Let $C$ be the constant in the excess decay statement, fix $\rho$ such that
$C\rho^{2}\leq\rho$ and fix $\tau_{0}$ and $\varepsilon_{0}$ accordingly.
Letting $r_{k}:=\rho^{k}$, the first inequality of Eq. 9.8 gives that

$\|P_{k+1}-P_{k}\|\leq C\sqrt{\bm{\mathrm{E}}_{1}(x,r_{k})}.$

Iterating we get that

$\|P_{\ell}-P_{k}\|\leq
C(n)\sum_{j=k}^{\ell-1}\sqrt{\bm{\mathrm{E}}_{1}(x,r_{j})}\leq
C(n)\sqrt{\bm{\mathrm{E}}_{1}(x,r_{k})}(1+\rho^{1/2}+\rho+\dots)\leq
C\sqrt{\bm{\mathrm{E}}_{1}(x,r_{k})}$

as long as $\bm{\mathrm{E}}_{1}(x,r_{j})>\frac{\varepsilon^{2}}{r_{j}^{2}}$
for $j=0,\dots,\ell-1$ and $r_{\ell}\geq M\varepsilon=:\overline{r}$ where $M$
is a large constant that we will fix at the end. Hence, if we call
$r_{k_{1}}>\dots>r_{k_{N}}\geq\overline{r}$ the possible radii where
$\bm{\mathrm{E}}_{1}(x,r_{k_{i}})\leq\frac{\varepsilon^{2}}{r_{k_{i}}^{2}}$,
we deduce that

$\displaystyle\|P_{\ell}-P_{0}\|$ $\displaystyle\leq
C\max\\{\sqrt{\bm{\mathrm{E}}_{1}(x,r_{0})},\sqrt{\bm{\mathrm{E}}_{1}(x,r_{k_{1}})},\dots,\sqrt{\bm{\mathrm{E}}_{1}(x,r_{k_{N}})}\\}$
$\displaystyle\leq
C\left[\sqrt{\bm{\mathrm{E}}_{1}(x,r_{0})}+\frac{\varepsilon}{\overline{r}}\right]$
$\displaystyle\leq
C\left[\sqrt{\bm{\mathrm{E}}_{1}(x,r_{0})}+\frac{1}{M}\right].$

Also, $P_{0}$ can be assumed arbitrarily close to $\mathbb{R}^{n-2}$ by a
simple compactness argument (similar to the proof of Proposition 5.3). Since
$\sqrt{\bm{\mathrm{E}}_{1}(x,r_{0})}$ and $1/M$ can be taken arbitrarily
small, the claim follows. ∎

The same proof gives the following.

######  Proposition 9.4.

For any $x\in Z\cap B_{3/4}^{n}$ and $r\in[C(n)\varepsilon,\frac{1}{8}]$, we
have

$\bm{\mathrm{E}}_{1}(u,\nabla,B_{r}(x),\mathbb{R}^{n-2})\leq
C(n)\bm{\mathrm{E}}_{1}(u,\nabla,B_{1}(0),\mathbb{R}^{n-2})+C(n)\frac{\varepsilon^{2}}{r^{2}}.$

We now prove Corollary 1.6.

###### Proof of Corollary 1.6.

We have already seen in Proposition 5.3 that the energy on $B_{R}$ is
asymptotic to $2\pi R^{n-2}$. We can then apply Proposition 9.3: for any
$\gamma>0$ we have

$\|P_{S(0,R)}-P_{S(0,R^{\prime})}\|\leq\gamma$

for $R<R^{\prime}$ large enough (we use Proposition 9.3 after scaling the
picture down by a factor $(R^{\prime})^{-1}$). We deduce that the limit

$\lim_{R\to\infty}S(0,R)$

exists. ∎

######  Proposition 9.5.

Up to a rotation, the vorticity set $\tilde{Z}:=Z\cap[B_{1/2}^{2}\times
B_{1/2}^{n-2}]$ is included in a $C(n,\gamma)\varepsilon$-neighborhood of the
graph of a $C^{1}$ map

$f:B_{1/2}^{n-2}\to B_{\gamma}^{2}$

with $\operatorname{Lip}(f)\leq\gamma$, if we assume that $\tau_{0}$ and
$\varepsilon_{0}$ are small enough (depending on $n,\gamma$).

###### Proof.

Indeed, as seen in the proof of Proposition 5.3, for $\varepsilon$ small
enough we have $u(\cdot,z)\neq 0$ on $\partial B_{1/2}^{2}$, for all $z\in
B_{1/2}^{n-2}$, and the degree of $(u/|u|)(\cdot,z)$ is $\pm 1$ on this
circle. Hence, each slice $B_{1/2}^{2}\times\\{z\\}$ intersects the zero set.

Moreover, using Lemma 5.4 on $B_{1}(0)$, we see that $\tilde{Z}\subseteq
B_{\gamma}^{2}\times B_{1/2}^{n-2}$. Also, Lemma 5.4 implies that for all
$x\in Z\cap B_{3/4}^{n}$ and $r\in[C(n,\gamma)\varepsilon,\frac{1}{8}]$ we
have

(9.10) $\displaystyle Z\cap B_{r}(x)\subseteq B_{\gamma r}(x+S(x,r)),$

where $B_{\gamma r}(x+S(x,r))$ is the $\gamma r$-neighborhood of the affine
plane $x+S(x,r)$. We now take a collection of points $\\{z_{k}\\}\subset
B_{1/2}^{n-2}$ with pairwise distance at least $C(n,\gamma)\varepsilon$ and
$B_{1/2}^{n-2}\subseteq\bigcup_{k}B_{5C(n,\gamma)\varepsilon}^{n-2}(z_{k})$.
For each $k$, we fix a point $x_{k}=(y_{k},z_{k})\in\tilde{Z}$. We then see
that

$|y_{k}-y_{j}|\leq C\gamma|x_{k}-x_{j}|,$

thanks to the previous observation applied with $r:=2|x_{k}-x_{j}|$ and the
fact that $S(x,r)$ is $\gamma$-close to $\mathbb{R}^{n-2}$ (for
$|x_{k}-x_{j}|>\frac{1}{16}$, this follows just from Lemma 5.4). Hence, the
assignment $z_{k}\mapsto y_{k}$ defines a $C(n)\gamma$-Lipschitz function,
which we can extend to a $C(n)\gamma$-Lipschitz function $f:B_{1/2}^{n-2}\to
B_{\gamma}^{2}$. It is easy to check that (a regularization of) $f$ satisfies
the desired conclusion, completing the proof. ∎

We are now in position to prove 1.7.

###### Proof of 1.7.

Let $\eta>0$ small such that Proposition 6.4 applies. We first remark that the
previous points $x_{k}$ can be taken such that $u(x_{k})=0$ and
$z_{k}\in\mathcal{G}^{\eta}$. Indeed, by Proposition 9.4, we have

$\bm{\mathrm{E}}_{1}(u,\nabla,B_{r}(x),\mathbb{R}^{n-2})\leq c(n)\eta^{2}$

for all points $x\in Z\cap B_{3/4}^{n}$ and radii
$r\in[C(n)\varepsilon,\frac{1}{8}]$ (by taking $\varepsilon_{0},\tau_{0}$
suitably small). We can apply this with $r:=M\varepsilon$; by Proposition 9.5
and exponential decay of energy away from $Z$, we have

$r^{2-n}\int_{B_{r/2}^{n-2}(z)}(\bm{\mathrm{E}}_{1})_{z}\leq
c(n)\eta^{2}+e^{-K/M}\leq 2c(n)\eta^{2}$

once we take $M=C(n)$ large enough, for any $z\in B_{1/2}^{n-2}$. Once we take
$c(n)$ small enough, by the weak $L^{1}$ bound we can then find

$z^{\prime}\in B_{r/2}^{n-2}(z)\cap\mathcal{G}^{\eta}$

(where we use slices of radius $\frac{1}{2}$ in the definition of
$\mathcal{G}^{\eta}$), showing the claim.

As a consequence of Lemma A.1, we deduce that

$|h(z_{k})-y_{k}|=|h(z_{k})-h_{0}(z_{k})|\leq\varepsilon.$

We immediately deduce that $\tilde{Z}$ is included in a
$C(n)\varepsilon$-neighborhood of the graph of $h$, which is the only
consequence of the claim which we will use in the sequel.

Now let $\overline{\rho}:=\max\\{M\varepsilon,\varepsilon^{1/(1+\alpha)}\\}$
(with $M$ as in Eq. 9.7) and consider _another_ finite collection of points
$\\{z_{k}\\}\subset B_{1/2}^{n-2}$ such that the balls
$B_{\overline{\rho}}^{n-2}(z_{k})$ are disjoint and the dilated balls
$B_{4{\overline{\rho}}}^{n-2}(z_{k})$ cover $B_{1/2}^{n-2}$. Let
$x_{k}=(y_{k},z_{k})$ be a point in $\tilde{Z}$ for each $k$.

On $B_{10\overline{\rho}}^{n}(x_{k})$ we consider the Lipschitz approximation
$h_{k}$ built with respect to the rotated picture, obtained as a graph over
$S_{k}:=x_{k}+S(x_{k},10{\overline{\rho}})$. When viewed as a graph over
$\mathbb{R}^{n-2}$, it becomes a function $\tilde{h}_{k}$ defined on the
slightly smaller ball $B_{5{\overline{\rho}}}^{n-2}(z_{k})$.

