analysis of the preceding section to give Proposition 4.1 in full generality,
from which Theorem 1.4 follows.

Since the measures $\frac{W(u_{\varepsilon}(x))}{\varepsilon^{2}}\,dx$
converge to an absolutely continuous measure with respect to
$\mathcal{H}^{n-2}\mathbin{\vrule
height=6.88889pt,depth=0.0pt,width=0.55974pt\vrule
height=0.55974pt,depth=0.0pt,width=5.59721pt}P$ (by Theorem 1.1), where the
plane $P=\varmathbb{R}^{n-2}\times\\{0\\}$, the estimate Eq. 5.1 also gives

$\limsup_{\varepsilon\to 0}\int_{B_{1/2}^{n-2}(0)\times
D_{1}^{2}(0)}\frac{2W(u_{\varepsilon})}{\varepsilon^{2}}\leqslant\frac{\pi\omega_{n-2}}{2^{n-2}}(1+\delta).$

In particular, this implies that

$\displaystyle\frac{\pi\omega_{n-2}}{2^{n-2}}(1+\delta)$
$\displaystyle\geqslant\limsup_{\varepsilon\to
0}\int_{\mathcal{G}_{\varepsilon,\delta_{\varepsilon}}\times
D_{1}^{2}}\frac{2W(u_{\varepsilon})}{\varepsilon^{2}}$
$\displaystyle\geqslant\limsup_{\varepsilon\to
0}\left(|\mathcal{G}_{\varepsilon,\delta_{\varepsilon}}|\cdot\inf_{y\in\mathcal{G}_{\varepsilon,\delta_{\varepsilon}}}\int_{\\{y_{\varepsilon}\\}\times
D_{1}^{2}}\frac{2W(u_{\varepsilon})}{\varepsilon^{2}}\right),$

which together with Eq. 3.1 implies the existence of
$y_{\varepsilon}\in\mathcal{G}_{\varepsilon,\delta_{\varepsilon}}$ for which

$\limsup_{\varepsilon\to 0}\int_{\\{y_{\varepsilon}\\}\times
D_{1}^{2}}\frac{2W(u_{\varepsilon})}{\varepsilon^{2}}\leqslant\frac{(\pi\omega_{n-2}/2^{n-2})(1+\delta)}{(\omega_{n-2}/2^{n-2})-(C/K)}.$

We now fix $K$ large enough (e.g.,
$K=K(\delta)=\frac{4C}{\delta}\frac{2^{n-2}}{\omega_{n-2}}$), in such a way
that the previous estimate becomes

(5.2) $\displaystyle\limsup_{\varepsilon\to
0}\int_{\\{y_{\varepsilon}\\}\times
D_{1}^{2}}\frac{2W(u_{\varepsilon})}{\varepsilon^{2}}\leqslant\pi(1+2\delta).$

Applying Propositions 3.1 and 3.2, we deduce that for these $y_{\varepsilon}$
such that (5.2) holds, the set

$U_{\delta,\varepsilon}=\\{x\in
P_{y_{\varepsilon}}^{\perp}:|u_{\varepsilon}(x)|<1-\delta\\}$

is contained in a disjoint collection of disks
$D_{C\varepsilon}(p_{1}^{\varepsilon}),\ldots,D_{C\varepsilon}(p_{k}^{\varepsilon})$
such that the degrees $\kappa_{j}^{\varepsilon}$ of $u_{\varepsilon}$ around
$\partial D_{C\varepsilon}(p_{j}^{\varepsilon})$ satisfy

$\limsup_{\varepsilon\to
0}\frac{\pi}{2}(1-5\delta){\textstyle\sum_{j=1}^{k}}|\kappa_{j}^{\varepsilon}|\leqslant\pi(1+2\delta).$

In particular, since $\kappa_{j}^{\varepsilon}\in\varmathbb{Z}$, taking
$\delta$ small enough it follows that

${\textstyle\sum_{j=1}^{k}}|\kappa_{j}^{\varepsilon}|\leqslant 2$

for $\varepsilon$ sufficiently small.

On the other hand, by Proposition 4.4, we know that

$|\kappa|=|{\textstyle\sum_{j=1}^{k}}\kappa_{j}^{\varepsilon}|=1,$

so we see by parity that the case
${\textstyle\sum_{j=1}^{k}}|\kappa_{j}^{\varepsilon}|\in\\{0,2\\}$ is
impossible, so we must have

${\textstyle\sum_{j=1}^{k}}|\kappa_{j}^{\varepsilon}|=1.$

In other words, up to relabeling
$p_{1}^{\varepsilon},\ldots,p_{k}^{\varepsilon}$ and possibly replacing
$u_{\varepsilon}$ with the conjugate solution $\overline{u}_{\varepsilon}$,
for $\varepsilon$ sufficiently small, we must have

(5.3) $\kappa_{1}^{\varepsilon}=1,\
\kappa_{2}^{\varepsilon}=\cdots=\kappa_{k}^{\varepsilon}=0.$

Finally, combining this with Proposition 3.4 immediately gives the following
conclusion.

######  Lemma 5.1.

Suppose that the hypotheses of Theorem 1.4 and the potential bound Eq. 5.1 are
both satisfied. Then the limiting density $\theta=1$.

By a simple contradiction and scaling argument, we can recast the result in
the following ‘quantitative’ form.

######  Proposition 5.2.

For any $\gamma>0$, there exists $\delta_{2}(\gamma)>0$ such that if
$u_{\varepsilon}$ solves the Ginzburg–Landau equation on a ball $B_{2r}(x)$
(where $x=(y,z)\in\varmathbb{R}^{n-2}\times\varmathbb{R}^{2}$) with
$\varepsilon\leqslant\delta_{2}r$, and satisfies

$(r/2)^{2-n}\int_{B_{r/2}(x)}\frac{2W(u_{\varepsilon})}{\varepsilon^{2}}\leqslant\pi\omega_{n-2}+\delta_{2},$
$\mathcal{V}(u_{\varepsilon})\cap B_{r}(x)\neq\varnothing,$

and (to ensure that all energy concentrates along the $(n-2)$-plane $P+x$)

$r^{2-n}\int_{B_{2r}(x)\setminus
B_{\delta_{2}r}(P+x)}e_{\varepsilon}(u_{\varepsilon})\leqslant\delta_{2}\log(r/\varepsilon),$

then

$\left|\frac{r^{2-n}}{\log(r/\varepsilon)}\int_{B_{r}^{n-2}(y)\times
D_{r}^{2}(z)}e_{\varepsilon}(u_{\varepsilon})-\pi\omega_{n-2}\right|<\gamma.$

##  6\. Solutions concentrating with prescribed density
$\theta\in\\{1\\}\cup[2,\infty)$

In this section, we explain how to use the entire solutions of the
Ginzburg–Landau equations constructed in [11] to prove Theorem 1.10. More
precisely, we prove the following proposition.

######  Proposition 6.1.

For each integer $\kappa\geqslant 2$ and $\tau\in[0,1)$, there exists a family
of solutions
$(u^{\tau}_{\varepsilon})_{\varepsilon\in(0,\varepsilon_{0}(\kappa,\tau))}$ in
the unit $3$-ball $B^{3}_{1}(0)\subset\varmathbb{R}^{3}$ with energy
concentrating along the line $P=\\{0\\}\times\varmathbb{R}$, degree
$\kappa=\deg(u_{\varepsilon},\frac{1}{2}S^{1}\times\\{0\\})$ and limiting
energy measure

$\lim_{\varepsilon\to
0}\frac{e_{\varepsilon}(u_{\varepsilon})}{\pi\lvert\log\varepsilon\rvert}=\theta(\kappa,\tau)\mathcal{H}^{1}\mathbin{\vrule
height=6.88889pt,depth=0.0pt,width=0.55974pt\vrule
height=0.55974pt,depth=0.0pt,width=5.59721pt}P,$

where

$\theta(\kappa,\tau):=\kappa+\kappa(\kappa-1)\frac{\tau}{1+\tau}\in\left[\kappa,\frac{\kappa(\kappa+1)}{2}\right).$

It is straightforward to check that Theorem 1.10 follows from Proposition 6.1,
since we have
$\bigcup_{\kappa=2}^{\infty}[\kappa,\frac{\kappa(\kappa+1)}{2})=[2,\infty)$.
The solutions described in Proposition 6.1 are obtained by rescaling families
of entire solutions with $\kappa$ helical vortex filaments constructed in
[11]. Namely, we rely on the following result. (In what follows, we make the
identifications $\varmathbb{R}^{2}\cong\varmathbb{C}$ and
$S^{1}\cong\varmathbb{R}/2\pi\varmathbb{Z}$.)

