# Asymptotics for minimizers of the Ginzburg-Landau energy with optimal
regularity of the boundary data and applications

Paul Laurain, Romain Petrides  Institut de Mathématiques de Jussieu,
Université de Paris, Bâtiment Sophie Germain, Case 7052, 75205 Paris Cedex 13,
France & DMA, Ecole normale supérieure, CNRS, PSL Research University, 75005
Paris<EMAIL_ADDRESS>Institut de Mathématiques de Jussieu,
Université de Paris, Bâtiment Sophie Germain, 75205 Paris Cedex 13, France
<EMAIL_ADDRESS>
###### Abstract.

We perform the classical asymptotic analysis for the Ginzburg-Landau energy
which originates from the celebrated paper by Bethuel, Brezis, Hélein for
nonsmooth boundary data. More precisely, we give optimal regularity
assumptions on the boundary curve of planar domains and Dirichlet boundary
data on them. When the Dirichlet boundary data is the tangent vector field of
the boundary curve, our framework allows us to define a natural energy
minimizing frame for simply connected domains enclosing Weil-Petersson curves.

The so-called simplified Ginzburg-Landau functional on a domain $\Omega$

$E_{\varepsilon}\left(u\right)=\frac{1}{2}\int_{\Omega}\left|\nabla
u\right|^{2}+\frac{1}{4\varepsilon^{2}}\int_{\Omega}\left(1-|u|^{2}\right)^{2}$

and all its variations were intensively studied after the celebrated paper by
Bethuel, Brezis, Hélein (BBH) [1], since energies of this type are useful to
study phase transition problems. From this seminal work, a lot of attention
has been given to the behavior of the critical points: see e.g. Pacard and
Rivière [31], Sandier and Serfaty [38], Farina and Mironescu [14], Millot and
Pisante [28], Ignat, Nguyen, Slastikov and Zarnescu [21] and the references
therein.

One other well known usefulness of this energy is to build kinds of “Dirichlet
energy minimizing” maps with singularities adapted to the strong topological
constraints given by some geometrical problems. This is our central
motivation. For instance, we cannot find any smooth nor $H^{1}$ unit vector
field extensions with finite energy of maps
$\partial{\Omega}\to\mathbb{S}^{1}$ of any topological degree. Therefore, we
have to relax the constraint on the set of admissible vector fields in the
variational problem. Vector fields do not necessary have unit norm anymore,
but as a compensation, we add a term to the energy which drastically penalizes
vector fields that do not have unit norm as $\varepsilon\to 0$. Classical
works by [1], [41], prove the convergence as $\varepsilon\to 0$ of minimizers
of $E_{\varepsilon}$ to harmonic maps with singularities that are minimizers
of the so-called renormalized energy. In the same spirit, studying the Allen-
Cahn equation is a valuable technique to build new minimal surfaces, see [17]
and [12].

In the current paper, we aim at weakening the known regularity assumptions on
the boundary data for the asymptotic analysis as $\varepsilon\to 0$ for
minimizers of the Ginzburg-Landau energy $E_{\varepsilon}$ on simply connected
domains $\Omega$ and $u:\Omega\to\mathbb{R}^{2}$ satisfying a Dirichlet
boundary condition $u=g$ on $\Gamma:=\partial\Omega$ for some functions
$g:\Gamma\to\mathbb{S}^{1}$ with a prescribed degree. Not only the weakenings
hold for the regularity of the boundary data $g$ but also for the regularity
of the domain $\Omega$. Thanks to Jerrad and Sandier, see [22], [36] and [37],
energy estimates on the Ginzburg-Landau energy allow $H^{\frac{1}{2}}$
boundary data but to our knowledge, the classical BBH asymptotic analysis
which originates from the paper [1] has never been completely done with
domains $\Omega$ with lower regularity than Lipschitz.

Before stating our result, we would like to give some motivations for these
weakenings of assumptions. One by-product of the BBH analysis is to produce
$u_{*}:\Omega\rightarrow S^{1}$ which is harmonic with $u_{*}=g$ on the
boundary (of course $u_{*}$ is singular if $\mathrm{deg}(g)\not=0$). In the
special case $g=\tau$ being the tangent unit vector field of the boundary, one
can easily produce a family of such harmonic maps by considering a
uniformization of the domain. Indeed, see proposition 4.3, if
$f_{a}:\mathbb{D}\rightarrow\Omega$ is a biholomorphic map such that
$f(0)=a\in\Omega$, then
$u_{a}=\frac{\partial_{\theta}f}{\left|\partial_{\theta}f\right|}$ is a
desired harmonic map. As we show in Theorem 0.3, using the BBH renormalized
energy, $u_{*}$ is the one that maximizes $|f_{a}^{\prime}(0)|$. More
precisely, the expression of the renormalized energy associated to a
uniformization $f$ has the form

$\int_{\mathbb{D}}\left|\frac{f^{\prime\prime}}{f^{\prime}}\right|^{2}+2\pi\ln\left|f^{\prime}(0)\right|,$

which is closely related to the Loewner energy of the boundary curve, see
section 1 for more details. Hence our goal is to make the BBH asymptotic
analysis for domains whose boundary curves have the minimal regularity that
makes this energy finite. Those curves are known as Weil-Petersson curves:
they are chord-arc curves whose unit tangent vector field lies in
$H^{\frac{1}{2}}$ (see theorem-definition 2.2 due to Bishop). More details, on
the geometric motivations of our result are given in section 1.

Weil-Petersson curves are not necessarily $C^{1}$ curves because there are
non-continuous $H^{\frac{1}{2}}$ functions. However, contrary to Lipschitz
curves they cannot have corners see section 2.1 of [35]. Nevertheless, it is
important to remark that they a priori not bound Lipschitz domain, since the
slow log-spiral, $t\mapsto te^{i\log(\log(1/t))}$, is a Weil-Petersson curve.

Figure 1. A domain with slow log-spiral

Let’s schematize all the regularity assumptions on the Jordan curve $\Gamma$
which borders $\Omega$ we mention in the paper (definitions are given in
Section 2):

$\begin{array}[]{ccccc}&&\text{Weil-Petersson curves}&&\\\
&\mathrel{\rotatebox[origin={c}]{45.0}{$\subsetneq$}}&&\mathrel{\rotatebox[origin={c}]{-45.0}{$\subsetneq$}}&\\\
\mathcal{C}^{1}\text{ curves}&&&&\text{ Chord-arc curves}\subsetneq\text{
Quasicircles }\\\
&\mathrel{\rotatebox[origin={c}]{-45.0}{$\subsetneq$}}&&\mathrel{\rotatebox[origin={c}]{45.0}{$\subsetneq$}}&\\\
&&\text{Lipschitz curves}&&\end{array}$

Coming back to the analysis, we first state an optimal result for the weakest
regularity of $\Omega$ such that the classical definition of
$H^{\frac{1}{2}}(\Gamma)$ makes sense:

######  Theorem 0.1.

Let $\Omega$ be a domain such that $\Gamma:=\partial\Omega$ is a chord-arc
curve and $g\in H^{\frac{1}{2}}(\Gamma,S^{1})$ with $\mathrm{deg}(g)=1$, then
there is $u_{\varepsilon}\in H^{1}(\Omega)$ which minimizes, $E_{\varepsilon}$
on the admissible set

$\mathcal{A}=\\{u\in H^{1}\left(\Omega,\mathbb{R}^{2}\right);u=g\text{ a.e on
}\Gamma\\}.$

Moreover, for any sequence $\varepsilon_{k}\to 0$, there are a subsequence
$\varepsilon_{k_{l}}\to 0$, $a\in\Omega$ and $u_{\star}\in
C^{\infty}\left(\Omega\setminus\\{a\\},S^{1}\right)\cap
H^{1}_{loc}(\overline{\Omega}\setminus\\{a\\})$ such that

  1. (i)

$u_{\star}$ is a harmonic map,

  2. (ii)

$u_{\star}=g$ on $\partial\Omega$,

  3. (iii)

$\mathrm{deg}\left(\left(u_{\star}\right)_{|\partial B(a,\rho)}\right)=1$ for
$\rho>0$ small enough,

  4. (iv)

$u_{\varepsilon_{k_{l}}}\rightarrow u_{\star}$ in
$C^{\infty}_{loc}(\Omega\setminus\\{a\\})$ as $l\to+\infty$,

  5. (v)

$u_{\varepsilon_{k_{l}}}\rightarrow u_{\star}$ in
$H^{1}_{loc}(\overline{\Omega}\setminus\\{a\\})$ as $l\to+\infty$,

  6. (vi)

$|u_{\varepsilon_{k_{l}}}|\to 1$ in
$L^{\infty}_{loc}\left(\overline{\Omega}\setminus\\{a\\}\right)$ as
$l\to+\infty$.

This theorem can be easily generalized to the case
$\mathrm{deg}(g)\in\mathbb{Z}^{*}$, since once the difficulty due to the weak
regularity of the data at the boundary is overcome for one bad disk, then it
is overcome for all. Hence, Theorems VI.1 and VI.2 of [2] hold true in our
setting. In the case $\mathrm{deg}(g)=0$, it is even easier, since we do not
need to remove any singularity and section 3.4 is not necessary. The energy
estimates on the Ginzburg-Landau energy given by Jerrad and Sandier allow
$H^{\frac{1}{2}}$ boundary data and Lipschitz domains, as done in [29]. For
$\mathcal{C}^{1}$ domains with $H^{\frac{1}{2}}$ data at the boundary, the
case $\mathrm{deg}(g)=0$ was simply solved in [14]. Notice that not only
Theorem 0.1 extends the known results to much lower regularity but we also use
and extend the natural and simple techniques due to the originated paper by
BBH.

The assumptions of this theorem are optimal in the sense that the presence of
the Dirichlet energy requires the $H^{\frac{1}{2}}$ condition on $g$, and
$H^{\frac{1}{2}}\left(\Gamma\right)$ only makes sense if $\Gamma$ is a chord-
arc curve. However, in the proof we only use that the pullback of $g$ by a
uniformization of the domain belongs to
$H^{\frac{1}{2}}(\mathbb{S}^{1},\mathbb{S}^{1})$. Hence, taking this as a
definition of $H^{\frac{1}{2}}(\Gamma,\mathbb{S}^{1})$ and defining
$\mathcal{A}=\\{v\circ f^{-1}\\}+H_{0}^{1}\left(\Omega,\mathbb{R}^{2}\right)$,
where $v$ is the harmonic extension of $g\circ f$ for some uniformization
$f:\mathbb{D}\rightarrow\Omega$, we can even weaken the regularity of the
boundary.

######  Theorem 0.2.

Let $\Omega$ be a quasidisk and $g\in H^{\frac{1}{2}}(\Gamma,\mathbb{S}^{1})$
in the sense of definition 2.3, then the conclusion of Theorem 0.1 holds.

Indeed, one main idea to handle the low regularity of $\Omega$ is to pull back
all the problem on the unit disk $\mathbb{D}$ thanks to one uniformization map
$f:\mathbb{D}\to\Omega$. The unit disk is then a pleasant smooth domain, but
we have to pay one price : the pullback functions $\tilde{u}=u\circ f$ for
$u\in\mathcal{A}$ now belong to the $H^{1}$ space associated to the Riemannian
metric $\left|f^{\prime}(z)\right|^{2}\left(dx^{2}+dy^{2}\right)$ which is the
weighted space of functions having the following norm finite:

$\left\|u\right\|_{H^{1}\left(\Omega\right)}^{2}=\left\|\tilde{u}\right\|_{H^{1}\left(\mathbb{D},\left|f^{\prime}\right|^{2}\right)}^{2}:=\int_{\mathbb{D}}\tilde{u}^{2}\left|f^{\prime}(z)\right|^{2}dz+\int_{\mathbb{D}}\left|\nabla\tilde{u}\right|^{2}dz\hskip
2.84544pt.$

If $\Omega$ is a smooth domain, the weight $\left|f^{\prime}(z)\right|^{2}$
satisfies a Harnack inequality up to the boundary, but if $\Omega$ has low
regularity, $\left|f^{\prime}(z)\right|^{2}$ may go to $+\infty$ or $0$ as
$\left|z\right|\to 1$ so that the geometry might be wildly different to the
Euclidean one close to the boundary. In our setting, we assume that $\Omega$
is a quasidisk. It is a sufficient regularity for the domain in order to make
the behavior of $\left|f^{\prime}(z)\right|^{2}$ close to the boundary good
enough to perform a BBH analysis on sequences of minimizers $u_{\varepsilon}$.
One reason is because the shape of disks $\mathbb{D}_{r}(x)$ centered in
$x\in\Omega$ is not too deformed after the pullback by $f$, even when $x$ is
close to $\Gamma$. This is because uniformization maps of quasidisks can be
extended to quasiconformal maps on $\mathbb{C}$.

For instance, one striking and fundamental lemma in the classical Ginzburg-
Landau analysis is the uniform bound of $\varepsilon\left|\nabla
u_{\varepsilon}\right|$ (see Step B.1 [1]) on $\Omega$. It is primordial to
control the behavior of bad disks (small disks centered on points $x$ such
that $|u_{\varepsilon}(x)|\leq\frac{1}{2}$). Indeed, this estimate leads to
the $\eta$-compactness (there is a quantum of energy around points such that
$|u_{\varepsilon}|\leq\frac{1}{2}$), see Lemma 24 of [33]. Classically, this
estimate is a consequence of the elliptic equation satisfied by
$u_{\varepsilon}$ as a critical point of $E_{\varepsilon}$:

$-\Delta
u_{\varepsilon}=\frac{1}{\varepsilon^{2}}\left(1-\left|u_{\varepsilon}\right|^{2}\right)u_{\varepsilon}\text{
in }\Omega\hskip 2.84544pt.$

The argument relies on the bound $\left\|u_{\varepsilon}\right\|_{\infty}\leq
1$ coming from a maximum principle argument and then on a rescaling argument
on the equation. However, with low regularity on $\Omega$, we cannot perform
any maximum principle argument nor any rescaling argument close to the
boundary. We first need to assume that $u_{\varepsilon}$ is a minimizer of the
energy in order to have the uniform bound
$\left\|u_{\varepsilon}\right\|_{\infty}\leq 1$, and then, a rescaling
argument only allows that $\varepsilon\left|\nabla u_{\varepsilon}(x)\right|$
is uniformly bounded on a set of points $x\in\Omega$ such that
$\frac{d(x,\partial\Omega)}{\varepsilon}$ is uniformly lower bounded. The main
novelty of our paper (see Proposition 3.3) is that there is $\eta_{0}>0$ such
that for the points $x_{\varepsilon}\in\Omega$ satisfying
$d(x_{\varepsilon},\partial\Omega)\leq\eta_{0}\varepsilon$, we have
$\left|u_{\varepsilon}(x_{\varepsilon})\right|\geq\frac{1}{2}$. We crucially
use the quasidisk assumption here. Thanks to this proposition, the classical
“bad disks” cannot intersect the boundary of $\Omega$ and the classical steps
of the BBH asymptotic analysis can be managed.

Notice that Theorem 0.1 is only stated for simply connected domains and in
Theorem 0.3 below, we give a link between the harmonic map $u_{*}$ and the
uniformization of domains. However, our analysis provides all the tools for
domains $\Omega$ such that $\partial\Omega$ is a disjoint union of chord-arc
closed curves, and $H^{\frac{1}{2}}$ data of any degree on these closed
curves. Then it would be interesting to give new constructions of
uniformizations in the spirit of [46], [23], see also [30].

As already remarked, a uniformization $f:\mathbb{D}\to\Omega$ defined a
harmonic map. Here we give a more precise description in terms of Coulomb
frame. We set $p=f(0)$ and we let

$u=\frac{f_{\theta}}{\left|f_{\theta}\right|}\circ f^{-1}\text{ and
}v=\frac{f_{r}}{\left|f_{r}\right|}\circ f^{-1}$

the pushforward in $\Omega$ of the normalized angular and radial derivatives
of $f$. Let $\omega=\langle du,v\rangle$ the Cartan form associated to
$(v,u)$. We prove in Proposition 4.3 that this frame is Coulomb and that

$\star\omega=d\left(\mu+G_{p}\right)\hbox{ in }\Omega\hskip 2.84544pt,$

where $G_{p}$ is the Dirichlet Green function with respect to $p$ and that

$\left|\frac{f^{\prime\prime}}{f^{\prime}}\right|^{2}=\left|\nabla\left(\mu\circ
f\right)\right|^{2}\hbox{ in }\mathbb{D}\text{ and then
}\int_{\mathbb{D}}\left|\frac{f^{\prime\prime}}{f^{\prime}}\right|^{2}dz=\int_{\Omega}\left|\nabla\mu\right|^{2}dz\hskip
2.84544pt.$

By definition, the last integral is finite if the boundary curve is Weil-
Petersson. As briefly explained above, looking at the case $g=\tau$, where
$\tau$ is the tangent vector field on $\Gamma$ such that the frame
$(\nu,\tau)$ is a direct frame, where $\nu$ is the out-pointing normal of the
bounded simply connected domain $\Omega$, the limit $u_{*}$ in Theorem 0.1
satisfies the following result, proved in section 4.3:

######  Theorem 0.3.

We assume that $\Gamma:=\partial\Omega$ is a Weil-Petersson curve. Let
$u_{*}:\Omega\to\mathbb{S}^{1}$ be given by Theorem 0.1 with Dirichlet
condition $u_{*}=\tau$ on $\Gamma$. Then $u_{*}=u_{a}$ is the pushforward on
$\Omega$ of normalized angular derivative of uniformization maps $f_{a}$ such
that $f_{a}(0)=a$, where $a$ is the singularity of $u_{*}$. Moreover, the
uniformization maps $f_{a}$ such that $f_{a}(0)=a$ maximize
$\left|f^{\prime}(0)\right|$ among all the uniformization maps $f$.

From this theorem, we deduced that the renormalized energy of frames in a
sense inspired by the BBH renormalized energy is closely related to the
Loewner energy of the domain, see [44] for a detailed presentation about the
Loewner Energy. However, in some sense, only a half part of the Loewner energy
appears in our construction since the frame is only defined on $\Omega$.
Therefore, in [27], Michelat and Wang give some reinterpretation of the
Loewner energy of a domain in terms of this new frame energy setting. Their
work is complementary to ours.

We also believe that our techniques are perfectly adapted to the analysis of
the Ginzburg-Landau functional on surfaces. Some very interesting works are
already given on closed immersed surfaces (see [20]). In fact, the present
article, with the one of [27], are the first stones to define a renormalized
frame energy in the case of surfaces with boundary but also with low
regularity, ideally some weak immersion with boundary as the one defined by
Rivière for closed surface, see [34].