By a scaled version of Proposition 6.4, we have

$\int_{B_{30{\overline{\rho}}/4}^{n}(x_{k})\cap S_{k}}|dh_{k}|^{2}\leq
C{\overline{\rho}}^{n-2}\bm{\mathrm{E}}_{1}(u,\nabla,B_{10{\overline{\rho}}}^{n}(x_{k}),S(x_{k},10{\overline{\rho}}))\leq
C{\overline{\rho}}^{n-2+2\alpha}.$

In particular, by Poincaré,

$\int_{B_{30{\overline{\rho}}/4}^{n}(x_{k})\cap S_{k}}|h_{k}-(h_{k})|^{2}\leq
C{\overline{\rho}}^{n+2\alpha}.$

Now, as in Eq. 8.7, we observe that

$|\tilde{h}_{k}(z)-h_{k}(z)-A_{k}(z)|\leq
C{\overline{\rho}}\sqrt{\bm{\mathrm{E}}_{1}(u,\nabla,B_{10{\overline{\rho}}}^{n}(x_{k}),S(x_{k},10{\overline{\rho}}))}\leq
C{\overline{\rho}}^{1+\alpha}$

for a suitable affine function $A_{k}$ (where, with abuse of notation,
$h_{k}(z)$ means $h_{k}$ composed with the isometry $\mathbb{R}^{n-2}\to
S_{k}$). Combining these two bounds, we get

$\int_{B_{5{\overline{\rho}}}^{n-2}(z_{k})}|\tilde{h}_{k}-A_{k}^{\prime}|^{2}\leq
C{\overline{\rho}}^{n+2\alpha}$

for another affine function $A_{k}^{\prime}$.

We now take a partition of unity $\varphi_{k}$ subordinated to the cover
$\\{B_{4\overline{\rho}}^{n-2}(z_{k})\\}$ and we let

$f:=\sum_{k}\varphi_{k}\tilde{h}_{k}.$

Since the zero set is within a $C\varepsilon$-neighborhood of the graph of
$\tilde{h}_{k}$ (on the set $B_{1/2}^{2}\times
B_{5{\overline{\rho}}}^{n-2}(z_{k})$), we deduce that

$|\tilde{h}_{k}-\tilde{h}_{k^{\prime}}|\leq C\varepsilon$

on $B_{5{\overline{\rho}}}^{n-2}(z_{k})\cap
B_{5{\overline{\rho}}}^{n-2}(z_{k^{\prime}})$. Thus, we also have

$|A_{k}-A_{k^{\prime}}|\leq C\varepsilon+C{\overline{\rho}}^{1+\alpha}$

whenever $B_{4{\overline{\rho}}}^{n-2}(z_{k})\cap
B_{4{\overline{\rho}}}^{n-2}(z_{k^{\prime}})\neq\emptyset$. Since
$\varepsilon\leq\overline{\rho}^{1+\alpha}$, this allows us to conclude that

$\int_{B_{2{\overline{\rho}}}^{n-2}(z)}|f-A_{z}|^{2}\leq
C{\overline{\rho}}^{n+2\alpha}$

for any $z\in B_{1/2}^{n-2}$, for a suitable affine function $A_{z}$ depending
on $z$.

Thus, taking a standard mollifier $\chi_{\overline{\rho}}$, setting

$g:=\chi_{\overline{\rho}}*f$

and using the previous bound, we deduce that

$|dg-dA_{z}|\leq C\overline{\rho}^{\alpha}\quad\text{on
}B_{\overline{\rho}}^{n-2}(z),$

and in fact

$[dg]_{C^{0,\alpha}(B_{\overline{\rho}}^{n-2}(z))}\leq C.$

Finally, recalling that $dA_{k}$ is the slope of the plane
$P_{S(x_{k},10\overline{\rho})}$, we also have

$|dA_{k}-dA_{k^{\prime}}|\leq C|z_{k}-z_{k^{\prime}}|^{\alpha}$

by the Hölder continuity Eq. 9.9, while

$|dA_{z}-dA_{k}|\leq C\overline{\rho}^{\alpha}\quad\text{for $z\in
B_{4\overline{\rho}}^{n-2}(z_{k})$.}$

From these bounds, we easily deduce that

$|dg(z)-dg(z^{\prime})|\leq C|z-z^{\prime}|^{\alpha}\quad\text{for
$|z-z^{\prime}|\geq\overline{\rho}$},$

completing the proof of the $C^{1,\alpha}$ regularity of $g$. Since $|g-f|\leq
C\overline{\rho}$, it follows that the vorticity set is included in a
$C\overline{\rho}$-neighborhood of the graph of $g$.

It is clear from the proof that we can actually make $[dg]_{C^{0,\alpha}}$
arbitrarily small, up to decreasing $\tau_{0}$ and $\varepsilon_{0}$. ∎

##  10\. Constructing competitors for local minimizers: a good gauge

In this section we prepare the ground to construct competitors for minimizers
and to show that the full excess decays as long as it is above
$\varepsilon^{\beta}$, for any $\beta>0$, giving a proof of 1.10. To
investigate minimizers, we construct competitors with the same boundary
conditions and compare the energies to show that the excess $\bm{\mathrm{E}}$
is effectively approximated by the Dirichlet energy of the harmonic
approximation.

To do this, we need to construct competitors modeled on graphs in the interior
and then glue them to the boundary condition, while controlling the error
terms. We pullback the $\varepsilon$-rescaled degree-one solution along the
graph of the Lipschitz approximation, as obtained in Proposition 6.4. Then we
gauge fix in balls of size $\varepsilon|\log\varepsilon|$ and, using the
estimates at that scale, we define a global gauge by a partition of unity. In
this gauge we can interpolate between the intial pair and the new one with
good estimates.

###  10.1. The pullback pair

Here we introduce the pullback pair $(u_{f},\nabla_{f})$ whose zero set is
prescribed to be the graph of a Lipschitz function $f:B^{n-2}\rightarrow
B^{2}_{1}$. We prove that the excess of these pairs are well approximated by
the Dirichlet energy of $f$, as in the following proposition.

######  Proposition 10.1 (Constructing the pullback pair).

There exist $\eta_{0}(n),\varepsilon_{0}(n)>0$ small enough with the following
property. Given any $\varepsilon\leq\varepsilon_{0}$ and a Lipschitz function
$f:B^{n-2}_{1}\rightarrow B^{2}_{1/2}$ with $Lip(f)=\eta\leq\eta_{0}$, there
exists a pair $(u_{f},\nabla_{f})$ obeying the following estimate:

$\displaystyle\frac{1}{2\pi}\int_{B^{2}_{1}\times
B_{1}^{n-2}}e_{\varepsilon}(u_{f},\nabla_{f})=|B^{n-2}_{1}|+(1+O(\eta^{2}))\int_{B^{n-2}_{1}}\frac{|df|^{2}}{2}+O(e^{-{K}/{\varepsilon}}),$

with implicit constants $C(n)$. Moreover, we have that

$\displaystyle u_{f}^{-1}(0)=\operatorname{graph}(f).$

###### Proof.

To construct $(u_{f},\nabla_{f})$ we pull back the planar degree-one solution
of Taubes [51], via the map $Q_{\varepsilon}:B^{2}_{1}\times
B^{n-2}_{1}\rightarrow\mathbb{R}^{2}$ given by

$\displaystyle
Q_{\varepsilon}(x)=\frac{(x_{1},x_{2})-f(x_{3},\dots,x_{n})}{\varepsilon}.$

Then we define $(u_{f},\nabla_{f})$ by

(10.1)
$\displaystyle(u_{f},\nabla_{f}):=Q_{\varepsilon}^{*}(u_{0},\nabla_{0}),$

where $(u_{0},\nabla_{0})$ is the degree-one solution in [51] with
$u_{0}(0)=0$ (unique up to change of gauge). First, we note that, since
$dQ_{\varepsilon}(x)[e_{k}]=-\partial_{e_{k}}f_{1}(x_{3},\dots,x_{n})e_{1}-\partial_{e_{k}}f_{2}(x_{3},\dots,x_{n})e_{2}$,
we have the following identities for $k=3,\dots,n$:

(10.2) $\displaystyle\begin{aligned}
|(\nabla_{f})_{k}u_{f}|^{2}(x)&=\varepsilon^{-2}|\partial_{e_{k}}f_{1}(\nabla_{0})_{e_{1}}u_{0}+\partial_{e_{k}}f_{2}(\nabla_{0})_{e_{2}}u_{0}|^{2}\left(Q_{\varepsilon}(x)\right)\\\
&=\varepsilon^{-2}\frac{|\partial_{k}f|^{2}}{2}|\nabla_{0}u_{0}|^{2}\left(Q_{\varepsilon}(x)\right),\end{aligned}$

where we omitted the argument of $f$ and we used the fact that
$(\nabla_{0})_{e_{2}}u_{0}=i(\nabla_{0})_{e_{1}}u_{0}$ for solutions of the
vortex equations Eq. 4.13. We also have

(10.3)
$\displaystyle|(\nabla_{f})_{e_{1}}u_{f}|^{2}(x)+|(\nabla_{f})_{e_{2}}u_{f}|^{2}(x)=\varepsilon^{-2}|\nabla_{0}u_{0}|^{2}\left(Q_{\varepsilon}(x)\right).$

We also compute for the curvature term
$-i\omega_{f}:=F_{\nabla_{f}}=F_{Q_{\varepsilon}^{*}(\nabla_{0})}=Q_{\varepsilon}^{*}(F_{\nabla_{0}})$
that