######  Theorem 6.2.

[11, Theorem 1] For $\kappa\in\\{2,3,\ldots\\}$ and
$\varepsilon<\varepsilon_{0}(\kappa)$ sufficiently small, there exists a
solution $v_{\varepsilon}:\varmathbb{R}^{2}\times S^{1}\to\varmathbb{R}^{2}$
of the Ginzburg–Landau equations

(6.1) $\varepsilon^{2}\Delta v_{\varepsilon}=DW(v_{\varepsilon})$

satisfying

$\left|v_{\varepsilon}(z,t)-{\textstyle\prod_{j=1}^{\kappa}}w(\varepsilon^{-1}[z-f_{j}^{\varepsilon}(t)])\right|\leqslant\frac{C(\kappa)}{\lvert\log\varepsilon\rvert},$

where $w:\varmathbb{R}^{2}\to\varmathbb{R}^{2}$ is the radially symmetric
degree-one solution constructed in [13], and
$f_{j}^{\varepsilon}:S^{1}\to\varmathbb{R}^{2}$ satisfies

$\lim_{\varepsilon\to
0}\left|\sqrt{\lvert\log\varepsilon\rvert}f_{j}^{\varepsilon}(t)-\sqrt{n-1}e^{it}e^{2i(j-1)\pi/\kappa}\right|=0.$

Moreover, these $v_{\varepsilon}$ have the additional symmetry

(6.2) $v_{\varepsilon}(z,t)=e^{\kappa it}\tilde{v}_{\varepsilon}(e^{-it}z)$

for some map $\tilde{v}_{\varepsilon}:\varmathbb{R}^{2}\to\varmathbb{R}^{2}$.

By looking closely at the construction of these solutions and keeping track of
a few key estimates in [11], we are able to check that the following estimate
holds.

######  Lemma 6.3.

For every $\tau\in[0,1)$, there exists a constant $C(\kappa)<\infty$ such that
the solutions $v_{\varepsilon}:\varmathbb{R}^{2}\times
S^{1}\to\varmathbb{R}^{2}$ from Theorem 6.2 satisfy

(6.3) $\int_{D_{\varepsilon^{-\tau}}\times
S^{1}}e_{\varepsilon}(v_{\varepsilon})\leqslant
C(\kappa)\log(1/\varepsilon^{\tau+1})$

for $\varepsilon<\varepsilon_{0}(\kappa,\tau)$, and, for every
$\delta\in(0,1)$, there exists moreover a constant
$C(\delta,\kappa,\tau)<\infty$ such that

(6.4) $\int_{[D_{\varepsilon^{-\tau}}\setminus
D_{\delta\varepsilon^{-\tau}}]\times
S^{1}}e_{\varepsilon}(v_{\varepsilon})\leqslant C(\delta,\kappa,\tau)$

for $\varepsilon<\varepsilon_{1}(\delta,\kappa,\tau)$.

We postpone the proof of Lemma 6.3 to the end of the section; next, we show
how the results of Theorem 6.2 and Lemma 6.3 can be used to prove Proposition
6.1.

###### Proof of Proposition 6.1.

As in [11], we identify the solutions $v_{\varepsilon}:\varmathbb{R}^{2}\times
S^{1}\to\varmathbb{R}^{2}$ given by Theorem 6.2 with solutions on
$\varmathbb{R}^{3}$ that are $2\pi$-periodic in the third variable. Under this
identification, note that Lemma 6.3 gives

$\displaystyle\int_{B^{3}_{\varepsilon^{-\tau}}}e_{\varepsilon}(v_{\varepsilon})$
$\displaystyle\leqslant\int_{D_{\varepsilon^{-\tau}}\times[-\varepsilon^{-\tau},\varepsilon^{-\tau}]}e_{\varepsilon}(v_{\varepsilon})$
$\displaystyle\leqslant
C\varepsilon^{-\tau}\int_{D_{\varepsilon^{-\tau}}\times
S^{1}}e_{\varepsilon}(v_{\varepsilon})$ $\displaystyle\leqslant
C(\kappa)\varepsilon^{-\tau}\log(\varepsilon^{-\tau-1}),$

and similarly

$\displaystyle\int_{B^{3}_{\varepsilon^{-\tau}}(0)\setminus
B_{\delta\varepsilon^{-\tau}}(P)}e_{\varepsilon}(v_{\varepsilon})$
$\displaystyle\leqslant
C\varepsilon^{-\tau}\int_{[D_{\varepsilon^{-\tau}}\setminus
D_{\delta\varepsilon^{-\tau}}]\times S^{1}}e_{\varepsilon}(v_{\varepsilon})$
$\displaystyle\leqslant C(\delta,\kappa,\tau)\varepsilon^{-\tau}.$

It is then straightforward to see that, for
$\tilde{\varepsilon}=\varepsilon^{1+\tau}$, the rescaled maps

$u_{\tilde{\varepsilon}}^{\tau}:\varmathbb{R}^{3}\to\varmathbb{R}^{2},\quad
u^{\tau}_{\tilde{\varepsilon}}(x):=v_{\varepsilon}(x/\varepsilon^{\tau})$

solve the $\tilde{\varepsilon}$-Ginzburg–Landau equations

$\tilde{\varepsilon}^{2}\Delta
u_{\tilde{\varepsilon}}^{\tau}=DW(u_{\tilde{\varepsilon}}^{\tau})$

on $\varmathbb{R}^{3}$, and satisfy

$\displaystyle\int_{B_{1}^{3}(0)}e_{\tilde{\varepsilon}}(u^{\tau}_{\tilde{\varepsilon}})$
$\displaystyle=\varepsilon^{\tau}\int_{B_{\varepsilon^{-\tau}}^{3}(0)}e_{\varepsilon}(v_{\varepsilon})\leqslant
C(\kappa,\tau)\log(1/\tilde{\varepsilon})$

and, for any $\delta>0$,

$\displaystyle\int_{B_{1}^{3}(0)\setminus
B_{\delta}(P)}e_{\tilde{\varepsilon}}(u_{\tilde{\varepsilon}}^{\tau})$
$\displaystyle=\varepsilon^{\tau}\int_{B^{3}_{\varepsilon^{-\tau}}(0)\setminus
B_{\delta\varepsilon^{-\tau}}(P)}e_{\varepsilon}(v_{\varepsilon})\leqslant
C(\delta,\kappa,\tau).$

In particular, it follows that the maps

$u^{\tau}_{\tilde{\varepsilon}}:B_{1}^{3}(0)\to\varmathbb{R}^{2},\quad\text{for
}\tilde{\varepsilon}\in(0,\varepsilon_{0}(\kappa)^{1+\tau}),$

give a family of solutions to the Ginzburg–Landau equations on $B_{1}^{3}(0)$
(or similarly, any fixed compact subset of $\varmathbb{R}^{3}$) with energy of
order $\log(1/\tilde{\varepsilon})$ concentrating along $P$ as
$\tilde{\varepsilon}\to 0$; i.e. (up to subsequences),

$\lim_{\tilde{\varepsilon}\to
0}\frac{e_{\tilde{\varepsilon}}(u_{\tilde{\varepsilon}}^{\tau})}{\pi\lvert\log\tilde{\varepsilon}\rvert}\,dx=\theta\mathcal{H}^{1}\mathbin{\vrule
height=6.88889pt,depth=0.0pt,width=0.55974pt\vrule
height=0.55974pt,depth=0.0pt,width=5.59721pt}P$

for some $\theta>0$.