Acknowledgements : This paper is part of a common project between Michelat and
Wang and the two authors of the current paper, on the links between the
Loewner energy and the Willmore energy. We would like to thank both of them
for all the fruitfull discussions on the topic - more precisely, Yilin for her
explanation of the Loewner Energy and Bishop’s work, and Alexis for our
conversations about Ginzburg-Landau - especially, Section 3.4 owes him a lot.

##  1\. Motivations

By an idea coming from Chern [11] (see also section 5.4 of [18]), using moving
frames in the Cartan formalism is more flexible in order to build conformal
coordinates. This point of view is particularly interesting in higher
dimension, where we can replace the absence of conformal coordinates by the
existence of Coulomb frames. We briefly remind why searching an optimal frame
gives rise to conformal coordinates in dimension $2$. The main idea consists
in building a Coulomb frame in a neighborhood $U$ of a point on a surface
$\Sigma$, i.e. an orthonormal family $\vec{e}_{1},\vec{e}_{2}\in\Gamma(TU)$
such that

(1.1) $d*(\langle\vec{e}_{1},d\vec{e}_{2}\rangle)=0.$

This could be done by minimizing the Dirichlet energy of the frame, assuming
that locally the total curvature is small enough. Then we deduce from (1.1)
that there is $\lambda\in C^{\infty}(U)$ such that

(1.2) $[e^{\lambda}\vec{e}_{1},e^{\lambda}\vec{e}_{2}]=0.$

We obtain the desired conformal coordinates after integration.

It is important to note that a priori, this construction is purely local for
two reasons. The first one is that if $\Sigma$ has some topology then the
space of frames can be empty. The second is that to get some control on the
frame we need that the total curvature does not exceed a certain level (see
[18] and [39] for more details). Of course, on planar domains, there are no
such problem. In this case, the minimizing Coulomb frame is given by the
trivial frame but it doesn’t see any geometry of the domain. However, since
the boundary has a natural frame given by a tangent and a normal unit vector
field, it seems interesting to consider it as Dirichlet boundary data for the
Coulomb frame.

Let us consider a simply connected domain $\Omega$. A tangent unit vector
field of $\Gamma:=\partial\Omega$ can be seen as a map from $\Gamma$ to
$\mathbb{S}^{1}$ of degree one. Then, it does not have any regular extension
in $\Omega$ which still takes values into $\mathbb{S}^{1}$. Anyway, if we are
able to construct a (singular) Coulomb frame with the tangent and exterior
normal unit vector fields as Dirichlet boundary data, then by integrating it,
we find a conformal map from the disk to our domain, which is nothing but a
uniformization of the domain.

Figure 2. A generic uniformization Figure 3. The uniformazition given by the
’minimizing’ frame

From this remark, we observe in proposition 4.3 that one natural global moving
frame $(v,u)$ is defined by the pushforward of the orthonormal polar
coordinates on the disk by a uniformization map $f:\mathbb{D}\to\Omega$
defined by

(1.3) $u=\frac{f_{\theta}}{|f_{\theta}|}\circ f^{-1}\text{ and
}v=\frac{f_{r}}{|f_{r}|}\circ f^{-1},$

where $f_{\theta}$ and $f_{r}$ denotes respectively the angular and radial
derivatives of $f$. As expected, this frame has a singularity, located at
$f(0)$. Note that for any choice of $a\in\Omega$, we can associate such a
frame having the singularity $a$: it suffices to choose a uniformization
$f:\mathbb{D}\to\Omega$ such that $f(0)=a$ by transitivity of the Mobius group
in the disk. This uniformization is unique up to a rotation in the disk. The
advantage of the frame approach is that since the frame is built by
minimization then it should give a “best” Coulomb frame and hence a “best”
uniformization.

Nevertheless, as already pointed above, the space of smooth (even $H^{1}$)
frames is empty by some degree obstruction. The problem of finding harmonic
extensions of vector fields despite some topological obstruction is exactly
the goal of the BBH analysis. Thanks to their seminal work, we know that on a
smooth simply connected domain $\Omega$ there exists $u\in
C^{\infty}(\Omega\setminus\\{a_{0}\\})$ such that $u$ is tangent to the
boundary and $(-iu,u)$ is a coulomb frame. Moreover $a_{0}$ minimizes the
renormalized energy

$W(a)=\lim_{\delta\to
0}\left(\int_{\Omega\setminus\mathbb{D}_{\delta}(a)}\left|\nabla
w\right|^{2}-2\pi\ln\frac{1}{\delta}\right),$

where $w$ is the canonical harmonic vector field (see [2], Chapter I for
details) matching with $u$ on the boundary and having its singularity at $a$.
Hence among all the uniformizations, the “best” one is the one such that
$f(0)=a_{0}$. In fact, $a_{0}$ is not necessary unique, but it is unique if
the domain is not far from a disk, see [9] for the convex case. Since we can
conversely associate a harmonic vector field to some uniformization, thanks to
(1.3), we can associate the following energy to each uniformization, see
proposition 4.20,

(1.4)
$W(f)=\int_{\mathbb{D}}\left|\frac{f^{\prime\prime}}{f^{\prime}}\right|^{2}+2\pi\ln\left|f^{\prime}(0)\right|.$

Then remarking that

$\int_{\mathbb{D}}\left|\frac{f^{\prime\prime}}{f^{\prime}}\right|^{2}+4\pi\ln\left|f^{\prime}(0)\right|$

is independent of the uniformization, the “best” one can be defined as the one
that maximizes $|f^{\prime}(0)|$. Of course, our method is not the first to
determine the “best” conformal representation of a simply connected domain.
For instance, in section 4.2 of [15], Flucher looks for an optimal conformal
rearrangement by studying the Robin function of the domain. His motivation was
the study of extremal functions for the Moser-Trudinger inequality. To our
knowledge, it is the first time that the point of view of optimal frames is
adopted, especially for domains with very low regularity.

In our construction of a “best” uniformization presented above, we used the
classical BBH analysis which is only valid for smooth domains. Having a
careful look to the classical proofs, it seems that the regularity of the
boundary can be reduced to $C^{2}$ but not much less since those proofs make
use of the Pohozaev identity which requires a control on the derivative of the
boundary data, that is to say here the derivative of the tangent unit vector
field.

Hence, our main motivation is to be able to have the existence of the limit
$u_{\star}$ for the minimizers of the Ginzburg-Landau functional as soon as
(1.4) makes sense. In fact (1.4) is very similar to the Loewner energy of a
Jordan curve $\Gamma$, given by

$E_{L}(\Gamma)=\frac{1}{\pi}\left(\int_{\mathbb{D}}\left|\frac{f^{\prime\prime}}{f^{\prime}}\right|^{2}+\int_{\mathbb{D}}\left|\frac{g^{\prime\prime}}{g^{\prime}}\right|^{2}+4\pi\ln\left|\frac{f^{\prime}(0)}{g^{\prime}(\infty)}\right|\right),$

where $f:\mathbb{D}\rightarrow\Omega$, $g:\mathbb{D}^{c}\rightarrow\Omega^{c}$
are two uniformization with $\partial\Omega=\Gamma$. It is well-known that the
class of domains to consider is the one enclosed by a Weil-Petersson curve
(see next section for definition). Those curves are curves whose tangent unit
vector field belongs to $H^{\frac{1}{2}}$.

Finally, to get the most general result for the asymptotic analysis of the
Ginzburg-Landau energy, we decouple the regularity of the boundary
$\partial\Omega$ and the regularity of the boundary data $g$ and we are able
to prove the existence of a limit $u_{*}$ as soon as the boundary is chord-arc
(not necessarily Weil-Petersson) and the boundary data $H^{\frac{1}{2}}$ in a
classical sense. Moreover, giving a natural sense to the notion of
$H^{\frac{1}{2}}$ maps for quasicircles (see next section), we are able to
assume only that the boundary is a quasicircle. It is clear that this result
is optimal, since the $H^{\frac{1}{2}}$ assumption cannot be weakened due to
the presence of the Dirichlet energy, and Weil-Petersson curves are
quasicircles. We then give the existence of a best uniformization in this weak
setting.

To conclude, we want to notice that the uniformization was not exactly our
first motivation. As explained in the introduction, this work has started by
some discussions with Michelat and Wang on the way to rely quantitatively the
Willmore energy to the Loewner Energy. To that purpose, we realized that we
need the existence of a limit $u_{*}$ for the minimizers of the Ginzburg-
Landau energy on a surface with a Weil-Petersson boundary. The present paper
is then devoted to the case of a planar domains. It is interesting in itself
since we have some purely BBH-type result with optimal regularity and a nice
application to the problem of finding a best uniformization. From their side,
Michelat and Wang have established some clear link of the frame approach and
the Loewner-Energy. Indeed, they prove the existence of a consistent
generalization of the frame energy (1.4) on $\Omega$ to a new frame energy
defined on frames $(v_{1},u_{1})$ and $(v_{2},u_{2})$ defined on both sides of
the curve $\Gamma$ viewed on the sphere by stereographic projection, denoted
here by $W_{v,u}(\Gamma)$ (see [27] for a precise definition of this frame
energy):

######  Theorem 1.1 (Michelat, Wang [27], Theorem A).

Let $\Gamma\subset\mathbb{S}^{2}$ be a Weil-Petersson curve. Let
$\Omega_{1},\Omega_{2}$ be the two connected components of
$\Omega\setminus\Gamma$. For $p_{1}\in\Omega_{1}$ and $p_{2}\in\Omega_{2}$,
there are harmonic frames $(v_{1},u_{1}):\Omega_{1}\setminus\\{p_{1}\\}\to
U\Omega_{1}$ and $(v_{2},u_{2}):\Omega_{2}\setminus\\{p_{2}\\}\to U\Omega_{2}$
such that for conformal maps $f_{1}:\mathbb{D}\to\Omega_{1}$ and
$f_{2}:\mathbb{D}\to\Omega_{2}$ such that $f_{1}(0)=p_{1}$ and
$f_{2}(0)=p_{2}$, then

$E_{L}(\Gamma)=\frac{1}{\pi}W_{v,u}(\Gamma)+2\ln\left|\nabla
f_{1}(0)\right|+2\ln\left|\nabla f_{2}(0)\right|-12\ln(2),$

where $W_{v,u}(\Gamma)$ is the renormalized energy of the frames
$(v_{1},u_{1})$ and $(v_{2},u_{2})$.

##  2\. Notations and definitions

  * -

$\mathbb{D}$ denotes the open unit disk of the complex plane $\mathbb{C}$
identified with $\mathbb{R}^{2}$.

  * -

$\mathbb{D}_{r}=r\mathbb{D}$ and $\mathbb{D}_{r}(x)$ the open disk centered at
$x$ with radius $r$.

  * -

$S_{r}=\partial\mathbb{D}_{r}$

  * -

$A_{r_{1},r_{2}}=\mathbb{D}_{r_{2}}\setminus\mathbb{D}_{r_{1}}$

  * -

$A^{c}$ is the complement of a given set $A$.

We remind some definition about quasiconformal maps, see [19] for more
details.

######  Theorem-Definition 2.1 (quasiconformal maps).

Let $U,V$ two open sets of $\mathbb{C}$ and $K\geq 1$. A mapping
$f:U\rightarrow V$ is $K$-quasiconformal if it is a homeomorphism and

  1. i)

$f\in H^{1}_{loc}(U)$,

  2. ii)

$\left|\frac{\partial
f}{\partial\bar{z}}\right|\leq\frac{K-1}{K+1}\left|\frac{\partial f}{\partial
z}\right|$ almost everywhere.

It is equivalent to be $L$-quasisymmetric of modulus $\eta$ that is to say,
there exists $\eta:\mathbb{R}^{+}\rightarrow\mathbb{R}^{+}$ a homeomorphism
(depending only on $K$) such that for any three distinct points $x,y,z\in U$
we have

$\left|\frac{f(x)-f(y)}{f(x)-f(z)}\right|\leq\eta\left(\left|\frac{x-y}{x-z}\right|\right).$

######  Definition 2.1 (quasidisk, quasicircle).

A quasidisk is the image of $\mathbb{D}$ by a quasiconformal map from
$\mathbb{C}$ into itself. A quasicircle is the boundary of a quasidisk.

Quasicircle can be very rough since it can even be a Julia set. Nevertheless,
it can’t be any Jordan curves since the cross-ratio must be bounded as shown
by the following proposition

######  Proposition 2.1.

A Jordan curve $\Gamma\subset\mathbb{C}$ containing $1$ is a quasicircle if
and only if there exists a constant $C$ such that for any
$z_{1},z_{2},z_{3}\in\Gamma$ that appears in that order111Which makes sense
since $1\in\Gamma$. on $\Gamma$, we have

(2.1) $|z_{1}-z_{2}|\leq C|z_{1}-z_{3}|.$

We will make a central use of the following theorem which permits to extend
uniformization of quasidisks to quasiconformal maps from $\mathbb{C}$ into
itself.

######  Theorem 2.1.

Any uniformization map from $\mathbb{D}$ to a quasidisk admits a
$K$-quasiconformal extension from $\mathbb{C}$ to itself.

Now we are restricting our attention to a special class of quasicircles, the
class of Weil-Petersson curves. Recently Bishop, see [5], gives no less than
26 equivalent definitions for Weil-Petersson curves. The most adapted one to
our setting is the following, which makes him said that ’roughly speaking,
Weil-Petersson curves are to $L^{2}$ as quasicircles are to $L^{\infty}$’.

######  Theorem-Definition 2.2 (Definition 1 of [5]).

A Jordan curve $\Gamma$ enclosing a domain $\Omega$ is Weil-Petersson if it is
a quasicircle and it admits a uniformization map
$f:\mathbb{D}\rightarrow\Omega$ such that

$\int_{\mathbb{D}}\left|\frac{f^{\prime\prime}}{f^{\prime}}\right|^{2}\,dz<\infty.$

In particular it is rectifiable and a chord-arc curve, i.e. there exists $C$
such that for any subarc $\gamma\subset\Gamma$, we have

$l(\gamma)\leq C|x-y|,$

where $x,y$ are the end points of $\gamma$.

######  Definition 2.2 ($H^{\frac{1}{2}}$ maps for chord-arc curves).

Let $n\geq 1$ and $\Gamma$ be a chord-arc curves, in particular rectifiable.
If $g:\Gamma\rightarrow\mathbb{R}^{n}$ satisfies

$\int_{\Gamma}\int_{\Gamma}\left|\frac{g(x)-g(y)}{|x-y|}\right|^{2}\,d\mu(x)\,d\mu(y)<\infty,$

where $\mu$ is the measure of $\Gamma$, we say that $g\in
H^{\frac{1}{2}}(\Gamma,\mathbb{R}^{n})$ and we denote
$\|g\|_{H^{\frac{1}{2}}(\Gamma)}$ the left-hand side of the previous
inequality. In particular, $g\in H^{\frac{1}{2}}(\Gamma,\mathbb{S}^{1})$ if
$g\in H^{\frac{1}{2}}(\Gamma,\mathbb{R}^{2})$ and $g(x)\in\mathbb{S}^{1}$ for
a.e. $x\in\Gamma$.

In the previous definition we could have only assumed that $\Gamma$ is
rectifiable, but we will need the following important proposition. It is due
to Bishop, see page 16 of [5], and it relies on the fact that quasisymmetric
maps preserve the $H^{\frac{1}{2}}$-norm by Ahlfors and Beurling, see page 129
of [3].

######  Theorem 2.2.

Let $\Gamma$ be a chord-arc curve enclosing a domain $\Omega$ and
$f:\mathbb{D}\rightarrow\Omega$ be a uniformization, then there exists $C>0$
depending only on $\Gamma$, such that for every $g\in
H^{\frac{1}{2}}(\Gamma,\mathbb{R}^{n})$ we have

$\frac{1}{C}\|g\|_{H^{\frac{1}{2}}(\Gamma)}\leq\|g\circ
f\|_{H^{\frac{1}{2}}(\mathbb{S}^{1})}\leq C\|g\|_{H^{\frac{1}{2}}(\Gamma)}.$

This theorem leads us to the following definition of
$H^{\frac{1}{2}}\left(\Gamma,\mathbb{S}^{1}\right)$ maps

######  Definition 2.3 ($H^{\frac{1}{2}}(\Gamma,\mathbb{S}^{1})$ maps).

Let $\Gamma$ be a quasicircle enclosing a domain $\Omega$, and
$f:\mathbb{D}\rightarrow\Omega$ a uniformization. We say that
$g:\Gamma\rightarrow\mathbb{S}^{1}$ is in
$H^{\frac{1}{2}}(\Gamma,\mathbb{S}^{1})$ if $g\circ f\in
H^{\frac{1}{2}}(\mathbb{S}^{1},\mathbb{S}^{1})$ and we set
$\|g\|_{H^{\frac{1}{2}}(\Gamma)}=\|g\circ
f\|_{H^{\frac{1}{2}}(\mathbb{S}^{1})}$.

Notice that there are subtleties with this definition. Indeed, let $v$ be the
harmonic extension of $g\circ f\in
H^{\frac{1}{2}}(\mathbb{S}^{1},\mathbb{S}^{1})$. Then, knowing both $v\in
H^{1}(\mathbb{D},\mathbb{R}^{2})$ and $v\circ f^{-1}\in
H^{1}(\Omega,\mathbb{R}^{2})$ makes our definition consistent. We explain why
in the appendix B.

Now we define the degree of a weak maps, see [7] and the appendix of [6].

######  Theorem 2.3.

Let $g\in H^{\frac{1}{2}}(\mathbb{S}^{1},\mathbb{S}^{1})$, then there exists
an integer $n$ and $\phi\in H^{\frac{1}{2}}(\mathbb{S}^{1},\mathbb{R})$,
unique up to a multiple of $2\pi$, such that $g=z^{n}e^{i\phi}$. The integer
$n$ is called the degree of $g$, and is denoted $\mathrm{deg}(g)$, moreover we
have222This formula is well defined since $g\in H^{\frac{1}{2}}$ and
$g_{\theta}\in H^{-\frac{1}{2}}$.

(2.2) $\mathrm{deg}(g)=\frac{1}{2\pi}\int_{\mathbb{S}^{1}}g\wedge
g_{\theta}\,d\theta.$

Thanks to Theorem 2.2, we have

######  Theorem-Definition 2.3.

For $\Gamma$ a chord-arc curve and $g\in
H^{\frac{1}{2}}(\Gamma,\mathbb{S}^{1})$, we define
$\mathrm{deg}(g,\Gamma)=\mathrm{deg}(g\circ f)$, where $f$ is any
uniformization map of the enclosing domain of $\Gamma$.

In particular, see property 5 and corollary 8 of [7], we have

######  Theorem 2.4.

Let $\Omega$ a domain whose boundary consists of a finite number of
quasicircle $(\Gamma_{i})_{i=1}^{k}$, then if $u\in
H^{1}(\Omega,\mathbb{S}^{1})$ then

$\sum_{i=1}^{k}\mathrm{deg}(u_{|\Gamma_{i}})=0.$

In particular, for $a\in\Omega$ and $u\in
H^{1}(\Omega\setminus\\{a\\},\mathbb{S}^{1})$, we set
$\mathrm{deg}(u,a)=\mathrm{deg}(u_{|\partial B(a,r)})$ for $r>0$ small enough.