(10.4)
$\displaystyle\sum_{j=1,2}\varepsilon^{2}\omega_{f}(e_{j},e_{k})^{2}(x)=\varepsilon^{-2}\omega_{0}(e_{1},e_{2})^{2}\left(Q_{\varepsilon}(x)\right)|\partial_{e_{j}}f|^{2}\quad\text{for
$j=1,2$ and $k=3,\dots,n$},$

and moreover

(10.5) $\displaystyle\varepsilon^{2}\omega_{f}(e_{1},e_{2})^{2}(x)$
$\displaystyle=\varepsilon^{-2}\omega_{0}^{2}(e_{1},e_{2})\left(Q_{\varepsilon}(x)\right),$

as well as

(10.6) $\displaystyle\varepsilon^{2}\omega_{f}(e_{k},e_{\ell})^{2}(x)$
$\displaystyle\leq\varepsilon^{-2}|df|^{4}\omega_{0}(e_{1},e_{2})^{2}\left(Q_{\varepsilon}(x)\right)\quad\text{for
$3\leq k<\ell\leq n$}.$

To compute $E_{\varepsilon}(u_{f},\nabla_{f})$, we use Eq. 10.2–Eq. 10.6 to
see that

$\displaystyle\int_{B^{2}_{1}\times B^{n-2}_{1}}e_{\varepsilon}(u,\nabla)$
$\displaystyle=\int_{B^{n-2}_{1}}\left[\int_{B^{2}_{1}}\varepsilon^{-2}e_{1}(u_{0},\nabla_{0})(Q_{\varepsilon}(x))\left(1+\frac{|df|^{2}}{2}+O(|df|^{4})\right)+O(e^{-{K}/{\varepsilon}})\right]$
$\displaystyle=2\pi\left[|B^{n-2}_{1}|+\int_{B_{1}^{n-2}}\frac{|df|^{2}}{2}+O(|df|^{4})\right]+O(e^{-{K}/{\varepsilon}}).$

In the above display we used the exponential decay from [36, Chapter III,
Theorem 8.1]:

$\displaystyle\int_{\mathbb{R}^{2}\setminus
B_{1/(2\varepsilon)}^{2}}e_{1}(u_{0},\nabla_{0})=O(e^{-{K}/{\varepsilon}}).$

Recalling that $|df|\leq\eta$ we get the desired estimate. ∎

###  10.2. Constructing the interpolation gauge

In this section we find a gauge transformation
$(u,\nabla)\mapsto(e^{i\xi}u,\nabla-id\xi)$ for which the new pair is
$L^{2}$-close to the pullback pair $(u_{h},\nabla_{h})$ constructed in
Proposition 10.1, where $h:B^{n-2}_{1}\rightarrow B^{2}_{1/2}$ is the
Lipschitz approximation built in Proposition 6.4. Since this is the most
technical part of the paper, we provide an overview of the arguments.

Step 1. We cover the vortex set with cylinders
$\\{B_{5C|\varepsilon\log\varepsilon|}^{2}(y_{k})\times
B^{n-2}_{5C\varepsilon}(x_{k})\\}_{k=1}^{N}$ with
$x_{k}=(y_{k},z_{k})\in\mathbb{R}^{2}\times\mathbb{R}^{n-2}$ such that
$B^{n-2}_{5C\varepsilon}(z_{k})$ is a Vitali cover of $B^{n-2}_{1}$. Then we
name a cylinder good if the excess on it is small, and bad otherwise. We also
define a partition of unity, subordinate to this cover, with derivatives at
most $C\varepsilon^{-1}$.

Step 2. In each cylinder we pass to the Coulomb gauge, via a function
$\xi_{k}$ with mean equal to the mean of $\theta_{h}-\theta$ on an appropriate
annulus away from the vortex set (note that $\theta-\theta_{h}$ is well-
defined far from the vortex set of both $u$ and $u_{h}$) Then we use Gaffney-
and Poincaré-type inequalities from Appendix B to derive estimates for
$e^{i\xi_{k}}u-u_{h}$ and $(\alpha+d\xi_{k})-\alpha_{h}$, where we write
$\nabla=d-i\alpha$. Far from the vorticity set we modify the gauge so that
$e^{i\xi_{k}}u$ and $u_{h}$ have the same phase. Then we use the exponential
decay away from the vortex set, which is where the error $\varepsilon^{\beta}$
comes from; however, this will be enough to show the classification result in
all dimensions, since we are free to take $\beta$ arbitrarily large.

Step 3. We use the estimates on $(\alpha+d\xi_{k})-\alpha_{h}$ (and the mean
condition) and Poincaré–Gaffney-type inequalities from Appendix B to bound
$\xi_{j}-\xi_{k}$ on overlapping cylinders.

Step 4. We patch together the $\xi_{k}$’s with the partition of unity defined
in the first step to obtain the function $\xi$. Then we use the bounds on
$\xi_{j}-\xi_{k}$ to derive estimates on $(\alpha+d\xi)-\alpha_{h}$ and
$e^{i\xi}u-u_{h}$.

######  Proposition 10.2 (The interpolation gauge).

For any $\beta>0$ there exist $\tau_{0}(n,\beta),\varepsilon_{0}(n,\beta)>0$
and $C_{0}(n,\beta)>0$ with the following property. Let $(u,\nabla)$ be a
critical pair for $E_{\varepsilon}$ on $\mathbb{R}^{n}$, with $u(0)=0$,
$\varepsilon\leq\varepsilon_{0}$, and the energy bound

$\frac{1}{|B^{n-2}_{2}|}\int_{B_{2}^{n}}e_{\varepsilon}(u,\nabla)\leq
2\pi+\tau_{0}.$

Moreover, let $h:B_{1}^{n-2}\rightarrow B^{2}_{1/2}$ be the Lipschitz
approximation defined in Proposition 6.4 (for a suitable $\eta$ chosen later
on) and let $(u_{h},\nabla_{h})$ be the pullback pair constructed in
Proposition 10.1. Then, on a given annulus

$\mathcal{A}_{s,\delta}:=B^{2}_{1}\times(B^{n-2}_{s+\delta}\setminus\overline{B}^{n-2}_{s})$

with $\delta\in[C_{0}\varepsilon,\frac{1}{16}]$ and $s\leq\frac{3}{4}$, we can
find a gauge transformation

$\displaystyle(u,\nabla)\mapsto(e^{i\xi}u,\nabla-id\xi),$

via a smooth function $\xi:\mathcal{A}_{s,\delta}\rightarrow\mathbb{R}$ such
that:

  1. (i)

letting $P:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n-2}$ be the projection onto
the last $n-2$ coordinates, we have

$\displaystyle\int_{\mathcal{A}_{s,\delta}}[\varepsilon^{-2}|e^{i\xi}u-u_{h}|^{2}+|(\alpha+d\xi)-\alpha_{h}|^{2}]$
$\displaystyle\leq
C(n,\beta)|\log\varepsilon|^{8}\int_{P(\mathcal{A}_{s-\delta,3\delta})}[\bm{\mathrm{E}}_{z}+\bm{1}_{\mathcal{G}^{\eta}}\bm{\mathrm{E}}_{z}|\log\bm{\mathrm{E}}_{z}|^{2}]+\varepsilon^{\beta};$

  2. (ii)

letting

$\displaystyle Z:=\\{|u|\leq 3/4\\},\quad
Z_{C_{0}\varepsilon|\log\varepsilon|}:=\\{x=(y,z)\,:\,\operatorname{dist}(x,Z\cap(B_{1}^{2}\times\\{z\\}))\leq
C_{0}\varepsilon|\log\varepsilon|\\},$

the function $e^{i\xi}u\to\mathbb{C}\setminus\\{0\\}$ has the same phase as
$u_{h}$ far from the vortex set, i.e.,

$\displaystyle\frac{e^{i\xi}u}{u_{h}}\in\mathbb{R}^{+}\quad\text{on
}\mathcal{A}_{s,\delta}\setminus Z_{5C_{0}|\varepsilon\log\varepsilon|}.$

###### Proof.

First we choose $\varepsilon_{0},\tau_{0}$ small enough so that 1.7 applies
(for $\alpha=0$). We divide the proof into several steps.

Covering arguments and a partition of unity. To begin with, by 1.7, we can
find a collection of points

$\\{x_{k}=(y_{k},z_{k})\\}_{k=1}^{N}\subset Z\cap\mathcal{A}_{s,\delta}$

satisfying the following.

  1. (i)

The projected collection $\\{z_{k}=P(x_{k})\\}_{k=1}^{N}\subset
B^{n-2}_{s+\delta}\setminus B^{n-2}_{s}$ gives a Vitali covering of the
projected annulus $P(\mathcal{A}_{s,\delta})$:

(10.7) $\displaystyle\begin{aligned}
&P(\mathcal{A}_{s,\delta})=B^{n-2}_{s+\delta}\setminus\overline{B}^{n-2}_{s}\subseteq\bigcup_{k=1}^{N}B^{n-2}_{5C_{0}\varepsilon}(z_{k}),\\\
&B_{C_{0}\varepsilon}^{n-2}(z_{j})\cap
B_{C_{0}\varepsilon}^{n-2}(z_{k})=\emptyset\quad\text{for all }j\neq
k.\end{aligned}$

This last line shows in particular that $N\leq C(n)\varepsilon^{2-n}$, where
by now $C_{0}$ depends only on $n$.

