_{k}(\Sigma))=\left\\{\begin{aligned}
&(k-1)(g_{N}-1)+\frac{km_{N}}{2}\quad(k=4,6,\ldots),\\\
&g_{N}\quad(k=2),\end{aligned}\right.$

where $g_{N}$ (resp. $m_{N}$) is the number of genera (resp. distinct cusps)
of $\Sigma$, given by (2.5)–(2.6). We seek integer solution $(k,N)$ to the
equation

(4.2) $\dim(\mathcal{S}_{k}(\Sigma))=1.$

For $N\geq 3,\,k=2$, (4.2) reduces to

(4.3) $\frac{N-6}{12}m_{N}=0.$

The only solution to (4.3) is $N=6$ and, for $k=2$, this gives $\deg E=12$.

###  4.1. Setup

Recall $X^{s}=\mathcal{H}^{s}\times\vec{\mathcal{H}}^{s}$ is the space of
(section, connection)-pairs of order $s$, defined in (1.13), and by convention
$X^{0}=\mathcal{L}^{2}\times\vec{\mathcal{L}}^{2}$.

Define a nonlinear map $F$ as

(4.4)
$\begin{array}[]{ccrcl}{F}&\colon&{X^{s}\times\mathbb{R}}&\longrightarrow&{X^{s-2}}\\\
\mbox{}&\mbox{}&{(\psi,\alpha,r)}&\longmapsto&{\left(-\Delta_{a^{b}+\alpha}\psi+\kappa^{2}(\left\lvert\psi\right\rvert^{2}-r)\psi,d^{*}d\alpha-
PJ(\psi,\alpha)\right)}\end{array},$

where

(4.5) $\displaystyle
J(\psi,\alpha):=\operatorname{Im}(\bar{\psi}\nabla_{a^{b}+\alpha}\psi)\text{
is the r.h.s. of the second equation in \eqref{GL},}$ (4.6) $\displaystyle
P:\vec{\mathcal{H}}^{s}\to\vec{\mathcal{H}}^{s}\text{ is the projection onto
the space of co-closed $1$-forms}.$

The map $F$ is the central object in the remaining sections.

######  Remark 11.

By definition (4.4) and direct computation, one finds that the rescaled
equation (2.15) has solution $\left(\psi,a^{b}+\alpha,r\right)$ if and only if

(4.7) $F(\psi,\alpha,r)=0,$

and $J(\psi,\alpha)$ is co-closed. The latter holds so long as the first GL
equation in (2.15) holds. A proof of this fact is found in [MR4049917,
Prop.5.1]. Hence it suffices to consider (4.7) only.

Next, by (3.1), eq. (4.7) has the trivial solution
$(\psi,\alpha,r)=(0,0,b/\kappa^{2})$:

(4.8) $\displaystyle F(0,0,b/\kappa^{2})=0,$

Below, we seek non-trivial solutions to (4.7), i.e. solutions to (4.7) with
variable curvature, in a small neighbourhood around the trivial one. We write

(4.9) $\displaystyle u:=(\psi,\alpha)\in X^{s}.$

Recall also that $K=\operatorname{Null}(-\Delta_{a^{b}}-b)$, c.f. (3.19), and
the critical threshold $\kappa_{c}(r)$ is defined in (1.12).

The main result of this section is the following:

######  Proposition 4.1.

Suppose $b\neq 1/2$, $\dim K=1$, and $r>0$ satisfies (c.f. (1.15))

(4.10) $\displaystyle 0<\left\lvert\kappa^{2}r-b\right\rvert\ll 1.$

Then there exists a family of non-trivial solutions

$(u_{s},r_{s}),\qquad s\in\mathbb{R},\ 0<\left\lvert s\right\rvert\ll 1,$

to (4.7) in a small neighbourhood of $(0,b/\kappa^{2})$ in
$X^{k}\times\mathbb{R}_{>0}$, $k\geq 2$.

The solution $u_{s}$ is unique, up to a gauge symmetry, in a small
neighbhourhood $U\subset X^{k}$ around $0$.

Furthermore, $r_{s}$ has the following expansion,

(4.11) $\displaystyle r_{s}=b/\kappa^{2}+O(\left\lvert s\right\rvert^{2}),$

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 $\mathcal{H}^{s}$ with a real Banach
space through
$\psi\leftrightarrow(\operatorname{Re}\psi,\operatorname{Im}\psi)$ and view
$F:X^{s}\times\mathbb{R}\to X^{s-2}$ as a map between real Banach spaces. From
now till the end of this paper, we assume $s\geq 2$ in (4.4).

######  Lemma 4.2.

We have

  1. (1)

The map $F$ is $C^{2}$ from the real Banach spaces $X^{s}\times\mathbb{R}$ to
$X^{s-2}$, $s\geq 2$;

  2. (2)

$F$ has gauge symmetry in the sense that for every $\theta\in\mathbb{R}$, we
have

(4.12)
$\displaystyle[F(e^{i\theta}s,t,r)]_{\psi}=e^{i\theta}[F(s,t,r)]_{\psi},\quad[F(e^{i\theta}s,t,r)]_{\alpha}=[F(s,t,r)]_{\alpha};$

  3. (3)

$F(u,r)=0$ has the trivial branch of solution $(0,r),\,r>0$.

###### Proof.

The first claim follows from the fact that the map $F$ is a polynomial in
$\psi$, $\bar{\psi}$, $\alpha$ 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. (3.2)–(3.3))

(4.13)
$d_{u}F(0,r)=\operatorname{diag}(-\Delta_{a^{b}}-\kappa^{2}r,d^{*}d):X^{s}\to
X^{s-2}.$

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 $X^{s}\to X^{s-2}$
for $s\geq 2$.

The operator (4.13) enters the l.h.s. of the linearized equations (3.2)–(3.3).
Put

(4.14) $N_{r}:=\operatorname{Null}d_{u}F(0,r).$

By Proposition 3.1 and Corollary 3.4, we have

(4.15) $N\equiv N_{b/\kappa^{2}}=K\times\Omega,$

where $\Omega$ 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 $0<\left\lvert\kappa^{2}r-b\right\rvert\ll 1$.

###  4.2. Lyapunov-Schmidt Reduction

In this subsection, we use the Lyapunov-Schmidt reduction to reduce the
infinite-dimensional problem (4.7) to a finite-dimensional one.

Define a linear operator $Q:X^{s}\to X^{s}$ by

(4.16) $Q:=\left.\begin{aligned} &\frac{1}{2\pi i}\oint R(z)\,dz\oplus
Q^{\prime}.\end{aligned}\right.$

Here, the operator $Q^{\prime}:\vec{\mathcal{H}}^{s}\to\vec{\mathcal{H}}^{s}$
is the orthogonal projection onto $\Omega$ defined in (3.5), which can be
identified with the space of equivariant harmonic $1$-forms. $R(z)$ is the
resolvant of $-\Delta_{a^{b}}-b$ at $0$, which is well-defined if $b\neq 1/2$.

By construction, $Q$ given in (4.16) is an orthogonal projection onto the
space $N\equiv N_{b/\kappa^{2}}$ from (4.15).

Define $v=Qu,\,w=Q^{\perp}u$, where $Q^{\perp}=1-Q$ is the projection onto
$N^{\perp}\subset X^{s}$. Then the key equation (4.7) is equivalent to the
following two equations,

(4.17) $\displaystyle QF(v+w,r)$ $\displaystyle=0,$ (4.18) $\displaystyle
Q^{\perp}F(v+w,r)$ $\displaystyle=0.$

######  Lemma 4.3.

Suppose $b\neq 1/2$. For every $(v,r)\in N\times\mathbb{R}_{>0}$ with

$\left\lVert v\right\rVert_{X^{s}}+\left\lvert\kappa^{2}r-b\right\rvert\ll 1,$

eq. (4.18) has a unique solution
$w=w(v,r)\in\operatorname{ran}Q^{\perp}\subset X^{s}$ which satisfies

(4.19) $w=O(\|v\|_{X^{s}}^{2}),$

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 $w=w(v,r)\in N$ in a small neighbourhood of
$(v,r)=(0,b/\kappa^{2})$ provided the partial Fréchet derivative
$d_{w}Q^{\perp}F(0,b/\kappa^{2}):N^{\perp}\subset X^{s}\to X^{s-2}$ is
invertible.

The partial Fréchet derivative $d_{w}Q^{\perp}F$ evaluated at
$(0,b/\kappa^{2})$ is given by the diagonal operator

(4.20)
$d_{w}Q^{\perp}F(0,b/\kappa^{2})=Q^{\perp}\operatorname{diag}(-\Delta_{a^{b}}-b,d^{*}d).$

We first consider the $\psi$-component of this operator.

By assumption, the ground state energy $b\neq 1/2$. Then by Theorem 3.2, Parts
(c)-(d), $b$ is an isolated eigenvalue of $-\Delta_{a^{b}}$ (see Remark 10).
In this case, by elementary spectral theory (e.g. [MR1361167, Thm. 6.7]), the
operator $-\Delta_{a^{b}}-b$ is invertible on $K^{\perp}$.

It remains to consider the $\alpha$-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 $1$-forms, proved in e.g. [MR729759, Prop. 5.1]), we have
$\inf\sigma_{\rm{ess}}(d^{*}d|_{\Omega^{\perp}})\geq\frac{1}{4}$. Thus
$d^{*}d|_{\Omega^{\perp}}$ is invertible.

It follows that $d_{w}Q^{\perp}F(0,b/\kappa^{2})$ is invertible on $N^{\perp}$
as desired.

2\. To prove estimate (4.19), consider the equation (4.18) satisfied by $w$,
which we rewrite as

(4.21) $Q^{\perp}F(v,r)+\bar{L}_{v,r}w+N_{v,r}(w)=0,$

where $\bar{L}_{v,r}:=Q^{\perp}d_{u}F(v,r)Q^{\perp}$, and $N_{v,r}(w)$ is the
nonlinearity defined through this relation.

The explicit expansion for $\bar{L}_{0,r}$ follows from (4.13). By Proposition
3.1 and Theorem 3.2, the operator $\bar{L}_{0,b/\kappa^{2}}\geq\delta
Q^{\perp}$ for some $\delta>0$ and is therefore invertible. By this fact and
elementary perturbation theory, we find that $\bar{L}_{v,r}$ is invertible for

(4.22) $\displaystyle\left\lVert
v\right\rVert_{X^{s}}+\left\lvert\kappa^{2}r-b\right\rvert\ll 1,$

with

(4.23) $\left\lVert\bar{L}_{v,r}^{-1}\right\rVert_{X^{s-2}\to X^{s}}\lesssim
1.$

Direct computation using the definition (4.4) of $F$ and the fact $v\in N$
(see (4.14)–(4.15)) shows that

(4.24)
$\displaystyle\left\lVert[F(v,r)]_{\psi}\right\rVert_{\mathcal{H}^{s-2}}$
$\displaystyle\lesssim\left\lVert[v]_{\psi}\right\rVert_{\mathcal{H}^{s}}(\left\lVert[v]_{\alpha}\right\rVert_{\vec{\mathcal{H}}^{s}}+\left\lVert[v]_{\psi}\right\rVert_{\mathcal{H}^{s}}^{2}),$
(4.25)
$\displaystyle\left\lVert[F(v,r)]_{\alpha}\right\rVert_{\vec{\mathcal{H}}^{s-2}}$
$\displaystyle\lesssim\left\lVert[v]_{\psi}\right\rVert_{\mathcal{H}^{s}}^{2}(1+\left\lVert[v]_{\alpha}\right\rVert_{\vec{\mathcal{H}}^{s}}),$
(4.26) $\displaystyle\left\lVert N_{v,r}(w)\right\rVert_{X^{s-2}}$
$\displaystyle\lesssim\left\lVert w\right\rVert_{X^{s}}^{2}.$

Here and below, for a vector $u\in X^{s}$ we write
$u=([u]_{\psi},[u]_{\alpha})$.