##  3\. Proof of Theorem 0.1

###  3.1. Upper and lower log estimates for the energy

Let $u_{\epsilon}\in\mathcal{A}$ be such that
$E_{\varepsilon}(u_{\varepsilon})=\inf_{u\in\mathcal{A}}E_{\varepsilon}\left(u\right)$.
Such a minimizer exists because the functional $E_{\varepsilon}$ is coercive.
Since $u_{\varepsilon}$ comes from a minimization problem, by a test function
argument, we have the following upper bound on the energy, see Theorem III.1
of [2]. There exists $D_{0}>0$ such that, for any $\varepsilon>0$,

(3.1)
$E_{\varepsilon}(u_{\varepsilon})\leq\pi\left|\ln\varepsilon\right|+D_{0}.$

Then we want to prove the following key estimate,

(3.2)
$\frac{1}{4\varepsilon^{2}}\int_{\Omega}\left(1-|u_{\varepsilon}|^{2}\right)^{2}dx\leq
C$

for some constant $C$ independent from $\varepsilon$, depending only on
$\Omega$ and $g$.

In their seminal work Bethuel, Brezis and Hélein proved this estimate using a
Pohožaev identity, but in our case since the boundary data is only
$H^{\frac{1}{2}}$ we can’t apply this idea.

Our argument, inspired from [13], is based on an extension of the sequence of
minimizers $u_{\varepsilon}:\Omega\to\mathbb{R}^{2}$ to some bigger smooth
domain $\Omega^{\prime}\supset\Omega$ whose boundary data is smooth on
$\partial\Omega^{\prime}$.

######  Lemma 3.1 (extension lemma).

Let $\Omega$ a domain with a quasicircle as boundary and $g\in
H^{\frac{1}{2}}\left(\partial\Omega,\mathbb{S}^{1}\right)$ there is a smooth
simply connected domain $\Omega^{\prime}$ such that
$\overline{\Omega}\subset\Omega^{\prime}$ and a function
$\hat{u}:\Omega^{\prime}\setminus\Omega\to\mathbb{R}^{2}$ such that

  * •

$\hat{u}=g$ on $\partial\Omega$,

  * •

$\hat{u}\in\mathcal{C}^{\infty}\left(\overline{\Omega^{\prime}}\setminus\overline{\Omega}\right)$,

  * •

$\hat{u}\in H^{1}\left(\Omega^{\prime}\setminus\Omega\right)$ and whose energy
depends only on $\Omega$ and $\|g\|_{H^{\frac{1}{2}}}$,

  * •

$\left|\hat{u}\right|=1$ on $\overline{\Omega^{\prime}}\setminus\Omega$,

  * •

$\mathrm{deg}(\hat{u}_{|\partial\Omega^{\prime}})=\mathrm{deg}(g)$.

###### Proof.

Let $h:\mathbb{D}^{c}\rightarrow\Omega^{c}$ be a uniformization map, since
$\Omega$ is a quasidisk, thanks to Theorem 2.1, $h$ admits a quasiconformal
extension from $\mathbb{C}$ to itself, we still denote $h$. We set
$\tilde{g}:\mathbb{S}^{1}\rightarrow\mathbb{S}^{1}$ by $\tilde{g}=g\circ h$,
by conformal invariance, see Theorem 2.2, $\tilde{g}$ is in
$H^{\frac{1}{2}}(\mathbb{S}^{1},\mathbb{S}^{1})$. Let $\tilde{u}\in
H^{1}(\mathbb{D}^{c})\cap C^{\infty}(\mathbb{D}^{c},\mathbb{D})$ be the
harmonic extension of $\tilde{g}$, finally we set $v=\tilde{u}\circ h^{-1}$.
Since $h$ and $\tilde{u}$ are smooth outside $\overline{\mathbb{D}}^{c}$ then
$v$ is smooth on $\overline{\Omega}^{c}$. Since $h$ is conformal on
$\overline{\mathbb{D}}^{c}$, the energy of $v$ is bounded and depends only on
$\|\tilde{g}\|_{H^{\frac{1}{2}}}$, i.e on $\Omega$ and $g$ by Theorem 2.2 .
Finally, thanks to Lemma A.2, there exists $\delta>0$ such that
$\frac{3}{4}\leq|\tilde{u}|\leq\frac{5}{4}$ on
$\mathbb{D}_{1+\delta}\setminus\mathbb{D}$. Then $\hat{u}=\frac{v}{|v|}$
satisfies the desired properties on $\Omega^{\prime}=g(\mathbb{D}_{1+\delta})$
since it has the same property than $v$ and the last property is given by
Theorem 2.4. ∎

Then, thanks to Lemma 3.1, the function
$\hat{u}_{\varepsilon}:\Omega^{\prime}\to\mathbb{R}^{2}$ defined as an
extension of $u_{\varepsilon}$ such that

$\hat{u}_{\varepsilon}=u_{\varepsilon}\text{ in }\Omega\text{ and
}\hat{u}_{\varepsilon}=\hat{u}\text{ in }\Omega^{\prime}\setminus\Omega$

can be used as a test function for the classical Ginzburg-Landau problem on
the smooth domain $\Omega^{\prime}$, minimizing the energy

$E^{\Omega^{\prime}}_{\varepsilon}(v)=\frac{1}{2}\int_{\Omega^{\prime}}\left|\nabla
v\right|^{2}+\frac{1}{4\varepsilon^{2}}\int_{\Omega^{\prime}}\left(1-\left|v\right|^{2}\right)^{2}$

on the set of admissible functions $\mathcal{A}^{\Omega^{\prime}}=\\{v\in
W^{1,2}\left(\Omega^{\prime}\right);v=\hat{u}\text{ on
}\partial\Omega^{\prime}\\}$. We recall that $\hat{u}$ is a smooth function on
$\partial\Omega^{\prime}$. Since

$E_{2\varepsilon}(u_{\varepsilon})=E^{\Omega^{\prime}}_{2\varepsilon}(\hat{u}_{\varepsilon})-\frac{1}{2}\int_{\Omega^{\prime}\setminus\Omega}\left|\nabla\hat{u}\right|^{2}\geq\inf_{v\in\mathcal{A}^{\Omega^{\prime}}}E^{\Omega^{\prime}}_{2\varepsilon}(v)-\frac{1}{2}\int_{\Omega^{\prime}\setminus\Omega}\left|\nabla\hat{u}\right|^{2}$

and by the classical lower bound on the energy, see estimate (1.9) in [13], we
finaly get

(3.3)
$E_{2\varepsilon}(u_{\varepsilon})\geq\pi\left|\ln\varepsilon\right|-D_{1}$

for some constant $D_{1}$ depending only on $\Omega$ and $\tau$.

Now, using (3.1) and (3.3) we obtain

$\frac{1}{4\varepsilon^{2}}\int_{\Omega}\left(1-\left|u_{\varepsilon}\right|^{2}\right)^{2}=\frac{4}{3}\left(E_{\varepsilon}(u_{\varepsilon})-E_{2\varepsilon}(u_{\varepsilon})\right)\leq\frac{4}{3}\left(D_{0}+D_{1}\right).$

Setting $C=\frac{4}{3}\left(D_{0}+D_{1}\right)$ ends the proof of (3.2).

###  3.2. A priori estimates for sequences of minimizers

In this section we prove two fundamental estimate, the first one, proposition
3.3, asserts that $u_{\varepsilon}$ can’t be too small near the boundary, and
the second one, proposition , is known as $\eta$-compactness. Their proof does
not require that $u_{\varepsilon}$ is a minimizer but they only need as
assumption the uniform boundness of $u_{\varepsilon}$. This boundness is an
easy consequence of the maximum principle in the smooth case, but in our our
weak setting we are not able to prove it without assuming that the
$u_{\varepsilon}$ is a minimizer. Which leads us to ask the following
question:

Open question : are there sequences of critical point of $E_{\varepsilon}$
with fixed boundary data in $H^{\frac{1}{2}}(\partial\Omega,\mathbb{S}^{1})$
which are not uniformly bounded ?

Any critical point $u_{\varepsilon}$ of $E_{\varepsilon}$ satisfies the Euler-
Lagrange equation

(3.4) $\begin{cases}-\Delta
u_{\varepsilon}=\frac{1}{\varepsilon^{2}}\left(1-\left|u_{\varepsilon}\right|^{2}\right)u_{\varepsilon}&\text{
in }\Omega\\\ u_{\varepsilon}=_{a.e}g&\text{ on }\partial{\Omega}\end{cases}$

and we have those very first estimates on $u_{\varepsilon}$, which are true as
soon as $g\in H^{\frac{1}{2}}\left(\Gamma,\mathbb{S}^{1}\right)$. In fatc in
[14] they assume that boundary is Lipschitz but the argument works as soon as
we have a minimizer.

######  Proposition 3.1 (Lemma 3.2 of [14]).

If $u_{\varepsilon}$ is a minimizer of $E_{\varepsilon}$, then

(3.5) $\left\|u_{\varepsilon}\right\|_{L^{\infty}\left(\Omega\right)}\leq 1.$

Notice that (3.5) is true for any critical solution $u_{\varepsilon}$ to
$E_{\varepsilon}$ if $\Omega$ is smooth enough to apply the maximum principle
argument, see proposition 2 of [1]. The next proposition is a classical
elliptic estimate true independently of the regularity of $\Omega$.

######  Proposition 3.2 (Lemma A.1 [1]).

There is a constant $C_{1}>0$ such that for any $\varepsilon>0$, and for any
$x\in\Omega$,

(3.6) $\left|\nabla u_{\varepsilon}\right|(x)\leq
C_{1}\|u_{\varepsilon}\|_{\infty}\left(\frac{1}{\varepsilon}+\frac{1}{dist(x,\partial\Omega)}\right).$

In particular, if (3.5) holds, $\|u_{\varepsilon}\|_{\infty}$ can be removed.

In the following proposition, we prove that if $u_{\varepsilon}$ is uniformly
bounded then $\left|u_{\varepsilon}(x)\right|$ has to be far from $0$ as soon
as $x$ is close to $\partial\Omega$ at the scale $\varepsilon$. This
hypothesis of uniform boundness of $u_{\varepsilon}$ is satisfies in the case
of minimizers thanks to proposition 3.1.

######  Proposition 3.3.

Let $u_{\varepsilon}$ be solutions of (3.4) which are uniformly bounded with
respect to $\varepsilon$, then there are $\varepsilon_{0},\eta_{0}>0$
depending only on $\sup_{\varepsilon}\|u_{\varepsilon}\|_{\infty}$, $\Omega$
and $g$ such that for any $0<\varepsilon\leq\varepsilon_{0}$, for any
$x\in\Omega$

(3.7) $dist(x,\partial\Omega)\leq\eta_{0}\varepsilon\Rightarrow\hskip
8.53581pt\left|u_{\varepsilon}(x)\right|\geq\frac{1}{2}.$

###### Proof.

First of all, let us remind that thanks to our assumption and (3.2), there
exists $C>0$ such that

$\frac{1}{4\varepsilon^{2}}\int_{\Omega}\left(1-|u_{\varepsilon}|^{2}\right)^{2}dx\leq
C\text{ and }\|u_{\varepsilon}\|_{\infty}\leq C.$

We aim at finding $\eta_{0}$ and $\varepsilon_{0}$ such that for any
$0<\varepsilon\leq\varepsilon_{0}$ such that for any
$y_{\varepsilon}\in\partial\Omega$,

(3.8) $\inf_{x\in
D(y_{\varepsilon},\eta_{0}\varepsilon)\cap\Omega}\left|u_{\varepsilon}(x)\right|\geq\frac{1}{2}.$

Let $f:\mathbb{D}\to\Omega$ be a uniformization of $\Omega$. Since
$\Gamma=\partial\Omega$ is a quasicircle, there is a $K$-quasiconformal
extension $f:\mathbb{C}\to\mathbb{C}$ of $f$, still denoted $f$. We let
$v_{\varepsilon}=u_{\varepsilon}\circ f:\mathbb{D}\to\mathbb{R}^{2}$ be the
pullback of $u_{\varepsilon}$ by $f$. We have by conformal invariance, and
because $u_{\varepsilon}\in H^{\frac{1}{2}}\left(\Gamma\right)$ that

$\begin{cases}-\Delta
v_{\varepsilon}=\left|f^{\prime}\right|^{2}\frac{1-\left|v_{\varepsilon}\right|^{2}}{\varepsilon^{2}}v_{\varepsilon}&\text{
in }\mathbb{D}\\\ v_{\varepsilon}=g\circ f\in
H^{\frac{1}{2}}\left(\mathbb{S}^{1}\right)&\text{ on
}\mathbb{S}^{1}\end{cases}$

Let $\lambda_{0}>0$ we shall fix later, we set
$D_{\varepsilon}=f^{-1}\left(\mathbb{D}_{\lambda_{0}\varepsilon}(y_{\varepsilon})\cap\Omega\right)$.
We split $v_{\varepsilon}=a_{\varepsilon}+b_{\varepsilon}+h$ so that

$\displaystyle\begin{cases}-\Delta
a_{\varepsilon}=\mathbf{1}_{D_{\varepsilon}}\left|f^{\prime}\right|^{2}\frac{1-\left|v_{\varepsilon}\right|^{2}}{\varepsilon^{2}}v_{\varepsilon}&\text{
in }\mathbb{D}\\\ a_{\varepsilon}=0&\text{ on }\mathbb{S}^{1}\end{cases}$
$\displaystyle\begin{cases}-\Delta
b_{\varepsilon}=\left(1-\mathbf{1}_{D_{\varepsilon}}\right)\left|f^{\prime}\right|^{2}\frac{1-\left|v_{\varepsilon}\right|^{2}}{\varepsilon^{2}}v_{\varepsilon}&\text{
in }\mathbb{D}\\\ b_{\varepsilon}=0&\text{ on }\mathbb{S}^{1}\end{cases}$
$\displaystyle\begin{cases}-\Delta h=0&\text{ in }\mathbb{D}\\\
h=v_{\varepsilon}=g\circ f&\text{ on }\mathbb{S}^{1}\end{cases}$

Claim 1: For $\lambda_{0}$ small enough
$\|a_{\varepsilon}\|_{\infty}\leq\frac{1}{8}$.

By Lemma C.1, we have

(3.9) $\left\|a_{\varepsilon}\right\|_{L^{\infty}}+\left\|\nabla
a_{\varepsilon}\right\|_{L^{2}}\leq
C_{0}\left(\left\|\left(1-\left|z\right|^{2}\right)^{2}\Delta
a_{\varepsilon}\right\|_{L^{\infty}}+\left\|\Delta
a_{\varepsilon}\right\|_{L^{1}}\right).$

We aim at estimating the right-hand terms. By Koebe distortion theorem, see
Corollary 1.4 [32], we have that

(3.10)
$\frac{dist(f(z),\partial\Omega)}{1-\left|z\right|^{2}}\leq\left|f^{\prime}(z)\right|\leq
4\frac{dist(f(z),\partial\Omega)}{1-\left|z\right|^{2}}.$

In particular, if $z\in D_{\varepsilon}$,

(3.11)
$\left|f^{\prime}(z)\right|^{2}\frac{1-\left|v_{\varepsilon}(z)\right|^{2}}{\varepsilon^{2}}\left|v_{\varepsilon}(z)\right|\leq
4C\frac{\lambda_{0}^{2}}{(1-\left|z\right|^{2})^{2}}.$

We also have that, by change of variables,

(3.12)
$\begin{split}\int_{D_{\varepsilon}}\left|f^{\prime}\right|^{2}\frac{1-\left|v_{\varepsilon}\right|^{2}}{\varepsilon^{2}}\left|v_{\varepsilon}\right|\,dz&=\int_{f(D_{\varepsilon})}\frac{1-\left|u_{\varepsilon}\right|^{2}}{\varepsilon^{2}}\left|u_{\varepsilon}\right|\,dz\\\
&\leq\left(\int_{\Omega}\frac{\left(1-\left|u_{\varepsilon}\right|^{2}\right)^{2}}{\varepsilon^{2}}\,dz\right)^{\frac{1}{2}}\left(C\frac{\left|f(D_{\varepsilon})\right|}{\varepsilon^{2}}\right)^{\frac{1}{2}}\\\
&\leq 2C\sqrt{2\pi}\lambda_{0}.\end{split}$

Combining (3.9), (3.11) and (3.12), we obtain

(3.13) $\left\|a_{\varepsilon}\right\|_{L^{\infty}}+\left\|\nabla
a_{\varepsilon}\right\|_{L^{2}}\leq
C\left(4\lambda_{0}^{2}+2\sqrt{2\pi}\lambda_{0}\right).$

Choosing $\lambda_{0}$ small enough such that
$C\left(4\lambda_{0}^{2}+2\sqrt{2\pi}\lambda_{0}\right)\leq\frac{1}{8}$, we
obtain that

(3.14) $\left\|a_{\varepsilon}\right\|_{L^{\infty}}\leq\frac{1}{8}.$

Claim 2: There exists $r_{0}>0$ such that
$\|b_{\varepsilon}\|_{L^{\infty}(f^{-1}\left(\mathbb{D}_{r_{0}\lambda_{0}\varepsilon}(y_{\varepsilon})\cap\Omega\right))}\leq\frac{1}{8}$.