  2. (ii)

As a direct consequence of 1.7, we can guarantee that

(10.8) $\displaystyle
Z_{C_{0}\varepsilon|\log\varepsilon|}\cap\mathcal{A}_{s,\delta}\subseteq\bigcup_{k=1}^{N}B_{5C_{0}\varepsilon|\log\varepsilon|}^{2}(y_{k})\times
B^{n-2}_{5C_{0}\varepsilon}(z_{k}).$

  3. (iii)

We say that a point $x_{k}=(y_{k},z_{k})$ is a good point if

$\displaystyle\sup_{\varepsilon\leq r\leq
10C_{0}\varepsilon}r^{2-n}\bm{\mathrm{E}}_{1}(u,\nabla,B^{2}_{1}\times
B_{r}^{n-2}(z_{k}),\mathbb{R}^{n-2})\leq\eta_{0}^{2},$

where with a certain abuse of notation we have set

(10.9) $\displaystyle\bm{\mathrm{E}}_{1}(u,\nabla,B^{2}_{1}\times
B_{r}^{n-2}(z),\mathbb{R}^{n-2})=\int_{B^{2}_{1}\times
B_{r}^{n-2}(z)}\left[\sum_{k=3}^{n}|\nabla_{e_{k}}u|^{2}+\varepsilon^{2}\sum_{(j,k)\neq(1,2)}\omega(e_{j},e_{k})^{2}\right]$

(note the absence of normalization) Let the set of good indices be $G$. We
also denote the set of bad ones by $B:=\\{1,\dots,N\\}\setminus G$.

  4. (iv)

Again as a direct consequence of 1.7 we get that

(10.10) $\displaystyle|u_{h}|,|u|>\frac{3}{4}\quad\text{on
}\bigcup_{k=1}^{N}(B^{2}_{5C_{0}\varepsilon|\log\varepsilon|}(y_{k})\setminus
B^{2}_{C_{0}\varepsilon|\log\varepsilon|}(y_{k}))\times
B^{n-2}_{5C_{0}\varepsilon}(z_{k}).$

Since both $u$ and $u_{h}$ have degree $1$ on each of the previous domains (up
to conjugating $(u,\nabla)$ on $\mathbb{R}^{n}$), the _difference of phases_
$\theta-\theta_{h}$ is well-defined on these domains.

  5. (v)

We also define a partition of unity $\\{\phi_{k}\\}_{k=1}^{N}$, subordinate to
the cylinders:

(10.11) $\displaystyle\phi_{k}\in
C^{1}_{c}(B^{2}_{5C_{0}\varepsilon|\log\varepsilon|}(y_{k})\times
B^{n-2}_{5C_{0}\varepsilon}(z_{k})),\quad 0\leq\phi_{k}\leq 1,$

and

$\displaystyle\sum_{k=1}^{N}\phi_{k}=1\quad\text{on
}Z_{C_{0}\varepsilon|\log\varepsilon|}\cap\mathcal{A}_{s,\delta}.$

We also require that $|d\phi_{k}|\leq\tilde{C}\varepsilon^{-1}$ for all
$k=1,\dots,N$.

  6. (vi)

Up to modifying the $\phi_{k}$’s, we can define $\phi_{0}\in
C^{1}_{c}(\mathcal{A}_{s,\delta}\setminus
Z_{C_{0}|\varepsilon\log\varepsilon|})$ with $0\leq\phi_{0}\leq 1$ and

$\displaystyle\phi_{0}=1\quad\text{on }\mathcal{A}_{s,\delta}\setminus
Z_{5C_{0}|\varepsilon\log\varepsilon|},$
$\displaystyle\phi_{0}+\sum_{k=1}^{N}\phi_{k}=1\quad\text{on
}\mathcal{A}_{s,\delta}.$

Lastly, $|d\phi_{0}|\leq\tilde{C}|\varepsilon\log\varepsilon|^{-1}$.

Note that $C_{0}>0$ is some large enough constant that we are still free to
choose later on ($\tilde{C}$ above depends on $C_{0}$). In the sequel we also
use the following notation for the excess on each ball:

$\displaystyle\bm{\mathrm{E}}(k):=\bm{\mathrm{E}}(u,\nabla,B^{2}_{1}\times
B^{n-2}_{10C_{0}\varepsilon}(z_{k}),\mathbb{R}^{n-2}).$

We define $\bm{\mathrm{E}}(k)=\bm{\mathrm{E}}_{1}(k)+\bm{\mathrm{E}}_{2}(k)$
similarly. Since the balls $B^{n-2}_{C_{0}\varepsilon}(z_{k})$ are disjoint,
the dilated balls have bounded overlap (i.e., at most $C(n)$ of them
intersect), giving

$\displaystyle\sum_{k=1}^{N}\bm{\mathrm{E}}(k)\leq
C\int_{P(\mathcal{A}_{s-\delta,3\delta})}\bm{\mathrm{E}}_{z},$

and the same holds for $\bm{\mathrm{E}}_{1}$ and $\bm{\mathrm{E}}_{2}$. With
abuse of notation, we also write

(10.12)
$|\log\bm{\mathrm{E}}|^{2}\bm{\mathrm{E}}(k):=\begin{cases}\int_{B^{n-2}_{5C_{0}\varepsilon}(z_{k})}|\log\bm{\mathrm{E}}_{z}|^{2}\bm{\mathrm{E}}_{z}&\text{if
}k\in G,\\\
\int_{B^{n-2}_{5C_{0}\varepsilon}(z_{k})}\bm{\mathrm{E}}_{z}&\text{if }k\in
B.\end{cases}$

######  Remark 10.3.

We will often use the following observation implicitly. For all $z\in
B_{1}^{n-2}$, the results in the previous section show that

$r^{2-n}\bm{\mathrm{E}}_{1}(u,\nabla,B_{1}^{2}\times
B_{r}^{n-2}(z),\mathbb{R}^{n-2})\leq\tilde{\eta}_{0}^{2}$

for an (arbitrarily) small $\tilde{\eta}_{0}$ and all radii
$\Lambda\varepsilon\leq r\leq\frac{1}{2}$, for some $\Lambda$ depending on
$n,\tilde{\eta}_{0}$. If $k\in G$ then, for all $z\in
B_{5C_{0}\varepsilon}^{n-2}(z_{k})$, we have

$(C_{0}\varepsilon)^{{2-n}}\bm{\mathrm{E}}_{1}(u,\nabla,B_{1}^{2}\times
B_{C_{0}\varepsilon}^{n-2}(z),\mathbb{R}^{n-2})\leq 10^{n-2}\eta_{0}^{2}.$

As a consequence of Lemma 5.7, if we take $C_{0}\geq C(n)$ large and
$\eta_{0}^{2}$ very small, we have

$r^{2-n}\bm{\mathrm{E}}_{1}(u,\nabla,B_{1}^{2}\times
B_{r}^{n-2}(z),\mathbb{R}^{n-2})\leq\tilde{\eta}_{0}^{2}$

also for $r\in(0,\Lambda\varepsilon)$, since we have $C^{1}$ control on the
pair at this scale. We now fix $\tilde{\eta}_{0}$ such that we actually apply
Proposition 6.4 for $\eta:=\tilde{\eta}_{0}$, so that _every $z\in
B_{5C_{0}\varepsilon}^{n-2}(z_{k})$ gives a good slice_.

Gauge fixing on each small cylinder with bounds. Fix $k\in\\{1,\dots,n\\}$ and
consider the unique solution $\xi_{k}$ to the following Neumann boundary value
problem:

(10.13)
$\displaystyle\begin{cases}\Delta\xi_{k}=d^{*}(\alpha-\alpha_{h})&\text{in
}\mathcal{C}_{k}:=B_{5C_{0}\varepsilon|\log\varepsilon|}^{2}(y_{k})\times
B^{n-2}_{5C_{0}\varepsilon}(z_{k}),\\\
\partial_{\nu}\xi_{k}=-(\alpha-\alpha_{h})(\nu)&\text{at
}\partial\mathcal{C}_{k},\\\
\int_{\mathcal{A}_{k}}[(\theta+\xi_{k})-\theta_{h}]=0&\text{where
}\mathcal{A}_{k}:=(B^{2}_{5C_{0}\varepsilon|\log\varepsilon|}(y_{k})\setminus
B^{2}_{C_{0}\varepsilon|\log\varepsilon|}(y_{k}))\times
B^{n-2}_{5C_{0}\varepsilon}(z_{k}).\end{cases}$

Recall that $\theta-\theta_{h}$ is well-defined on $\mathcal{A}_{k}$. Then we
perform the following gauge transformation in $\mathcal{C}_{k}$: writing
$\nabla=d-i\alpha$, we transform

$\displaystyle(u,\alpha)\mapsto(e^{i\xi_{k}}u,\alpha+d\xi_{k}).$

Since $d^{*}[(\alpha+d\xi_{k})-\alpha_{h}]=0$ and
$[(\alpha+d\xi_{k})-\alpha_{h}](\nu)=0$, we can use the Gaffney–Poincaré-type
inequality in Lemma B.1:

(10.14)
$\displaystyle\int_{\mathcal{C}_{k}}|(\alpha+d\xi_{k})-\alpha_{h}|^{2}\leq
C(n,C_{0})|\varepsilon\log\varepsilon|^{2}\int_{\mathcal{C}_{k}}|d\alpha-d\alpha_{h}|^{2}.$

We bound separately the contributions of the good set and the bad set, using
Lemma 10.4 and Lemma 10.5 below, obtaining

(10.15)
$\displaystyle\int_{\mathcal{C}_{k}}|(\alpha+d\xi_{k})-\alpha_{h}|^{2}\leq
C|\log\varepsilon|^{2}|\log\bm{\mathrm{E}}|^{4}\bm{\mathrm{E}}{(k)}+C\varepsilon^{\beta+3n}$

for some $C=C(n,C_{0},\beta)=C(n,\beta)$.