Now we rewrite (4.21) as the fixed point equation

(4.27) $w=-\bar{L}_{v,r}^{-1}(Q^{\perp}F(v,r)+N_{v,r}(w)).$

Applying (4.24)–(4.26) to (4.27) and using triangle inequality, we find

(4.28)
$\displaystyle\left\lVert[w]_{\psi}\right\rVert_{\mathcal{H}^{s}}\lesssim\left\lVert[v]_{\psi}\right\rVert_{\mathcal{H}^{s}}(\left\lVert[v]_{\alpha}\right\rVert_{\vec{\mathcal{H}}^{s}}+\left\lVert[v]_{\psi}\right\rVert_{\mathcal{H}^{s}}^{2}),$
(4.29)
$\displaystyle\left\lVert[w]_{\alpha}\right\rVert_{\vec{\mathcal{H}}^{s}}\lesssim\left\lVert[v]_{\psi}\right\rVert_{\mathcal{H}^{s}}^{2}(1+\left\lVert[v]_{\alpha}\right\rVert_{\vec{\mathcal{H}}^{s}}),$

provided $\left\lVert
v\right\rVert_{X^{s}}+\left\lvert\kappa^{2}r-b\right\rvert\ll 1$. 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. ∎

###  4.3. The Bifurcation Equation

In this section we prove Proposition 4.1, by solving the bifurcation equation
(4.17).

###### Proof of Proposition 4.1.

1\. Using Lemma 4.2, we can now plug the solution $w=w(v,r)$ to (4.18) back
into (4.17), and get the bifucation equation

(4.30) $QF(v+w(v,r),r)=0.$

Recall here $Q$ is the orthogonal projection onto
$N:=\operatorname{Null}d_{u}F(0,b/\kappa^{2})$ 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).

2\. Let

(4.31) ${\xi\quad\text{and}\quad\eta_{k},\,1\leq k\leq\dim\Omega}$

be some orthogonal bases of $K$ and $\Omega$, defined respectively in (3.19)
and (3.5), with
$\langle\left\lvert\xi\right\rvert^{2}\rangle\equiv\frac{1}{\left\lvert\Sigma\right\rvert}\int\left\lvert\xi\right\rvert^{2}=1$.

Here $K\subset\mathcal{H}^{s}$ is a complex vector subspace with dimension $1$
by assumption, while $\Omega\subset\vec{\mathcal{H}}^{s}$ is real vector space
with finite dimensions [MR729759].

Let $s$ and $t=(t_{1},\ldots,t_{\dim_{\Omega}})$ respectively be the complex
and real coefficients of vectors in $K$ and $\Omega$ w.r.t. the bases (4.31).
For $v=(\phi,\gamma)\in K\times\Omega$, we will use the parametrization

(4.32) $\phi=\phi(s)\equiv s\xi\in
K,\quad\gamma=\gamma(t)\equiv\sum_{k=1}^{\dim\Omega}t_{k}\eta_{k}\in\Omega.$

3\. Next, let $V$ be a sufficiently small neighbourhood of
$(s,t,r)=(0,0,b/\kappa^{2})$ in
$\mathbb{R}\times\mathbb{R}^{\dim\Omega}\times\mathbb{R}_{>0}$ (note that
hereafter, $s$ is real-valued in $V$). For every $(s,t,r)\in V$, we consider

(4.33) $\displaystyle u_{str}:=(\psi_{str},\alpha_{str})=v_{st}+w_{str},$

where $v_{st}=(\phi(s),\gamma(t))$ is parametrized as (4.32) and
$w_{str}=w(v_{st},r)$ is the solution found in Lemma 4.3, satisfying, by
estimates (4.28)–(4.29),

(4.34) $\displaystyle w_{str}=(O(\left\lvert s\right\rvert^{3})+O(\left\lvert
s\right\rvert\left\lvert t\right\rvert),O(\left\lvert s\right\rvert^{2})),$

and similarly for its derivatives in $s,t,r$ on $V$.

By Lemma 4.2, Part (1), we can expand

$F(s,t,r)\equiv F(u_{str},r)$

as a $C^{2}$ map from
$V\subset\mathbb{R}\times\mathbb{R}^{\dim\Omega}\times\mathbb{R}_{>0}$ to the
real Banach space $X^{s-2}$, $s\geq 2$.

By the gauge symmetry (4.12) of $F$, it is not hard to check that any solution
to the equation

(4.35) $\displaystyle F(s,t,r)=0$

in $V$ gives rise to a circle of solutions to the equation $F(u,r)=0$, with
$u=v_{e^{i\theta}s,t}+w_{e^{i\theta}s,t,r}$ and any $\theta\in\mathbb{R}$.
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 (4.34), together with definition (4.33), yields

(4.36) $\displaystyle\psi_{str}=$ $\displaystyle\phi(s)+O(\left\lvert
s\right\rvert^{3})+O(\left\lvert s\right\rvert\left\lvert t\right\rvert),$
(4.37) $\displaystyle\alpha_{str}=$ $\displaystyle\gamma(t)+O(\left\lvert
s\right\rvert^{2}),$

with corresponding estimates for derivatives in $s,t,r$ on $V$. Plugging
(4.36)–(4.37) into (4.4), observing that the nonlinear term in the first
component of (4.4) is cubic in $\psi$, and using that

(4.38)
$\displaystyle(\Delta_{a^{b}+{\alpha_{str}}}-\Delta_{a^{b}})\xi=O(\left\lvert
t\right\rvert)+O(\left\lvert s\right\rvert^{2}),$

we obtain

(4.39) $\displaystyle[F(s,t,r)]_{\psi}$
$\displaystyle=s(-\Delta_{a^{b}}-\kappa^{2}r)\xi+O(\left\lvert
s\right\rvert\left\lvert t\right\rvert){+O(\left\lvert s\right\rvert^{3}),}$

with suitable estimates on the error terms.

Next, since
$J(\psi,\alpha)=\operatorname{Im}(\bar{\psi}\nabla_{a^{b}+\alpha}\psi)$ (see
(4.5)) is quadratic in $\psi$, we find, with $\psi_{str}$ as in (4.36),

(4.40) $\displaystyle J(\psi_{str},\alpha)=\left\lvert
s\right\rvert^{2}J(\xi,0)-\left\lvert
s\right\rvert^{2}\left\lvert\xi\right\rvert^{2}\alpha+O(\left\lvert
s\right\rvert^{2}\left\lvert t\right\rvert^{2})+O(\left\lvert
s\right\rvert^{4}\left\lvert t\right\rvert)+O(\left\lvert s\right\rvert^{6}).$

For $\alpha=\alpha_{str}$ as in (4.37), expansion (4.40) becomes

(4.41) $\displaystyle J(\psi_{str},\alpha_{str})=\left\lvert
s\right\rvert^{2}J(\xi,0)-\sum_{k}t_{k}\left\lvert
s\right\rvert^{2}\left\lvert\xi\right\rvert^{2}\eta_{k}+O(\left\lvert
s\right\rvert^{2}\left\lvert t\right\rvert^{2})+O(\left\lvert
s\right\rvert^{4}).$

Plugging (4.41) into (4.4) yields

(4.42) $\displaystyle[F(s,t,r)]_{\alpha}=d^{*}d\alpha_{str}-\left\lvert
s\right\rvert^{2}J(\xi,0)-\sum_{k}t_{k}\left\lvert
s\right\rvert^{2}\left\lvert\xi\right\rvert^{2}\eta_{k}+O(\left\lvert
s\right\rvert^{2}\left\lvert t\right\rvert^{2})+O(\left\lvert
s\right\rvert^{4}),$

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 $\left\lvert
s\right\rvert$ and $\left\lvert t\right\rvert$ 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 $G=([G]_{\psi},[G]_{\alpha})$. Our goal now is to derive a more
explicit formula for $G$ in terms of the parametrization (4.32).

Firstly, by Lemma 4.2 part (1) and the chain rule for Fréchet derivatives,
$G(s,t,r)$ is a $C^{2}$ map from
$V\subset\mathbb{R}\times\mathbb{R}^{\dim\Omega}\times\mathbb{R}_{>0}$ to the
finite-dimensional real vector space $N\subset X^{s-2}$, $s\geq 2$. Secondly,
by definition, we have $-\Delta_{a^{b}}-b|_{K}=0$. Lastly, so long as $\xi$
solves (3.2), i.e. $\xi\in K$, by [MR4049917, Props. 5.1, 5.3], the
supercurrent $J(\xi,0)$ is both co-closed and co-exact. By the
characterization of the space $\Omega$ in Proposition 3.1, this implies
$Q^{\prime}d^{*}d\alpha_{str}=Q^{\prime}J(\xi,0)=0$.

Hence, using these facts, we can rewrite (4.39) and (4.42) as

(4.43) $\displaystyle{[G(s,t,r)]_{\psi}}$
$\displaystyle=s\left(\kappa^{2}r-b\right)\xi{+O(\left\lvert
s\right\rvert\left\lvert t\right\rvert){+O(\left\lvert s\right\rvert^{3})}},$
(4.44) $\displaystyle[G(s,t,r)]_{\alpha}$ $\displaystyle=\left\lvert
s\right\rvert^{2}{\sum_{k}t_{k}Q^{\prime}\left(\left\lvert\xi\right\rvert^{2}\eta_{k}\right)+O(\left\lvert
s\right\rvert^{2}\left\lvert t\right\rvert^{2})+O(\left\lvert
s\right\rvert^{4})},$

with corresponding estimates for derivatives in $s,t,r$.

5\. Now we investigate the equations

(4.45) $\displaystyle{\frac{1}{s}[G(s,t,r)]_{\psi}=}$ $\displaystyle 0,$
(4.46) $\displaystyle\frac{1}{\left\lvert
s\right\rvert^{2}}[G(s,t,r)]_{\alpha}=$ $\displaystyle 0.$

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. $\xi$ and $\eta_{k}$, we can write these equations as

(4.47) $\displaystyle{\kappa^{2}r-b}{+R_{\psi}(s,t,r)}$ $\displaystyle=0,$
(4.48) $\displaystyle\sum_{l=1}^{\dim\Omega}B_{kl}t_{l}{+R_{\alpha,k}(s,t,r)}$
$\displaystyle=0,\quad k=1,\ldots,\dim\Omega.$

Here, the remainders $R_{\psi}$ and $R_{\alpha,k}$ are $C^{2}$ functions from
$V$ to $\mathbb{C}$ and $\mathbb{R}^{\dim\Omega}$, respectively, since $G$ is
a $C^{2}$ map from $V$ to $N\subset X^{s-2},\,s\geq 2$. The remainders satisfy
the estimates

(4.49) $\displaystyle R_{\psi}(s,t,r)=$ $\displaystyle{O(\left\lvert
t\right\rvert)+O(\left\lvert s\right\rvert^{2})},$ (4.50) $\displaystyle
R_{\alpha,k}(s,t,r)=$ $\displaystyle{O(\left\lvert
t\right\rvert^{2})+O(\left\lvert s\right\rvert^{2})},$

and similarly for their derivatives, and

(4.51)
${B_{kl}:=\left\langle\eta_{k},\,\xi\eta_{l}\right\rangle_{\vec{L}^{2}}.}$

To obtain (4.48), we use self-adjointness of the orthogonal projection
$Q^{\prime}:\vec{\mathcal{H}}^{s}\to\vec{\mathcal{H}}^{s}$, and the fact that
$\eta_{k}\in\Omega=\operatorname{ran}Q^{\prime}$ for every $k$.