Let

$\delta_{\varepsilon}=\frac{\mathrm{dist}(f^{-1}(y_{\varepsilon}),\partial(f^{-1}(\mathbb{D}_{\lambda_{0}\varepsilon}(y_{\varepsilon}))))}{2},$

then

(3.15) $\mathbb{D}_{\delta_{\varepsilon}}(f^{-1}(y_{\varepsilon}))\subset
f^{-1}\left(\mathbb{D}_{\lambda_{0}\varepsilon}(y_{\varepsilon})\right)$

We set
$\tilde{b}_{\varepsilon}(z)=b_{\varepsilon}(f^{-1}(y_{\varepsilon})+\delta_{\varepsilon}z)$.
Then, let
$B_{\varepsilon}=\frac{\mathbb{D}-f^{-1}(y_{\varepsilon})}{\delta_{\varepsilon}}\cap\mathbb{D}$,
we have that

$\begin{cases}\Delta\tilde{b}_{\varepsilon}=0&\text{ in }B_{\varepsilon}\\\
\tilde{b}_{\varepsilon}=0&\text{ on }\partial
B_{\varepsilon}\cap\mathbb{D}.\end{cases}$

By standard elliptic estimates, there is a constant $D>0$ independent from
$\varepsilon>0$ such that

(3.16)
$\left\|\nabla\tilde{b}_{\varepsilon}\right\|_{L^{\infty}(B_{\varepsilon}\cap\mathbb{D}_{\frac{1}{2}})}\leq
D\left\|\tilde{b}_{\varepsilon}\right\|_{L^{\infty}\left(B_{\varepsilon}\right)}.$

We also know by $\eqref{eqLinftysmall}$, the fact that $v_{\varepsilon}$ is
bounded and that $h\in\mathbb{D}$ as harmonic extension of a $\mathbb{S}^{1}$
valued function, that

(3.17)
$\left\|\tilde{b}_{\varepsilon}\right\|_{L^{\infty}\left(B_{\varepsilon}\right)}\leq\left\|b_{\varepsilon}\right\|_{L^{\infty}}\leq\left\|v_{\varepsilon}\right\|_{L^{\infty}}+\left\|a_{\varepsilon}\right\|_{L^{\infty}}+\left\|h\right\|_{L^{\infty}}\leq
C+\frac{1}{8}+1\leq C+2.$

In particular, since we have that $\tilde{b}_{\varepsilon}=0$ on $\partial
B_{\varepsilon}\cap\mathbb{D}$, thanks to (3.16) and (3.17), there is a radius
$0<\rho_{0}<\frac{1}{2}$ such that

(3.18)
$\left\|b_{\varepsilon}\right\|_{L^{\infty}\left(\mathbb{D}\cap\mathbb{D}_{\rho_{0}\delta_{\varepsilon}}(f^{-1}(y_{\varepsilon}))\right)}=\left\|\tilde{b}_{\varepsilon}\right\|_{L^{\infty}\left(B_{\varepsilon}\cap\mathbb{D}_{\rho_{0}}\right)}\leq\frac{1}{8}.$

Finally since $f^{-1}$ is also quasiconformal, see Corollary 4.5.10 of [19],
thanks to Theorem 2.1, for every
$x\in\partial\mathbb{D}_{\lambda_{0}\varepsilon}(y_{\varepsilon})$ and
$y\in\mathbb{D}_{r_{0}\lambda_{0}\varepsilon}(y_{\varepsilon})$ we have

$\frac{|f^{-1}(y)-f^{-1}(y_{\varepsilon})|}{|f^{-1}(x)-f^{-1}(y_{\varepsilon})|}\leq\eta(r_{0})$

which gives

$|f^{-1}(y)-f^{-1}(y_{\varepsilon})|\leq\eta(r_{0})(2\eta(1)\delta_{\varepsilon})$

hence there is a constant $r_{0}>1$, smaller than $1$, depending only on the
quasiconformal constant $K$ of $f^{-1}$, that is to say only on $\Omega$, such
that

(3.19)
$f^{-1}\left(\mathbb{D}_{r_{0}\lambda_{0}\varepsilon}(y_{\varepsilon})\cap\Omega\right)\subset\mathbb{D}_{\rho_{0}\delta_{\varepsilon}}(f^{-1}(y_{\varepsilon}))$

hence (3.18) and (3.19) give that

(3.20)
$\left\|\tilde{b}_{\varepsilon}\right\|_{L^{\infty}\left(f^{-1}\left(\mathbb{D}_{r_{0}\lambda_{0}\varepsilon}(y_{\varepsilon})\cap\Omega\right)\right)}\leq\frac{1}{8},$

which achieves the proof of claim 2.

Now, we set $\eta_{0}=r_{0}\lambda_{0}$. To complete the proof it sufficies to
proves that
$\|h\|_{L^{\infty}(f^{-1}\left(\mathbb{D}_{\eta_{0}\varepsilon}(y_{\varepsilon})\cap\Omega\right)}\geq\frac{3}{4}$.
Notice that the $H^{\frac{1}{2}}\left(\mathbb{S}^{1}\right)$ norm of
$h=v_{\varepsilon}=u_{\varepsilon}\circ f=g\circ f$ is independent of
$\varepsilon$, hence the result follows because $h$ is uniformly proper by
Lemma A.2. ∎

The following proposition is an easy consequence of Proposition 3.2 and
Proposition 3.3.

######  Proposition 3.4.

Let $u_{\varepsilon}$ be solutions of (3.4) which are uniformly bounded with
respect to $\varepsilon$, then there are
$\varepsilon_{0},\eta_{0},\delta_{0}>0$ depending only on
$\sup_{\varepsilon}\|u_{\varepsilon}\|_{\infty}$, $\Omega$ and $g$ such that
for any $0<\varepsilon\leq\varepsilon_{0}$, for any $x\in\Omega$

(3.21)
$\begin{split}\left|u_{\varepsilon}(x)\right|\leq\frac{1}{2}\Rightarrow\frac{1}{4\varepsilon^{2}}\int_{\mathbb{D}_{\eta_{0}\varepsilon}(x)}\left(1-|u_{\varepsilon}|^{2}\right)^{2}dx\geq\delta_{0}.\end{split}$

###  3.3. Weak convergence to $u_{*}$

From now to the end of the subsection, we follow the lines of part 3 in [41].
First, the following claim is a very simple consequence of (3.2), Lemma 3.1
and Proposition 3.4.

######  Claim 3.1.

There are $\eta_{0}>0$ and $J_{0}\in\mathbb{N}$ depending only on $\Omega,g$
such that for any disjoint collection of balls
$\left(\mathbb{D}_{\eta_{0}\varepsilon}(x_{j}^{\varepsilon})\right)_{1\leq
j\leq J}$ such that
$\left|u_{\varepsilon}(x_{j}^{\varepsilon})\right|\leq\frac{1}{2}$, then
$J\leq J_{0}$.

By a classical Vitali’s covering lemma, we can find a family of disjoint balls
$\left(\mathbb{D}_{\eta_{0}\varepsilon}(x_{j}^{\varepsilon})\right)_{1\leq
j\leq J_{\varepsilon}}$ such that
$\left|u_{\varepsilon}(x_{j}^{\varepsilon})\right|\leq\frac{1}{2}$ for any
$1\leq j\leq J_{\varepsilon}$ and

$\left\\{x\in\Omega;\left|u_{\varepsilon}(x)\right|\leq\frac{1}{2}\right\\}\subset\bigcup_{1\leq
j\leq J_{\varepsilon}}\mathbb{D}_{5\eta_{0}\varepsilon}(x_{j}^{\varepsilon}).$

By Claim 3.1, $J_{\varepsilon}\leq J_{0}$ and for some radius $r>0$, we denote
by

${\Omega^{\prime}}_{\varepsilon,r}=\Omega^{\prime}\setminus\bigcup_{1\leq
j\leq J_{\varepsilon}}\mathbb{D}_{r}(x_{j}^{\varepsilon})\text{ and
}\Omega_{\varepsilon,r}={\Omega^{\prime}}_{\varepsilon,r}\cap\Omega$

where we recall that $\Omega^{\prime}$ is the extension of $\Omega$ given by
Lemma 3.1. ${\Omega^{\prime}}_{\varepsilon,r}$ is defined for technical
reasons since we have the property $\bigcup_{1\leq j\leq
J_{\varepsilon}}\mathbb{D}_{r}(x_{j}^{\varepsilon})\subset\Omega^{\prime}$ for
$r$ small enough.

Notice that by definition, $\left|u_{\varepsilon}(x)\right|\geq\frac{1}{2}$
for $x\in{\Omega}_{\varepsilon,r}$ if $r\geq 5\nu_{0}\varepsilon$ and that it
is also true for the extension $\hat{u}_{\varepsilon}$ given by Lemma 3.1
then:

$\left|\hat{u}_{\varepsilon}(x)\right|\geq\frac{1}{2}\text{ for
}x\in{\Omega^{\prime}}_{\varepsilon,r}\text{ if }r\geq 5\nu_{0}\varepsilon$

Let us remind the following general proposition orginally due to Brezis, Merle
and Rivière [8].

######  Proposition 3.5 ([41] Proposition 3.4 and [42]).

Let $0<r_{0}\leq r_{1}\leq r_{2}$ and $C>0$. Then for any $v\in
H^{1}\left(A_{r_{0},r_{2}}\right)$ such that
$\frac{1}{2}\leq\left|v\right|\leq 1$ a.e on $A_{r_{0},r_{1}}$ and such that
$v$ has degree $d$ on every circle $S_{r}$ for $r_{1}\leq r\leq r_{2}$, and
such that

$\frac{1}{r_{1}^{2}}\int_{A_{r_{0},r_{2}}}\left(1-|v|^{2}\right)^{2}\leq
C\text{ and }\int_{A_{r_{0},r_{2}}}\left|\nabla v\right|^{2}\leq C|\ln
r_{0}|,$

then for any $r_{1}\leq r\leq r_{2}$,

$\int_{A_{r_{1},r}}\left|\nabla v\right|^{2}\geq 2\pi
d^{2}\left(\ln\left(\frac{r}{r_{1}}\right)-D\right)-D$

where $D$ only depends on $C$.

The next proposition is a consequence of (3.1), Proposition 3.5 applied on the
extended function $\hat{u}_{\varepsilon}$ on an accurate disjoint union of
annuli centered at points in the set $\left(x_{j}^{\varepsilon}\right)_{1\leq
j\leq J_{\varepsilon}}$ almost filling
$\left(\Omega^{\prime}\right)_{\varepsilon,r}$ and the fact that the degree of
$\hat{u}_{\varepsilon}$ is $1$ on $\partial\Omega^{\prime}$.

######  Proposition 3.6 ([41] Proposition 3.3).

There are constants $\kappa>0$, $C$ depending only on $\Omega$ and $g$ such
that for any $\varepsilon<r<\delta$,

(3.22) $\int_{\Omega^{\prime}_{\varepsilon,r}}\left|\nabla
u_{\varepsilon}\right|^{2}\leq 2\pi|\ln r|+C$

######  Claim 3.2 (Remark that follows Proposition 3.3 of [41]).

Let $\varepsilon_{k}\to 0$ be a sequence,

Then there is a subsequence $\varepsilon_{k_{l}}$ such that
$\left(J_{\varepsilon_{k_{l}}}\right)_{l\in\mathbb{N}}$ is constant and such
that for any $1\leq j\leq J:=J_{\varepsilon_{k_{l}}}$ there is
$a_{j}\in\overline{\Omega}$ such that $x_{j}^{\varepsilon_{k_{l}}}\to a_{j}$
as $l\to+\infty$. We denote

$S=\\{a_{j};1\leq j\leq J\\}$

Moreover there is $a\in S$ and $u_{\star}\in
H^{1}_{loc}(\overline{\Omega}\setminus\\{a\\})\cap\mathcal{C}^{\infty}\left(\Omega\setminus\\{a\\}\right)$
such that $u_{\varepsilon_{k_{l}}}\to u_{\star}$ in
$\mathcal{C}_{loc}^{\infty}\left(\Omega\setminus S\right)$ as $l\to+\infty$.
Moreover, we have that

(3.23) $\int_{\Omega\setminus\mathbb{D}_{r}(a)}\left|\nabla
u_{\star}\right|^{2}\leq 2\pi\ln\left(\frac{1}{r}\right)+\kappa.$ (3.24)
$\int_{\mathbb{D}_{r_{2}}(a)\setminus\mathbb{D}_{r_{1}}(a)}\left|\nabla
u_{\star}\right|^{2}\leq
2\pi\ln\left(\frac{r_{2}}{r_{1}}\right)+\kappa^{\prime}.$

and that $u_{\star}$ is a harmonic map into $\mathbb{S}^{1}$.

Notice that combining proposition 3.5 and Claim 3.2, $a$ is a degree one
singularity for $u_{\star}$ and the other points of $S\setminus\\{a\\}$ are
degree zero singularities. In the next subsections, we aim at proving that
$a\in\Omega$ is an interior point, that $S\setminus\\{a\\}=\emptyset$ and that
$u_{\varepsilon_{k_{l}}}\to u_{\star}$ in
$H^{1}_{loc}\left(\overline{\Omega}\setminus\\{a\\}\right)$ as $l\to+\infty$.
These properties will complete the proof of (i)-(v) of Theorem 0.1. The last
point of Theorem 0.1 is a direct consequence of Proposition 7.1 [14].

###  3.4. The degree one singularity is an interior point of the domain

We aim at proving that $a\in\Omega$. Let us assume by contradiction that
$a\in\partial\Omega$ and without loss of generality that $a=0$.

As in the proof of of the convergence of $u_{\varepsilon}$, $u_{*}$ is
extended to a smooth domain $\Omega^{\prime}$ thanks to Lemma 3.1. In order to
light the notation, we will denote its extension $u_{*}$. Moreover we can also
apply Lemma 3.1 to $\Omega^{c}$ in order get the following claim

######  Claim 3.3.

Let $\Gamma$ a quasicircle and $g\in H^{\frac{1}{2}}(\Gamma,\mathbb{S}^{1})$
then there exists a smooth neighborhood $\Omega^{\prime\prime}$ of $\Gamma$
diffeomorphic to an annulus and $u_{0}\in
H^{1}(\Omega^{\prime\prime},\mathbb{S}^{1})$ such that $u=g$ on $\Gamma$.

We can assume that
$\Omega^{\prime\prime}\setminus\Omega=\Omega^{\prime}\setminus\Omega$. Let
$t>0$ small enough such that $\mathbb{D}_{t}\subset\Omega^{\prime\prime}$.

We first proof that on $S_{t}\cap\Omega^{\prime}$ there is always an arc with
angle uniformly bounded from below.

######  Claim 3.4.

there exists $\delta>0$ such that $\mathbb{D}_{t}\cap\Omega^{\prime}$ contains
a connected arc of arc length greater than $\delta t$, we denote it
$A_{t}^{\prime}$.

###### Proof.

Figure 4. Uniform control on $S_{t}\cap\Omega^{\prime}$

We choose some orientation on $\Gamma$, let $a^{-}_{t}$ the last point of
$S_{t}\cap\Gamma$ before $0$ and $a_{t}^{+}$ the first point of
$S_{t}\cap\Gamma$ after $0$. We denote by $\Gamma_{-}$ the open arc of
$\Gamma$ from $a^{+}_{t}$ to $0$. Thanks to (2.1), there is no point of
$\Gamma^{-}$ in $B(a^{+}_{t},\frac{t}{C})\cap\Gamma$. let $A_{t}^{1}$ and
$A_{t}^{2}$ the two open connected component of
$S_{t}\setminus\\{a_{t}^{-},a_{t}^{+}\\}$, and let $a_{t}^{1}$ (resp.
$a_{t}^{2}$) the closest point to $a_{t}^{+}$ of $\Gamma_{-}\cap A_{t}^{1}$
(resp. $\Gamma_{-}\cap A_{t}^{2}$). By the previous remark, the distance of
those points to $a_{t}^{+}$ is bigger than $\frac{t}{2C}$, Hence there is the
two open arc starting at the $a^{i}_{t}$ of arc length $\frac{t}{2C}$ which
contains no point of $\Gamma$, one of this two is included in
$\Omega^{\prime}$. ∎

By (2.2) and the fact $0$ is a degree one singularity, we have

$\displaystyle 1=\mathrm{deg}(u_{*},0)$
$\displaystyle=\frac{1}{2\pi}\int_{S_{t}}u_{*}\wedge{u_{*}}_{\theta}\,d\theta$
$\displaystyle=\frac{1}{2\pi}\int_{A_{t}^{\prime}}u_{*}\wedge{u_{*}}_{\theta}\,d\theta+\frac{1}{2\pi}\int_{\Gamma\setminus
A_{t}^{\prime}}u_{*}\wedge{u_{*}}_{\theta}\,d\theta.$

Then, we have

$\frac{1}{2\pi}\left|\int_{A_{t}^{\prime}}u_{*}\wedge{u_{*}}_{\theta}\,d\theta\right|\leq\frac{1}{2\pi}\|u_{0}\|_{H^{\frac{1}{2}}(S_{t})}^{2}\leq
C\|u_{0}\|_{H^{1}(A_{t})}^{2},$

where $A_{t}=\mathbb{D}_{2t}\setminus\mathbb{D}_{\frac{t}{2}}$, and

$\frac{1}{2\pi}\left|\int_{\Gamma\setminus
A_{t}^{\prime}}u_{*}\wedge{u_{*}}_{\theta}\,d\theta\right|\leq\frac{1}{2\pi}\int_{\Gamma\setminus
A_{t}^{\prime}}|{u_{*}}_{\theta}|\,d\theta\leq\frac{1}{2\pi}\left(\int_{\Gamma\setminus
A_{t}^{\prime}}|{u_{*}}_{\theta}|^{2}\,d\theta\right)^{\frac{1}{2}}\sqrt{(2\pi-\delta)t}$

Here we use the fact that the length of $A_{t}^{t}$ is at least $\delta t$.
Then, we have

$1-C\|\nabla
u_{0}\|^{2}_{L^{2}(A_{t})}\leq\frac{\sqrt{(2\pi-\delta)t}}{2\pi}\left(\int_{S_{t}}|{u_{*}}_{\theta}|^{2}\,d\theta\right)^{\frac{1}{2}},$

hence by squaring this identity

$\frac{4\pi^{2}}{(2\pi-\delta)t}(1-2C\|\nabla
u_{0}\|^{2}_{L^{2}(A_{t})})\leq\int_{S_{t}}|{u_{*}}_{\theta}|^{2}\,d\theta,$

But, thanks to Fubini Theorem, we have

$\displaystyle\int_{r_{1}}^{r_{2}}\frac{\|\nabla
u_{0}\|_{L^{2}(A_{t})}^{2}}{t}\,dt$
$\displaystyle=\int_{r_{1}}^{r_{2}}\frac{1}{t}\int_{A_{\frac{r_{1}}{2},2r_{2}}}|\nabla
u_{0}|^{2}\boldsymbol{1}_{\frac{t}{2}\leq|x|\leq 2t}\,dx\,dt$
$\displaystyle=\int_{A_{\frac{r_{1}}{2},2r_{2}}}|\nabla
u_{0}|^{2}\int_{r_{1}}^{r_{2}}\frac{1}{t}\,dt\,dx$
$\displaystyle=\ln(4)\int_{A_{\frac{r_{1}}{2},2r_{2}}}|\nabla u_{0}|^{2}\,dx.$

Finally we get

$\frac{4\pi^{2}}{(2\pi-\delta)t}\ln\left(\frac{r_{2}}{r_{1}}\right)-C\|\nabla
u_{0}\|_{L^{2}\left(A_{\frac{r_{1}}{2},2r_{2}}\right)}\leq\int_{A_{r_{1},r_{2}}}|\nabla
u_{*}|^{2}\,dx.$

But thanks to (3.24), we know that

$\int_{A_{r_{1},r_{2}}}|\nabla u_{*}|^{2}\,dx\leq
2\pi\ln\left(\frac{r_{2}}{r_{1}}\right)+C,$

which leads to a contradiction taking $\frac{r_{2}}{r_{1}}$ small enough.