Gauge fixing far from the vortex set with bounds. Far from the vortex set, in
the set $\mathcal{A}_{s,\delta}\setminus
Z_{C_{0}|\varepsilon\log\varepsilon|}$, we gauge fix via a function $\xi_{0}$
such that $e^{i\xi_{0}}u/u_{h}$ is real-valued. Hence, we define

(10.16) $\displaystyle\xi_{0}:=\theta_{h}-\theta\quad\text{on
}\mathcal{A}_{s,\delta}\setminus Z_{C_{0}|\varepsilon\log\varepsilon|};$

note that a priori $\xi_{0}$ is well-defined only in the quotient
$\mathbb{R}/2\pi\mathbb{Z}$, but this is enough to have a well-defined gauge
transformation (in fact, since the vorticity set is included in a
$C\varepsilon$-neighborhood of a graph, we can check that $\theta_{h}-\theta$
is a well-defined real number).

We can estimate $e^{i\xi_{0}}u-u_{h}$ and $(\alpha+d\xi_{0})-\alpha_{h}$ in
this domain using the exponential decay (Proposition 4.5), as follows:

$\displaystyle\int_{\mathcal{A}_{s,\delta}\setminus
Z_{C_{0}|\varepsilon\log\varepsilon|}}[\varepsilon^{-2}|e^{i\xi_{0}}u-u_{h}|^{2}+|(\alpha+d\xi_{0})-\alpha_{h}|^{2}]$
$\displaystyle\leq\int_{\mathcal{A}_{s,\delta}\setminus
Z_{C_{0}|\varepsilon\log\varepsilon|}}[\varepsilon^{-2}||u|-|u_{h}||^{2}+2|\alpha-d\theta|^{2}+2|\alpha_{h}-d\theta_{h}|^{2}]$
$\displaystyle\leq C(n)\varepsilon^{-2}e^{-K(n)C_{0}|\log\varepsilon|}.$

In the last inequality, we used the following observation: since each slice
intersects the zero set and $|\log\varepsilon|\geq C(n)$, using Lemma 5.7 we
see that on $B_{1}^{2}\times\\{z\\}$ the distance from $\\{|u|\leq 3/4\\}$ is
comparable with the distance from $(B_{1}^{2}\times\\{z\\})\cap\\{|u|\leq
3/4\\}$.

Taking $C_{0}$ large enough, we see that

(10.17) $\displaystyle\int_{\mathcal{A}_{s,\delta}\setminus
Z_{C_{0}|\varepsilon\log\varepsilon|}}[\varepsilon^{-2}|e^{i\xi_{0}}u-u_{h}|^{2}+|(\alpha+d\xi_{0})-\alpha_{h}|^{2}]\leq\varepsilon^{\beta+3n}.$

Difference of local gauges in the overlap regions. Fix $1\leq j<k\leq N$ such
that

$\displaystyle\Omega_{j,k}:=\mathcal{C}_{j}\cap\mathcal{C}_{k}\neq\emptyset.$

Notice that we can bound the $L^{2}$ norm of the difference
$d\xi_{k}-d\xi_{j}$ as follows:

$\displaystyle\int_{\Omega_{j,k}}|d\xi_{j}-d\xi_{k}|^{2}\leq
2\int_{\mathcal{C}_{j}}|(\alpha+d\xi_{j})-\alpha_{h}|^{2}+2\int_{\mathcal{C}_{k}}|(\alpha+d\xi_{k})-\alpha_{h}|^{2}.$

By Eq. 10.15 we then have

(10.18) $\displaystyle\int_{\Omega_{j,k}}|d(\xi_{j}-\xi_{k})|^{2}\leq
C|\log\varepsilon|^{4}(|\log\bm{\mathrm{E}}|^{2}\bm{\mathrm{E}}{(j)}+|\log\bm{\mathrm{E}}|^{2}\bm{\mathrm{E}}{(k)})+C\varepsilon^{\beta+3n}.$

Our goal is to apply a Poincaré-type inequality on $\Omega_{j,k}$ to estimate
$\xi_{k}-\xi_{j}$. To this aim, we first look at $\xi_{j},\xi_{k}$ on an
appropriate annulus. By the definition of $\mathcal{C}_{j},\mathcal{C}_{k}$
and the structure of the vortex set in 1.7 we can see that $|x_{j}-x_{k}|\leq
20C_{0}\varepsilon$. We name the midpoint
$x_{j,k}=(y_{j,k},z_{j,k}):=\frac{x_{j}+x_{k}}{2}$ and we see that

$\displaystyle\mathcal{A}_{j,k}:=[B^{2}_{3C_{0}\varepsilon|\log\varepsilon|}(y_{j,k})\setminus
B^{2}_{2C_{0}\varepsilon|\log\varepsilon|}(y_{j,k})]\times[B^{n-2}_{5C_{0}\varepsilon}(z_{j})\cap
B^{n-2}_{5C_{0}\varepsilon}(z_{k})]\subseteq\mathcal{A}_{j}\cap\mathcal{A}_{k}\subseteq\Omega_{j,k}.$

So we can compute that

$\displaystyle\int_{\mathcal{A}_{j,k}}|\xi_{j}-\xi_{k}|^{2}\leq
2\int_{\mathcal{A}_{j}}|(\theta+\xi_{j})-\theta_{h}|^{2}+2\int_{\mathcal{A}_{k}}|(\theta+\xi_{k})-\theta_{h}|^{2}.$

We know that $(\theta+\xi_{j})-\theta_{h}$ and $(\theta+\xi_{k})-\theta_{h}$
have zero mean on $\mathcal{A}_{j}$ and $\mathcal{A}_{k}$, respectively.
Hence, we can apply Lemma B.3 on each annulus to see that

$\displaystyle\int_{\mathcal{A}_{j,k}}\varepsilon^{-2}|\xi_{j}-\xi_{k}|^{2}$
$\displaystyle\leq
C\int_{\mathcal{A}_{j}}|d(\theta+\xi_{j})-d\theta_{h}|^{2}+C\int_{\mathcal{A}_{k}}|d(\theta+\xi_{k})-d\theta_{h}|^{2},$

and we can bound

$|d(\theta+\xi_{j})-d\theta_{h}|\leq|\alpha-d\theta|+|\alpha_{h}-d\theta_{h}|+|\alpha+d\xi_{j}-\alpha_{h}|.$

As before, on $\mathcal{A}_{j}$ we have

$|\alpha-d\theta|^{2}+|\alpha_{h}-d\theta_{h}|^{2}\leq|u|^{-2}|\nabla
u|^{2}+|u_{h}|^{-2}|\nabla_{h}u_{h}|^{2}\leq\varepsilon^{\beta+3n}$

by exponential decay, and the same holds for $k$. Together with Eq. 10.15 we
thus estimate

(10.19)
$\displaystyle\int_{\mathcal{A}_{j,k}}\varepsilon^{-2}|\xi_{j}-\xi_{k}|^{2}\leq
C|\log\varepsilon|^{4}(|\log\bm{\mathrm{E}}|^{2}\bm{\mathrm{E}}{(j)}+|\log\bm{\mathrm{E}}|^{2}\bm{\mathrm{E}}{(k)})+C\varepsilon^{\beta+3n}.$

By Eq. 10.18–Eq. 10.19, using Lemma B.3 and B.4, we arrive at

(10.20)
$\displaystyle\int_{\Omega_{j,k}}\varepsilon^{-2}|\xi_{j}-\xi_{k}|^{2}\leq
C|\log\varepsilon|^{4}(|\log\bm{\mathrm{E}}|^{2}\bm{\mathrm{E}}{(j)}+|\log\bm{\mathrm{E}}|^{2}\bm{\mathrm{E}}{(k)})+\varepsilon^{\beta+3n}.$

We also need to estimate the difference of $\xi_{k}-\xi_{0}$ for all $1\leq
k\leq N$. Defining

$\displaystyle\Omega_{0,k}:=\mathcal{C}_{k}\cap[\mathcal{A}_{s,\delta}\setminus
Z_{C_{0}|\varepsilon\log\varepsilon|}],$

we see that

$\displaystyle\Omega_{0,k}\subseteq\mathcal{A}_{k}^{\prime}:=[B^{2}_{5C_{0}\varepsilon|\log\varepsilon|}(y_{k})\setminus
B^{2}_{(C_{0}/2)\varepsilon|\log\varepsilon|}(y_{k})]\times
B^{n-2}_{5C_{0}\varepsilon}(z_{k}).$

Note that by Eq. 10.16 we have $\xi_{0}=\theta_{h}-\theta$ in $\Omega_{0,k}$.
Hence, we can apply Lemma B.3 and compute that

$\displaystyle\varepsilon^{-2}\int_{\Omega_{0,k}}|\xi_{k}-\xi_{0}|^{2}$
$\displaystyle\leq\varepsilon^{-2}\int_{\mathcal{A}_{k}^{\prime}}|(\theta+\xi_{k})-\theta_{h}|^{2}$
$\displaystyle\leq\int_{\mathcal{A}_{k}^{\prime}}|d(\theta+\xi_{k})-d\theta_{h}|^{2}$
$\displaystyle\leq\int_{\mathcal{A}_{k}^{\prime}}[|\alpha-d\theta|^{2}+|\alpha_{h}-d\theta_{h}|^{2}+|(\alpha+d\xi_{k})-\alpha_{h}|^{2}],$

where again we used the fact that $(\theta+\xi_{k})-\theta_{h}$ has zero mean
on $\mathcal{A}_{k}\subset\mathcal{A}_{k}^{\prime}$. Summing the previous
bounds and noting that there are at most $(C\varepsilon^{2-n})^{2}$ pairs
$(j,k)$, while any point belongs to at most $C=C(n,C_{0})$ domains
$\Omega_{j,k}$, we arrive at

(10.21) $\displaystyle\sum_{0\leq j<k\leq
N}\int_{\Omega_{j,k}}\varepsilon^{-2}|\xi_{j}-\xi_{k}|^{2}\leq
C|\log\varepsilon|^{4}\int_{P(\mathcal{A}_{s-\delta,3\delta})}|\log\bm{\mathrm{E}}_{z}|^{2}\bm{\mathrm{E}}_{z}+C\bm{\mathrm{E}}+C\varepsilon^{\beta+1},$

for some $C=C(n,\beta)$ (recall Eq. 10.12).