To summarize, we are now left with solving (4.47)–(4.48) in
$V\subset\mathbb{R}\times\mathbb{R}^{\dim\Omega}\times\mathbb{R}_{>0}$, which
amounts to solving the bifurcation equation (4.30).

Now, we view $s$ as a parameter and solve the algebraic system (4.47)–(4.48)
for $(t,r)\in\mathbb{R}^{\dim\Omega}\times\mathbb{R}_{>0}$.

######  Lemma 4.4.

For every $s\in\mathbb{R}$ with $\left\lvert s\right\rvert\ll 1$, there exists
a unique $C^{2}$ solution $(t,r)$ to (4.47)–(4.48) in a small neighbourhood of
$(0,b/\kappa^{2})\in\mathbb{R}^{\dim\Omega}\times\mathbb{R}_{>0}$.

Moreover, this solution satisfies

(4.52) $\displaystyle t=O(\left\lvert s\right\rvert^{2}),\quad
r=\frac{b}{\kappa^{2}}+O(\left\lvert s\right\rvert^{2}),\quad\text{ as
}\left\lvert s\right\rvert\to 0,$

and similarly for their derivatives.

This proposition is proved in Section 5.

6\. Lemma 4.4, together with Lemma 4.3, implies that there exists a family

(4.53) $\displaystyle(u_{s},r_{s}),\qquad s\in\mathbb{R},\quad\left\lvert
s\right\rvert\ll 1,$

with $r_{s}$ satisfying (4.52), that uniquely solves (4.7) in a small
neighbourhood of the trivial solution $(u,r)=(0,b/\kappa^{2})$.

This completes the proof of Proposition 4.1. ∎

##  5\. Solvability of the Bifurcation Equation

In this section, we solve the bifurcation equations (4.47)–(4.48) by proving
Lemma 4.4.

###### Proof of Lemma 4.4.

1\. We first solve (4.48). This equation has the trivial solution
$(s,t,r)=(0,0,b/\kappa^{2})$ (see (4.8)), and the l.h.s. of (4.48) is $C^{2}$
in a small neighbourhood
$V\subset\mathbb{R}\times\mathbb{R}^{\dim\Omega}\times\mathbb{R}_{>0}$ around
this zero.

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

(5.1) $B=[B_{kl}],$

where $B_{kl}$ is as in (4.51) and $1\leq k,l\leq\dim\Omega.$

Let ${t}=(t_{1},\dots t_{\dim\Omega})$ be an arbitrary non-zero real vector.
We compute, for any
$\omega=\sum_{i=1}^{\dim\Omega}t_{i}\eta_{i}\in\Omega\setminus\left\\{0\right\\}$,

$\displaystyle\left\langle Bt,\,t\right\rangle=$
$\displaystyle\sum_{l,k=1}^{\dim\Omega}\left\langle\eta_{k},\,\left\lvert\xi\right\rvert^{2}\eta_{l}\right\rangle_{\vec{\mathcal{L}}^{2}}t_{l}t_{k}$
(5.2) $\displaystyle=$
$\displaystyle\left\langle\omega,\,\left\lvert\xi\right\rvert^{2}\omega\right\rangle_{\vec{\mathcal{L}}^{2}}>0.$

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, $\mathbb{R}\times\mathbb{R}^{\dim\Omega}\times\mathbb{R}$. By
the gauge symmetry (4.12), the functions $R_{\alpha,k}$ depend only on
$\left\lvert s\right\rvert$ for all $k$. Hence, by the invertibility of $B$
and the Implicit Function Theorem, eq. (4.48) has a unique $C^{2}$ solution
$t=(t_{1},\ldots,t_{\dim\Omega})$ in a small neighbourhood around
$(s,r)=(0,b/\kappa^{2})$ in $\mathbb{R}\times\mathbb{R}_{>0}$, satisfying

(5.3) $t(s,r)=O(\left\lvert s\right\rvert^{2}),$

and similarly for its derivatives.

We note that, by the second relation in (4.12), the solution $t$ depends only
on $\left\lvert s\right\rvert$ and $r$.

2\. Next, we define the l.h.s. of (4.47) after plugging the solution $t$ found
in Step 1 to (4.48) as

(5.4) $\displaystyle\tilde{G}(s,r)=\kappa^{2}r-b{+\tilde{R}_{\psi}(s,r)},$

where $\tilde{R}_{\psi}(s,r):=R_{\psi}(s,t(s,r),r)$ satisfies, by (4.49) and
(5.3),

(5.5) $\displaystyle\tilde{R}_{\psi}(s,r)=$ $\displaystyle{O(\left\lvert
s\right\rvert^{2})},$

and similarly for its derivatives.

We look for solution in a small neighbourhood of $(s,r)=(0,b/\kappa^{2})$ to
the equation

(5.6) $\displaystyle\tilde{G}(s,r)=0.$

By (4.8) and definition (5.4), eq. (5.6) has the trivial solution
$(s,r)=(0,b/\kappa^{2})$. Moreover, by the $C^{2}$ regularity of the solution
$t=t(s,r)$ found in Step 1 above, the map $\tilde{G}$ is $C^{2}$ around this
zero, and $\partial_{r}\tilde{G}(0,b/\kappa^{2})=\kappa^{2}>0$ by estimate
(5.3) and similarly for $\partial_{r}\tilde{R}_{\psi}$. Thus, by the Implicit
Function Theorem, we obtain a unique solution $r=r(s)$ to (5.6) around $s=0$,
satisfying

(5.7) $r(s)=\frac{b}{\kappa^{2}}+O(\left\lvert s\right\rvert^{2}),$

and similarly for its derivatives. This proves Lemma 4.4.

∎

##  6\. Precise Asymptotics of the Non-trivial branch

In this section we finish our proof of Theorem 1.2.

######  Proposition 6.1.

Let the conditions of Proposition 4.1 hold and let

$(u_{s},r_{s}),\quad\left\lvert s\right\rvert\ll 1$

be the nontrivial branch of solution to (4.7), constructed in Proposition 4.1.
Then, for $u_{s}=(\psi_{s},\alpha_{s})$, we have

(6.1) $\displaystyle\psi_{s}$ $\displaystyle=\phi(s)+O(\left\lvert
s\right\rvert^{3}),$ (6.2) $\displaystyle\alpha_{s}$
$\displaystyle=\gamma(t(s))+O(\left\lvert s\right\rvert^{4}).$

Here $\phi$ and $\gamma$ are as in (4.32) and satisfy

(6.3) $\displaystyle(-\Delta_{a^{b}}-b)\phi=0,$ (6.4) $\displaystyle
d\gamma=\frac{1}{2}*\left(1-\left\lvert\phi\right\rvert^{2}\right).$

Moreover, we can take $s\in\mathbb{R}_{\geq 0}$ and if, in addition, $r$
satisfies (c.f. (1.15))

(6.5) $\displaystyle(\kappa-\sqrt{b/r})(\kappa-\kappa_{c})>0$

where $\kappa_{c}=\kappa_{c}(1)$ is given by (1.12) , then, the equation
$r=r(s)$ can be solved for s to obtain $s=s(r)$ with

(6.6) $\displaystyle{s}^{2}$
$\displaystyle=\frac{\kappa^{2}r-b}{(\kappa^{2}-\tfrac{1}{2})\beta+\tfrac{1}{2}}+O\left(\left(\frac{\kappa^{2}r-b}{(\kappa^{2}-\tfrac{1}{2})\beta+\tfrac{1}{2}}\right)^{2}\right),$

where $\beta=\beta(1)$ is the Abrikosov function given by (1.11).

###### Proof.

The solution branch $u_{s}$ from (4.53) is given by

(6.7) $u_{s}=v_{s}+w(v_{s},r_{s}),$

where $v_{s}=(\phi(s),\gamma(t(s)))$ with $\phi$ and $\gamma$ given in (4.32),
$t(s)=t(s,r(s))$ and $r_{s}=r(s)$ given in (5.3) and (5.7), and with $w$
satisfying (4.34). This implies (6.1)–(6.2). Eqs. (6.3)–(6.4) are implicit in
the proof of Proposition 4.1. These parts are analogous to the proofs of
[MR4049917, Prop.5.5, eqs. (5.26)–(5.27)].

It remains to prove (6.6). Observe that by gauge symmetry (4.12),
$\tilde{R}_{\psi}(s,r)$ in (5.4) depends on $s$ through $\left\lvert
s\right\rvert$. Thus, we can assume $s\in\mathbb{R}$. 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

(6.8) $r=\frac{b}{\kappa^{2}}+R{s}^{2}+O({s}^{4}),$

for some $R\in\mathbb{R}$, and with corresponding estimates on derivatives in
the remainder.

Take $\xi\in K$ as in (4.31). Using the expansion (6.1)–(6.2), together with
the (6.8), we find

(6.9)
$\displaystyle\left\langle\xi,\,(-\Delta_{a^{b}+\alpha}-\kappa^{2}r)\psi\right\rangle$
$\displaystyle=$
$\displaystyle\left\langle\xi,\,(-\Delta_{a^{b}}-b)\psi\right\rangle+{s}^{2}(\left\langle\xi,\,2i\eta\cdot\nabla_{a^{b}}\psi\right\rangle-\kappa^{2}R\left\langle\xi,\,\psi\right\rangle)+O({s}^{4})$
$\displaystyle=$
$\displaystyle{s}^{2}(\left\langle\xi,\,2i\eta\cdot\nabla_{a^{b}}\psi\right\rangle-\kappa^{2}R\left\langle\xi,\,\psi\right\rangle)+O({s}^{4}).$

Here, $\eta:={s}^{-2}\gamma(t(s))$, and we move to the last line using the
fact that $-\Delta_{a^{b}}$ is self-adjoint, and the equation (6.3) satisfied
by $\xi$.