To compute $\theta$, we appeal to Proposition 3.4, together with Theorem 6.2.
Fixing a small (but arbitrary) $\delta\in(0,\frac{1}{2})$, consider
$\delta_{\tilde{\varepsilon}}\to 0$ and
$t_{\tilde{\varepsilon}}\in\mathcal{G}_{\tilde{\varepsilon},\delta_{\tilde{\varepsilon}}}\subseteq(-\frac{1}{2},\frac{1}{2})$
a family of $\delta_{\tilde{\varepsilon}}$-good slices for
$u_{\tilde{\varepsilon}}^{\tau}$ as in the preceding sections. By virtue of
the symmetry Eq. 6.2 of $v_{\varepsilon}$, note that we can simply take
$t_{\tilde{\varepsilon}}=0$. Following the analysis of the preceding section,
consider the set

$U_{\delta,\tilde{\varepsilon}}:=\\{z\in\varmathbb{R}^{2}:|u_{\tilde{\varepsilon}}^{\tau}(z,0)|<1-\delta\\}.$

By Theorem 6.2, we see that if $|v_{\varepsilon}(z,0)|<1-\delta$, then

$\left|{\textstyle\prod_{j=1}^{\kappa}}w(\varepsilon^{-1}[z-f_{j}^{\varepsilon}(0)])\right|\leqslant\frac{C(\kappa)}{\lvert\log\varepsilon\rvert}+1-\delta\leqslant
1-\frac{\delta}{2}$

for $\varepsilon$ sufficiently small, and since the model single-vortex
solution $w$ satisfies $|w(z)|\to 1$ as $|z|\to\infty$, it follows that

$z\in\bigcup_{j=1}^{\kappa}B_{C(\delta)\varepsilon}(f_{j}^{\varepsilon}(0)).$

Moreover, note that for $1\leqslant j<l\leqslant\kappa$, Theorem 6.2 gives

$\displaystyle|f_{j}^{\varepsilon}(0)-f_{l}^{\varepsilon}(0)|$
$\displaystyle\geqslant\frac{\sqrt{n-1}}{\sqrt{\lvert\log\varepsilon\rvert}}|e^{2i(j-1)\pi/\kappa}-e^{2i(l-1)\pi/\kappa}|+o(\lvert\log\varepsilon\rvert^{-1/2})$
$\displaystyle\geqslant\frac{c(\kappa)}{\sqrt{\lvert\log\varepsilon\rvert}}$

for $\varepsilon$ sufficiently small and $c(\kappa)>0$, and a similar upper
bound also holds. In particular, it follows that the balls
$B_{C(\delta)\varepsilon}(f_{j}^{\varepsilon}(0))$ are mutually disjoint for
$\varepsilon$ sufficiently small, and it follows from the $C^{0}$ closeness

$\left|v_{\varepsilon}(z,0)-{\textstyle\prod_{j=1}^{k}}w(\varepsilon^{-1}[z-f_{j}^{\varepsilon}(t)])\right|\leqslant\frac{C(\kappa)}{\lvert\log\varepsilon\rvert}$

that

$\deg(v_{\varepsilon},\partial
D_{C(\delta)\varepsilon}(f_{j}^{\varepsilon}(0)))=1.$

In particular, for our rescaled solutions
$u_{\tilde{\varepsilon}}^{\tau}(z,0)=v_{\varepsilon}(z/\varepsilon^{\tau},0),$
writing

$p_{j}^{\tilde{\varepsilon}}:=\varepsilon^{\tau}f_{j}^{\varepsilon}(0),$

it follows that

$\displaystyle U_{\delta,\tilde{\varepsilon}}$
$\displaystyle=\\{(\varepsilon^{\tau}z,0):|v_{\varepsilon}(z,0)|<1-\delta\\}\subseteq\bigcup_{j=1}^{\kappa}B_{C(\delta)\varepsilon^{1+\tau}}(\varepsilon^{\tau}f_{j}^{\varepsilon}(0))=\bigcup_{j=1}^{\kappa}B_{C(\delta)\tilde{\varepsilon}}(p_{j}^{\tilde{\varepsilon}}),$

where the balls
$B_{C(\delta)\tilde{\varepsilon}}(p_{j}^{\tilde{\varepsilon}})$ are mutually
disjoint, $u_{\tilde{\varepsilon}}^{\tau}$ has degree

$\kappa_{j}^{\tilde{\varepsilon}}:=\deg(u_{\tilde{\varepsilon}}^{\tau},\partial
D_{C(\delta)\tilde{\varepsilon}}(\tilde{p}_{j}^{\varepsilon}))=1,$

and

$\frac{c(\delta,\kappa)\tilde{\varepsilon}^{\tau/(1+\tau)}}{\sqrt{\lvert\log\varepsilon\rvert}}=\frac{c(\delta,\kappa)\varepsilon^{\tau}}{\sqrt{\lvert\log\varepsilon\rvert}}\leqslant|\tilde{p}_{j}^{\varepsilon}-\tilde{p}_{l}^{\varepsilon}|\leqslant\frac{C(\delta,\kappa)\varepsilon^{\tau}}{\sqrt{\lvert\log\varepsilon\rvert}}=\frac{C(\delta,\kappa)\tilde{\varepsilon}^{\tau/(1+\tau)}}{\sqrt{\lvert\log\varepsilon\rvert}}$

for $1\leqslant j<l\leqslant\kappa$. Thus, applying Proposition 3.4, we deduce
that

$\displaystyle\theta$ $\displaystyle=\lim_{\tilde{\varepsilon}\to
0}\left({\textstyle\sum_{j=1}^{\kappa}}1+2{\textstyle\sum_{j<l}}1\cdot
1\cdot\frac{\lvert\log|p_{j}^{\tilde{\varepsilon}}-p_{l}^{\tilde{\varepsilon}}|\rvert}{\lvert\log\tilde{\varepsilon}\rvert}\right)$
$\displaystyle=\kappa+\kappa(\kappa-1)\frac{\tau}{1+\tau},$

completing the proof of Proposition 6.1. ∎

It remains now to prove Lemma 6.3, verifying that natural energy growth
conditions hold for the solutions constructed in [11].

###### Proof of Lemma 6.3.

For simplicity, we specialize to the case $\kappa=2$, for which the
construction in [11] is carried out in detail. In this case, the solutions
$v_{\varepsilon}:\varmathbb{R}^{2}\times S^{1}\to\varmathbb{R}^{2}$ of Theorem
6.2 have the form

$v_{\varepsilon}(z,t)=e^{2it}V_{\varepsilon}(e^{-it}z/\varepsilon),$

for a map $V_{\varepsilon}$ of the form

$V_{\varepsilon}(z):=w(z-\tilde{d}_{\varepsilon})w(z+\tilde{d}_{\varepsilon})[\eta\cdot(1+i\psi_{\varepsilon})+(1-\eta)\cdot
e^{i\psi_{\varepsilon}}]$

(see [11, eq. (3.2)]), where $w:\varmathbb{R}^{2}\to\varmathbb{R}^{2}$ is the
radially symmetric solution of

$\Delta v+(1-|v|^{2})v=0$

with degree one constructed in [13],
$\tilde{d}_{\varepsilon}\in\varmathbb{R}^{2}$ are points in the plane with

$|\tilde{d}_{\varepsilon}|\leqslant\frac{C}{\varepsilon\sqrt{\lvert\log\varepsilon\rvert}},$

$\eta$ is a cutoff function of the form

$\eta(z)=\eta_{1}(|z-\tilde{d}_{\varepsilon}|)+\eta_{1}(|z+\tilde{d}_{\varepsilon}|),$

with $\eta_{1}(t)=1$ for $t\leqslant 1$ and $\eta_{1}(t)=0$ for $t\geqslant
2$, and $\psi_{\varepsilon}:\varmathbb{R}^{2}\to\varmathbb{C}$ is an unknown
function whose implicit construction (with estimates) is the content of the
proof of Theorem 6.2.