###  3.5. Final convergence arguments

######  Proposition 3.7.

With the notations of Claim 3.2, we have that $u_{\varepsilon_{k_{l}}}\to
u_{\star}$ in $H^{1}_{loc}\left(\overline{\Omega}\setminus\\{a\\}\right)$ and
for any compact set $K\Subset\overline{\Omega}\setminus\\{a\\}$,

(3.25)
$\frac{1}{4\varepsilon_{k_{l}}^{2}}\int_{K}\left(1-|u_{\varepsilon_{k_{l}}}|^{2}\right)^{2}\to
0\text{ as }l\to+\infty$

###### Proof.

Let $r_{0}>0$ small enough be such that
$\mathbb{D}_{r_{0}}(a)\cap\left(\partial\Omega\cup S\right)=\emptyset$. Since
$u_{\varepsilon_{k_{l}}}$ is bounded in
$H^{1}\left(\Omega\setminus\mathbb{D}_{r_{0}}(a)\right)$,
$u_{\varepsilon_{k_{l}}}$ weakly converges to some function $u_{\star}\in
H^{1}\left(\Omega\setminus\mathbb{D}_{r_{0}}(a)\right)$. Moreover, $u_{\star}$
is a harmonic map into $\mathbb{S}^{1}$ and $u_{\star}$ satisfies

(3.26) $\int_{\Omega\setminus\mathbb{D}_{r_{0}}(a)}\left|\nabla
u_{\star}\right|^{2}\leq\liminf_{l\to+\infty}\int_{\Omega\setminus\mathbb{D}_{r_{0}}(a)}\left|\nabla
u_{\varepsilon_{k_{l}}}\right|^{2}.$

From now to the end of the proof of the proposition we denote
$\varepsilon:=\varepsilon_{k_{l}}$.

Let $w_{\varepsilon}$ be the solution the following minimization problem

$\frac{1}{2}\int_{\Omega\setminus\mathbb{D}_{r_{0}}(a)}\left|\nabla
w_{\varepsilon}\right|^{2}=\inf_{u\in\mathcal{A}_{\varepsilon}}\frac{1}{2}\int_{\Omega\setminus\mathbb{D}_{r_{0}}(a)}\left|\nabla
u\right|^{2}$

on the admissible set

$\mathcal{A}_{\varepsilon}=\left\\{u\in
H^{1}\left(\overline{\Omega}\setminus\mathbb{D}_{r_{0}}(a),\mathbb{S}^{1}\right);u=\frac{u_{\varepsilon}}{\left|u_{\varepsilon}\right|}\text{
on }\partial\left(\Omega\setminus\mathbb{D}_{r_{0}}(a)\right)\right\\}.$

As explained in Claim 3.3, we can find $u^{0}_{\varepsilon}$ is defined a
tubular neighborhood $\Omega^{\prime\prime}$ of $\partial\Omega$ which is in
$C^{\infty}(\Omega^{\prime\prime}\setminus,\mathbb{S}^{1})\cap
H^{1}(\Omega^{\prime\prime})$ and such that
$u_{\varepsilon}^{0}=u_{\varepsilon}$ on $\partial\Omega$. Up to reduce
$r_{0}$, we can assume that
$\mathbb{D}(a,r_{0})\cap\Omega^{\prime\prime}=\emptyset$. Then we set
$\omega=\Omega\setminus\Omega^{\prime\prime}$. Let
$\Psi:[a,b]\times\mathbb{S}^{1}\to\overline{\omega}\setminus\mathbb{D}_{r_{0}}(a)$
be a smooth diffeomorphism such that
$\Psi(\\{a\\}\times\mathbb{S}^{1})=\partial\omega$ and
$\Psi(\\{b\\}\times\mathbb{S}^{1})=S(a,r_{0})$. For instance, $\Psi$ can be
build with a uniformization map from an annulus to
$\overline{\omega}\setminus\mathbb{D}_{r_{0}}(a)$. We set
$a_{\varepsilon}=u^{0}_{\varepsilon}\circ\Psi_{|\\{a\\}\times\mathbb{S}^{1}}$
and $b_{\varepsilon}=u_{\varepsilon}\circ\Psi_{|\\{b\\}\times\mathbb{S}^{1}}$.
Since $deg(a_{\varepsilon})=deg(b_{\varepsilon})$, there is a smooth homotopy
$H_{\varepsilon}:[a,b]\times\mathbb{S}^{1}\to\mathbb{S}^{1}$ such that
$H_{\varepsilon}(a,.)=a_{\varepsilon}$ and
$H_{\varepsilon}(b,.)=b_{\varepsilon}$ and the map

$v_{\varepsilon}:=\begin{cases}H_{\varepsilon}\circ\Psi^{-1}\text{ in
}\omega\setminus\mathbb{D}_{r_{0}}(a)\\\ u_{\varepsilon}^{0}\text{ in
}\Omega\setminus\omega\end{cases}$

satisfies $v_{\varepsilon}\in\mathcal{A}_{\varepsilon}$.

It is clear that since $w_{\varepsilon}\to u_{\star}$ in
$H^{\frac{1}{2}}\left(\partial\left(\Omega\setminus\mathbb{D}_{r_{0}}(a)\right)\right)$
as $l\to+\infty$, $w_{\varepsilon_{k_{l}}}$ converges in
$H^{1}\left(\Omega\setminus\mathbb{D}_{r_{0}}(a)\right)$ to some function
$w_{\star}$ such that $w_{\star}=u_{\star}$ on
$\partial\left(\Omega\setminus\mathbb{D}_{r_{0}}(a)\right)$ and solution of

$\frac{1}{2}\int_{\Omega\setminus\mathbb{D}_{r_{0}}(a)}\left|\nabla
w_{\star}\right|^{2}=\inf_{u\in\mathcal{A}_{\star}}\frac{1}{2}\int_{\Omega\setminus\mathbb{D}_{r_{0}}(a)}\left|\nabla
u\right|^{2}$

on the admissible set

$\mathcal{A}_{\star}=\\{u\in
H^{1}\left(\overline{\Omega}\setminus\mathbb{D}_{r_{0}}(a),\mathbb{S}^{1}\right);u=u_{\star}\text{
on }\partial\left(\Omega\setminus\mathbb{D}_{r_{0}}(a)\right)\\}.$

In particular $u_{\star}\in\mathcal{A}_{\star}$ and

(3.27) $\int_{\Omega\setminus\mathbb{D}_{r_{0}}(a)}\left|\nabla
w_{\star}\right|^{2}\leq\int_{\Omega\setminus\mathbb{D}_{r_{0}}(a)}\left|\nabla
u_{\star}\right|^{2}$

We let $f_{\varepsilon}$ be the solution of

$\begin{cases}\Delta f_{\varepsilon}=0&\text{ in
}\Omega\setminus\mathbb{D}_{r_{0}}(a)\\\ f_{\varepsilon}=1&\text{ on
}\partial\Omega\\\ f_{\varepsilon}=|u_{\varepsilon}|&\text{ on
}S(a,r_{0})\end{cases}$

Testing the function

$h_{\varepsilon}:=\begin{cases}f_{\varepsilon}w_{\varepsilon}\text{ in
}\Omega\setminus\mathbb{D}_{r_{0}}(a)\\\ u_{\varepsilon}\text{ in
}\mathbb{D}_{r_{0}}(a)\end{cases}$

for the Ginzburg-Landau energy, we have that

$E_{\varepsilon}(u_{\varepsilon})\leq E_{\varepsilon}(h_{\varepsilon}).$

so that since $h_{\varepsilon}=u_{\varepsilon}$ in $\mathbb{D}_{r_{0}}(a)$ and
$|h_{\varepsilon}|=f_{\varepsilon}$ in $\Omega\setminus\mathbb{D}_{r_{0}}(a)$,

(3.28)
$\begin{split}\frac{1}{2}\int_{\Omega\setminus\mathbb{D}_{r_{0}}(a)}&\left|\nabla
u_{\varepsilon}\right|^{2}+\frac{1}{4\varepsilon^{2}}\int_{\Omega\setminus\mathbb{D}_{r_{0}}(a)}\left(1-\left|u_{\varepsilon}\right|^{2}\right)^{2}\\\
&\leq\frac{1}{2}\int_{\Omega\setminus\mathbb{D}_{r_{0}}(a)}f_{\varepsilon}^{2}\left|\nabla
w_{\varepsilon}\right|^{2}+\frac{1}{2}\int_{\Omega\setminus\mathbb{D}_{r_{0}}(a)}\left|\nabla
f_{\varepsilon}\right|^{2}+\frac{1}{4\varepsilon^{2}}\int_{\Omega\setminus\mathbb{D}_{r_{0}}(a)}\left(1-f_{\varepsilon}^{2}\right)^{2}\end{split}$

We set $\varphi_{\varepsilon}=\frac{1-f_{\varepsilon}}{\varepsilon^{2}}$.
Knowing that $f_{\varepsilon}\leq 1$ by the maximum principle, we then easily
obtain that

(3.29)
$\begin{split}\frac{1}{2}\int_{\Omega\setminus\mathbb{D}_{r_{0}}(a)}&f_{\varepsilon}^{2}\left|\nabla
w_{\varepsilon}\right|^{2}+\frac{1}{2}\int_{\Omega\setminus\mathbb{D}_{r_{0}}(a)}\left|\nabla
f_{\varepsilon}\right|^{2}+\frac{1}{4\varepsilon^{2}}\int_{\Omega\setminus\mathbb{D}_{r_{0}}(a)}\left(1-f_{\varepsilon}^{2}\right)^{2}\\\
\leq&\frac{1}{2}\int_{\Omega\setminus\mathbb{D}_{r_{0}}(a)}\left|\nabla
w_{\varepsilon}\right|^{2}+\varepsilon^{2}\left(\frac{\varepsilon^{2}}{2}\int_{\Omega\setminus\mathbb{D}_{r_{0}}(a)}\left|\nabla\varphi_{\varepsilon}\right|^{2}+\int_{\Omega\setminus\mathbb{D}_{r_{0}}(a)}\varphi_{\varepsilon}^{2}\right)\end{split}$

Now, we aim at proving that $\varphi_{\varepsilon}$ is a non negative
uniformly bounded function in $H^{1}\cap
L^{\infty}(\Omega\setminus\mathbb{D}_{r_{0}}(a))$. Indeed,
$\varphi_{\varepsilon}$ is a harmonic function, uniformly bounded in
$\mathcal{C}^{\infty}(S(a,r_{0}))$ since we know by Claim 3.2 that

$\frac{1-|u_{\varepsilon}|^{2}}{\varepsilon^{2}}\to\left|\nabla
u_{\star}\right|^{2}\text{ on }\mathcal{C}^{\infty}(S(a,r_{0})).$

Setting $\tilde{\varphi}_{\varepsilon}=\varphi_{\varepsilon}\circ f$, where
$f:\mathbb{D}\to\Omega$ is a uniformization map, we have that

$\begin{cases}\Delta\tilde{\varphi}_{\varepsilon}=0&\text{ in
}\mathbb{D}\setminus f^{-1}(\mathbb{D}_{r_{0}}(a))\\\
\tilde{\varphi}_{\varepsilon}=0&\text{ on }\mathbb{S}^{1}\\\
\tilde{\varphi}_{\varepsilon}=\frac{1-|u_{\varepsilon}\circ
f|}{\varepsilon^{2}}&\text{ on }f^{-1}\left(S(a,r_{0})\right)\end{cases}$

and by standard elliptic regularity, $\tilde{\varphi}_{\varepsilon}$ is
uniformly bounded. Coming back to
$\varphi_{\varepsilon}=\tilde{\varphi}_{\varepsilon}\circ f^{-1}$, using the
conformal invariance of the energy, we obtain the expected result.

Now, using (3.28), (3.29) and the uniform boundedness of
$\varphi_{\varepsilon}$ in $H^{1}\cap L^{\infty}$, we obtain

(3.30)
$\begin{split}\frac{1}{2}\int_{\Omega\setminus\mathbb{D}_{r_{0}}(a)}&\left|\nabla
u_{\varepsilon}\right|^{2}+\frac{1}{4\varepsilon^{2}}\int_{\Omega\setminus\mathbb{D}_{r_{0}}(a)}\left(1-\left|u_{\varepsilon}\right|^{2}\right)^{2}\\\
&\leq\frac{1}{2}\int_{\Omega\setminus\mathbb{D}_{r_{0}}(a)}\left|\nabla
w_{\varepsilon}\right|^{2}+O(\varepsilon^{2})\end{split}$

Passing to the limit on (3.30) and using (3.27) and (3.26), we obtain

$\begin{split}\frac{1}{2}\int_{\Omega\setminus\mathbb{D}_{r_{0}}(a)}\left|\nabla
u_{\star}\right|^{2}\leq&\liminf_{\varepsilon\to
0}\int_{\Omega\setminus\mathbb{D}_{r_{0}}(a)}\left|\nabla
u_{\varepsilon}\right|^{2}\\\ \leq&\limsup_{\varepsilon\to
0}\left(\int_{\Omega\setminus\mathbb{D}_{r_{0}}(a)}\left|\nabla
u_{\varepsilon}\right|^{2}+\frac{1}{4\varepsilon^{2}}\int_{\Omega\setminus\mathbb{D}_{r_{0}}(a)}\left(1-\left|u_{\varepsilon}\right|^{2}\right)^{2}\right)\\\
\leq&\frac{1}{2}\int_{\Omega\setminus\mathbb{D}_{r_{0}}(a)}\left|\nabla
w_{\star}\right|^{2}\leq\frac{1}{2}\int_{\Omega\setminus\mathbb{D}_{r_{0}}(a)}\left|\nabla
u_{\star}\right|^{2}\end{split}$

so that choosing $r_{0}$ as small as necessary, we obtain strong
$H^{1}_{loc}\left(\Omega\setminus\\{a\\}\right)$ convergence of
$u_{\varepsilon}$ and (3.25). ∎

We obtain from Proposition 3.7 and proposition 3.4 that
$S\setminus\\{a\\}=\emptyset$, so that the proof of Theorem 0.1 is complete.

##  4\. Minimization of the renormalized energy and the example of tangential
boundary data

###  4.1. Renormalized energy of canonical harmonic maps

Let $\Omega$ be a quasidisk and $\Gamma$ its boundary. In this section, we
give an $H^{1}$ version of the definition of canonical harmonic maps with a
prescribed degree $1$ singularity $a\in\Omega$ and of its renormalized energy.
The following definition is an $H^{1}$ version of section I.3 of [2].

######  Theorem-Definition 4.1 (Canonical harmonic map).

Let $a\in\Omega$. Let $g\in H^{\frac{1}{2}}(\Gamma,\mathbb{S}^{1})$ and
$deg(g,\Gamma)=1$. There is a unique map
$u:\Omega\setminus\\{a\\}\to\mathbb{S}^{1}$ such that

  * •

$u\in H^{1}_{loc}(\overline{\Omega}\setminus\\{a\\})$,

  * •

$deg(u,a)=1$,

  * •

$u=g$ on $\Gamma$,

  * •

$\star\omega=d\Phi$, where $\omega=\left\langle du,iu\right\rangle$ and
$\Phi=\tilde{\Phi}\circ f^{-1}$, where $f:\mathbb{D}\to\Omega$ is a
uniformization map such that $f(0)=a$ and $\tilde{\Phi}$ is the unique
solution (up to a constant) of

(4.1) $\begin{cases}\Delta\tilde{\Phi}=2\pi\delta_{0}&\text{ in
}\mathbb{D},\\\
\partial_{r}\tilde{\Phi}=\left\langle\partial_{\theta}\left(g\circ
f\right),i\left(g\circ f\right)\right\rangle&\text{ on
}\mathbb{S}^{1}.\end{cases}$

Moreover $u:\Omega\setminus\\{a\\}\to\mathbb{S}^{1}$ is a harmonic map. We say
that $u$ is the canonical harmonic map of degree $1$ associated to the data
$(g,a)$.

###### Proof.

Step 1 : Uniqueness of $u$. Let $u_{1}$ and $u_{2}$ satisfying all the
properties. Let $U$ be a simply connected domain such that
$U\Subset\Omega\setminus\\{a\\}$. Then, we can write $u_{1}=e^{i\psi_{1}}$ and
$u_{2}=e^{i\psi_{2}}$ for $\psi_{1},\psi_{2}\in H^{1}(U)$. Then we have that

$d\left(\psi_{1}-\psi_{2}\right)=\left\langle
du_{1},iu_{1}\right\rangle-\left\langle du_{2},iu_{2}\right\rangle=\star
d\Phi-\star d\Phi=0$

Therefore there is a constant $\alpha_{U}$ such that
$\left|\alpha_{U}\right|=1$ and $u_{2}=\alpha_{U}u_{1}$ in $U$. Since $\Omega$
is connected, we get that $\alpha:=\alpha_{U}$ does not depend on $U$ and
$u_{2}=\alpha u_{1}$ in $\Omega$. We finally get that $\alpha=1$ since on
$\Gamma$ we have $u_{1}=g=u_{2}$.

Step 2 : $\tilde{\Phi}$ is well defined by the Green representation formula.
Let $\mathcal{G}_{x}(y)$ the Green function such that

(4.2) $\begin{cases}\Delta\mathcal{G}_{x}=2\pi\delta_{x}&\text{ in
}\mathbb{D}\\\ \partial_{r}\mathcal{G}_{x}=0&\text{ on }\mathbb{S}^{1}\\\
\int_{\mathbb{S}^{1}}\mathcal{G}_{x}=0\text{ on }\mathbb{S}^{1}\end{cases}$

Then for $x\in\mathbb{D}$, since $g\circ f\in
H^{\frac{1}{2}}\left(\mathbb{S}^{1}\right)$, $\partial_{\theta}\left(g\circ
f\right)\in H^{-\frac{1}{2}}$ and $\mathcal{G}_{x}\in\mathcal{C}^{\infty}$

$\tilde{\Phi}(x)=\mathcal{G}_{0}(x)+\int_{\mathbb{S}^{1}}\mathcal{G}_{x}\left\langle\partial_{\theta}\left(g\circ
f\right),i\left(g\circ f\right)\right\rangle$

is a solution of (4.1).

Step 3 : Construction of $u$. We pullback all the expected properties of $u$
by $f:\mathbb{D}\to\Omega$, a uniformization map such that $f(0)=a$. We set

$\tilde{g}=g\circ f\hskip 14.22636pt\tilde{u}=u\circ f\hskip
14.22636pt\tilde{\omega}=\left\langle d\tilde{u},i\tilde{u}\right\rangle$

Then, $u$ is nothing but $u=\tilde{u}\circ f^{-1}$, where $\tilde{u}$ is the
canonical harmonic map of degree $1$ on the disk $\mathbb{D}$ associated to
the data $(g\circ f,0)$ which satisfies the desired property thanks to section
1.3 of [2].