Since, by construction, the pair $(\psi,a^{b}+\alpha)$ solves the rescaled
Ginzburg-Landau equations (2.15), we have

(6.10)
$(-\Delta_{a^{b}+\alpha}-\kappa^{2}r)\psi+\kappa^{2}\left\lvert\psi\right\rvert^{2}\psi=0.$

Then taking the inner product w.r.t. ${s}^{-2}\xi$ on both sides of (6.10),
using the expansion (6.9), and then taking $s\to 0$, we find

(6.11)
$\left\langle\xi,\,2i\eta\cdot\nabla_{a^{b}}\xi\right\rangle-\kappa^{2}R\left\lVert\xi\right\rVert_{\mathcal{L}^{2}}^{2}+\kappa^{2}\left\lVert\xi\right\rVert_{\mathcal{L}^{4}}^{4}=0.$

Next, recalling the definition $\langle
f\rangle=\frac{1}{\left\lvert\Sigma\right\rvert}\int_{\Sigma}f$, we have the
identity (see the equation after [MR4049917, eq. (5.32)])

(6.12)
$\displaystyle\frac{1}{\left\lvert\Sigma\right\rvert}\left\langle\xi,\,2i\eta\cdot\nabla_{a^{b}}\xi\right\rangle=$
$\displaystyle{\frac{1}{2}\left(\langle\left\lvert\xi\right\rvert^{2}\rangle^{2}-\langle\left\lvert\xi\right\rvert^{4}\rangle\right)}.$

Substituting (6.12) into (6.11) and solving for $R$, we find

(6.13)
$\displaystyle{R=\frac{1}{2\kappa^{2}}+\left(1-\frac{1}{2\kappa^{2}}\right)\langle\left\lvert\xi\right\rvert^{4}\rangle\quad\text{since
}\langle\left\lvert\xi\right\rvert^{2}\rangle=1.}$

Now, rearranging (6.8), we find

(6.14) $\displaystyle{s}^{2}=\frac{\kappa^{2}r-b}{\kappa^{2}R}+O({s}^{4}),$

which, together with (6.13) and the fact that
${\beta=\langle\left\lvert\xi\right\rvert^{4}\rangle}$ in the non-degenerate
case with $\dim K=1$, yields

(6.15)
$\displaystyle{s}^{2}=\frac{\kappa^{2}r-b}{\left(\kappa^{2}-\frac{1}{2}\right){\langle\left\lvert\xi\right\rvert^{4}\rangle}{+\frac{1}{2}}}+O({s}^{4}).$

With this, the equation $r=r(s^{2})$ can be solved for $s^{2}$ to yield
$s^{2}=s^{2}(r)$, with (6.6).

Note that, since
$(\kappa^{2}-\tfrac{1}{2})\beta+\tfrac{1}{2}=\beta(\kappa^{2}-\kappa_{c}^{2})$
by definitions (1.11)–(1.12), condition (6.5) ensures that the leading term in
(6.15) is strictly positive and so (6.6) is self-consistent for small $s$
(c.f. Remark 1).

This completes the proof of Proposition 6.1. ∎

######  Remark 12.

It follows from (6.6) and condition (1.15) that $r(s)$ is locally parabolic;
i.e., $r\sim Cs^{2}$. Hence there can be at most two branches, $\pm s(r)$.
However, since these would be related by a gauge transformation, up to gauge
change, there will locally near $(r,s)=(b/\kappa^{2},0)$ be only one branch
$s(r)$.

###### Proof of Theorem 1.2.

For every $s=s(r)$ with $r$ satisfying (1.15), the solution
$u_{s}\equiv(\psi_{s},\alpha_{s})$ constructed in Proposition 4.1 gives a
solution to the rescaled Ginzburg-Landau equation (2.15) on $(\Sigma,h_{1})$
as $(\psi_{s},a^{b}+\alpha_{s})$. By relation (3.19), eq. (6.3) implies that
$\xi$ is a holomorphic section of $E$ corresponding to $a^{b}$.

Undoing the rescaling (2.12), the asymptotics (1.18)–(1.20) follow from (5.3)
and (6.1)–(6.4), and the expansion (1.21) follows from (6.6). This completes
the proof of Theorem 1.2. ∎

###### Proof of Corollary 1.6.

We first prove that, with $(\psi,a)\equiv(\psi_{s(r)},a_{s(r)})$ solving
(2.15) and rescaled GL energy functional $\mathcal{E}_{r}(\psi,a)$ from
(2.11), we have

(6.16)
$\mathcal{E}_{r}(\psi,a)=\mathcal{E}_{r}(0,a^{b})-\frac{{\left\lvert\Sigma\right\rvert}}{4}\frac{\left\lvert\kappa^{2}r-b\right\rvert^{2}}{(\kappa^{2}-\tfrac{1}{2}){\langle\left\lvert\xi\right\rvert^{4}\rangle}+\tfrac{1}{2}}+O(\left\lvert\kappa^{2}r-b\right\rvert^{3}),$

for $0<\left\lvert\kappa^{2}r-b\right\rvert\ll 1$. Here $\xi\in
K\equiv\operatorname{Null}(-\Delta_{a^{b}}-b)$,
${\langle\left\lvert\xi\right\rvert^{2}\rangle=1}$ is as in (4.31).

First, taking the inner product of the first equation in (2.15) with $\psi$,
and then integrating by parts, we find

$\int\left\lvert\nabla_{a}\psi\right\rvert=\int\kappa^{2}\left(|\psi|^{2}-|\psi|^{4}\right).$

Substituting this into the rescaled GL energy (2.11), we get

(6.17)
$\mathcal{E}_{r}(\psi,a)=\frac{1}{2}\left(\frac{\kappa^{2}r^{2}}{2}\left\lvert\Sigma\right\rvert+\|da\|^{2}_{\mathcal{L}^{2}}-\frac{\kappa^{2}}{2}\|\psi\|^{4}_{\mathcal{L}^{4}}\right).$

This gives

(6.18)
$\displaystyle\mathcal{E}_{r}(0,a^{b})=\frac{1}{2}\left(\frac{\kappa^{2}r^{2}}{2}+b^{2}\right)|\Sigma|\quad\text{with
}b=\frac{2\pi n}{\left\lvert\Sigma\right\rvert}.$

Next, we choose $\eta:=\left\lvert s\right\rvert^{-2}\gamma(t(s))$ according
to expansion (6.2). Then $\eta\in\Omega$ by construction and therefore
$\left\langle d\eta,\,d\beta\right\rangle=0$ for any $1$-form $\beta$, see
Proposition 3.1. Using this, the expansion $a=a^{b}+\left\lvert
s\right\rvert^{2}\eta+O(\left\lvert s\right\rvert^{4})$ (see (6.2)), together
with the fact that $da^{b}=b\omega$, we find

(6.19) $\displaystyle\|da\|_{\mathcal{L}^{2}}^{2}$
$\displaystyle=\|da^{b}\|_{\mathcal{L}^{2}}^{2}+2\left\lvert
s\right\rvert^{2}\langle da^{n},d\eta\rangle+\left\lvert
s\right\rvert^{4}\|d\eta\|_{\mathcal{L}^{2}}^{2}+O(\left\lvert
s\right\rvert^{6})$
$\displaystyle=b^{2}\left\lvert\Sigma\right\rvert+\left\lvert
s\right\rvert^{4}\|d\eta\|^{2}_{\mathcal{L}^{2}}+O(\left\lvert
s\right\rvert^{6}).$

With the choice (4.31) for $\xi$, expansion (6.1) becomes
$\psi=s\xi+O(\left\lvert s\right\rvert^{3})$. Plugging this into formula

(6.20) $\displaystyle\frac{1}{\left\lvert\Sigma\right\rvert}\left\lVert
d\eta\right\rVert_{\mathcal{L}^{2}}^{2}=$
$\displaystyle-\frac{1}{4}(\langle\left\lvert\xi\right\rvert^{2}\rangle^{2}-\langle\left\lvert\xi\right\rvert^{4}\rangle),$

proved in [MR4049917, Prop. 5.5, eq. (5.32)], and using that
$\langle\left\lvert\xi\right\rvert^{2}\rangle\equiv\frac{1}{\left\lvert\Sigma\right\rvert}\int\left\lvert\xi\right\rvert^{2}=1$,
we find

(6.21) $\displaystyle\left\lvert
s\right\rvert^{4}\|d\eta\|^{2}_{\mathcal{L}^{2}}-\frac{\kappa^{2}}{2}\|\psi\|^{4}_{\mathcal{L}^{4}}=\frac{{\left\lvert\Sigma\right\rvert}}{2}\left\lvert
s\right\rvert^{4}\left(\left(\frac{1}{2}-\kappa^{2}\right){\langle\left\lvert\xi\right\rvert^{4}\rangle}-\frac{1}{2}\right)+O(\left\lvert
s\right\rvert^{6}).$

Plugging (6.19)–(6.21) into (6.17), we conclude that

(6.22)
$\mathcal{E}_{r}(\psi,a)=\mathcal{E}_{r}(0,a^{b})-\frac{{\left\lvert\Sigma\right\rvert}}{4}\left\lvert
s\right\rvert^{4}\left(\left(\kappa^{2}-\frac{1}{2}\right){\langle\left\lvert\xi\right\rvert^{4}\rangle}+\frac{1}{2}\right)+O(\left\lvert
s\right\rvert^{6}).$

This, together with expansion (6.15) for $s$, gives (6.16).

By definition (1.11) and the assumption $\dim K=1$, we have
${\beta=\langle\left\lvert\xi\right\rvert^{4}\rangle}$. This, together with
(6.16), the rescaling (2.14), and the relation
$\left\lvert\Sigma\right\rvert=r^{-1}\left\lvert\Sigma\right\rvert$, gives
(1.27). ∎

##  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 $\xi_{j}\,\,\,1\leq j\leq D$ for
$K$. Define $v=v_{st}=(\phi(s),\gamma(t))$, where $s=(s_{1},\ldots,s_{D})$,
$t=(t_{1},\ldots,t_{\dim\Omega})$, $\phi(s)=\sum_{j=1}^{D}s_{j}\xi_{j}$, and
$\gamma(t)=\sum_{k=1}^{\dim\Omega}t_{k}\eta_{k}$ as in (4.32).

Taking inner products of the bifurcation equation (4.30) respectively with
$\xi_{j}$ and $\eta_{k}$, this equation can be rewritten in terms of the
coordinates $s,t,r$ as

(7.1) $\displaystyle(\kappa^{2}r-b)s_{j}{+H_{\psi,j}(s,t,r)}$
$\displaystyle=0,\quad j=1,\ldots,D,$ (7.2)
$\displaystyle\sum_{l=1}^{\dim\Omega}B_{kl}(s)t_{l}+H_{\alpha,k}(s,t,r)$
$\displaystyle=0,\quad k=1,\ldots,\dim\Omega,$

where
$B_{k,\ell}(s)=\langle\eta_{k},|\phi(s)|^{2}\eta_{\ell}\rangle_{\vec{L}^{2}}.$
The remainders satisfy the estimates

(7.3) $\displaystyle H_{\psi,j}(s,t,r)=$ $\displaystyle O(\left\lvert
s\right\rvert\left\lvert t\right\rvert)+O(\left\lvert s\right\rvert^{3}),$
(7.4) $\displaystyle H_{\alpha,k}(s,t,r)=$ $\displaystyle O(\left\lvert
s\right\rvert^{2}\left\lvert t\right\rvert^{2})+O(\left\lvert
s\right\rvert^{4}),$

and similarly for their derivatives.

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.

######  Theorem 7.1 ([MR660633, Chapt. 4 ]).

Let $E:\mathbb{R}^{n+1}\to\mathbb{R}^{n}$ be $C^{2}$ of the form

(7.5) $\displaystyle E(\ell,s)$ $\displaystyle=$ $\displaystyle\ell
s+H(\ell,s)$

where $s\in\mathbb{R}^{n},\ell\in\mathbb{R}$ and $H(\ell,s)=O(|s|^{2})$ as
$|s|\to 0$ uniformly in a neighborhood of $\ell=0$. Assume that $n$ is odd;
then, $(\ell,s)=(0,0)$ is a bifurcation point of $E(\ell,s)$; i.e., there are
solutions $(\ell,s)$ with $s\neq 0$ of the equation $E(\ell,s)=0$ in every
neighborhood of $(\ell,s)=(0,0)$.