It follows from [11, Proposition 6.1] (see also [11, p. 18] for the definition
of $\|\psi\|_{*}$) that the real part $\Re(\psi_{\varepsilon})$ of
$\psi_{\varepsilon}$ satisfies

(6.5)
$|\Re(\psi_{\varepsilon})|\leqslant\frac{C}{\lvert\log\varepsilon\rvert}\quad\text{where
}\min\\{|z\pm\tilde{d}_{\varepsilon}|\\}>2$

and, for any fixed $\sigma\in(0,1]$, the imaginary part satisfies111Note that
the first occurrence of $\varepsilon^{\sigma-2}$ in the definition of
$\|\psi_{2}\|_{2,*}$ from [11, p. 18] should read $\varepsilon^{2-\sigma}$.

(6.6) $|\Im(\psi_{\varepsilon})|\leqslant
C(\sigma)(|z-\tilde{d}_{\varepsilon}|^{\sigma-2}+|z+\tilde{d}_{\varepsilon}|^{\sigma-2}+\varepsilon^{2-\sigma})\quad\text{where
}\min\\{|z\pm\tilde{d}_{\varepsilon}|\\}>2.$

In particular, where $\min\\{|z\pm\tilde{d}_{\varepsilon}|\\}>2$, we have

$\displaystyle|1-|V_{\varepsilon}(z)||$
$\displaystyle=|1-|w(z-\tilde{d}_{\varepsilon})||w(z+\tilde{d}_{\varepsilon})|e^{-\Im(\psi_{\varepsilon})}|$
$\displaystyle\leqslant|1-|w(z-\tilde{d}_{\varepsilon})||+|1-|w(z+\tilde{d}_{\varepsilon})||+|1-e^{-\Im(\psi_{\varepsilon})}|$
$\displaystyle\leqslant
C(\sigma)(|z-\tilde{d}_{\varepsilon}|^{\sigma-2}+|z+\tilde{d}_{\varepsilon}|^{\sigma-2}+\varepsilon^{2-\sigma}),$

where we used that the model single-vortex solution $w(z)$ satisfies
$0\leqslant 1-|w(z)|\leqslant\frac{C}{|z|^{2}}$ (cf. [11, Lemma 7.1]). In
particular, scaling back down to the solutions
$v_{\varepsilon}=e^{2it}V_{\varepsilon}(e^{-it}z/\varepsilon)$, it follows
that

$\displaystyle|1-|v_{\varepsilon}(z,t)||$ $\displaystyle\leqslant
C(\sigma)\varepsilon^{2-\sigma}(|z-e^{it}\varepsilon\tilde{d}_{\varepsilon}|^{\sigma-2}+|z+e^{it}\varepsilon\tilde{d}_{\varepsilon}|^{\sigma-2}+1)$

where $|z\pm e^{it}\varepsilon\tilde{d}_{\varepsilon}|>2\varepsilon$. Also,
from the bound Eq. A.2 in the appendix (and a trivial rescaling), we have
$|u(x)|\leqslant 1+C\varepsilon^{2}R^{-2}$ on $B_{R/2}(0)$, for all
$R\geqslant 1$, which implies that $|v_{\varepsilon}|\leqslant 1$ everywhere.

In particular, by the preceding estimates, setting

$\rho_{\varepsilon}(z,t):=\min\\{|z-e^{it}\varepsilon\tilde{d}_{\varepsilon}|,|z+e^{it}\varepsilon\tilde{d}_{\varepsilon}|\\},$

we see that for $\sigma,\tau\in(0,1)$

$\displaystyle\int_{D_{2\varepsilon^{-\tau}}\times
S^{1}}\frac{(1-|v_{\varepsilon}|^{2})^{2}}{\varepsilon^{2}}$
$\displaystyle\leqslant\int_{\\{\rho_{\varepsilon}\leqslant
2\varepsilon\\}}\frac{(1-|v_{\varepsilon}|^{2})^{2}}{\varepsilon^{2}}$
$\displaystyle\quad+\int_{\\{4\varepsilon^{-\tau}\geqslant\rho_{\varepsilon}>2\varepsilon\\}}\frac{(1-|v_{\varepsilon}|^{2})^{2}}{\varepsilon^{2}}$
$\displaystyle\leqslant
C\varepsilon^{2}\cdot\frac{C}{\varepsilon^{2}}+C(\sigma)\int_{\\{4\varepsilon^{-\tau}\geqslant\rho_{\varepsilon}>2\varepsilon\\}}\frac{1}{\varepsilon^{2}}[\varepsilon^{2-\sigma}(\rho_{\varepsilon}^{\sigma-2}+1)]^{2}$
$\displaystyle\leqslant
C+\frac{C(\sigma)}{\varepsilon^{2}}\cdot\varepsilon^{2(2-\sigma)}(\varepsilon^{2\sigma-2}+\varepsilon^{-2\tau})$
$\displaystyle\leqslant C(\sigma)(1+\varepsilon^{2(1-\sigma-\tau)}),$

where we used the coarea formula to bound
$\int_{\\{4\varepsilon^{-\tau}\geqslant\rho_{\varepsilon}>2\varepsilon\\}}\rho_{\varepsilon}^{2(\sigma-2)}\leqslant
C(\sigma)\varepsilon^{2\sigma-2}$ (as each level set
$\\{\rho_{\varepsilon}=r\\}$ has length at most $Cr$); hence, taking
$\sigma=1-\tau$ gives

(6.7) $\int_{D_{2\varepsilon^{-\tau}}\times
S^{1}}\frac{(1-|v_{\varepsilon}|^{2})^{2}}{\varepsilon^{2}}\leqslant C(\tau)$

for all $\tau\in(0,1)$, and thus also for all $\tau\in[0,1)$.

With the bound Eq. 6.7 in place, it follows from Proposition A.4 (after a
suitable rescaling) that

(6.8) $\int_{D_{\varepsilon^{-\tau}}\times
S^{1}}|d|v_{\varepsilon}||^{2}\leqslant C(\tau)$

as well, so to obtain the desired energy estimates for $v_{\varepsilon}$, it
remains to estimate the contribution from

$|dv_{\varepsilon}|^{2}-|d|v_{\varepsilon}||^{2}=|v_{\varepsilon}|^{-2}|jv_{\varepsilon}|^{2},$

recalling that

$jv_{\varepsilon}=v_{\varepsilon}^{*}(r^{2}\,d\theta)=v_{\varepsilon}^{1}\,dv_{\varepsilon}^{2}-v_{\varepsilon}^{2}\,dv_{\varepsilon}^{1}.$

To start, observe that on $\\{\rho_{\varepsilon}\leqslant 4\varepsilon\\}$, we
have

(6.9) $\int_{\\{\rho_{\varepsilon}\leqslant
4\varepsilon\\}}|dv_{\varepsilon}|^{2}\leqslant
C\varepsilon^{2}\cdot\frac{C}{\varepsilon^{2}}\leqslant C,$

so we only need to estimate the energy contribution from the region

$A:=[D_{\varepsilon^{-\tau}}\times S^{1}]\cap\\{\rho_{\varepsilon}\geqslant
4\varepsilon\\}.$

To this end, for $R\in[2\varepsilon,2\varepsilon^{-\tau}]$, consider the
annular regions

$\Omega_{R}:=\\{R<\rho_{\varepsilon}(z,t)<5R\\}$

and

$\Omega_{R}^{\prime}:=\\{2R\leqslant\rho_{\varepsilon}(z,t)\leqslant 4R\\},$

so that, for $\varepsilon<\varepsilon_{0}(\tau)$ sufficiently small (since
$|e^{it}\varepsilon\tilde{d}_{\varepsilon}|\leqslant
C/\sqrt{\lvert\log\varepsilon\rvert}\leqslant\varepsilon^{-\tau}$), we have

(6.10)
$A\subseteq\bigcup_{j=1}^{J_{\varepsilon,\tau}}\Omega_{2^{j}\varepsilon}^{\prime}$

where $J_{\varepsilon,\tau}:=\lceil\log(\varepsilon^{-\tau-1})/\log 2\rceil$
and, for $\varepsilon<\varepsilon_{1}(\delta,\tau)$ (since
$|e^{it}\varepsilon\tilde{d}_{\varepsilon}|\leqslant
C/\sqrt{\lvert\log\varepsilon\rvert}\leqslant\frac{1}{2}\delta\varepsilon^{-\tau}$),