∎

######  Definition 4.1 (Renormalized energy of canonical harmonic maps).

Let $u:\Omega\to\mathbb{S}^{1}$ be the canonical harmonic map with data
$a\in\Omega$, $g\in H^{\frac{1}{2}}\left(\Gamma,\mathbb{S}^{1}\right)$: the
renormalized energy is well defined by

$W(a)=\lim_{\delta\to
0}\left(\int_{\Omega\setminus\mathbb{D}_{\delta}(a)}\left|\nabla
u\right|^{2}-2\pi\ln\frac{1}{\delta}\right)$

###  4.2. Energy minimizing Ginzburg-Landau solutions minimize the
renormalized energy

Let $u:=u_{\star}$ a minimizing solution of the Ginzburg-Landau problem given
by Theorem 0.1. We set $v=-iu$, so that
$v:\Omega\setminus\\{a\\}\to\mathbb{S}^{1}$ is a unit vector field such that,
for any point $x\in\Omega\setminus\\{a\\}$, $(v(x),u(x))$ is an direct
orthonormal basis. We set

$\omega=\left\langle du,v\right\rangle=-\left\langle dv,u\right\rangle$

the Cartan form in $\Omega$.

Let $G$ be the Dirichlet Green function with respect to $a$ in $\Omega$, i.e
satisfying

(4.3) $\begin{cases}\Delta G=2\pi\delta_{a}&\text{in}\ \Omega\\\
G=0&\text{on}\ \Gamma\hskip 2.84544pt.\\\ \end{cases}$

######  Proposition 4.1.

There is a harmonic function $\mu$ in $\Omega$, such that

(4.4) $\star\omega=d\left(\mu+G\right)\hskip 2.84544pt.$

Then $u$ is the canonical harmonic map of degree $1$ with respect to the data
$(g,a)$.

###### Proof.

We know that $du\in L^{p}(\Omega)$ for any $1<p<2$ (see e.g. [41], page 1620),
so that $\omega\in L^{p}(\Omega)$. We have that

$\star d\omega=\star\left(\left\langle du\wedge
dv\right\rangle\right)=\left\langle\nabla^{\perp}u,\nabla v\right\rangle=0$

in $\Omega\setminus\\{a\\}$ since $\left(v,u\right)=i\left(-u,v\right)$. Then
the support of the distribution $\star d\omega$ is $\\{a\\}$. Since $\omega\in
L^{p}$, we can write

(4.5) $d\omega=\alpha\delta_{a}\hskip 2.84544pt.$

Let’s compute $\alpha$.

(4.6)
$\alpha=\int_{\Omega}d\omega=\int_{\partial\Omega}\omega=\int_{\mathbb{S}^{1}}\tilde{\omega}=\int_{\partial\Omega}\left\langle\tilde{u}_{\theta},\tilde{v}\right\rangle=2\pi
deg(g)=2\pi\hskip 2.84544pt.$

We then have that

(4.7) $d\left(\omega-\star dG\right)=0$

Moreover, we prove that there is $\beta\in\mathbb{R}$ such that

(4.8) $d\left(\star\omega\right)=\beta\delta_{a}$

in the sense of distributions in $\Omega$. Indeed, again, since $\omega\in
L^{p}(\Omega)$, we just have to prove that the support of the distribution
$d\left(\star\omega\right)$ is $\\{a\\}$. Indeed, we first have that

$\star d\left(\star\omega\right)=\left\langle\Delta
u,v\right\rangle+\star\left(\left\langle\star du\wedge
dv\right\rangle\right)=\left\langle\Delta u,v\right\rangle+\left\langle\nabla
u,\nabla v\right\rangle$

in $\Omega\setminus\\{a\\}$. Since $u:\Omega\setminus\\{a\\}\to\mathbb{S}^{1}$
is a harmonic map we have that

$\left\langle\Delta u,v\right\rangle=0$

in $\Omega\setminus\\{a\\}$. Moreover, we have that

(4.9) $0=\Delta\left(\left\langle u,v\right\rangle\right)=\left\langle\Delta
u,v\right\rangle+\left\langle u,\Delta v\right\rangle+2\left\langle\nabla
u,\nabla v\right\rangle=2\left\langle\nabla u,\nabla v\right\rangle\text{ in
}\Omega\setminus\\{a\\}$

since $\left\langle u,v\right\rangle=0$ and
$\left(v,u\right)=i\left(-u,v\right)$. We obtain (4.8). Then we have that

(4.10) $d\left(\star\omega-dG\right)=\beta\delta_{a}$

which gives that $h=\star\omega-dG-\beta\star dG$ satisfies

(4.11) $dh=0\hbox{ and }d\left(\star h\right)=0\hskip 2.84544pt.$

Then $h$ is a smooth harmonic form in $\Omega$. There is a smooth function
$\tilde{\mu}$ such that $h=d\tilde{\mu}$. To complete the proof of Claim 4.1,
we finally prove that $\beta=0$. From 3.23 we know that there is a constant
$C>0$ such that for $\delta<\frac{dist(a,\Gamma)}{2}$,

$\int_{\Omega\setminus B_{a}(\delta)}\left|\nabla u\right|^{2}\leq
2\pi\ln\left(\frac{1}{\delta}\right)+C\hskip 2.84544pt.$

Knowing that $\left|\nabla u\right|^{2}=\left|\left\langle\nabla
u,v\right\rangle\right|^{2}=\left|\omega\right|^{2}$, we have

(4.12) $2\pi\ln\left(\frac{1}{\delta}\right)+C\geq\int_{\Omega\setminus
B_{a}(\delta)}\left|\omega\right|^{2}=\int_{\Omega\setminus
B_{a}(\delta)}\left|\nabla G+\beta\nabla^{\perp}G+\nabla\mu\right|^{2}\hskip
2.84544pt.$

Since $\mu$ is smooth on $\Omega$ and $\nabla G$ is bounded in $L^{p}$ for any
$p<2$, we have a constant $C^{\prime}$ independent from
$\delta<\frac{d(a,\Gamma)}{2}$ such that

$\left|\int_{\Omega\setminus B_{a}(\delta)}2\left\langle\nabla
G+\beta\nabla^{\perp}G,\nabla\mu\right\rangle+\left|\nabla\mu\right|^{2}\right|\leq
C^{\prime}$

Then

$\int_{\Omega\setminus B_{a}(\delta)}\left|\nabla
G+\beta\nabla^{\perp}G\right|^{2}=\left(1+\beta^{2}\right)\int_{\Omega\setminus
B_{a}(\delta)}\left|\nabla G\right|^{2}+2\beta\int_{\Omega\setminus
B_{a}(\delta)}\left\langle\nabla G,\nabla^{\perp}G\right\rangle$

Since $\left\langle\nabla G,\nabla^{\perp}G\right\rangle=0$, we then can write
(4.12) as

(4.13)
$2\pi\ln\left(\frac{1}{\delta}\right)+C\geq\left(1+\beta^{2}\right)\int_{\Omega\setminus
B_{a}(\delta)}\left|\nabla G\right|^{2}-C^{\prime}\geq
2\pi\left(1+\beta^{2}\right)\ln\left(\frac{1}{\delta}\right)-C^{\prime\prime}\hskip
2.84544pt,$

for a constant $C^{\prime\prime}$ independent from $\delta$, since
$G(x)=\ln\left|x-a\right|+w(x)$ on $\Omega$, where $w$ is a harmonic function.
Dividing by $\ln\left(\frac{1}{\delta}\right)$ and letting $\delta\to 0$ gives
$\beta=0$.

Now it is easy to check that $u$ is the canonical harmonic map of degree $1$
with respect to the data $(g,a)$ and the proof of Claim 4.1 is complete. ∎

Finally, we also have, in our setting, the fact that the singularity of the
minimizer minimizes the renormalized energy, the proof is exactly the same as
the one of Theorem VIII.1 [2].

######  Proposition 4.2.

The singularity $a$ of the minimizing Ginzburg-Landau solution $u$ is a
minimizer of the renormalized energy $W(a)$.

###  4.3. The special case of tangential boundary data

In this section we prove the Proposition 0.3, when the boundary data $g=\tau$,
where $\tau$ is the tangent vector field on $\Gamma$ such that the frame
$(\nu,\tau)$ is a direct frame, where $\nu$ is the out-pointing normal of
$\Omega$. We have $\tau=-i\nu$.

In the following proposition 4.3 we aim at proving that canonical harmonic
maps with respect to tangential and normal data are exactly push forwards in
$\Omega$ of the polar orthonormal frame of the disk (with singularity at $0$)
by uniformization maps.

Let $f:\mathbb{D}\to\Omega$ be a uniformization map of $\Omega$. We set
$u:\Omega\to\mathbb{S}^{1}$

$u=\frac{f_{\theta}}{\left|f_{\theta}\right|}\circ f^{-1}$

and $v:\Omega\to\mathbb{S}^{1}$

$v=\frac{f_{r}}{\left|f_{r}\right|}\circ f^{-1}$

where $f_{r}$ and $f_{\theta}$ are the radial and angular derivatives of $f$
defined by

$f_{r}:=\partial_{r}f=\frac{xf_{x}+yf_{y}}{\sqrt{x^{2}+y^{2}}}=\frac{xf_{x}+yf_{y}}{r}$

where $r=\sqrt{x^{2}+y^{2}}$ is the radial coordinate and

$f_{\theta}:=\partial_{\theta}f=-yf_{x}+xf_{y}$

where $f_{x}:=\partial_{x}f$ and $f_{y}:=\partial_{y}f$ are the partial
derivatives with respect to the coordinates of $z=x+iy$.

######  Proposition 4.3.

Let $a\in\Omega$. For any uniformization map $f:\mathbb{D}\to\Omega$ such that
$f(0)=a$, $u=\frac{f_{\theta}}{\left|f_{\theta}\right|}\circ f^{-1}$ and
$v=\frac{f_{r}}{\left|f_{r}\right|}\circ f^{-1}$ are the canonical harmonic
maps of degree $1$ associated to the data $(\tau,a)$ and $(\nu,a)$.

###### Proof.

Let $f:\mathbb{D}\to\Omega$ be a uniformization map such that $f(0)=a$. Notice
that since $f$ is a conformal map, $(v,u)$ is an orthonormal frame in $\Omega$
and notice also that on $\Gamma$, $u=\tau$ and $v=\nu$. We now set

$\tilde{\omega}=\left\langle d\tilde{u},\tilde{v}\right\rangle=-\left\langle
d\tilde{v},\tilde{u}\right\rangle$

the Cartan form of $(\tilde{v},\tilde{u})$ where

$\tilde{v}=\frac{f_{r}}{\left|f_{r}\right|}\text{ and
}\tilde{u}=\frac{f_{\theta}}{\left|f_{\theta}\right|}$

and

$\tilde{\mu}=\ln\left|f_{x}\right|=\ln\left|f_{y}\right|=\ln\left|f_{r}\right|=\ln\left(\frac{\left|f_{\theta}\right|}{r}\right)$

so that

$\begin{cases}f_{r}=e^{\tilde{\mu}}\tilde{v}\\\
f_{\theta}=e^{\tilde{\mu}+\tilde{G}}\tilde{u}\end{cases}$

and

$\begin{cases}\tilde{v}=\left(xf_{x}+yf_{y}\right)e^{-\tilde{\mu}-\tilde{G}}\\\
\tilde{u}=\left(-yf_{x}+xf_{y}\right)e^{-\tilde{\mu}-\tilde{G}}\end{cases}$

where $\tilde{G}(x,y)=\ln r$ is the Green function of $\mathbb{D}$ with
Dirichlet boundary condition with respect to the singularity $0$.

Let’s prove the following equation:

(4.14) $\star\tilde{\omega}=d\left(\tilde{\mu}+\tilde{G}\right)$

in $\mathbb{D}$. We have that

$\begin{split}\left\langle\tilde{u}_{x},\tilde{v}\right\rangle&=\left\langle-
yf_{xx}+f_{y}+xf_{xy},xf_{x}+yf_{y}\right\rangle
e^{-2\tilde{\mu}-2\tilde{G}}\\\
&=\left(-xyf_{xx}.f_{x}-y^{2}f_{xx}.f_{y}+xf_{y}.f_{x}+y\left|f_{y}\right|^{2}+x^{2}f_{xy}.f_{x}+xyf_{xy}.f_{y}\right)e^{-2\tilde{\mu}-2\tilde{G}}\\\
&=\left(-\frac{xy}{2}\left(\left|f_{x}\right|^{2}\right)_{x}+\frac{y^{2}}{2}\left(\left|f_{y}\right|^{2}\right)_{y}+y\left|f_{y}\right|^{2}+\frac{x^{2}}{2}\left(\left|f_{x}\right|^{2}\right)_{y}+\frac{xy}{2}\left(\left|f_{y}\right|^{2}\right)_{x}\right)e^{-2\mu-2\tilde{G}}\\\
&=\left(-xy\tilde{\mu}_{x}+y^{2}\tilde{\mu}_{y}+y+x^{2}\tilde{\mu}_{y}+xy\tilde{\mu}_{x}\right)e^{-2\tilde{G}}\\\
&=\tilde{\mu}_{y}+\tilde{G}_{y}\end{split}$

where we used that $f_{xx}=-f_{yy}$ since $f$ is harmonic $\Delta f=0$. We
also have that

$\begin{split}\left\langle\tilde{u}_{y},\tilde{v}\right\rangle&=\left\langle-
f_{x}-yf_{xy}+xf_{yy},xf_{x}+yf_{y}\right\rangle
e^{-2\tilde{\mu}-2\tilde{G}}\\\
&=\left(-x\left|f_{x}\right|^{2}-yf_{x}.f_{y}-xyf_{xy}.f_{x}-y^{2}f_{xy}.f_{y}+x^{2}f_{yy}.f_{x}+xyf_{yy}.f_{y}\right)e^{-2\tilde{\mu}-2\tilde{G}}\\\
&=\left(-x\left|f_{x}\right|^{2}-\frac{xy}{2}\left(\left|f_{x}\right|^{2}\right)_{y}-\frac{y^{2}}{2}\left(\left|f_{y}\right|^{2}\right)_{x}-\frac{x^{2}}{2}\left(\left|f_{x}\right|^{2}\right)_{x}+\frac{xy}{2}\left(\left|f_{y}\right|^{2}\right)_{y}\right)e^{-2\tilde{\mu}-2\tilde{G}}\\\
&=\left(-x-xy\tilde{\mu}_{y}-y^{2}\tilde{\mu}_{x}-x^{2}\tilde{\mu}_{x}+xy\tilde{\mu}_{y}\right)e^{-2\tilde{G}}\\\
&=-\left(\tilde{\mu}_{x}+\tilde{G}_{x}\right)\end{split}$

We proved (4.14). Moreover, since $(\tilde{u},\tilde{v})$ is an orthonormal
frame, we have

$\star d\tilde{\omega}=\star\left(\left\langle d\tilde{u}\wedge
d\tilde{v}\right\rangle\right)=\left\langle\nabla^{\perp}\tilde{u},\nabla\tilde{v}\right\rangle=0\text{
in }\mathbb{D}\setminus\\{0\\}$

since $\left(\tilde{u},\tilde{v}\right)=i\left(\tilde{v},-\tilde{u}\right)$.
Then the support of the distribution $\star d\tilde{\omega}$ is $\\{0\\}$.
Since by (4.14) $\tilde{\omega}\in L^{p}_{loc}\left(\mathbb{D}\right)$ for
$1\leq p<2$, we can write

(4.15) $d\tilde{\omega}=\alpha\delta_{0}\hskip 2.84544pt,$

where $\delta_{0}$ is the Dirac mass supported in $\\{0\\}$. Let’s compute
$\alpha$.

(4.16)
$\alpha=\int_{\mathbb{D}}d\tilde{\omega}=\int_{\mathbb{S}^{1}}\tilde{\omega}=\int_{\mathbb{S}^{1}}\left\langle\tilde{u}_{\theta},\tilde{v}\right\rangle=2\pi
deg(\tilde{u},\mathbb{S}^{1})=2\pi\hskip 2.84544pt.$

This degree is well defined since $\tilde{u}\in
H^{\frac{1}{2}}(\mathbb{S}^{1})$ because $\Gamma$ is Weil-Petersson. We then
have that

(4.17) $\Delta\tilde{\mu}=d\left(\tilde{\omega}-\star
d\tilde{G}\right)=0\hskip 2.84544pt.$

In $\Omega$, we have $u=\tilde{u}\circ f^{-1}$ and $v=\tilde{v}\circ f^{-1}$.
We set

$\omega=\left\langle
du,v\right\rangle=\left(f^{-1}\right)^{\prime}\tilde{\omega}\circ f^{-1}$

the Cartan form on $\Omega$ and $G=\tilde{G}\circ f^{-1}$ the Dirichlet-Green
function with respect to $a:=f(0)$. We obtain from (4.14)

(4.18) $\star\omega=d\left(\mu+G\right)$

denoting $\mu=\tilde{\mu}\circ f^{-1}$, where we have that
$\tilde{\Phi}=\left(\mu+G\right)\circ f$ is a solution of (4.1).

Notice that $\Gamma$ is a Weil-Petersson curve if and only if

$\left|\nabla\tilde{\mu}\right|^{2}=\left|\frac{f^{\prime\prime}}{f^{\prime}}\right|^{2}\in
L^{1}(\mathbb{D})$

if and only if $\left|\nabla\mu\right|^{2}\in L^{1}\left(\Omega\right)$ where
$\mu=\tilde{\mu}\circ f^{-1}$ by conformal invariance of the Dirichlet energy.
In particular by (4.18), $u\in
H^{1}_{loc}\left(\overline{\Omega}\setminus\\{a\\}\right)$ and $\omega\in
L^{2}_{loc}\left(\overline{\Omega}\setminus\\{a\\}\right)$. Therefore, we have
all the necessary regularity to have $u=\tau\in H^{\frac{1}{2}}(\Gamma)$ and
the proof of the proposition is complete. ∎

######  Remark 4.1.

Notice that by Proposition 4.1, $\tilde{\mu}=\mu\circ f$ satisfies the
following equation

(4.19) $\begin{cases}\Delta\tilde{\mu}=0&\text{ in }\ \mathbb{D}\\\
\partial_{r}\tilde{\mu}=\left\langle\partial_{\theta}\tilde{u},\tilde{v}\right\rangle-1&\text{
on }\ \partial{\mathbb{D}}\hskip 2.84544pt,\end{cases}$

where $\partial_{r}$ and $\partial_{\theta}$ denote the radial and angular
derivatives on the disk, and $\tilde{u}=u\circ f$, $\tilde{v}=v\circ f$.
Equation (4.19) is nothing but the Liouville equation up to the boundary,
since $\left\langle u_{\tau},v\right\rangle$ is the curvature of $\Gamma$,see
[10] for more details about prescribing geodesic curvature.