######  Remark 13.

We note that (7.5) is the standard normal form for a nonlinear problem whose
linearization has $\ell=0$ as an isolated eigenvalue of multiplicity $n\geq
1$.

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
$(b/\kappa^{2},0)$, 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, $s=(x_{1},x_{2},y_{2},\dots x_{D},y_{D})$, which
coordinatizes $\mathbb{R}^{2D-1}$. 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
$\ell=\kappa^{2}r-b$ and substituting the solution $t=t(r,s)$ found in Lemma
7.2 to (7.1), the equation takes the vector form

(7.6) $\displaystyle\tilde{G}(\ell,s)$ $\displaystyle=$ $\displaystyle\ell
s+H(s,r)=0$

where $H(s,r)=O(|s|^{2})$. Since $\tilde{G}(\ell,s)$ 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 $s\hat{E}(s)=0$. So
for each $s\neq 0$, $\hat{E}(s)=0$. It follows that the unique $C^{2}$ curve
$\ell(s)$ 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 $\kappa^{2}r(s)-b$, determining a unique $r(s)$ that
satisfies

(7.7) $r(s)=\frac{b}{\kappa^{2}}+O(\left\lvert s\right\rvert^{2}).$

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),

(7.8) $\displaystyle\Phi(s,t)$ $\displaystyle=$ $\displaystyle\langle
B(s)t,t\rangle$ $\displaystyle=$
$\displaystyle\sum_{k,\ell=1}^{\dim\Omega}t_{k}t_{\ell}B_{k\ell}(s)$ (7.9)
$\displaystyle=$
$\displaystyle\langle\omega,|\phi(s)|^{2}\omega\rangle_{\vec{L}^{2}}$

It is manifest from this that $\Phi$ is real valued and, away from $s=0$,
defined, differentiable and positive definite as a quadratic form in ${t}$.

To understand $\Phi$ 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
$\mathbb{R}^{n}\times\mathbb{P}^{n-1}$ with $\mathbb{R}^{n}$ coordinatized by
the affine coordinates $\vec{w}=(\vec{x},\vec{y})$ and $\mathbb{P}^{n-1}$
coordinatized by the homogeneous coordinates $\vec{z}=[z_{1},\dots z_{n}]$.
The blow-up of $\mathbb{R}^{n}$ at the origin, which we’ll denote by $M$, is
the closed subset of $\mathbb{R}^{n}\times\mathbb{P}^{n-1}$ defined by the
equations $\\{w_{i}z_{j}-w_{j}z_{i}=0\,\,|\,\,i,j=1,\dots,n\\}$. One has a
natural map $\pi:M\to\mathbb{R}^{n}$ induced by projection onto the first
factor. It is fairly straightforward to check that away from $\vec{w}=0$,
$\pi$ is a diffeomorphism. However, $\pi^{-1}(0)\simeq\mathbb{P}^{n-1}$; it
consists of all points of the form $0\times[z_{1},\dots z_{n}]$. Now to see
that points of $\pi^{-1}(0)$ are in 1:1 correspondence with the set of lines
through the origin in $\mathbb{R}^{n}$, note that a line $L$ through the
origin is parametrically given by $w_{i}=c_{i}\sigma$ where the $c_{i}$ are
not all zero. Then consider the lift of this line to $\tilde{L}=\pi^{-1}(L-0)$
in $M-\pi^{-1}(0)$ whose parametrization is
$w_{i}=c_{i}\sigma,z_{i}=c_{i}\sigma$. However, since $z_{i}$ are homogeneous
coordinates, one may as well take this parametrization to be
$w_{i}=c_{i}\sigma,z_{i}=c_{i}$. These equations are well-defined for
$\sigma=0$, defining the closure of $\tilde{L}$ in $M$ which meets
$\pi^{-1}(0)$ in the point $[c_{1},\dots,c_{n}]\in\mathbb{P}^{n-1}$. This
defines the mapping, $L\to[c_{1},\dots,c_{n}]$ that gives the 1:1
correspondence between lines through the origin and points of $\pi^{-1}(0)$.
For further explanation and details we refer the reader to [MR0463157] on
which the above is based.

Applying this to $\Phi$, we set
$A_{jm}=\int_{F_{\Sigma}}\xi_{m}\bar{\xi}_{j}\omega\wedge*\omega=\mu_{jm}+i\nu_{jm}$.
Then one has a representation of $\Phi$ as

(7.10) $\displaystyle\Phi(s,t)$ $\displaystyle=$ $\displaystyle\sum_{1\leq
j\leq m\leq
D}\left(A_{jm}s_{j}\bar{s}_{m}+\bar{A}_{jm}\bar{s}_{j}s_{m}\right)$
$\displaystyle=$ $\displaystyle
2\mu_{jm}(x_{j}x_{m}+y_{j}y_{m})-2\nu_{jm}(y_{j}x_{m}-x_{j}y_{m}).$

To lift $\Phi$ to $M$ we set $\vec{c}=[a_{1},a_{2},b_{2},\dots a_{D},b_{D}]$.
Applying the corresponding parametrization in $\Phi$ one has

(7.11) $\displaystyle\Phi(s,t)$ $\displaystyle=$ $\displaystyle
2\sigma^{2}\left(\mu_{jm}(a_{j}a_{m}+b_{j}b_{m})-\nu_{jm}(b_{j}a_{m}-a_{j}b_{m})\right).$

Under the blow-up change of coordinates to $(\vec{c},\sigma,r)$, the higher
order terms scale similarly but involve powers of $\sigma$ greater than 2.
Consequently, the terms in $(1/\sigma^{2})$ 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
$\sigma$. Hence the positivity seen in (7.9) is inherited by
$(1/\sigma^{2})\Phi(s,t)$ from (7.11). But the latter is defined just in terms
of $\vec{c}$ which are coordinates on a compact projective space. So by
continuity, $(1/\sigma^{2})\Phi(s,t)$ realizes its infimum at some point in
$\mathbb{P}^{n-1}$. 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 $O(\sigma^{2})$ in these
coordinates and so vanishes in a continuous fashion as $\sigma\to 0$. Hence
$t$ on the blowup $M$ can be pushed forward under $\pi$ to $t(s,r)$ in the
original $s$-space. Similarly for the first derivatives of $t(s,r)$ and the
second derivatives of $st(s,r)$. ∎

###  7.2. Energy estimates

It is possible to extend the energy estimate proved in Corollary 1.6 to the
degenerate case. In general, if one drops the second part in condition (1.14),
then, for the solution $(\psi_{s(r)},a_{s(r)})$ constructed in (1.22), we
have,

(7.12)
${\mathcal{E}(\psi_{s(r)},a_{s(r)},h_{r})\geq\mathcal{E}(0,a^{b_{r}},h_{r})}{}-\frac{{\left\lvert\Sigma\right\rvert_{r}}}{4}\frac{\left\lvert\kappa^{2}-{b_{r}}\right\rvert^{2}}{(\kappa^{2}-\tfrac{1}{2})\beta(r)+\tfrac{1}{2}}+O(\left\lvert\kappa^{2}-{b_{r}}\right\rvert^{3})$

for $\kappa\geq 1/\sqrt{2}$, and the opposite inequality for
$\kappa<1/\sqrt{2}$.

Indeed, for $\dim K>1$, we have
$\beta(r)\leq\langle\left\lvert\xi\right\rvert^{4}\rangle$ for all $\phi\in
K,\,\langle\left\lvert\phi\right\rvert^{2}\rangle=1$. This, together with
(6.16), gives (7.12) depending on the sign of $\kappa^{2}-\frac{1}{2}$.

One can also introduce the ‘upper Abrikosov function’,

(7.13)
$\displaystyle\beta_{+}(r):={\max\left\\{{\langle\left\lvert\xi\right\rvert^{4}\rangle}:\xi\in
K(r),\langle\left\lvert\xi\right\rvert^{2}\rangle=1\right\\}.}$

which leads to the upper energy bound, for $\kappa^{2}\geq 1/2$,

(7.14)
${\mathcal{E}(\psi_{s(r)},a_{s(r)},h_{r})\leq\mathcal{E}(0,a^{b_{r}},h_{r})}{}-\frac{{\left\lvert\Sigma\right\rvert_{r}}}{4}\frac{\left\lvert\kappa^{2}-{b_{r}}\right\rvert^{2}}{(\kappa^{2}-\tfrac{1}{2})\beta_{+}(r)+\tfrac{1}{2}}+O(\left\lvert\kappa^{2}-{b_{r}}\right\rvert^{3}).$

For $\kappa_{c+}(r):=\sqrt{\frac{1}{2}\left(1-\frac{1}{\beta_{+}(r)}\right)}$,
estimate (7.14) ensures that
$\mathcal{E}(\psi_{s(r)},a_{s(r)},h_{r})<\mathcal{E}(0,a^{b_{r}},h_{r})$ for
$\kappa>\kappa_{c+}(r)$.

## Acknowledgment

J. Zhang is supported by Danish National Research Foundation grant CPH-GEOTOP-
DNRF151, The Niels Bohr Grant of The Royal Danish Academy of Sciences and
Letters, and NSERC Grant No. NA7901. The research of IMS is supported in part
by NSERC Grant No. NA7901.

## Declarations

  * •

Conflict of interest: The Authors have no conflicts of interest to declare
that are relevant to the content of this article.

  * •

Data availability: Data sharing is not applicable to this article as no
datasets were generated or analysed during the current study.

##  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 $\Sigma$,
with fibers isomorphic to $\mathbb{C}$ 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, $\nabla_{a}$, on sections of $E$, which can be written locally as
$\nabla_{a}\psi=\nabla\psi-ia\psi$. Here $\nabla$ 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

(A.1) $\displaystyle da=b\omega,$

for some $b\in\mathbb{R}$, where $\omega$ is the standard hyperbolic $2$-form
on $\Sigma$.

By [MR4049917, Lem. 3.2], a constant curvature connection, $a$, solves the
(static) Maxwell equation

(A.2) $d^{*}da=0.$

Let $x^{i},\,i=1,2$ be a local coordinate on $\Sigma$. With the summation
convention understood, we can write a connection (gauge field), $a$, and the
covariant derivative, $\nabla_{a}$, as $a=a_{i}dx^{i}$ and
$\nabla_{a}=\nabla_{i}dx^{i}$, where $\nabla_{j}:=\partial_{i}-ia_{j}$. Let
$d$ be the exterior derivative on $\Sigma$. Then, for a section $\psi$ and
1-form $B=B_{i}dx^{i}$,

$d_{a}\psi=\nabla_{i}\psi dx^{i},\ dB=\partial_{i}B_{j}dx^{i}\wedge dx^{j},\
d^{*}B=-\partial_{i}B_{i}$

and the curvature, $F_{a}=da$, of $a$ is given by

(A.3) $\displaystyle F_{a}=\big{[}\nabla_{i},\nabla_{j}\big{]}dx^{i}\wedge
dx^{j},\
\quad\big{[}\nabla_{i},\nabla_{j}\big{]}=\partial_{i}a_{j}-\partial_{j}a_{i}.$

###  A.2. Automorphy factor

Let $\Sigma=\mathbb{H}/\Gamma$ be a non-compact Riemann surface of the form
(2.1). Let $\pi_{1}(\Sigma)$ be the first fundamental group of $\Sigma$, which
we identify with $\Gamma$. A map

$\rho(\gamma,z):\pi_{1}(\Sigma)\times\mathbb{H}\cong\Gamma\times\mathbb{H}\to
U(1)$

is called an automorphy factor if it satisfies the important co-cycle
condition,

(A.4)
$\rho(\gamma\gamma^{\prime},z)=\rho(\gamma,\gamma^{\prime}z)\rho(\gamma^{\prime},z)\quad\left(\gamma,\gamma^{\prime}\in\Gamma,\,z\in\mathbb{H}\right).$

For fixed $\gamma\in\Gamma$, we often write
$\rho_{\gamma}(z)\equiv\rho(\gamma,z)$ as a function from $\mathbb{H}$ to
$U(1)$.