(6.11) $[D_{\varepsilon^{-\tau}}\setminus D_{\delta\varepsilon^{-\tau}}]\times
S^{1}\subseteq\bigcup_{j=I_{\varepsilon,\tau,\delta}}^{J_{\varepsilon,\tau}}\Omega_{2^{j}\varepsilon}^{\prime}$

where
$I_{\varepsilon,\tau,\delta}:=\lfloor\log(\delta\varepsilon^{-\tau-1})/\log(2)\rfloor-3$.
Now, given $R\in[2\varepsilon,2\varepsilon^{-\tau}],$ let $\chi_{R}$ be a
cutoff function such that

$0\leqslant\chi_{R}\in C_{c}^{\infty}(\Omega_{R}),\quad\chi_{R}\equiv 1\text{
on }\Omega_{R}^{\prime},\quad|d\chi_{R}|\leqslant\frac{C}{R}.$

Next, observe that where $\rho_{\varepsilon}\geqslant 2\varepsilon$,
$v_{\varepsilon}$ has the form

$v_{\varepsilon}(z,t)=e^{2it}w(e^{-it}\varepsilon^{-1}z-\tilde{d}_{\varepsilon})w(e^{-it}\varepsilon^{-1}z+\tilde{d}_{\varepsilon})e^{i\psi_{\varepsilon}(e^{-it}\varepsilon^{-1}z)},$

and since the model single-vortex solution $w$ satisfies
$\frac{w(z)}{|w(z)|}=\frac{z}{|z|}$, it follows that

$\frac{v_{\varepsilon}(z,t)}{|v_{\varepsilon}|(z,t)}=\frac{z-e^{it}\varepsilon\tilde{d}_{\varepsilon}}{|z-e^{it}\varepsilon\tilde{d}_{\varepsilon}|}\cdot\frac{z+e^{it}\varepsilon\tilde{d}_{\varepsilon}}{|z+e^{it}\varepsilon\tilde{d}_{\varepsilon}|}\cdot
e^{i\varphi_{\varepsilon}},$

where we set

$\varphi_{\varepsilon}(z,t):=\Re(\psi_{\varepsilon}(e^{-it}\varepsilon^{-1}z)).$

It is straightforward to check that

$\left\lvert d\left(\frac{z\pm e^{it}\varepsilon\tilde{d}_{\varepsilon}}{|z\pm
e^{it}\varepsilon\tilde{d}_{\varepsilon}|}\right)\right\rvert\leqslant\frac{C}{|z\pm
e^{it\varepsilon}\tilde{d}_{\varepsilon}|},$

and as a consequence,

$\displaystyle|j(v_{\varepsilon}/|v_{\varepsilon}|)-d\varphi_{\varepsilon}|$
$\displaystyle=|j(e^{-i\varphi_{\varepsilon}}v_{\varepsilon}/|v_{\varepsilon}|)|\leqslant\frac{C}{\rho_{\varepsilon}}$

where $\rho_{\varepsilon}\geqslant 2\varepsilon$. Moreover, recall that, since
$v_{\varepsilon}$ solves the Ginzburg–Landau equations, we have as always
$d^{*}jv_{\varepsilon}=0$; as a consequence, for any
$R\in[2\varepsilon,2\varepsilon^{-\tau}]$, we see that

$\int\chi_{R}^{2}\langle
jv_{\varepsilon},d\varphi_{\varepsilon}\rangle=-\int\varphi_{\varepsilon}\langle
jv_{\varepsilon},d(\chi_{R}^{2})\rangle,$

and therefore

$\displaystyle\int\chi_{R}^{2}|v_{\varepsilon}|^{-2}|jv_{\varepsilon}|^{2}$
$\displaystyle=\int\chi_{R}^{2}\langle
jv_{\varepsilon},j(v_{\varepsilon}/|v_{\varepsilon}|)\rangle$
$\displaystyle=\int\chi_{R}^{2}\langle
jv_{\varepsilon},j(v_{\varepsilon}/|v_{\varepsilon}|)-d\varphi_{\varepsilon}\rangle+\int\chi_{R}^{2}\langle
jv_{\varepsilon},d\varphi_{\varepsilon}\rangle$
$\displaystyle\leqslant\int\chi_{R}^{2}|jv_{\varepsilon}|\cdot\frac{C}{\rho_{\varepsilon}}-\int\varphi_{\varepsilon}\langle
jv_{\varepsilon},d(\chi_{R}^{2})\rangle$ $\displaystyle\leqslant
C\|\chi_{R}\,jv_{\varepsilon}\|_{L^{2}}\left(\int\rho_{\varepsilon}^{-2}\chi_{R}^{2}+\|\varphi_{\varepsilon}\,d\chi_{R}\|_{L^{\infty}}^{2}\cdot|\Omega_{R}|\right)^{1/2}.$

Now, since $\chi_{R}$ is supported on the set
$\Omega_{R}=\\{R<\rho_{\varepsilon}<5R\\}$, whose area is $\leqslant CR^{2}$,
we see that

$\int\rho_{\varepsilon}^{-2}\chi_{R}^{2}\leqslant\int_{\Omega_{R}}\frac{C}{R^{2}}\leqslant
C.$

Moreover, it follows from Eq. 6.5 that
$|\varphi_{\varepsilon}|\leqslant\frac{C}{\lvert\log\varepsilon\rvert}$ on
$\Omega_{R}$ for $R\geqslant 2\varepsilon$, so that

$\|\varphi_{\varepsilon}\,d\chi_{R}\|_{L^{\infty}}^{2}\cdot|\Omega_{R}|\leqslant\frac{C}{\lvert\log\varepsilon\rvert^{2}}\cdot\frac{C}{R^{2}}\cdot
CR^{2}\leqslant\frac{C}{\lvert\log\varepsilon\rvert^{2}}.$

Finally, since $|v_{\varepsilon}|\leqslant 1$, we have

$\|\chi_{R}\,jv_{\varepsilon}\|_{L^{2}}\leqslant\|\chi_{R}|v_{\varepsilon}|^{-1}\,jv_{\varepsilon}\|_{L^{2}},$

and putting together the preceding computations gives

$\|\chi_{R}|v_{\varepsilon}|^{-1}\,jv_{\varepsilon}\|_{L^{2}}^{2}\leqslant
C\|\chi_{R}|v_{\varepsilon}|^{-1}\,jv_{\varepsilon}\|_{L^{2}},$

and hence

(6.12)
$\int_{\Omega_{R}^{\prime}}|v_{\varepsilon}|^{-2}|jv_{\varepsilon}|^{2}\leqslant\int\chi_{R}^{2}|v_{\varepsilon}|^{-2}|jv_{\varepsilon}|^{2}\leqslant
C.$

Now, applying Eq. 6.10, it follows that

$\displaystyle\int_{D_{\varepsilon^{-\tau}}\times
S^{1}}|v_{\varepsilon}|^{-2}|jv_{\varepsilon}|^{2}$
$\displaystyle=\int_{\\{\rho_{\varepsilon}\leqslant
4\varepsilon\\}}|v_{\varepsilon}|^{-2}|jv_{\varepsilon}|^{2}+\int_{A}|v_{\varepsilon}|^{-2}|jv_{\varepsilon}|^{2}$
$\displaystyle\leqslant
C+{\textstyle\sum_{j=1}^{J_{\varepsilon,\tau}}}\int_{\Omega_{2^{j}\varepsilon}^{\prime}}|v_{\varepsilon}|^{-2}|jv_{\varepsilon}|^{2}$
$\displaystyle\leqslant C+CJ_{\varepsilon,\tau}$ $\displaystyle\leqslant
C\log(1/\varepsilon^{\tau+1}),$

and since we have already shown (in Eq. 6.7 and Eq. 6.8) that

$\int_{D_{\varepsilon^{-\tau}}\times
S^{1}}(e_{\varepsilon}(u_{\varepsilon})-|v_{\varepsilon}|^{-2}|jv_{\varepsilon}|^{2})\leqslant
C(\tau),$

it follows that

$\int_{D_{\varepsilon^{-\tau}}\times
S^{1}}e_{\varepsilon}(u_{\varepsilon})\leqslant
C\log(1/\varepsilon^{\tau+1})+C(\tau)\leqslant C\log(1/\varepsilon^{\tau+1})$

for $\varepsilon<\varepsilon_{0}(\tau)$ sufficiently small, as claimed.