Constructing the global gauge via the partition of unity. Recall the
definition of the partition of unity in Eq. 10.11. We define the global gauge
transformation function as follows:

(10.22) $\displaystyle\xi:=\sum_{k=0}^{N}\phi_{k}\xi_{k}\quad\text{on
}\mathcal{A}_{s,\delta}.$

Then we estimate

$\displaystyle\int_{\mathcal{A}_{s,\delta}}\varepsilon^{-2}\left|e^{i\xi}u-\sum_{k=0}^{N}\phi_{k}e^{i\xi_{k}}u\right|^{2}$
$\displaystyle\leq\varepsilon^{-2}\int_{\mathcal{A}_{s,\delta}}\sum_{k=0}^{N}\phi_{k}|\xi-\xi_{k}|^{2}$
$\displaystyle\leq
2\sum_{j<k}\int_{\Omega_{j,k}}\varepsilon^{-2}|\xi_{j}-\xi_{k}|^{2}$
$\displaystyle\leq
C|\log\varepsilon|^{4}\int_{P(\mathcal{A}_{s-\delta,3\delta})}|\log\bm{\mathrm{E}}_{z}|^{2}\bm{\mathrm{E}}_{z}+C\bm{\mathrm{E}}+C\varepsilon^{\beta+1}.$

In particular,

(10.23) $\displaystyle\begin{aligned}
\int_{\mathcal{A}_{s,\delta}}\varepsilon^{-2}|e^{i\xi}u-u_{h}|^{2}&\leq
2\int_{\mathcal{A}_{s,\delta}}\left[\varepsilon^{-2}\left|e^{i\xi}u-\sum_{k=0}^{N}\phi_{k}e^{i\xi_{k}}u\right|^{2}+\sum_{k=0}^{N}\phi_{k}|e^{i\xi_{k}}u-u_{h}|^{2}\right]\\\
&\leq
C|\log\varepsilon|^{8}\int_{P(\mathcal{A}_{s-\delta,3\delta})}|\log\bm{\mathrm{E}}_{z}|^{2}\bm{\mathrm{E}}_{z}+C\bm{\mathrm{E}}+C\varepsilon^{\beta+1},\end{aligned}$

where we used Lemma 10.4 and Lemma 10.5 to estimate the term involving
$e^{i\xi_{k}}u-u_{h}$.

Moreover, for the connection part, we can bound

(10.24) $\displaystyle\begin{aligned}
&\int_{\mathcal{A}_{s,\delta}}|(\alpha+d\xi)-\alpha_{h}|^{2}\\\ &\leq
2\int_{\mathcal{A}_{s,\delta}}\left[\sum_{k=0}^{N}\phi_{k}|(\alpha+d\xi_{k})-\alpha_{h}|^{2}+\left|\sum_{k=0}^{N}d\phi_{k}\xi_{k}\right|^{2}\right].\end{aligned}$

The first term is bounded by Eq. 10.15 and Eq. 10.17. We are left to bound the
last term. Since the functions $\phi_{k}$ form a partition of unity, we have

$\displaystyle\sum_{k=0}^{N}d\phi_{k}(z)=0.$

We can then write

$\sum_{k=0}^{N}d\phi_{k}\xi_{k}=\sum_{j,k=0}^{N}\phi_{j}d\phi_{k}(\xi_{k}-\xi_{j}).$

Since $|d\phi_{k}|\leq C\varepsilon^{-1}$, the last term above is bounded by

$C\varepsilon^{-2}\sum_{j<k}\int_{\Omega_{j,k}}|\xi_{j}-\xi_{k}|^{2},$

which is a quantity that we already estimated. Combining these bounds, we see
that

$\displaystyle\int_{\mathcal{A}_{s,\delta}}[\varepsilon^{-2}|e^{i\xi}u-u_{h}|^{2}+|(\alpha+d\xi)-\alpha_{h}|^{2}]\leq
C|\log\varepsilon|^{8}\int_{P(\mathcal{A}_{s-\delta,3\delta})}[\bm{\mathrm{E}}_{z}+\bm{1}_{\mathcal{G}^{\eta}}\bm{\mathrm{E}}_{z}|\log\bm{\mathrm{E}}_{z}|^{2}]+\varepsilon^{\beta}.$

This is indeed the desired conclusion. ∎

We now turn to the bounds which were postponed in the previous proof.

######  Lemma 10.4.

Assume that $k\in G$. Then

$\int_{\mathcal{C}_{k}}\varepsilon^{-2}|e^{i\xi_{k}}u-u_{h}|^{2}\leq
C|\log\varepsilon|^{8}|\log\bm{\mathrm{E}}|^{2}\bm{\mathrm{E}}(k)+C\varepsilon^{\beta+3n}$

and

$\int_{\mathcal{C}_{k}}\varepsilon^{2}|d\alpha-d\alpha_{h}|^{2}\leq
C|\log\varepsilon|^{4}\eta^{-2}|\log\bm{\mathrm{E}}|^{2}\bm{\mathrm{E}}(k)+C\varepsilon^{\beta+3n}.$

###### Proof.

We bound each part separately.

Estimating $d\alpha-d\alpha_{h}$. Recalling the definition of
$\bm{\mathrm{E}}_{1}$ in Eq. 10.9 and the construction of $\alpha_{h}$, we
have

$\int_{\mathcal{C}_{k}}\varepsilon^{2}|d\alpha-d\alpha_{h}|^{2}\leq\int_{\mathcal{C}_{k}}\varepsilon^{2}|d\alpha(e_{1},e_{2})-d\alpha_{h}(e_{1},e_{2})|^{2}+C\bm{\mathrm{E}}_{1}(k)+C\int_{B_{5C_{0}\varepsilon}^{n-2}(z_{k})}|dh|^{2}.$

We are going to use some estimates from [31] which are slightly more refined
than Eq. 4.14. Compared to the main result of [31], these hold under some
additional assumptions, which are however satisfied on good slices: in
particular, for any $z\in B_{5C_{0}\varepsilon}^{n-2}(z_{k})$, the function
$u$ vanishes linearly at a unique point along the slice
$B_{1}^{2}\times\\{z\\}$. We will often compare $u$ with the function
$u_{h_{0}}$, where $h_{0}$ is the function built in Proposition 6.6, whose
graph approximates the zero set; along the good slice, this function vanishes
at the same point as $u$, and is just a translation of the standard degree-one
planar solution.

Specifically, using an $\varepsilon$-rescaling of [31, Theorem 3.1] and [31,
Corollary 3.3] (applied with $N=1$; see also the part below [31, Corollary
3.3] for the choice of functions to which this result is applied), we have the
following estimate:

(10.25) $\displaystyle\begin{aligned}
&\int_{B^{2}_{1}\times\\{z\\}}[\varepsilon^{-2}||u|-|u_{h_{0}}||^{2}+|u_{h_{0}}|^{2+1/2}|(\alpha-d\theta)_{(1,2)}-(\alpha_{h_{0}}-d\theta_{h_{0}})_{(1,2)}|^{2}\\\
&\phantom{\int_{B^{2}_{1}\times\\{z\\}}[}+\varepsilon^{2}|d\alpha(e_{1},e_{2})-d\alpha_{h_{0}}(e_{1},e_{2})|^{2}]\\\
&\leq C\bm{\mathrm{E}}_{z},\end{aligned}$

for an absolute constant $C$, where the subscript $(1,2)$ means that we
restrict the one-form along the slice.

Now by the construction in Proposition 10.1 we can see that
$(u_{h},\alpha_{h})$, along the slice $B^{2}_{1}\times\\{z\\}$, is equal to
$(u_{h_{0}},\alpha_{h_{0}})$ translated to vanish at the barycenter
$\Phi_{\chi(x_{1},x_{2})}(z)$. As shown in Lemma A.1, the translation is by a
vector $v$ with $|v|\leq
C\varepsilon|\log\bm{\mathrm{E}}_{z}|\sqrt{\bm{\mathrm{E}}_{z}}+e^{-K/\varepsilon}$.
By the mean value theorem, we then have

(10.26) $\displaystyle\begin{aligned}
&\int_{B_{5C_{0}\varepsilon|\log\varepsilon|}^{2}(y_{k})\times\\{z\\}}[\varepsilon^{2}|d\alpha_{h}(e_{1},e_{2})-d\alpha_{h_{0}}(e_{1},e_{2})|^{2}+|(\alpha_{h}-\alpha_{h_{0}})_{(1,2)}|^{2}\\\
&\phantom{\int_{B_{5C_{0}\varepsilon|\log\varepsilon|}^{2}(y_{k})\times\\{z\\}}[}+\varepsilon^{-2}|u_{h}-u_{h_{0}}|^{2}]\\\
&\leq C\varepsilon^{2}|\log\varepsilon|^{2}\cdot|v|^{2}\cdot
C\varepsilon^{-4}\\\ &\leq
C|\log\varepsilon|^{2}|\log\bm{\mathrm{E}}_{z}|^{2}\bm{\mathrm{E}}_{z}+e^{-K/\varepsilon},\end{aligned}$

since $\bm{\mathrm{E}}_{z}$ is bounded on good slices and the differential of
each quantity (such as $\varepsilon d\alpha_{h}(e_{1},e_{2})$ and so on) is
bounded by $C\varepsilon^{-2}$. The claimed estimate follows by combining the
previous bounds (together with item (iv) from Proposition 6.4, which gives
$|dh|^{2}(z)\leq C\bm{\mathrm{E}}_{z}+e^{-K/\varepsilon}$).