For every $b\in\mathbb{R}$, we fix a canonical choice of automorphy factor
with weight $b$ as

(A.5)
$\rho_{b}(\gamma,z)\equiv\rho_{b,\gamma}(z):=\left(\frac{cz+d}{c\bar{z}+d}\right)^{b}\quad\left(\gamma=\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\in SL(2,\mathbb{R})\right).$

For results in this paper, we understood that $b\in\mathbb{Z}$, but (A.5) can
also be defined for non-integer values of $b$ with branch cuts.

The choice (A.5) is customary in the study of Maass forms in number theory,
see e.g. [MR1431508, Sec.2.1]. In Theorem B.1, we prove this choice of
$\rho(\gamma,z)$ satisfies the co-cycle condition (A.4). Note that
$\rho(\gamma,z)$ remains bounded as $z$ approaches the boundary
$\mathbb{R}\cup\left\\{\infty\right\\}$ of $\mathbb{H}$, which contains the
cusps of $\Sigma$.

Recall that the universal cover of the Riemann surface $\Sigma$ is the
Poincaré half-plane $\mathbb{H}$. Then using the co-cycle condition (A.4), one
can define a line bundle $E_{\rho}$ as

(A.6) $E_{\rho}:=(\mathbb{H}\times\mathbb{C})/\rho,\text{ with the action
}\rho_{\gamma}:(z,\psi)\mapsto(\gamma z,\rho_{\gamma}(z)\psi).$

In fact, (A.6) defines an one-to-one correspondence, $\rho\leftrightarrow
E_{\rho}$, between (equivalence classes of) automorphy maps for $\Sigma$ and
line $U(1)$-bundles over $\Sigma$. See [MR0132828] for details.

Through the correspondence (A.6), one can define a topological degree for a
unitary line bundle over $\Sigma$, which by the Chern-Weil correspondence is
equal to the Chern number, or the degree of $E$, as follows. Suppose
$E=E_{\rho}$ as in (A.6) for an automorphy factor $\rho$ satisfying (A.4).
Suppose the Fuchsian group $\Gamma$ has genus $g$ and $m$ cusps, with no
elliptic points. Then $\Gamma$ is generated by $2g$ hyperbolic transforms,
$A_{1},B_{1},\ldots,A_{g},B_{g}$, and $m$ parabolic transforms
$S_{1},\ldots,S_{m}$, satisfying the relation

(A.7) $\displaystyle A_{1}B_{1}A_{1}^{-1}B_{1}^{-1}\dots
A_{g}B_{g}A_{g}^{-1}B_{g}^{-1}S_{1}\dots S_{m}={\bf{1}}.$

Now, for every $\gamma\in\Gamma$, define a map
$\sigma=\sigma_{\gamma}:\mathbb{H}\to\mathbb{R}$ by the relation
$e^{i\sigma_{\gamma}(z)}:=\rho(\gamma,z)$. The first Chern number
$c_{1}(\sigma)$ is defined by

(A.8) $\displaystyle c_{1}(\sigma):=$
$\displaystyle\frac{1}{2\pi}\sum_{1}^{g}[\sigma_{A_{i}}(v_{i})-\sigma_{A_{i}}(B_{i}^{-1}v_{i})+\sigma_{B_{i}}(B_{i}^{-1}v_{i})-\sigma_{B_{i}}(A_{i}B_{i}^{-1}v_{i})],$

where $v_{i}:=B_{i}A_{i}^{-1}B_{i}^{-1}C_{i+1}\dots
C_{g}z_{0}=A_{i}^{-1}C_{i}\dots C_{g}z_{0}$, with
$C_{i}:=A_{i}B_{i}A_{i}^{-1}B_{i}^{-1}$. By [MR0457787, Theorem 2A],
$c_{1}(\sigma)$ is independent of $z_{0}$, and takes values in $\mathbb{Z}$.

Let $E$ and $E^{\prime}$ be the line bundles corresponding to two automorphy
factors $e^{i\sigma},\,e^{i\sigma^{\prime}}$ respectively. Then we say that
$E$ and $E^{\prime}$ are equivalent if and only if
$c_{1}(\sigma)=c_{1}(\sigma^{\prime}).$

Hence, by the one-to-one correspondence (A.4), $c_{1}(\sigma)$ is a
topological invariant for a unitary line bundle $E=E_{\rho}$ over
$\Sigma=\mathbb{H}/\Gamma$. It turns out (see [Gunning62]) that
$c_{1}(\sigma)$ is equal to the topological degree of $E$:

(A.9) $c_{1}(\sigma)=\deg E_{e^{i\sigma}}.$

(Recall that $\deg E$ is defined through multiplicity of zeroes of sections.)

###  A.3. Equivariant States

######  Definition A.1.

Let $\Gamma$ be a Fuchsian group. A (function, $1$-form)-pair
$(\Psi,A):\mathbb{H}\to\mathbb{C}\times\mathbb{R}^{2}$ is called an
equivariant state w.r.t. $\Gamma$ if and only if for all $\gamma\in\Gamma$,
$z\in\mathbb{H}$, the following relations hold:

(A.10) $\displaystyle\gamma^{*}\Psi(z)$ $\displaystyle=\rho(\gamma,z)\Psi(z),$
(A.11) $\displaystyle\gamma^{*}A(z)$
$\displaystyle=A(z)+i\rho(\gamma,z)^{-1}d\rho(\gamma,z),$

where $\gamma^{*}$ is the pull-back by the Möbius transform (2.2) associated
to $\gamma\in\Gamma$, and $\rho(\gamma,z)$ is an automorphy factor, satisfying
the co-cycle condition (A.4)

######  Remark 14.

For a standard lattice $\mathcal{L}\subset\mathbb{C}\cong\mathbb{R}^{2}$,
equivariant solutions to (GL) are known as the Abrikosov lattices, predicted
by A.A. Abrikosov in 1957 [Abrikosov]. (This discovery was recognized by a
Nobel prize.) These are ground state solutions to (GL) on the flat torus

(A.12) $\mathbb{T}=\mathbb{C}/\mathcal{L},$

which is a compact Riemann surface of genus $1$. Existence and stability
theory of solutions to (GL) on $\mathbb{T}$ are studied in [MR2560758,
MR2904275, MR3123370, MR3758428]. Physically, these solutions correspond to
regular arrays of vortices as seen in Type II superconductors.

In (A.12), the lattice $\mathcal{L}$ acts on the complex plane by translation,
and the action is commutative. In comparison, with the background geometry
(2.1), the action of $\Gamma$ on the Poincaré half plane $\mathbb{H}$ by
Möbius transforms is in general non-commutative. In this sense, equivariant
solutions to (GL) on (2.1) are non-commutative generalizations of the
Abrikosov lattice.

Equivariant states $(\Psi,A)$ are in one-to-one correspondence with (section,
connection)-pairs $(\psi,a)$ on the unitary complex line bundle $E\to\Sigma$
through the explicit correspondence (A.6). One can restrict $(\Psi,A)$ to a
fundamental domain $F\subset\mathbb{H}$ of $\Gamma$, with (A.10)–(A.11)
considered as boundary conditions. Although fundamental domain is not unique,
every two fundamental domains $F,F^{\prime}\subset\mathbb{H}$ of $\Gamma$ are
related by a Möbius transform $\gamma\in\Gamma$, which sends $F\to
F^{\prime}$. Hence, if $(\Psi,A)$ is an equivariant state on $F$ satisfying
(A.10)–(A.11), then the change of identification $F\to F^{\prime}$ amounts to
a gauge transform of the form

(A.13) $\displaystyle(\Psi,A)\longrightarrow(\rho\Psi,A+i\rho^{-1}d\rho).$

Since (GL) is invariant under (A.13), such change has no consequence when one
studies the solution to (GL).

The important correspondence formulated above allows us to interpret problems
posed in $X^{s}$ in terms of equivariant states. For example, given an
automorphy factor $\rho_{b}$ with weight $b$, let
$L^{2}(\mathbb{H}/\Gamma)\equiv L^{2}(\mathbb{H}/\Gamma,\rho_{b})$ be the
space of square integrable equivariant function satisfying (A.10) with
$\rho=\rho_{b}$ (see e.g. [MR1431508, Sect. 2.1] for details).

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 $L^{2}(\mathbb{H}/\Gamma)$ and prove inclusion
(3.45), using similar symmetrization argument as in [MR1942691], which studies
the same problem with $b=0$. This method was introduced by Selberg in 1950s,
and is well-known to the number theorists.

######  Proposition A.2.

Let $\Sigma=\mathbb{H}/\Gamma$ be a non-compact Riemann surface with $m$ cusps
and no elliptic points, with $\mathcal{S}(\Sigma)\neq\emptyset$. Then the
solution space to (3.2) in $L^{2}(\mathbb{H}/\Gamma)$ is spanned by the
vectors

(A.14) $\displaystyle\xi_{i}(z)$
$\displaystyle:=\sum_{\gamma\in\Gamma/\Gamma_{i}}\operatorname{Im}(\gamma_{i}\gamma
z)^{b}e^{\gamma_{i}\gamma 2\pi iz}\rho_{b}(\gamma,z)^{-1},\quad i=1,\ldots,m$

where each $\Gamma_{i}:=Stab(c_{i},\Gamma)$ denotes the stabilizer of a
distinct cusp $c_{i}$ of $\Gamma$, and $\gamma_{i}\in SL(2,\mathbb{R})$ is a
scaling matrix of $c_{i}$.

Moreover, there holds the asymptotics

(A.15) $\left\lvert\xi_{i}(\gamma_{i}^{-1}z)\right\rvert=O(e^{-2\pi
y})\quad(y\to\infty),$

which implies that $\xi_{i}(z)$ decays exponentially fast as $z$ approaches
the cusp $c_{i}$.

###### Proof.

Suppose a function $\phi:\mathbb{H}\to\mathbb{C}$ given by
$\phi(x,y)\equiv\phi(x+iy)$ satisfies

(A.16) $\phi(x+1,y)=\phi(x,y)\quad\text{ for every }y>0.$

Then for every cusp $c_{i}$, we can form the Poincaré series

(A.17)
$E_{c_{i},\phi}(z):=\sum_{\gamma\in\Gamma/\Gamma_{i}}\phi(\gamma_{i}\gamma
z)\rho^{-1}(\gamma,z).$

This series (3.1) converges absolutely if $\phi$ satisfies certain growth
condition in $y$.

The Poincaré series has two important properties:

  1. (a)

By construction, $E_{c_{i},\phi}$ satisfies the equivariance condition (A.10);

  2. (b)

If $L$ is an invariant operator acting on $\mathbb{H}/\Gamma$ in the sense
that

(A.18) $\Delta_{a^{b}}(\gamma^{*}\phi)=\rho(\gamma,z)(-\Delta_{a^{b}}\phi)$

for every $\gamma\in\Gamma,\,\phi\in\mathcal{H}^{s}$, and $L\phi=\lambda\phi$
for some $\lambda\in\mathbb{C}$, then $LE_{c_{i},\phi}=\lambda
E_{c_{i},\phi}$.