Moreover, for any $\delta\in(0,1)$ and $\varepsilon<\varepsilon_{0}(\delta)$
sufficiently small, it follows from Eq. 6.11 and Eq. 6.12 that

$\displaystyle\int_{[D_{\varepsilon^{-\tau}}\setminus
D_{\delta\varepsilon^{-\tau}}]\times
S^{1}}|v_{\varepsilon}|^{-2}|jv_{\varepsilon}|^{2}$
$\displaystyle\leqslant{\textstyle\sum_{j=I_{\varepsilon,\tau,\delta}}^{J_{\varepsilon,\tau}}}\int_{\Omega^{\prime}_{2^{j}\varepsilon}}|v_{\varepsilon}|^{-2}|jv_{\varepsilon}|^{2}$
$\displaystyle\leqslant{\textstyle\sum_{j=I_{\varepsilon,\tau,\delta}}^{J_{\varepsilon,\tau}}}C$
$\displaystyle=C(\lceil\log(\varepsilon^{-\tau-1})/\log
2\rceil-(\lfloor\log(\delta\varepsilon^{-\tau-1})/\log(2)\rfloor-3))$
$\displaystyle\leqslant C(5-\log(\delta)),$

hence

$\int_{[D_{\varepsilon^{-\tau}}\setminus D_{\delta\varepsilon^{-\tau}}]\times
S^{1}}e_{\varepsilon}(u_{\varepsilon})\leqslant C(\delta,\tau),$

completing the proof of the claim. ∎

## Appendix.

In this appendix we collect some fundamental estimates for maps
$u:B_{1}^{n}(0)\to\varmathbb{C}$ which are critical for the Ginzburg–Landau
energy

$E_{\varepsilon}(u)=\int_{B_{1}}\left(\frac{|du|_{g}^{2}}{2}+\frac{W(u)}{\varepsilon^{2}}\right)d\,\operatorname{vol}_{g},$

with respect to a smooth Riemannian metric $g$, defined on the closure
$\overline{B}_{1}$. Recall that $u$ solves the nonlinear elliptic equation

$\displaystyle\varepsilon^{2}\Delta_{g}u+(1-|u|^{2})u=0.$

In the proof of these results a central ingredient, which also appears in our
arguments, is the following _monotonicity formula_ (see, e.g., [25,
Proposition A.1]).

######  Proposition A.1.

For any $x\in B_{1}^{n}(0)$, denoting by $\mathcal{B}_{s}(x)$ the geodesic
ball with respect to $g$, we have

(A.1)
$\displaystyle\frac{d}{ds}\left(e^{C(g)s^{2}}\frac{E_{\varepsilon}(u;\mathcal{B}_{s}(x))}{s^{n-2}}\right)\geqslant\frac{1}{s^{n-2}}\int_{\partial\mathcal{B}_{s}(x)}|\partial_{\nu}u|_{g}^{2}+\frac{1}{s^{n-1}}\int_{\mathcal{B}_{s}(x)}\frac{2W(u_{\varepsilon})}{\varepsilon^{2}}$

for all $s\in(0,\operatorname{inj}_{g}(x))$, where we omit the volume element
of $g$. In particular,

$\displaystyle s\mapsto
e^{C(g)s^{2}}\frac{E_{\varepsilon}(u;\mathcal{B}_{s}(x))}{s^{n-2}}$

is an increasing function of the radius $s\in(0,\operatorname{inj}_{g}(x))$.

Note that the constant $C(g)\to 1$, when we let $g$ converge smoothly to the
Euclidean metric. We also record some useful pointwise bounds for $u$ and its
differential.

######  Proposition A.2.

Assuming $\varepsilon\leqslant 1$, on the smaller ball $B_{1/2}^{n}(0)$ we
have

(A.2) $\displaystyle|u(x)|\leqslant
1+C(g,n)\varepsilon^{2},\quad|du|_{g}\leqslant\frac{C(g,n)}{\varepsilon}.$

Also, if the energy
$E_{\varepsilon}(u)\leqslant\Lambda\lvert\log\varepsilon\rvert$, then on
$B_{1/2}^{n}(0)$

(A.3)
$\displaystyle|du|_{g}^{2}\leqslant\frac{1-|u|^{2}}{\varepsilon^{2}}+C(g,\Lambda,n).$

###### Proof.

The function $\rho:=|u|$ satisfies

$\displaystyle-\Delta_{g}\rho+\frac{(\rho+1)(\rho-1)\rho}{\varepsilon^{2}}\leqslant
0,$

while it is easy to check that, for any fixed $s\in(\frac{1}{2},1)$,
$b_{\gamma}(x):=1+\gamma\varepsilon^{2}\frac{s^{2}}{(s^{2}-|x|^{2})^{2}}$ is a
supersolution on $B_{s}=B_{s}^{n}(0)$, for
$\gamma\geqslant\overline{\gamma}(g,n)>0$ large enough.

On $B_{s}$, we have $\rho\leqslant b_{\gamma}$ for some least $\gamma\geqslant
0$. However, we cannot have $\gamma\geqslant\overline{\gamma}$, since then the
supersolution $b_{\gamma}$ would touch the subsolution $\rho$ from above (at
an interior point), violating the maximum principle for semilinear equations.

Thus, we must have $\rho\leqslant b_{\overline{\gamma}}$ on $B_{s}$, and
letting $s\to 1$ we get

$|u(x)|\leqslant 1+C(g,n)\frac{\varepsilon^{2}}{(1-|x|^{2})^{2}}$

on $B_{1}$, from which the first half of Eq. A.2 follows. Using also the
equation, it follows that $|u|\leqslant C(g,n)$ and
$|\Delta_{g}u|\leqslant\frac{C(g,n)}{\varepsilon^{2}}$ on $B_{15/16}$, which
easily imply the bound

$|du|_{g}\leqslant\frac{C(g,n)}{\varepsilon}$

on $B_{7/8}$. Indeed, the bounds
$|\tilde{u}|+|\Delta_{\tilde{g}}\tilde{u}|\leqslant C(g,n)$ for
$\tilde{u}(x):=u(\varepsilon x)$ on $B_{15/16\varepsilon}$ (with the rescaled
metric $\tilde{g}$) easily give the desired bound
$|d\tilde{u}|_{\tilde{g}}\leqslant C(g,n)$ on any ball $B_{1/8}(x)\subseteq
B_{7/8\varepsilon}(0)$, hence on $B_{7/8\varepsilon}$ (as
$\varepsilon\leqslant 1$).

It is interesting to observe that, even without assumptions on the energy of
$u$ on $B_{1}$, the previous inequalities give
$E_{\varepsilon}(u;B_{1/2})\leqslant\frac{C(g,n)}{\varepsilon^{2}}$ (which is
sharp, for the trivial unstable solution $u\equiv 0$).