Now, we can reformulate the renormalized energy in terms of the uniformization

######  Proposition 4.4.

Let $f:\mathbb{D}\to\Omega$ be a uniformization map such that $f(0)=a$. Then

(4.20) $W(a)=W_{0}-2\pi\ln\left|f^{\prime}(0)\right|$

where

$W_{0}=\int_{\mathbb{D}}\left|\frac{f^{\prime\prime}}{f^{\prime}}\right|^{2}+4\pi\ln\left|f^{\prime}(0)\right|$

does not depend on the choice of the uniformization $f:\mathbb{D}\to\Omega$
nor in $a$.

###### Proof.

The fact that $W_{0}$ only depends on $\Omega$ is well known, see lemma 3.4
page 86 of [43]. As in the proof of Proposition 4.3, we let
$\tilde{\mu}=\ln\left|f^{\prime}\right|$. Since $\Gamma$ is a Weil-Petersson
curve,

$\left|\nabla\tilde{\mu}\right|^{2}=\left|\frac{f^{\prime\prime}}{f^{\prime}}\right|^{2}\in
L^{1}(\mathbb{D})$

so that $\left|\nabla\mu\right|^{2}\in L^{1}\left(\Omega\right)$ where
$\mu=\tilde{\mu}\circ f^{-1}$ by conformal invariance of the Dirichlet energy.
In particular by (4.18), $\omega\in
L^{2}_{loc}\left(\overline{\Omega}\setminus\\{a\\}\right)$. Then by a simple
computation and by (4.18),

$\begin{split}\int_{\Omega\setminus
B_{\rho}(a)}\left|\omega\right|^{2}=&\int_{\Omega\setminus
B_{\rho}(a)}\left|\nabla G\right|^{2}+2\int_{\Omega\setminus
B_{\rho}(a)}\left\langle\nabla G,\nabla\mu\right\rangle+\int_{\Omega\setminus
B_{\rho}(a)}\left|\nabla\mu\right|^{2}\\\ =&-\int_{\partial
B_{\rho}(a)}G\partial_{\nu}G-2\int_{\partial
B_{\rho}(a)}G\partial_{\nu}\mu+\int_{\Omega\setminus
B_{\rho}(a)}\left|\nabla\mu\right|^{2}\\\ =&-\int_{\partial
B_{\rho}(a)}\left(\ln\left|x-a\right|+h\right)\partial_{\nu}\left(\ln\left|x-a\right|\right)+O\left(\rho\ln\rho\right)\\\
&+\int_{\Omega\setminus B_{\rho}(a)}\left|\nabla\mu\right|^{2}\\\
=&2\pi\ln\left(\frac{1}{\rho}\right)+\int_{\Omega\setminus
B_{\rho}(a)}\left|\nabla\mu\right|^{2}-2\pi h(a)+o(1)\end{split}$

as $\rho\to 0$, where $h(x)=G(x,a)-\ln\left|x-a\right|$ is a smooth harmonic
function in $\Omega$. Since $\ln\left|x\right|=G(f(x),a)$, we have that

(4.21) $\begin{split}h(a)&=h(f(0))\\\ &=\lim_{z\to
0}\left(G(f(z))-\ln\left|f(z)-a\right|\right)\\\ &=\lim_{z\to
0}\left(-\ln\left|\frac{f(z)-a}{z}\right|\right)\\\
&=-\ln\left|f^{\prime}(0)\right|\end{split}$

so that

$W(a)=\int_{\mathbb{D}}\left|\nabla\tilde{\mu}\right|^{2}+2\pi\ln\left|f^{\prime}(0)\right|\hskip
2.84544pt.$

We then have the expected identity (4.20). ∎

######  Remark 4.2.

A minimizer $a\in\Omega$ of $W$ is $a=f(0)$ where $f$ is a maximizer of the
following maximization problem

$\max_{f:\mathbb{D}\to\Omega}\left|f^{\prime}(0)\right|$

where the maximum stands on the set of uniformizing maps of $\Omega$. Or
equivalently, fixing one uniformization map $f_{0}:\mathbb{D}\to\Omega$, a
minimizer is $a=f_{0}(\omega_{0})$ where $\omega_{0}\in\mathbb{D}$ is chosen
as a maximizer for

(4.22)
$\left|f_{0}^{\prime}(\omega_{0})\right|\left(1-\left|\omega_{0}\right|^{2}\right)=\max_{\omega\in\mathbb{D}}\left|f_{0}^{\prime}(\omega)\right|\left(1-\left|\omega\right|^{2}\right)\hskip
2.84544pt.$

Indeed, we can write any uniformization map as $f=f_{0}\circ\psi^{-1}$, where

$\psi(z)=e^{i\theta}\frac{z-\omega}{\bar{\omega}z-1}$

for $\omega\in\mathbb{D}$ and $\theta\in\mathbb{R}$ so that since
$\psi(\omega)=0$

$\left|f^{\prime}(0)\right|=\left|f_{0}^{\prime}(\omega)\left(\psi^{-1}\right)^{\prime}(0)\right|=\left|\frac{f_{0}^{\prime}(\omega)}{\psi^{\prime}(\omega)}\right|=\left|f_{0}^{\prime}(\omega)\right|\left(1-\left|\omega\right|^{2}\right)\hskip
2.84544pt.$

As we can see in (4.21), $-\ln\left|f^{\prime}(0)\right|$ is nothing but the
mass of the Dirichlet-Green function of $\Omega$ at the point $a$ where $f$ is
defined as a uniformization map such that $f(0)=a$.

######  Remark 4.3.

We know by section 3.4 that a minimizer of the renormalized energy cannot be
realized on the boundary of $\Omega$. In the case of tangential data, it is a
simple consequence of the previous remark and Koebe’s distortion theorem.
Indeed, a point $\omega_{0}$ satisfying (4.22) must belong to $\mathbb{D}$
since

$\left|f_{0}^{\prime}(\omega)\right|\left(1-\left|\omega\right|^{2}\right)\leq
dist(f_{0}(\omega),\partial\Omega)\to 0\text{ as }\left|\omega\right|\to 1$

so that $a=f_{0}(\omega_{0})\in\Omega$.

##  5\. A new point of view on the Riemann mapping theorem

We assume in this section that $\Gamma$ is a smooth333 In fact we can assume
$C^{2}$ but not much less as explained in remark 5.1 below. Jordan curve. As a
side remark, we aim at proving the Riemann mapping theorem for the smooth
bounded domain $\Omega$ enclosing $\Gamma$.

Let $(v,u)$ be the frame defined in subsection 4.2 with tangential and normal
boundary data, and $\omega=\left\langle du,v\right\rangle$ be the associated
Cartan form. We let $\Phi=\mu+G$ where $G$ is defined by (4.3) and $\mu$ is
given by Proposition (4.1). Notice that since we assumed that $\Omega$ is
smooth, we never needed any uniformization map of $\Omega$ to build the moving
frame $(v,u)$, since the map $u$ is given from the classical Ginzburg-Landau
setting. We compute the following Lie bracket in $\Omega\setminus\\{a\\}$:

$\displaystyle\left[e^{\Phi}v,e^{\Phi}u\right]$ $\displaystyle=$
$\displaystyle
e^{2\Phi}\left\\{\left[v,u\right]+d\Phi\left(v\right)u-d\Phi\left(u\right)v\right\\}$
$\displaystyle=$ $\displaystyle
e^{2\Phi}\left\\{\nabla_{v}u-\nabla_{u}v+d\Phi\left(v\right)u-d\Phi\left(u\right)v\right\\}$
$\displaystyle=$ $\displaystyle
e^{2\Phi}\left\\{\left(\omega\left(u\right)+d\Phi\left(v\right)\right)u+\left(-d\Phi\left(u\right)+\omega\left(v\right)\right)v\right\\}\hskip
2.84544pt.$

Let $(\alpha,\beta)$ the dual basis of $(v,u)$ in $T^{\star}\Omega$, we have
that

$\star\omega=\star\left(\omega\left(v\right)\alpha+\omega\left(u\right)\beta\right)=\omega\left(v\right)\beta-\omega\left(u\right)\alpha\hskip
2.84544pt,$

where $\star\alpha=\beta$ and $\star\beta=-\alpha$. Therefore, we have that

$\left[e^{\Phi}v,e^{\Phi}u\right]=e^{2\Phi}\left\\{\left(-\star\omega\left(v\right)+d\Phi\left(v\right)\right)u+\left(-d\Phi\left(u\right)+\star\omega\left(u\right)\right)v\right\\}=0\hskip
2.84544pt,$

since $\star\omega=d\Phi$. We have the following result, which is a global
version of the Frobenius theorem in our context

######  Proposition 5.1.

There is a homeomorphism
$\psi:]-\infty,0]\times\mathbb{R}/\rho\mathbb{Z}\to\overline{\Omega}\setminus\\{a\\}$
such that

(5.1) $\begin{cases}\partial_{s}\psi=e^{\Phi\circ\psi}v\circ\psi\\\
\partial_{\theta}\psi=e^{\Phi\circ\psi}u\circ\psi\hskip 2.84544pt.\\\
\end{cases}$

In particular,
$\psi:]-\infty,0[\times\mathbb{R}/\rho\mathbb{Z}\to\Omega\setminus\\{a\\}$ is
a conformal diffeomorphism.

###### Proof.

We know by the Frobenius theorem (see for instance [45]), that for any
$x\in\Omega\setminus\\{a\\}$, there is a map
$\psi_{U}:U\to\Omega\setminus\\{a\\}$, defined on an open set $U$ such that
$0\in U$, $\psi_{U}(0)=x$ and $\psi(s,\theta)$ satisfies

(5.2)
$\begin{cases}\partial_{s}\psi_{U}=e^{\Phi\circ\psi_{U}}v\circ\psi_{U}\\\
\partial_{\theta}\psi_{U}=e^{\Phi\circ\psi_{U}}u\circ\psi_{U}\hskip
2.84544pt.\\\ \end{cases}$

Moreover, $\psi_{U}$ is unique in the sense that two such maps are equal on
the intersection of there open sets $U$ of definition. Notice also that such
an application $\psi_{U}$ is a local diffeomorphism. Let $(U_{x},\psi_{x})$ be
the maximal solution: $U_{x}$ is defined as the union of open sets $U$ such
that there is a map $\psi_{U}$ satisfying $\psi_{U}(0)=x$ and (5.2). By
uniqueness in the Frobenius theorem, $\psi_{x}(y)=\psi_{U}(y)$ is perfectly
defined independently of $U$ and satisfies $\psi_{x}(0)=x$ and (5.2). We aim
at proving that up to factorisation and translation on the domain of
definition, $\psi_{x}$ is the map we are looking for.

All along the proof we denote $\varphi_{X_{1}}$ and $\varphi_{X_{2}}$ the
flows along the vector fields $X_{1}:=e^{\Phi}v$ and $X_{2}:=e^{\Phi}u$.
Notice that the partial maps of $\psi_{x}$ are paths along the flow: if
$x_{0}=\psi_{x}(s_{0},\theta_{0})$,

$\psi_{x}(s_{0}+s,\theta_{0})=\varphi^{s}_{X_{1}}(x_{0})\hbox{ and
}\psi_{x}(s_{0},\theta_{0}+\theta)=\varphi^{\theta}_{X_{2}}(x_{0})\hskip
2.84544pt.$

Notice also that we can extend $X_{1}$ and $X_{2}$ by smooth vector-fileds on
an open set $\widetilde{\Omega}$ such that
$\overline{\Omega}\subset\widetilde{\Omega}$, and that there is no stationary
point on $\widetilde{\Omega}\setminus\\{a\\}$ for $\varphi_{X_{1}}$ and
$\varphi_{X_{2}}$. In particular, the flows are perfectly defined up to the
boundary in $\overline{\Omega}$. We recall that $\partial\Omega$ is a cycle of
$\varphi_{X_{2}}$ by assumption. Moreover, the only stationary point of
$\varphi_{X_{1}}$ and $\varphi_{X_{2}}$ is $a$ since by definition of $\Phi$
as a Green function on $a$, the only vanishing point of $e^{\Phi}$ is $a$.

Step 1 : We prove that for any $x$, $\psi_{x}$ is surjective

Notice that if $x,y\in\Omega\setminus\\{a\\}$, we have that

$\psi_{x}(U_{x})\cap\psi_{y}(U_{y})\neq\emptyset\Rightarrow\psi_{x}(U_{x})=\psi_{y}(U_{y})\hskip
2.84544pt.$

Indeed, assume that $z=\psi_{x}(u)=\psi_{y}(v)$ for $u\in U_{x}$ and $v\in
U_{y}$. By uniqueness in the Frobenius theorem, and by maximality of
$(U_{x},\psi_{x})$ and $(U_{y},\psi_{y})$, one has $U_{x}=U_{y}+\\{b-a\\}$ and
$\psi_{x}(z)=\psi_{y}(z+v-u)$ for any $z\in U_{x}$. Since for any
$y\in\Omega$, $\psi_{y}$ is a local diffeomorphism, then if
$y\in\psi_{x}(U_{x})$, there is an open set $U$ such that $\psi_{y}(U)$ is an
open neighbourhood of $y$ in $\psi_{x}(U_{x})$. Therefore $\psi_{x}(U_{x})$ is
open. Denoting $C=\\{C(x);x\in\Omega\setminus\\{a\\}\\}$, where

$C(x)=\\{y\in\Omega\setminus\\{a\\};\psi_{x}(U_{x})=\psi_{y}(U_{y})\\}\hskip
2.84544pt,$

we can write

$\Omega\setminus\\{a\\}=\bigcup_{C(x)\in C}\psi_{x}(U_{x})$

as a disjoint union of open sets. By connectedness, the cardinal of $C$ is $1$
and $\psi_{x}(U_{x})=\Omega\setminus\\{a\\}$ for any
$x\in\Omega\setminus\\{a\\}$.

From now on, we drop the index $x$ in the definition of $U:=U_{x}$ and
$\psi:=\psi_{x}$

Step 2 : We prove the existence of $\beta$ such that
$U=(-\infty,\beta)\times\mathbb{R}$ and such that we can extend $\psi$ on
$(-\infty,\beta]\times\mathbb{R}$ so that
$\psi\left(\\{\beta\\}\times\mathbb{R}\right)=\partial\Omega$ and $\psi$ is a
local homeomorphism up to the boundary.

We first prove that for any $(s_{0},\theta_{0})\in U$, we have
$\\{s_{0}\\}\times\mathbb{R}\subset U$. Indeed, let $(s_{0},\theta_{0})\in U$,
$x_{0}=\psi(s_{0},\theta_{0})$, and let $(\gamma,\delta)$ be the maximal
interval such that $\gamma<\theta_{0}<\delta$ and
$\\{s_{0}\\}\times(\gamma,\delta)\subset U$. Let’s prove that $\delta=+\infty$
(proving that $\gamma=-\infty$ is analogous).

Since $\overline{\Omega}$ is compact, up to a subsequence of
$\theta_{n}\to\delta$ one has
$\lim_{\theta_{n}\to\delta}\psi(s,\theta_{n})=l\in\overline{\Omega}$. If
$l\in\Omega\setminus\\{a\\}$ and $\delta<+\infty$, by the Frobenius theorem,
one can extend $\psi$ by a map $\hat{\psi}$ satisfying
$\hat{\psi}(s,\delta)=l$ and (5.2), defined on an open neighbourhood of
$(s,\delta)$ and we contradict the maximality of $\delta$. Therefore,
$l\in\partial\Omega\cup\\{a\\}$ or $\delta=+\infty$. Now, we recall that
$\psi(s_{0},\theta)=\varphi^{\theta-\theta_{0}}_{X_{2}}(x_{0})$ is the flow
associated to $X_{2}$. If $l=a$, then $\lim_{\theta\to\delta}\psi(s,\theta)=a$
and since $a$ is a stationary point, we have that $\delta=+\infty$. Similarly,
if $l\in\partial\Omega$, since $\partial\Omega$ is a cycle of
$\varphi_{X_{2}}$, we must have $\delta=+\infty$. In all the cases,
$\delta=+\infty$.

Now, let
$\beta=\sup\\{s\in\mathbb{R};\\{s\\}\times\mathbb{R}\subset\Omega\\}$.

We prove that $\beta<+\infty$ and that for any $\theta\in\mathbb{R}$,
$\lim_{z\to(\beta,\theta)}\psi(y)$ exists and belongs to $\partial\Omega$.

The vector field $X_{1}=e^{\Phi}v$ is transversal to $\partial\Omega$ (and
outward pointing on $\partial{\Omega}$). Let $y\in\partial\Omega$. Let
$F:]-\varepsilon,\varepsilon[\times]-\varepsilon,\varepsilon[\to V$ be a
straightening of $X_{1}$ at the neighbourhood $V$ of $y=F(0,0)$: ie $F$ is a
diffeomorphism and $F^{\star}(X_{1})=\partial_{x_{1}}$. We have that
$F(]-\varepsilon,0[\times]-\varepsilon,\varepsilon[)\subset\Omega$. Let
$(s_{0},\theta_{0})\in U$ be such that
$x_{0}=\psi(s_{0},\theta_{0})=F(-\frac{\varepsilon}{2},0)$. Recalling that
$\psi(s,\theta_{0})=\varphi_{X_{1}}^{s-s_{0}}(x_{0})$, we deduce the expected
result.

Now, we let
$\alpha=\inf\\{\tilde{\alpha}\in(-\infty,\beta);(\tilde{\alpha},\beta)\times\mathbb{R}\subset
U\\}$. With the same methods as previously thanks to the minimality of
$\alpha$ and using the flow associated to $X_{1}$ by
$\psi(s,\theta)=\varphi^{s-s_{0}}_{X_{1}}(x_{0})$ one can prove that
$\alpha=-\infty$. This ends the proof of STEP 2.

Step 3 : We prove the existence of $\rho>0$ such that
$\psi(s,\theta)=\psi(s,\theta+\rho)$ for any $(s,\theta)\in U$.

It is equivalent to prove that for any $x\in\psi(U)$, we have that
$\varphi_{X_{2}}(x)$ is a cycle.

Let $y\in\partial\Omega$. Since $\varphi_{X_{2}}(y)$ is a cycle, there is a
minimal $\rho>0$ such that $\varphi_{X_{2}}^{\rho}(y)=y$. We aim at proving
that $\varphi_{X_{2}}^{\rho}(x)=x$ for any $x\in\psi(U)$.