Estimating $e^{i\xi_{k}}u-u_{h}$. Writing formally $u=|u|e^{i\theta}$ and
using a similar notation for $u_{h}$ and $u_{h_{0}}$, recall that on the
annulus

$\mathcal{A}_{k,z}:=[B^{2}_{5C_{0}\varepsilon|\log\varepsilon|}(y_{k})\setminus
B^{2}_{C_{0}\varepsilon|\log\varepsilon|}(y_{k})]\times\\{z\\}$

the differences $\theta-\theta_{h}$, $\theta-\theta_{h_{0}}$ and
$\theta_{h}-\theta_{h_{0}}$ are well-defined. We record the following
estimate:

(10.27)
$\displaystyle\int_{\mathcal{A}_{k,z}}\varepsilon^{-2}|\theta_{h}-\theta_{h_{z}}|^{2}\leq
C|\log\bm{\mathrm{E}}_{z}|^{2}\bm{\mathrm{E}}_{z}.$

This holds again by the mean value theorem, since
$|(d\theta_{h})_{(1,2)}|(y)\leq C|y-y_{k}|^{-1}$. We are going to use the
Caffarelli–Kohn–Nirenberg-type inequality from Lemma B.2, which implies that

$\displaystyle\int_{B_{R}^{2}(y_{k})}|y-h_{0}(z)|^{2}|f(y)|^{2}\leq
CR^{3/2}\int_{B_{R}^{2}(y_{k})}|y-h_{0}(z)|^{2+1/2}|df(y)|^{2},\quad\text{for
$f\in C^{1}_{c}(B_{R}^{2}(y_{k}))$},$

with $R:=5C_{0}\varepsilon|\log\varepsilon|$ (since there exists a biLipschitz
transformation sending $B_{R}^{2}(y_{k})$ to itself and mapping the origin to
$h_{0}(z)$). Recalling that the standard degree-one solution vanishes linearly
at the origin, by the construction of $u_{h_{0}}$ in Proposition 10.1 we have

$\displaystyle
C^{-1}\leq\frac{|u_{h_{0}}|(y,z)}{\min\\{\varepsilon^{-1}|y-h_{0}(z)|,1\\}}\leq
C$

on the good slice, for some universal constant $C$. Moreover,

$\displaystyle
1\leq\frac{\varepsilon^{-1}|y-h_{0}(z)|}{\min\\{\varepsilon^{-1}|y-h_{0}(z)|,1\\}}\leq
C|\log\varepsilon|\quad\text{for all $y\in
B_{5C_{0}\varepsilon|\log\varepsilon|}^{2}(y_{k})$}.$

Hence, given a $C^{1}$ function $f$ on
$B^{2}_{5C_{0}\varepsilon|\log\varepsilon|}(y_{k})\times\\{z\\}$ vanishing
near the boundary, we can write

(10.28)
$\displaystyle\int_{B^{2}_{5C_{0}\varepsilon|\log\varepsilon|}(y_{k})\times\\{z\\}}|u_{h_{0}}|^{2}|f|^{2}$
$\displaystyle\leq
C\varepsilon^{2}|\log\varepsilon|^{4}\int_{B^{2}_{5C_{0}\varepsilon|\log\varepsilon|}(y_{k})\times\\{z\\}}|u_{h_{0}}|^{2+1/2}|df|^{2}.$

To estimate $ue^{i\xi_{k}}-u_{h_{0}}$, we first notice that $u$ and
$u_{h_{0}}$ have the same unique zero point (with the same degree around it),
and hence the difference $\theta-\theta_{h_{0}}$ gives a well-defined smooth
function on the full slice.

We define a cut-off $\chi:B^{2}_{1}\to\mathbb{R}$ with $\chi=1$ on
$B^{2}_{C_{0}\varepsilon|\log\varepsilon|}(y_{k})$ and $\chi=0$ outside of
$B^{2}_{5C_{0}\varepsilon|\log\varepsilon|}(y_{k})$, with $|d\chi|\leq
C|\varepsilon\log\varepsilon|^{-1}$. Then we use the first term of Eq. 10.25
to bound $|u|-|u_{h_{z}}|$ and Eq. 10.28 to see that

$\displaystyle\varepsilon^{-2}\int_{B^{2}_{5C_{0}\varepsilon|\log\varepsilon|}(y_{k})\times\\{z\\}}\chi^{2}|e^{i\xi_{k}}u-u_{h_{0}}|^{2}$
$\displaystyle\leq
C\int_{B^{2}_{5C_{0}\varepsilon|\log\varepsilon|}(y_{k})\times\\{z\\}}\chi^{2}\left[\frac{\left||u|-|u_{h_{0}}|\right|^{2}}{\varepsilon^{2}}+\frac{|u_{h_{0}}|^{2}}{\varepsilon^{2}}|(\theta+\xi_{k})-\theta_{h_{0}}|^{2}\right]$
$\displaystyle\leq C\bm{\mathrm{E}}_{z}+C(\bm{\mathrm{I}}+\bm{\mathrm{II}}),$

where

$\displaystyle\bm{\mathrm{I}}$
$\displaystyle:=|\log\varepsilon|^{4}\int_{B^{2}_{5C_{0}\varepsilon|\log\varepsilon|}(y_{k})\times\\{z\\}}|u_{h_{0}}|^{2+1/2}\chi^{2}|d(\theta+\xi_{k}-\theta_{h_{0}})_{(1,2)}|^{2},$
$\displaystyle\bm{\mathrm{II}}$
$\displaystyle:=|\log\varepsilon|^{4}\int_{B^{2}_{5C_{0}\varepsilon|\log\varepsilon|}(y_{k})\times\\{z\\}}|u_{h_{0}}|^{2+1/2}|d\chi|^{2}|\theta+\xi_{k}-\theta_{h_{0}}|^{2}.$

First we estimate $\bm{\mathrm{I}}$ using the second term in Eq. 10.25 and Eq.
10.26 (to replace $\alpha_{h_{0}}$ with $\alpha_{h}$):

$\displaystyle\bm{\mathrm{I}}$ $\displaystyle\leq
C|\log\varepsilon|^{6}|\log\bm{\mathrm{E}}_{z}|^{2}\bm{\mathrm{E}}_{z}+C|\log\varepsilon|^{4}\int_{B^{2}_{5C_{0}\varepsilon|\log\varepsilon|}(y_{k})\times\\{z\\}}\chi^{2}|(\alpha+d\xi_{k})-\alpha_{h}|^{2}.$

Then we estimate $\bm{\mathrm{II}}$: we note that $|d\chi|$ is supported in
$\mathcal{A}_{k,z}$ and $|d\chi|\leq C|\varepsilon\log\varepsilon|^{-1}$, and
hence we can use Eq. 10.27 to estimate

$\displaystyle\bm{\mathrm{II}}$ $\displaystyle\leq
C|\log\varepsilon|^{2}|\log\bm{\mathrm{E}}_{z}|^{2}\bm{\mathrm{E}}_{z}+C|\log\varepsilon|^{2}\int_{\mathcal{A}_{k,z}}\varepsilon^{-2}|\theta+\xi_{k}-\theta_{h}|^{2}.$

Putting $\bm{\mathrm{I}}$ and $\bm{\mathrm{II}}$ together and integrating over
$B^{n-2}_{5C_{0}\varepsilon}(z_{k})$, we see that

$\displaystyle\int_{\mathcal{C}_{k}}\varepsilon^{-2}|e^{i\xi_{k}}u-u_{h_{0}}|^{2}$
$\displaystyle\leq
C|\log\varepsilon|^{6}|\log\bm{\mathrm{E}}|^{2}\bm{\mathrm{E}}{(k)}+C|\log\varepsilon|^{4}\int_{\mathcal{C}_{k}}|\alpha+d\xi_{k}-\alpha_{h}|^{2}+C|\log\varepsilon|^{2}\int_{\mathcal{A}_{k}}\varepsilon^{-2}|\theta+\xi_{k}-\theta_{h}|^{2}$
$\displaystyle\leq
C|\log\varepsilon|^{8}|\log\bm{\mathrm{E}}|^{2}\bm{\mathrm{E}}{(k)}+C|\log\varepsilon|^{2}\int_{\mathcal{A}_{k}}\varepsilon^{-2}|\theta+\xi_{k}-\theta_{h}|^{2}+\varepsilon^{\beta+3n},$

where we used Eq. 10.15 (which uses only the previous bound on
$d\alpha-d\alpha_{h}$). Recalling that we imposed

$\displaystyle\int_{\mathcal{A}_{k}}(\theta+\xi_{k}-\theta_{h})=0,$

we can apply Lemma B.3 (suitably rescaled) and Eq. 10.15 another time to see
that