In order to improve on the previous pointwise bound for $|du|=|du|_{g}$, we
observe that

$\displaystyle\Delta_{g}\frac{|du|^{2}}{2}\geqslant\langle
du,d\Delta_{g}u\rangle+\operatorname{Ric}_{g}(du,du)\geqslant-\frac{1-|u|^{2}}{\varepsilon^{2}}|du|^{2}-\|\operatorname{Ric}_{g}\|_{L^{\infty}}|du|^{2}$

by Bochner’s formula, and

$\displaystyle\Delta_{g}\frac{1-|u|^{2}}{2\varepsilon^{2}}=\frac{|u|^{2}}{\varepsilon^{2}}\cdot\frac{1-|u|^{2}}{\varepsilon^{2}}-\frac{|du|^{2}}{\varepsilon^{2}}.$

As a consequence, the difference

$f:=\frac{|du|^{2}}{2}-(1+\varepsilon^{2}\|\operatorname{Ric}_{g}\|_{L^{\infty}})\frac{1-|u|^{2}}{2\varepsilon^{2}}$

satisfies

$\Delta_{g}f\geqslant\frac{2|u|^{2}}{\varepsilon^{2}}f.$

In particular, the positive part $f^{+}$ is subharmonic, and it follows that

$f\leqslant C(g,\Lambda,n)\lvert\log\varepsilon\rvert$

on $B_{7/8}$. Also, by the bound $|du|\leqslant\frac{C(g,n)}{\varepsilon}$ and
Proposition A.5 below, we have

$\displaystyle\int_{B_{3/4}\cap\\{|u|\leqslant\frac{1}{2}\\}}f^{+}\leqslant
C(g,n)\int_{B_{3/4}}\frac{W(u)}{\varepsilon^{2}}\leqslant C(g,\Lambda,n).$

On the other hand, the subequation for $f$ easily implies that

$\displaystyle\int_{B_{7/8}}\varphi^{2}\left[|df^{+}|^{2}+\frac{|u|^{2}}{\varepsilon^{2}}(f^{+})^{2}\right]\leqslant
C\int_{B_{7/8}}(f^{+})^{2}|d\varphi|^{2},$

for any $\varphi\in C^{\infty}_{c}(B_{7/8}^{n})$. In particular, by
Cauchy–Schwarz,

$\displaystyle\int_{B_{3/4}\cap\\{|u|\geqslant\frac{1}{2}\\}}f^{+}\leqslant
C(g,n)\left[\int_{B_{7/8}\cap\\{|u|\geqslant\frac{1}{2}\\}}|u|^{2}(f^{+})^{2}\right]^{1/2}\leqslant
C(g,\Lambda,n)\varepsilon\lvert\log\varepsilon\rvert,$

where we used the bound $0\leqslant f^{+}\leqslant
C(g,\Lambda,n)\lvert\log\varepsilon\rvert$ on $B_{7/8}$. Together with the
previous bound, using again the subharmonicity of $f^{+}$, we arrive at

$\displaystyle f\leqslant C(g,n)\int_{B_{3/4}}f^{+}\leqslant
C(g,\Lambda,n)(1+\varepsilon\lvert\log\varepsilon\rvert)$

on $B_{1/2}$, which gives Eq. A.3. ∎

In the asymptotic analysis, the most fundamental tool is the _clearing-out_
for the vorticity, which we state here for arbitrary metrics (the proof is a
simple localization of the arguments from [24, Section 4.3]).

######  Theorem A.3.

Given $\beta\in(0,1)$, there exist constants $\eta(\beta,n)$ and
$c(\beta,g,n)$ such that, for a geodesic ball $\mathcal{B}_{r}(x)\subseteq
B_{1}^{n}(0)$ with $\varepsilon\leqslant r\leqslant c$, if
$E_{\varepsilon}(u;\mathcal{B}_{r}(x))\leqslant\eta
r^{n-2}\log(r/\varepsilon)$, then $|u(x)|>\beta$.

As we saw in Eq. 2.5, the logarithmic growth of the energy exhibited by
typical solutions $u$ is caused solely by the angular part
$ju=u^{*}(r^{2}\,d\theta)=u^{1}\,du^{2}-u^{2}\,du^{1}$ of the differential.
This fact relies on two inequalities: first of all, we can bound the radial
part $d|u|$ in terms of the potential as follows (see, e.g., the argument from
[7, pp. 329–331], which readily generalizes to arbitrary metrics).

######  Proposition A.4.

On the smaller ball $B_{1/2}=B_{1/2}^{n}(0)$ we have

(A.4) $\displaystyle\int_{B_{1/2}}|d|u||^{2}\leqslant
C(g,n)\int_{B_{1}}\frac{(1-|u|^{2})^{2}}{4\varepsilon^{2}}+C(g,n)\varepsilon^{2},$

provided that $\varepsilon\leqslant 1$.

Also, we have the following sharp bound, which constitutes one of the main
contributions of [7], and allows to deduce the same bound for the previous
integral of $|d|u||^{2}$.

######  Proposition A.5.

On the smaller ball $B_{1/2}=B_{1/2}^{n}(0)$ we have

(A.5)
$\displaystyle\int_{B_{1/2}}\frac{(1-|u|^{2})^{2}}{4\varepsilon^{2}}\leqslant
C(g,n)\frac{E_{\varepsilon}(u;B_{1})}{\lvert\log\varepsilon\rvert}\log\left(2+\frac{E_{\varepsilon}(u;B_{1})}{\lvert\log\varepsilon\rvert}\right),$

provided that $\varepsilon\leqslant c$ and
$E_{\varepsilon}(u;B_{1})\leqslant\varepsilon^{-\alpha_{0}}$, for some
$c=c(g,n)$ and $\alpha_{0}=\alpha_{0}(n)$. In particular, assuming
$E_{\varepsilon}(u;B_{1})\leqslant\Lambda\lvert\log\varepsilon\rvert$, it
follows that

$\displaystyle\int_{B_{1/2}}\frac{(1-|u|^{2})^{2}}{4\varepsilon^{2}}\leqslant
C(g,\Lambda,n)$

for $\varepsilon$ small enough.

The proof relies on a covering argument using Theorem A.3 (see [7, pp.
323–328]222Note that (2.3) in [7, Proposition 2.2] should read
$|u_{\varepsilon}(x)|\leqslant
1+\frac{C\varepsilon^{2}}{\operatorname{dist}(x,\partial\Omega)^{2}}$ (which
follows from the bound Eq. A.2 in the present paper, by scaling) and that the
assumption in [7, Proposition 2.2 and Proposition 2.3] should be
$\operatorname{dist}(x,\partial\Omega)>\varepsilon$.), and adapts to arbitrary
metrics with straightforward modifications, using balls with respect to $g$ in
the statement of [7, Proposition 2.4] (see [12, Thereom 2.8.14] for a proof of
the Besicovitch covering theorem on Riemannian manifolds).

The conclusion then follows from an estimate off the vorticity set
$\\{|u|\leqslant 1-\sigma_{0}\\}$, for some $\sigma_{0}$ small enough (see [7,
Theorem 2.1]333We point out the following misprints: in (A.5),
$a=\frac{(1-\theta_{\varepsilon})(2-\theta_{\varepsilon})}{\varepsilon^{2}}\geqslant\frac{1}{2\varepsilon^{2}}$
(we assume $|u_{\varepsilon}|\geqslant\frac{1}{2}$); in equations
(A.11)–(A.12) some signs are wrong, but this does not affect the argument;
most importantly, in (A.21) the right-hand side is just
$C\|e_{\varepsilon}(u_{\varepsilon})\|_{L^{1}(B_{1})}^{1/2}$ but, assuming
(without loss of generality) $\frac{q}{q-2}\geqslant 2$, the last estimate on
p. 347 still implies (A.23) with
$\beta_{q}=(2-\alpha_{0})\frac{q-2}{q}\in(0,1)$, as well as (A.25) with the
same $\beta_{q}$ (by (A.20) with $q=2$).).

On an unrelated note, we also record the following useful Lorentz estimate for
a Riesz potential, which is used in the proof of Lemma 3.3. Recall that, for a
function $f:\varmathbb{R}^{m}\to\varmathbb{R}$, its $L^{2,\infty}$-quasinorm
is defined as

$\displaystyle\|f\|_{L^{2,\infty}(\varmathbb{R}^{m})}=\sup_{\lambda>0}\lambda|\\{|f|>\lambda\\}|^{1/2}.$

######  Proposition A.6.

If $f,g:\varmathbb{R}^{n}\to\varmathbb{R}$ satisfy

$\displaystyle|f|\leqslant\frac{1}{|x|^{n-1}}*|g|,$

then for any $y\in\varmathbb{R}^{n-2}$ we have

(A.6) $\displaystyle\|f(y,\cdot)\|_{L^{2,\infty}(\varmathbb{R}^{2})}\leqslant
C(n)\sup_{r>0}\frac{1}{r^{n-2}}\int_{B_{r}^{n-2}(y)\times\varmathbb{R}^{2}}|g|.$

Thus, the exponent $\frac{n}{n-1}$ in the classical Sobolev bound
$\|f\|_{L^{n/(n-1),\infty}}\leqslant C(n)\|g\|_{L^{1}}$ can be improved to $2$
(the exponent that we have on the plane), on a slice
$\\{y\\}\times\varmathbb{R}^{2}$, provided that we control the maximal
function on the right-hand side of Eq. A.6.