In view of these properties, to solve the eigenvalue problem (3.2) in
$L^{2}(\mathbb{H}/\Gamma)$, we first solve the following problem for some
$\phi\in C^{2}(\mathbb{H},\mathbb{C})$:

(A.19) $\displaystyle(-\Delta_{a^{b}}-b)\phi$ $\displaystyle=0,$ (A.20)
$\displaystyle\phi(x+1,y)$
$\displaystyle=\phi(x,y),\quad\lim_{y\to\infty}\phi(x,y)=0.$

In the boundary condition (A.20), the periodicity in $x$ ensures that the
symmetrization is well-defined, c.f. (A.16). The decay property in $y$ is
required to ensure convergence of the Poincaré series (A.17).

The solution to (A.19)–(A.20) is proportional to the function

$\phi:=y^{b}e^{-2\pi y}e^{i2\pi x}.$

This is calculated in e.g. [MR776146, MR870891]. Forming Poincaré series
(A.17) w.r.t. each of the cusps of $\Sigma$ 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.
[MR750670, Sect. 2], there holds the Fourier expansion

(A.21) $\xi_{i}(\gamma_{i}^{-1}z)=\sum_{k\neq
0}A_{k}W_{b\operatorname{sign}(k),b-\tfrac{1}{2}}(4\pi\left\lvert
n\right\rvert y)e^{2k\pi ix}.$

Here $W_{\beta,\mu}(y)=O(e^{-y/2})$ is the Whittaker function, a decaying
solution to the ODE

$W^{\prime\prime}(y)+\left(-\frac{1}{4}+\frac{\beta}{y}+\frac{1/4-\mu^{2}}{y^{2}}\right)W(y)=0.$

This $W_{\beta,\mu}(y)$ can be expressed in terms of the modified Bessel
function of the second kind [MR1773820]. 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 [MR870736, MR931205].

###### Proof of (3.45).

We use the same symmetrization method as in the proof of Proposition A.2.
First, we seek $C^{2}(\mathbb{H},\mathbb{C})$-solutions to the eigenvalue
problem

(A.22) $\displaystyle(-\Delta_{a^{b}}-\lambda)\psi$ $\displaystyle=0,$ (A.23)
$\displaystyle\psi(x+1,y)$ $\displaystyle=\psi(x,y).$

Notice here we do not require the decay condition in $y$, c.f. (A.20). The
problem (A.22)–(A.23) has has the following family of solutions:

(A.24) $\phi_{k}:=y^{1/2-ik}\quad(k\geq 0).$

These $\phi_{k}$’s are the generalized eigenfunctions, which correspond to the
spectral points

(A.25) $\lambda_{k}:=k^{2}+\frac{1}{4}+b^{2}\quad(k\geq 0).$

Next, fix a cusp $c_{i}$ of $\Sigma$ and some $k\geq 0$. For $n=1,2,\ldots,$
let $u_{n}:=\chi_{n}\phi_{k}$, where $\chi_{n}\in
C_{c}^{2}(\mathbb{R}_{>0},\mathbb{R}_{\geq 0})$ is a sequence of cut-off
functions with

(A.26) $\chi_{n}(y)\equiv\left\\{\begin{aligned} &1\quad(n\leq y\leq n+1),\\\
&0\quad(0<n-\tfrac{1}{n}\leq y\text{ or }y\geq
n+\tfrac{1}{n}).\end{aligned}\right.$

Now, we form the Poincaré series (A.17) w.r.t. $c_{i}$ and $u_{n}$. Since
$u_{n}$ has compact support, $E_{c_{i},u_{n}}$ converges absolutely for every
$n$. Moreover, each $E_{c_{i},u_{n}}$ is an incomplete Eisenstein series,
which is bounded and satisfies the equivariant condition (A.10). (See
[MR1942691, Sec. 3.2] for a discussion.) Hence
$E_{c_{i},u_{n}}\in\mathcal{L}^{2}$ and satisfies (A.19) except for on the two
bounded strips $\left\\{(n-1)/n<y<n\right\\}$ and
$\left\\{n<y<(n+1)/n\right\\}$.

Finally, let

$\tilde{u}_{n}:=\left\lVert
E_{c_{i},u_{n}}\right\rVert_{L^{2}}^{-1}E_{c_{i},u_{n}}.$

Then $\tilde{u}_{n}\in L^{2}(\mathbb{H}/\Gamma)$ forms a Weyl sequence for
$-\Delta_{a^{b}}$ and $\lambda_{k}$, c.f. (3.36). This shows that
$\lambda_{k}\in\sigma_{\rm ess}(-\Delta_{a^{b}})$. Varying $k$, we find

$[\tfrac{1}{4}+b^{2})\subset\sigma_{\rm ess}(-\Delta_{a^{b}}).$

∎

##  Appendix B Classification of $U(1)$-Automorphy Factors and Constant
Curvature $U(1)$-Connections

In this section, we reproduce with minor modifications results of [MR4049917]
on classification of automorphy factors and constant curvature connections.

Recall that a character of $\Gamma$ is a homomorphism $\chi:\Gamma\rightarrow
U(1)$. For other standard definitions, see Appendix A.

######  Theorem B.1 (classification of automorphy factor).

For any $\beta\in\mathbb{R}$, the map
$\rho_{\beta}:PGL(2,\mathbb{R})\times{\mathbb{H}}\rightarrow U(1)$, given by

(B.1) $\displaystyle\rho_{\beta}(\gamma,z)$
$\displaystyle=\biggr{[}\frac{c+d}{c\bar{z}+d}\biggl{]}^{\beta},\quad\gamma=\left(\begin{smallmatrix}a&b\\\
c&d\end{smallmatrix}\right)\in PGL(2,\mathbb{R}),$

satisfies the co-cycle condition

$\displaystyle\rho_{\beta}(s\cdot t,z)$ $\displaystyle=\rho_{\beta}(s,t\cdot
z)\rho_{\beta}(t,z),\quad\forall s,t\in PGL(2,\mathbb{R}).$

Consequently, for any Fuchsian group $\Gamma$ s.t.
$\Sigma:={\mathbb{H}}/\Gamma$ has finite area and $b:=\frac{2\pi
n}{|\Sigma|}$, and any character $\chi:\Gamma\rightarrow U(1)$, the map
$\rho_{b,\chi}:\Gamma\times{\mathbb{H}}\rightarrow U(1)$ given by

(B.2) $\displaystyle\rho_{b,\chi}(\gamma,z)$
$\displaystyle=\chi(\gamma)\rho_{b}(\gamma,z),\quad\gamma\in\Gamma,$

is also an automorphy factor. Two automorphy factors related as in (B.2) are
said to be equivalent.

Let $n\in\mathbb{Z}$ and $b:=\frac{2\pi n}{|\Sigma|}$. Then, the first Chern
number of $\rho_{b,\chi}$ is $c_{1}(\rho_{b,\chi})=n$. Hence any automorphy
factor $\rho:\Gamma\times{\mathbb{H}}\rightarrow U(1)$ of degree $n$ is
equivalent to $\rho_{b}$ as in (A.5).

###### Proof.

Let $s=\left(\begin{smallmatrix}a&b\\\ c&d\end{smallmatrix}\right),\
t=\left(\begin{smallmatrix}e&f\\\ g&h\end{smallmatrix}\right)\in\Gamma$. Using
(B.1), we compute for any $\beta$,

$\displaystyle\rho_{\beta}(s\cdot t,z)$
$\displaystyle=\biggr{[}\frac{(ce+dg)z+(cf+dh)}{(ce+dg)\overline{z}+(cf+dh)}\biggl{]}^{\beta}$
$\displaystyle=\biggr{[}\frac{c(ez+f)+d(gz+h)}{(c(e\overline{z}+f)+d(g\overline{z}+h)}\biggl{]}^{\beta}$
$\displaystyle=\biggr{[}\frac{c\frac{ez+f}{gz+h}+d}{c\frac{e\overline{z}+f}{g\overline{z}+h}+d}\biggl{]}^{\beta}\biggr{[}\frac{gz+h}{g\overline{z}+h}\biggl{]}^{\beta}$
$\displaystyle=\rho_{\beta}(s,t\cdot z)\rho_{\beta}(t,z).$

Using the formula for the Chern class, $c_{1}(\rho)$, of a co-cycle $\rho$
(see [MR0457787], Theorem 2a), we compute $c_{1}(\rho_{\beta})=n$, provided
$\beta=\frac{2\pi n}{\left\lvert\Sigma\right\rvert}$, where
$\left\lvert\Sigma\right\rvert$ is the area of $\Sigma$ w.r.t. to the standard
hyperbolic metric. ∎

######  Theorem B.2 (classification of constant curvature connections).

For any $b\in\mathbb{R}$, the connection

(B.3) $\displaystyle A^{b}=by^{-1}dx,$

on the trivial line bundle $\tilde{E}:={\mathbb{H}}\times\mathbb{C}$

(a) has a constant curvature with respect to the standard hyperbolic area form
on ${\mathbb{H}}$;

(b) is equivariant with respect to the automorphy factor (B.2) for any
Fuchsian group $\Gamma$ and any character $\chi:\Gamma\rightarrow U(1)$;

(c) is unique (as a connection on $\tilde{E}:={\mathbb{H}}\times\mathbb{C}$)
up to gauge transformations (1.2).

######  Remark 16.

The description in the above theorem does not depend at all on the complex
structure of the underlying Riemann surface. Hence, if $E_{n,\chi}$ is the
unitary line bundle corresponding to the automorphy factors (B.2), then the
projection of $A^{b}$ to $E_{n,\chi}$ gives the distinguished connection
$a^{b,\chi}$ on $E_{n,\chi}$.

###### Proof of Theorem B.2.

We begin with some preliminary constructions. We consider the trivial bundle,
$\tilde{E}:={\mathbb{H}}\times\mathbb{C}$ with the standard complex structure
on ${\mathbb{H}}$ associated to the standard hyperbolic metric
$\tilde{h}=(\operatorname{Im}z)^{-2}|dz|^{2}$.

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 $\mathbb{C}$ over the point $z\in{\mathbb{H}}$ to be
$k_{z}=(\operatorname{Im}z)^{-2}|dw|^{2}$, where $w$ is the coordinate on the
fiber $\mathbb{C}_{z}$.

Let the connection $A$ be given by $A:=A_{1}dx_{1}+A_{2}dx_{2}$. We decompose
the covariant derivative $\nabla_{A}$ into $(1,0)$ and $(0,1)$ parts as
$\nabla_{A}=\partial_{A}^{\prime}+{\partial_{A}^{\prime\prime}}$, where
$\partial_{A}^{\prime}$ and $\partial_{A}^{\prime\prime}$ are defined in
(3.9)–(3.10).

Recall from Section 3.3 that, in terms of $A_{c}$, the curvature is given by
$F_{A}=2\operatorname{Re}\bar{\partial}A_{c}$. Moreover, if $A_{c}$ satisfies
the equivariance relation
$s^{*}\bar{A}_{c}=\bar{A}_{c}-i\overline{\partial}\tilde{f}_{s}$, then $A$
satisfies $s^{*}A=A+df_{s}$, with $f_{s}$ satisfying
$df_{s}:=2\operatorname{Im}\overline{\partial}\tilde{f}_{s}$.