Let $x_{0}\in\psi(U)$ and $(s_{0},\theta_{0})\in U$ be such that
$x_{0}=\psi(s_{0},\theta_{0})$. From the computations we made before
Proposition 5.1, we have that $[X_{1},X_{2}]=0$ in $\Omega\setminus\\{a\\}$.
Therefore, the flows $\varphi_{X_{1}}$ and $\varphi_{X_{2}}$ commute. Then by
STEP 2,

$\varphi_{X_{2}}^{\rho}(x_{0})=\varphi_{X_{1}}^{s_{0}-s}\circ\varphi_{X_{2}}^{\rho}\circ\varphi_{X_{1}}^{s-s_{0}}(x_{0})\hskip
2.84544pt,$

for any $s_{0}\leq s<\beta$. By STEP 2 again, we can pass to the limit as
$s\to\beta$ and we get

(5.3)
$\varphi_{X_{2}}^{\rho}(x_{0})=\varphi_{X_{1}}^{s_{0}-\beta}\circ\varphi_{X_{2}}^{\rho}\circ\varphi_{X_{1}}^{\beta-
s_{0}}(x_{0})\hskip 2.84544pt.$

Since $\varphi_{X_{1}}^{\beta-s_{0}}(x_{0})\in\partial\Omega$, we have that
$\varphi_{X_{2}}^{\rho}\left(\varphi_{X_{1}}^{\beta-
s_{0}}(x_{0})\right)=\varphi_{X_{1}}^{\beta-s_{0}}(x_{0})$. We then get STEP 3
from (5.3).

Step 4 : We prove the Proposition

We assume that $\rho>0$ given by STEP 3 is minimal. Now we can factorize
$\psi$ into a surjective map we still denote
$\psi:(-\infty,\beta]\times\mathbb{R}/\rho\mathbb{Z}\to\overline{\Omega}\setminus\\{a\\}$.

Let’s prove that this map is now injective. Let $z_{1}=(s_{1},\theta_{1})$ and
$z_{2}=(s_{2},\theta_{2})$ such that $\psi(z_{2})=\psi(z_{1})$. Then
$\psi(z)=\psi(z+z_{2}-z_{1})$. Assume that $s_{1}\leq s_{2}$. We then have
that $\psi(\\{\beta-s_{2}+s_{1}\\}\times\mathbb{R})=\partial\Omega$. Then we
must have $s_{1}=s_{2}$. By minimality of $\rho$, we get that
$\theta_{1}=\theta_{2}$. Then $\psi$ is injective. Since $\psi$ is bijective
and a local homeomorphism, it is a global homeomorphism.

Moreover, since
$\psi:(-\infty,\beta)\times\mathbb{R}/\rho\mathbb{Z}\to\Omega\setminus\\{a\\}$
is bijective smooth and open, it is a diffeomorphism. By equation (5.2), it is
also a conformal diffeomorphism. Finally, we replace $\psi$ by
$\psi(z-(0,\beta))$ and we complete the proof of Proposition 5.1.

∎

We set a holomorphic function
$f:\mathbb{D}\setminus\\{0\\}\to\Omega\setminus\\{a\\}$, defined by

(5.4) $f(re^{i\theta})=\psi\left(\frac{\rho}{2\pi}\left(\ln
r,\theta\right)\right)\hskip 2.84544pt,$

where $\psi$ is the map given by claim (5.1). The implicit formula
$f(e^{\frac{2\pi w}{\rho}})=\psi(w)$ holds and one can directly extend $f$ to
a bi-holomorphic map $f:\mathbb{D}\to\Omega$ by $f(0)=a$. This ends the proof
of the Riemann mapping theorem.

######  Remark 5.1.

We cannot ask for low regularity of $\Omega$ to prove the result by this
method for two reasons : $\mathcal{C}^{1}$ regularity of $\Omega$ is needed to
use classical construction of Ginzburg-Landau minimizers (without using the
uniformization map as in the current paper for Weil-Petersson curves).
Moreover, we use that the unit tangent vector field on $\Gamma$ is Lipschitz
in order to apply the Cauchy-Lipschitz theorem and define the flow of vector
fields. Using some methods from dynamical systems, it may be possible to prove
the existence of a closed orbit in the interior, which would play the same
role as the boundary orbit and would permit to conclude.

##  Appendix A Technical lemmas

The result of this section are well known, we prove it in a way more adapted
to our paper for the sake of completeness. First we need a basic estimate on
the Poisson Kernel

######  Lemma A.1.

Let $M>0$, then

(A.1) $\lim_{r\rightarrow
1^{-}}(1-r)\int_{\delta}^{2\pi-M(1-r)}\frac{1}{1+r^{2}-2r\cos(\theta)}\,d\theta=8\arctan\left(\frac{1}{M}\right).$

###### Proof.

We fix $M>0$ and we set $\delta=M(1-r)$, then we have

$\displaystyle(1-r)\int_{\delta}^{2\pi-\delta}\frac{1}{1+r^{2}-2r\cos(\theta)}\,d\theta=$
$\displaystyle(1-r)\int_{-\pi+\delta}^{\pi-\delta}\frac{1}{1+r^{2}+2r\cos(\theta)}\,d\theta$
$\displaystyle=$ $\displaystyle
2(1-r)\int_{0}^{\pi-\delta}\frac{1}{1+r^{2}+2r\cos(\theta)}\,d\theta$
$\displaystyle=$ $\displaystyle
2(1-r)\int_{0}^{\tan(\frac{\pi-\delta}{2})}\frac{}{1+r^{2}+2r\frac{1-t^{2}}{1+t^{2}}}\,\frac{2}{1+t^{2}}dt$
$\displaystyle=$
$\displaystyle\frac{4(1-r)}{(1+r)^{2}}\int_{0}^{\tan(\frac{\pi-\delta}{2})}\frac{1}{1+(\frac{1-r}{1+r})^{2}t^{2}}\,dt$
$\displaystyle=$ $\displaystyle
4(1+r)\int_{0}^{\frac{1-r}{1+r}\tan(\frac{\pi-\delta}{2})}\frac{1}{1+u^{2}}\,du$
$\displaystyle=$ $\displaystyle
4(1+r)\arctan\left(\frac{1-r}{1+r}\tan\left(\frac{\pi-\delta}{2}\right)\right),$

finally the result follows from
$\tan\left(\frac{\pi}{2}-x\right)\sim_{0}\frac{1}{x}$. ∎

Then we are in position to prove that the harmonic extension of a map in
$H^{\frac{1}{2}}(\mathbb{S}^{1},\mathbb{S}^{1})$ is proper, more precisely

######  Lemma A.2.

Let $v\in H^{1/2}(\mathbb{S}^{1},\mathbb{S}^{1})$ and $\tilde{v}$ its harmonic
extension on $\mathbb{D}$. We denote, for $z\in\mathbb{D}$, $I_{z}$ the subarc
of $\mathbb{S}^{1}$ centred at $\frac{z}{|z|}$ of length $2(1-|z|)$, and for
subarc $I\subset\mathbb{S}^{1}$ we set

$v_{I}=\frac{1}{|I|}\int_{I}v\,d\theta.$

Then

(A.2) $\lim_{M\rightarrow+\infty}\lim_{r\rightarrow 1}\sup_{|z|\geq
r}|\tilde{v}(z)-v_{MI_{z}}|=0.$

Moreover for every $\epsilon>0$ there exists $M>0$ and $r<1$ such that if
$|z|\geq r$ there exists $S_{z}\subset MI_{z}$ such that

$|S_{z}|\geq\frac{|MI_{z}|}{2}$

and

$|\tilde{v}(z)-v(x)|\leq\epsilon\text{ for all }x\in S_{z},$

in particular $\tilde{v}$ is proper.

The result and proof are sketch at beginning of section 8 of [4], we give
details for the sake of completeness.

###### Proof.

Let us remind that, $H^{1/2}\subset\mathrm{VMO}\subset\mathrm{BMO}$, see §1.2
[7]444In this reference it is proved on $\mathbb{R}$ but it is also true on
$\mathbb{S}^{1}$ by conformal invariance of all those spaces, see corollary
VI.1.3 [16]. So we have, by Lemma 3 of [7], that, for every $M>0$,

(A.3) $\lim_{|z|\rightarrow
1}\frac{1}{|MI_{z}|}\int_{MI_{z}}|v-v_{MI_{z}}|\,d\theta=0\text{ uniformly in
}z.$

Moreover, $\tilde{v}$ is given by the Poisson formula,

$\tilde{v}(z)=\int_{0}^{2\pi}P_{z}(\theta)v(e^{i\theta})\,d\theta,$

where $P_{z}(\theta)=\frac{1}{2\pi}\frac{1-|z|^{2}}{|z-e^{i\theta}|^{2}}$.
Hence, let $M>0$, we have, using the fact that
$\int_{\mathbb{S}^{1}}P_{z}\,d\theta=1$ and $P_{z}\geq 0$, that

$|\tilde{v}(z)-v_{MI_{z}}|\leq\int_{MI_{z}}P_{z}|v-v_{MI_{z}}|\,d\theta+\int_{(MI_{z})^{c}}P_{z}|v-v_{MI_{z}}|\,d\theta$

Let $\epsilon>0$, thanks to (A.1) and the fact that $|v|=1$ and
$|v_{MI_{z}}|\leq 1$, we have, for $M$ large enough, that

$\lim_{|z|\rightarrow
1}\int_{(I_{MI_{z}})^{c}}P_{z}|v-v_{MI_{z}}|\,d\theta\leq\frac{\varepsilon}{4}$

Once $M$ is fixed, since $v$ is in VMO, we have, thanks to (A.3)), that

$\int_{I_{MI_{z}}}P_{z}|v-v_{MI_{z}}|\,d\theta\leq\frac{2M}{\pi|MI_{z}|}\int_{I_{MI_{z}}}|v-v_{MI_{z}}|\,d\theta\leq\frac{\epsilon}{4},$

which proves the first part of the lemma.

Moreover, $v$ satisfies the John-Nirenberg theorem, see (3’) of [24], there
exist $\beta,C>0$ independent of $v$, such for any arc
$I\subset\mathbb{S}^{1}$ we have

$\frac{1}{|I|}\int_{I}e^{\frac{\beta|v-v_{I}|}{\kappa}}\,d\theta\leq C,$

where

$\kappa=\sup_{I^{\prime}\subset I}\frac{1}{|I|}\int_{I}|v-v_{I}|\,d\theta,$

In particular,

$|\\{x\in MI_{z}\,|\,|v(x)-v_{MI_{z}}|\geq\frac{\epsilon}{2}\\}|\leq
C|MI_{z}|e^{-\frac{\beta\epsilon}{2\kappa}}.$

Hence by (A.3), for $1-|z|$ small enought, there exists $x\in M_{I}z$ such
that

(A.4) $|g_{MI_{z}}-g(x)|\leq\frac{\delta}{2}$

which achieves the proof of the lemma. ∎

##  Appendix B Note on the extended definition of
$H^{\frac{1}{2}}\left(\Gamma\right)$

In this section, we comment and justify the general definition of
$H^{\frac{1}{2}}\left(\Gamma,\mathbb{S}^{1}\right)$ (see definition 2.3).

First, let $\Omega\subset\mathbb{R}^{2}$ be a bounded open set and
$f:\mathbb{D}\rightarrow\Omega$ be a uniformization. Then for any
$u:\Omega\to\mathbb{R}^{n}$, we have

(B.1) $u\circ f\in
H^{1}_{0}\left(\mathbb{D},\mathbb{R}^{n}\right)\Leftrightarrow u\in
H^{1}_{0}\left(\Omega,\mathbb{R}^{n}\right)\hskip 2.84544pt.$

Notice that it is equivalent to
$H^{1}_{0}\left(\mathbb{D},\mathbb{R}^{n}\right)=H^{1}_{0}\left(\mathbb{D},\left|f^{\prime}\right|^{2},\mathbb{R}^{n}\right)$
by pullback of functions with $f$. In fact (B.1) is just an application of the
conformal invariance of the Dirichlet energy and the classical Poincaré
inequality on $\mathbb{D}$ and on $\Omega$:

$\int_{\mathbb{D}}\left|u\circ f\right|^{2}\leq
C_{\mathbb{D}}\int_{\mathbb{D}}\left|\nabla\left(u\circ
f\right)\right|^{2}=C_{\mathbb{D}}\int_{\Omega}\left|\nabla u\right|^{2}\leq
C_{\mathbb{D}}\left\|u\right\|_{H^{1}\left(\Omega\right)}$
$\int_{\Omega}\left|u\right|^{2}\leq C_{\Omega}\int_{\Omega}\left|\nabla
u\right|^{2}=C_{\Omega}\int_{\mathbb{D}}\left|\nabla\left(u\circ
f\right)\right|^{2}\leq C_{\Omega}\left\|u\circ
f\right\|_{H^{1}\left(\mathbb{D}\right)}$

for any $u\in\mathcal{C}^{\infty}_{c}(\Omega)$. Notice that the Poincaré
inequality in $\Omega$ can also be seen as the classical Hardy inequality
since
$\sup_{z\in\mathbb{D}}\left|f^{\prime}(z)\right|^{2}\left(1-\left|z\right|\right)^{2}<+\infty$
by Koebe’s theorem.

Now, we would ideally like to have
$H^{1}\left(\mathbb{D},\left|f^{\prime}(z)\right|^{2}\right)=H^{1}\left(\mathbb{D}\right)$
too in order to give a sense to definition 2.3. Indeed, we would like the
function $g:\Gamma\to\mathbb{R}^{n}$ to be some trace of a
$H^{1}\left(\Omega\right)$ function $u$. Pulling back to $\mathbb{D}$ by $f$,
this means that $g\circ f:\Gamma\to\mathbb{R}^{n}$ is some trace of the
function $u\circ f\in
H^{1}\left(\mathbb{D},\left|f^{\prime}(z)\right|^{2}\right)$. However, $g\circ
f:\Gamma\to\mathbb{R}^{n}$ is well-defined as a
$H^{\frac{1}{2}}\left(\mathbb{S}^{1}\right)$ trace only if it is a trace of a
$H^{1}\left(\mathbb{D}\right)$ function.

Then, we assume that $g\circ f$ admits some extension in
$H^{1}\left(\mathbb{D},\left|f^{\prime}(z)\right|^{2}\right)\cap
H^{1}\left(\mathbb{D}\right)$ since those sets are not a priori equal. We then
a priori ask more restrictions for being in
$H^{\frac{1}{2}}(\Gamma,\mathbb{R}^{n})$ than in definition 2.3:

######  Definition B.1 ($H^{\frac{1}{2}}(\Gamma,\mathbb{R}^{n})$ maps).

Let $n\geq 1$, $\Gamma$ be a quasicircle enclosing a domain $\Omega$, and
$f:\mathbb{D}\rightarrow\Omega$ a uniformization. We say that
$g:\Gamma\rightarrow\mathbb{R}^{n}$ is in
$H^{\frac{1}{2}}(\Gamma,\mathbb{R}^{n})$ if $g\circ f\in
H^{\frac{1}{2}}(\mathbb{S}^{1},\mathbb{R}^{n})$ and if $g\circ f$ admits an
extension $v\in H^{1}(\mathbb{D},\mathbb{R}^{n})$ such that

(B.2)
$\int_{\mathbb{D}}\left|v\right|^{2}(z)\left|f^{\prime}\right|^{2}(z)dz<+\infty$

that is to say $v\circ f^{-1}\in H^{1}\left(\Omega,\mathbb{S}^{1}\right)$.

Once we know that such an extension $v$ exists, then by (B.1), the set of
$H^{1}\left(\mathbb{D}\right)$ extensions of $g\circ f$:
$\\{v\\}+H^{1}_{0}\left(\mathbb{D}\right)$ is equal to the pullback of the
natural set of $H^{1}\left(\Omega\right)$ extensions of $g$: $\\{v\circ
f^{-1}\\}+H^{1}_{0}\left(\Omega\right)$. This makes the definition consistent.
In particular, the harmonic extension $v$ of $g\circ f$ satisfies $v\circ
f^{-1}\in H^{1}\left(\Omega\right)$.

In the special case where $g$ is bounded, then (B.2) is automatically
satisfied since $H^{1}\cap
L^{\infty}\left(\mathbb{D},\left|f^{\prime}(z)\right|^{2}\right)=H^{1}\cap
L^{\infty}\left(\mathbb{D}\right)$ and the harmonic extension $v$ of $g\circ
f$ is also bounded. This shows that definition 2.3 makes sense.

##  Appendix C Wente type estimate

We gives a proof of a Wente type lemma, which is inspired of Theorem A.4 of
[25] (see also Theorem 1.100 [40] and proof of Theorem 1.2 [26]).

######  Lemma C.1.

There exists $C>0$ such that if $f\in L^{1}(\mathbb{D})$ such that

$\sup_{\mathbb{D}}|f(x)(1-|x|)^{2}|<+\infty,$

then, there exists $\phi\in W_{0}^{1,2}(\mathbb{D})\cap
L^{\infty}(\mathbb{D})$ such that

$\Delta\phi=f,$

and

$\|\phi\|_{\infty}+\|\nabla\phi\|_{2}\leq
C(\|f\|_{1}+\|f(1-|x|)^{2}\|_{\infty}).$

###### Proof.

It is classical that, by integration by part, the $L^{\infty}$-estimate
implies the $L^{2}$-estimate. Hence we set

$\phi(x)=\int_{\mathbb{D}}G(x,y)f(y)\,dy,$

where $G$ is the Green function of the Laplacian. Then, we remark that the
singularity of the Green function can be decomposed as a sum of bump functions
with dyadic support, see Theorem A.4 of [25]: there exits $C>0$ such that

$-C\leq G(x,y)-l_{x}(y)\leq C,$

with

$l_{x}(y)=-\sum_{j=1}^{\infty}\Theta\left(\frac{2^{j}(x-y)}{1-|x|}\right),$

where $\Theta\in C^{\infty}_{c}(\mathbb{R},[0,1])$ is a bump function, such as
$\mathrm{supp}(\Theta)\subset[-\frac{1}{8},\frac{1}{8}]$ and $\Theta\equiv 1$
on $[-\frac{1}{16},\frac{1}{16}]$. Then

$|\phi(x)|\leq
C\|f\|_{1}+\sum_{j=1}^{\infty}\int_{\mathbb{D}}\Theta\left(\frac{2^{j}(x-y)}{1-|x|}\right)f(y)\,dy.$

Then we estimate each integral of the right hand

(C.1)
$\begin{split}\int_{\mathbb{D}}\Theta\left(\frac{2^{j}(x-y)}{1-|x|}\right)f(y)\,dy&\leq\int_{B(x,2^{-j-3}(1-|x|))}f(y)\,dy\\\
&\leq\|f(1-|x|)^{2}\|_{\infty}\int_{B(x,2^{-j-3}(1-|x|))}\frac{1}{(1-|y|)^{2}}\,dy\\\
&\leq\frac{\pi}{(2^{j+3}-1)^{2}}\|f(1-|x|)^{2}\|_{\infty}\end{split}$

Finally we just have to sum this inequality to get the result. ∎

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