The proof is presented in [18, Lemma A.2] when $n=3$, but it is
straightforward to adapt it to the case of general $n$.

Finally, we briefly show how one can obtain precise asymptotics for the
(local) Green function of $\Delta_{H}$, the Hodge Laplacian on $k$-forms, even
when the metric is not Euclidean. Let $U\subset\varmathbb{R}^{n}$ be a bounded
smooth domain ($n\geqslant 3$), together with a smooth metric $g$ on
$\overline{U}$. Let us fix an orthonormal frame $(\omega_{i})_{i\in I}$ for
the bundle of $k$-forms on $\overline{U}$.

######  Proposition A.7.

Given a compact subset $K\subset U$, there exists
$G_{i,p}\in\Omega^{k}(U\setminus\\{p\\})$ for every $p\in K$, satisfying

$\Delta_{H}G_{i,p}=\delta_{p}\cdot\omega_{i}(p)$

on $U$, in the distributional sense, and such that the difference

$H_{i,p}(q):=G_{i,p}(q)-\overline{G}(\operatorname{dist}(p,q))\,\omega_{i,p}(q)$

obeys the bounds

$|H_{i,p}(q)|\leqslant C\operatorname{dist}(p,q)^{3-n},\quad|\nabla
H_{i,p}(q)|\leqslant C\operatorname{dist}(p,q)^{2-n}$

for $q\in U$, for some constant $C=C(g,K,U)$, where
$\overline{G}(r):=\frac{1}{n(n-2)\omega_{n}r^{n-2}}$ is the standard Green
function on $\varmathbb{R}^{n}$ and $\operatorname{dist}(p,q)$ is the geodesic
distance induced by $g$ (the constant $C\to 0$ when $g$ converges to the
Euclidean metric in the smooth topology).

It is clear from the proof that $G_{i,p}(q)$ and $\nabla G_{i,p}(q)$ depend
continuously on the couple $(p,q)$, away from the diagonal $\\{p=q\\}$. With
this proposition in hand, we can then easily invert the Hodge Laplacian
(locally): given $\eta\in\Omega^{k}(\overline{U})$, the convolution

$\beta(q):={\textstyle\sum_{i\in
I}}\int_{K}G_{i,p}(q)\langle\eta(p),\omega_{i}(p)\rangle\,d\operatorname{vol}_{g}(p)$

then satisfies $\Delta_{H}\beta=\eta$ on the interior of $K$, and the previous
bounds for $H_{i,p}$ imply that $\beta$ resembles the usual convolution with
the Euclidean Green function (at small scales, or when $g$ is almost flat).

###### Proof.

For any (smooth) differential form $\omega\in\Omega^{k}(\overline{U})$ we can
find a unique $\alpha\in\Omega^{k}(\overline{U})$ such that
$\Delta_{H}\alpha=\omega$, with each component of $\alpha$ vanishing at
$\partial U$. Such $\alpha$ can be obtained by minimizing the energy

$\alpha\mapsto\int_{U}\left(\frac{|d\alpha|^{2}}{2}+\frac{|d^{*}\alpha|^{2}}{2}-\langle\alpha,\omega\rangle\right)$

in the space $W^{1,2}_{0}(U,\Lambda^{k}\varmathbb{R}^{n})$ (note that
$\|\alpha\|_{W^{1,2}}\leqslant
C(g,U)(\|d\alpha\|_{L^{2}}+\|d^{*}\alpha\|_{L^{2}})$ for $\alpha$ in this
space, by [15, Theorem 4.8] and a simple compactness and contradiction
argument). We have $\|\alpha\|_{W^{1,2}(U)}\leqslant
C(g,U)\|\omega\|_{L^{2}(U)}$ and, by standard elliptic regularity for systems,

(A.7) $\displaystyle\|\alpha\|_{L^{s}(U)}\leqslant
C(g,s,U)\|\omega\|_{L^{t}(U)}$

for all $s,t\in(1,\infty)$ such that $\frac{1}{s}>\frac{1}{t}-\frac{2}{n}$.

Fix a cutoff function $\chi\in C^{\infty}_{c}(U)$ with $\chi\equiv 1$ near
$K$, and let

$\tilde{G}_{i,p}(q):=\chi(q)\overline{G}(\operatorname{dist}(p,q))\,\omega_{i,p}(q)$

for any fixed $p\in K$, where $\omega_{i,p}\in\Omega^{k}(\overline{U})$ is
such that $\omega_{i,p}(p)=\omega_{i}(p)$ and $\nabla\omega_{i,p}(p)=0$. Using
normal coordinates centered at $p$, it is easy to check that

$|\Delta_{H}\tilde{G}_{i,p}|\leqslant C(g,K,U)\operatorname{dist}(p,q)^{2-n};$

hence, $\Delta_{H}\tilde{G}_{i,p}$ coincides with a $k$-form $\varphi_{i,p}\in
L^{t}(U)$ on $U\setminus\\{p\\}$, where $t\in(1,\frac{n}{n-2})$.

On the other hand, an integration by parts shows that

$\Delta_{H}\tilde{G}_{i,p}=\delta_{p}\cdot\omega_{i,p}(p)+\varphi_{i,p}=\delta_{p}\cdot\omega_{i}(p)+\varphi_{i,p}$

on $U$, in the distributional sense. As explained above, by approximating
$\varphi_{i,p}$ with smooth $k$-forms, we can then find $\alpha=\alpha_{i,p}$
such that $\Delta_{H}\alpha_{i,p}=\varphi_{i,p}$ and Eq. A.7 holds (with
$\omega:=\varphi_{i,p}$). To conclude the proof, we show that
$|\nabla\alpha(q)|\leqslant C\operatorname{dist}(p,q)^{2-n}$ for some
$C=C(g,K,U)$; the conclusion will follow by taking
$G_{i,p}:=\tilde{G}_{i,p}-\alpha_{i,p}$.

But indeed, considering the rescaled $k$-form $\alpha_{r}(x):=\alpha(p+rx)$,
we see that

$\|\Delta_{H}\alpha_{r}\|_{L^{\infty}(A)}\leqslant
Cr^{2}\|\varphi_{i,p}\|_{L^{\infty}(B_{r}\setminus B_{r/2})}\leqslant
Cr^{4-n},\quad\|\alpha_{r}\|_{L^{s}(A)}\leqslant C(s)r^{-n/s}$

whenever $\frac{1}{s}>1-\frac{4}{n}$, for the annular region
$A:=B_{1}\setminus B_{1/2}$, provided that $r$ is small enough (with $C$
depending also on $g,K,U$). By standard elliptic regularity, we then obtain
$|\nabla\alpha_{r}|\leqslant C(s)(r^{4-n}+r^{-n/s})$ on $A$, which gives

$|\nabla\alpha(q)|\leqslant
C(s)(\operatorname{dist}(p,q)^{3-n}+\operatorname{dist}(p,q)^{-1-n/s}).$

Taking $s$ sufficiently close to $\frac{n}{n-4}$ (if $n>4$, or to $\infty$ if
$n=4$) gives the claim for $n\geqslant 4$; when $n=3$, from
$\Delta_{H}\alpha=\varphi_{i,p}$ we can immediately conclude that
$|\alpha|\leqslant C$, and we can take $s:=\infty$ in the previous bound to
conclude that $|\nabla\alpha(q)|\leqslant C\operatorname{dist}(p,q)^{-1}$. ∎

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