According to (3.10), the complexification of the connection $A^{b}$ given in
the theorem is

$A_{c}^{b}=b\frac{1}{2\operatorname{Im}(z)}dz.$

In the remaining of this proof, we omit the superindex $b$ in $A^{b}$ and
$A_{c}^{b}$.

Proof of constant curvature. Using that
$\omega=\frac{i}{2}\operatorname{Im}(z)^{-2}dz\wedge d\overline{z}$, we find

$\displaystyle\bar{\partial}A_{c}$
$\displaystyle=\frac{\partial}{\partial\bar{z}}\frac{ib}{z-\overline{z}}d\bar{z}\wedge
dz=\frac{-ib}{(z-\overline{z})^{2}}dz\wedge
d\overline{z}=\frac{ib}{4\operatorname{Im}(z)^{2}}dz\wedge
d\overline{z}=\frac{b}{2}\omega.$

Since $F_{A}=2\operatorname{Re}\bar{\partial}A_{c}$, this gives the desired
result.

Proof of uniqueness. If $A$ and $B$, satisfy $dB=dA$ then $d(A-B)=0$. It
follows from the simple connectedness of $\mathbb{H}$ that $A-B=df$ for some
function $f:\mathbb{H}\rightarrow\mathbb{R}$ and $f$ is unique up to an
additive constant. So we can map $B$ to $A$ through a suitable
reparametrization. This completes the proof.

Proof of equivariance. For a generic isometry $s(z)=\frac{\alpha
z+\beta}{\gamma z+\delta}$, we have $\frac{\partial s(z)}{\partial z}=(\gamma
z+\delta)^{-2}$ and
$\operatorname{Im}(s(z))=\frac{\operatorname{Im}(z)}{|\gamma z+\delta|^{2}}$,
which gives

$\displaystyle s^{*}\bar{A}_{c}$
$\displaystyle=\frac{b}{2\operatorname{Im}(s(z))}\overline{\frac{\partial
s(z)}{\partial z}}d\overline{z}=\frac{k|\gamma
z+\delta|^{2}}{2\operatorname{Im}(z)}\frac{d\overline{z}}{(\gamma\overline{z}+\delta)^{2}}$
$\displaystyle=\frac{b}{2\operatorname{Im}(z)}\frac{\gamma
z+\delta}{\gamma\overline{z}+\delta}d\overline{z}=\frac{b}{2\operatorname{Im}(z)}d\overline{z}+\frac{b(\gamma
z-\gamma\overline{z})}{2\operatorname{Im}(z)(\gamma\overline{z}+\delta)}d\overline{z}$
$\displaystyle=\bar{A}_{c}+\frac{ib\gamma}{\gamma\overline{z}+\delta}d\overline{z}=:\bar{A}_{c}+\overline{\partial}\tilde{f}_{s},$

where $\tilde{f}_{s}$ is the function defined by the last relation, i.e.
$\overline{\partial}\tilde{f}_{s}=\frac{ib\gamma}{\gamma\overline{z}+\delta}d\overline{z}.$
Solving this equation, we find

$\tilde{f_{s}}=b\text{ln}(\gamma\overline{z}+\delta)+c_{s}.$

Now, we define $f_{s}:=2\operatorname{Re}\tilde{f_{s}}$ and use that
$\operatorname{Re}(\bar{\partial}\tilde{f_{s}})=d(\operatorname{Re}\tilde{f_{s}})$
(as can be checked by the direct computation:
$\frac{1}{b}\operatorname{Re}(\bar{\partial}\tilde{f_{s}})=-\frac{\gamma^{2}x_{2}}{|\gamma
z+\delta|^{2}}dx_{1}+\frac{\gamma(\gamma x_{1}+\delta)}{|\gamma
z+\delta|^{2}}dx_{2}$) to obtain $s^{*}A=A+df_{s}$, with

(B.4) $\displaystyle
f_{s}=2\operatorname{Re}(\tilde{f_{s}})=ib\text{ln}\biggr{[}\frac{\gamma\overline{z}+\delta}{\gamma
z+\delta}\biggl{]}+c_{s},\
\rho(s,z)=e^{if_{s}(z)}=e^{ic_{s}}\biggr{[}\frac{\gamma\overline{z}+\delta}{\gamma
z+\delta}\biggl{]}^{-b}.$

Here note that since $\mathbb{H}$ is the upper half plane, the complex
logarithm is well defined and $\gamma z+d$ is always non zero. ∎

######  Remark 17.

The function $\tilde{f}_{s}(z)$ appearing above gives the character
$\tilde{\rho}(z)=e^{\tilde{f}_{s}(z)}$, which is now $\mathbb{C}^{*}$-valued
instead of $U(1)$-valued.

######  Remark 18.

$A^{b}$ is $\mathbb{R}$ \- linear while $A^{c}$ is naturally $\mathbb{C}$ \-
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
$iA^{b}_{c}=i\frac{b}{2}\frac{1}{\operatorname{Im}(z)}dz$ 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.

##  Appendix C Chern-Weil Correspondence

###### Proof of Theorem 2.2.

The argument below, due to D. Chouchkov, is streamlined from classical results
from [MR0457787, Chap. II.4].

Let $F_{\Sigma}\subset\mathbb{H}$ be a fundamental domain of $\Gamma$. Let
$A_{i},B_{i},i=1,\dots g,$ and $S_{i},i=1,\dots m,$ be the hyperbolic and
parabolic generators of $\Gamma$, respectively (no elliptic ones by
assumption). Let
$\alpha_{i},\alpha^{\prime}_{i},\beta_{i},\beta^{\prime}_{i},i=1,\dots g,$ and
$\delta_{j},\delta_{j}^{\prime},j=1,\dots m$ be the sides of $F_{\Sigma}$.

As in the beginning of Section 3, we have the following relations:

$A_{i}^{*}\alpha_{i}=-\alpha_{i}^{\prime},B_{i}^{*}\beta_{i}=-\beta_{i}^{\prime},i=1,\ldots
g,\text{ and }S_{j}^{*}\delta_{j}=-\delta_{j}^{\prime},j=1,\dots m.$

as well as the decomposition

(C.1) $\displaystyle\partial
F_{\Sigma}=\sum_{i=1}^{g}(\alpha_{i}+\alpha^{\prime}_{i}+\beta_{i}+\beta^{\prime}_{i})+\sum_{j=1}^{m}(\delta_{j}+\delta_{j}^{\prime}).$

Here the plus sign denotes disjoint union.

Let $A$ be an equivariant $1$-form on $F_{\Sigma}$ that corresponds to the
connection $a$ on $E\to\Sigma$, as in Sections A.2–A.3. By definition (A.9),
the theorem is proved once we establish

$\frac{1}{2\pi}\int_{\partial F_{\Sigma}}A=c_{1}(\sigma).$

Since the $1$-form $A$ is gauge-equivariant, we have

(C.2) $\gamma^{*}A=A-i\rho_{\gamma}^{-1}d\rho_{\gamma},\
\forall\gamma\in\Gamma,\,z\in\partial F_{\Sigma}.$

This, together with the definition $\rho(\gamma,z)=e^{i\sigma_{\gamma}(z)}$,
implies

$\int_{\alpha_{i}+\alpha_{i}^{\prime}}A=\int_{\alpha_{i}}(A-A_{i}^{*}A)=-\int_{\alpha_{i}}d\sigma_{A_{i}}=\sigma_{A_{i}}(v_{i})-\sigma_{A_{i}}(v_{i+1})\quad(i=1,\dots
g),$

where $v_{i},v_{i+1}$ are the vertices of $F_{\Sigma}$ spanned by the side
$\alpha_{i}$. Similarly we proceed with $\int_{\beta_{i}+\beta_{i}^{\prime}}A$
and $\int_{\delta_{j}+\delta_{j}^{\prime}}A$. In the last case, since the
gauge exponent $\sigma_{S_{i}}(z)$ is independent of $z$, we have
$\int_{\delta_{j}+\delta_{j}^{\prime}}A=0$, so these term do not contribute.

Summing up these contributions and using (C.1), we arrive at

$\int_{\partial
F_{\Sigma}}A=\sum_{1}^{g}[\sigma_{A_{i}}(v_{i})-\sigma_{A_{i}}(v_{i+1})+\sigma_{B_{i}}(w_{i})-\sigma_{B_{i}}(w_{i+1})],$

where $v_{i},v_{i+1}$ and $w_{i},w_{i+1}$ are the vertices of $F_{\sigma}$
spanned by $\alpha_{i}$ and $\beta_{i}$, respectively. Note that the vertices
$v_{i},v_{i+1}$, $w_{i},w_{i+1}$ and $u_{i},u_{i+1}$ are related as
$w_{i}=v_{i+1},w_{i+1}=A_{i}v_{i+1}$ and $v_{i+1}=B_{i}^{-1}v_{i}$, which
gives

(C.3) $\frac{1}{2\pi}\int_{\partial
F_{\Sigma}}A=\frac{1}{2\pi}\sum_{1}^{g}[\sigma_{A_{i}}(v_{i})-\sigma_{A_{i}}(B_{i}^{-1}v_{i})+\sigma_{B_{i}}(B_{i}^{-1}v_{i})-\sigma_{B_{i}}(A_{i}B_{i}^{-1}v_{i})].$

By construction,

$v_{i}=B_{i}A_{i}^{-1}B_{i}^{-1}C_{i+1}\dots C_{g}z_{0}=A_{i}^{-1}C_{i}\dots
C_{g}z_{0},$

with $C_{i}:=A_{i}B_{i}A_{i}^{-1}B_{i}^{-1}\text{ and some }z_{0}\in
F_{\Sigma}.$ Hence, r.h.s. of (C.3) agrees with the definition of the first
Chern number $c_{1}$ in (A.8). This proves the theorem. ∎

##  Appendix D  Proof of Theorem 7.1

One reduces to a scalar bifurcation problem by considering the scalar-valued
function

(D.1) $\displaystyle\phi(\ell,s)$ $\displaystyle=$
$\displaystyle\frac{1}{|s|^{2}}\langle s,\ell s+H(\ell,s)\rangle.$

Manifestly, $\phi(0,0)=0$ and $\partial\phi(0,0)/\partial\ell=1$. It then
follows from the implicit function theorem that there exists a unique function
$\ell(s)$, for small $s$, such that $\phi(\ell(s),s)=0$. Set
$\hat{E}(s)=E(\ell(s),s)$, where, recall, $E(\ell,s)=\ell s+H(\ell,s)$, and
let $S_{\epsilon}:=\left\\{s\in\mathbb{R}^{n}:\left\lvert
s\right\rvert=\epsilon\right\\}$ be the $\epsilon$-sphere in $\mathbb{R}^{n}$.
Then $\hat{E}(s)$, regarded as a vector field on $\mathbb{R}^{n}$, is
everywhere tangent to $S_{\epsilon}$. To see this, simply note that

$\displaystyle 0$ $\displaystyle=$ $\displaystyle\phi(\ell(s),s)$
$\displaystyle=$ $\displaystyle\frac{1}{|s|^{2}}\langle
s,\ell(s)s+H(\ell(s),s)\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{|s|^{2}}\langle s,\hat{E}(s)\rangle,$

which precisely states that the vector field $\hat{E}(s)$ 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.
$\hat{E}(s)=0$ has a solution on $S_{\epsilon}$. The theorem follows.

## References