(10.29) $\displaystyle\begin{aligned}
&|\log\varepsilon|^{2}\int_{\mathcal{A}_{k}}\varepsilon^{-2}|\theta+\xi_{k}-\theta_{h}|^{2}\\\
&\leq
C|\log\varepsilon|^{2}\int_{\mathcal{A}_{k}}|d(\theta+\xi_{k})-d\theta_{h}|^{2}\\\
&\leq
C|\log\varepsilon|^{2}\int_{\mathcal{A}_{k}}[|\alpha-d\theta|^{2}+|\alpha_{h}-d\theta_{h}|^{2}+|(\alpha+d\xi_{k})-\alpha_{h}|^{2}]\\\
&\leq
C|\log\varepsilon|^{6}|\log\bm{\mathrm{E}}|^{2}\bm{\mathrm{E}}{(k)}+\varepsilon^{\beta+3n}\end{aligned}$

up to taking $C_{0}$ large enough (the last inequality follows from the
exponential decay of energy); combining these bounds with Eq. 10.26, we get
the desired bound for $e^{i\xi_{k}}u-u_{h}$. ∎

######  Lemma 10.5.

For $k\in B$ we have

$\int_{\mathcal{C}_{k}}[\varepsilon^{-2}|e^{i\xi_{k}}u-u_{h}|^{2}+\varepsilon^{2}|d\alpha-d\alpha_{h}|^{2}]\leq
C|\log\varepsilon|^{2}\bm{\mathrm{E}}_{1}{(k)}.$

###### Proof.

On the bad set we simply use $L^{\infty}$ bounds: we have

(10.30) $\displaystyle\begin{aligned}
&\int_{\mathcal{C}_{k}}[\varepsilon^{-2}|e^{i\xi_{k}}u-u_{h}|^{2}+\varepsilon^{2}|d\alpha-d\alpha_{h}|^{2}]\\\
&\leq C|\log\varepsilon|^{2}\varepsilon^{n-2}\\\ &\leq
C|\log\varepsilon|^{2}\bm{\mathrm{E}}_{1}{(k)},\end{aligned}$

where we used the definition of bad index in the last inequality. ∎

######  Corollary 10.6.

We have

$\displaystyle\int_{\mathcal{A}_{s,\delta}}[\varepsilon^{-2}|e^{i\xi}u-u_{h}|^{2}+|(\alpha+d\xi)-\alpha_{h}|^{2}]\leq
C(n,\beta)|\log\varepsilon|^{10}\int_{P(\mathcal{A}_{s-\delta,3\delta})}\bm{\mathrm{E}}_{z}+\varepsilon^{\beta}.$

###### Proof.

Recalling that for a good $z$ the sliced excess $\bm{\mathrm{E}}_{z}$ is
small, it suffices to split the integral of
$\bm{\mathrm{E}}_{z}|\log\bm{\mathrm{E}}_{z}|^{2}$ on the two sets
$\\{\bm{\mathrm{E}}_{z}\leq\varepsilon^{\beta+1}\\}$ and
$\\{\bm{\mathrm{E}}_{z}>\varepsilon^{\beta+1}\\}$. On the second one, we bound
$|\log\bm{\mathrm{E}}_{z}|^{2}\leq C|\log\varepsilon|^{2}$, while on the first
one we have $\bm{\mathrm{E}}_{z}|\log\bm{\mathrm{E}}_{z}|^{2}\leq
C\varepsilon^{\beta+1}|\log\varepsilon|^{2}$. ∎

##  11\. Proof of a stronger decay of tilt-excess for local minimizers

###  11.1. Strong approximation of the excess for minimizers

In this section we use variational arguments and the estimates from
Proposition 10.2 to construct competitors. As a consequence, we prove that the
full excess $\bm{\mathrm{E}}$ is well approximated by the Dirichlet energy of
a harmonic approximation $w$ built as in Proposition 7.2.

######  Proposition 11.1 (Strong harmonic approximation of minimizers).

For any $\nu,\beta>0$ and any radius $0<s<1$ there exist small constants
$\varepsilon_{0}(n,s,\nu,\beta),\tau_{0}(n,s,\nu,\beta),\eta_{0}(n,\nu,\beta)>0$
with the following properties. Let $(u,\nabla)$ be a minimizer of
$E_{\varepsilon}$ defined on $B^{n}_{2}(0)$, with
$\varepsilon\leq\varepsilon_{0}$ and the energy bound

$\frac{1}{|B_{2}^{n-2}|}\int_{B_{2}^{n}}e_{\varepsilon}(u,\nabla)\leq
2\pi+\tau_{0}.$

After a suitable rotation, let $h:B^{n-2}_{1}\to B^{2}_{1}$ be the Lipschitz
approximation defined in Proposition 6.4 with $\eta:=\eta_{0}$. Then the
following holds, assuming

$Ce^{-K/\varepsilon}\leq
C\varepsilon^{\beta}\leq\bm{\mathrm{E}}:=\bm{\mathrm{E}}(u,\nabla,B^{n}_{2},\mathbb{R}^{n-2})$

for some $C=C(n,\nu,\beta)$ and $K=K(n)$: there exists a harmonic function
$w:B_{1}^{n-2}\to\mathbb{R}^{2}$ such that

  1. (i)

$\int_{B_{1}^{n-2}}|dw|^{2}\leq C(n)$;

  2. (ii)

we have

$\int_{B_{1}^{n-2}}\left|\frac{h-(h)_{B_{1}^{n-2}}}{\sqrt{\bm{\mathrm{E}}}}-w\right|^{2}\leq\nu,$

where $(h)_{B_{1}^{n-2}}$ is the average;

  3. (iii)

most importantly, we have

$\int_{B_{s}^{n-2}}\bm{\mathrm{E}}_{z}\leq\bm{\mathrm{E}}\int_{B_{s}^{n-2}}\frac{|dw|^{2}}{2}+\nu\bm{\mathrm{E}}.$

###### Proof.

We prove the statement by compactness and contradiction. Fix $\nu,\beta,s$ and
assume that there exist sequences $\varepsilon_{k},\tau_{k}\to 0$ and a
sequence of minimizers $(u_{k},\nabla_{k})$ for $E_{\varepsilon_{k}}$ with the
previous energy bound for $\tau_{0}=\tau_{k}$, violating the conclusion.
Moreover, let $h_{k}:B^{n-2}_{1}\to B^{2}_{1}$ be the Lipschitz approximation
for the threshold $\eta_{0}$, to be chosen below.

_Lower bound on the energy of the given pair._ By item (i) in Proposition 6.4
we have

$\displaystyle\int_{B^{n-2}_{1}}|dh_{k}|^{2}\leq
C(n)\bm{\mathrm{E}}^{(k)},\quad\bm{\mathrm{E}}^{(k)}:=\bm{\mathrm{E}}(u_{k},\nabla_{k},B^{n}_{2},\mathbb{R}^{n-2}),$

for $k$ large enough. Hence, up to a subsequence, we can extract a weak limit

$\frac{h_{k}-(h_{k})_{B_{1}^{n-2}}}{\sqrt{\bm{\mathrm{E}}^{(k)}}}\rightharpoonup
w$

in $W^{1,2}$, so that

$\int_{B_{1}^{n-2}}|dw|^{2}\leq C(n).$

By Lemma 7.1 and Proposition 7.2, $w$ is harmonic with $w(0)=0$. This shows
that the first two conclusions hold, so we must have

(11.1)
$\displaystyle\int_{B_{s}^{n-2}}\bm{\mathrm{E}}_{z}^{(k)}>\bm{\mathrm{E}}^{(k)}\int_{B_{s}^{n-2}}\frac{|dw|^{2}}{2}+\nu\bm{\mathrm{E}}^{(k)}.$

By Lemma 5.8 and the bound
$Ce^{-K/\varepsilon}\leq\frac{\nu}{5}\bm{\mathrm{E}}^{(k)}$, this gives

$\frac{1}{2\pi}\int_{B_{1}^{2}\times
B_{s}^{n-2}}e_{\varepsilon_{k}}(u_{k},\nabla_{k})>|B_{s}^{n-2}|+\bm{\mathrm{E}}^{(k)}\int_{B_{s}^{n-2}}\frac{|dw|^{2}}{2}+\frac{4\nu}{5}\bm{\mathrm{E}}^{(k)}.$

Let $a,b\in(s,1)$ with $a<b$, which we write as $b=a+4\delta$. Calling
$\mathcal{G}^{k}$ the good set for $(u_{k},\nabla_{k})$, since the indicator
function $\bm{1}_{(B^{n-2}_{a}\setminus B^{n-2}_{s})\cap\mathcal{G}^{k}}\to 1$
strongly $L^{2}(B^{n-2}_{a}\setminus B^{n-2}_{s})$, we have

$\bm{1}_{(B^{n-2}_{a}\setminus
B^{n-2}_{s})\cap\mathcal{G}^{k}}\frac{dh_{k}}{\sqrt{\bm{\mathrm{E}}^{(k)}}}\rightharpoonup
dw$

weakly in this space, and hence

$\int_{B^{n-2}_{a}\setminus
B^{n-2}_{s}}\frac{|dw|^{2}}{2}\leq\liminf_{k\to\infty}\int_{(B^{n-2}_{a}\setminus
B^{n-2}_{s})\cap\mathcal{G}^{k}}\frac{|dh_{k}|^{2}}{2\bm{\mathrm{E}}^{(k)}}.$

Using item (iv) in Proposition 6.4 and the assumption
$\bm{\mathrm{E}}^{(k)}\geq C\varepsilon_{k}^{\beta}\geq
Ce^{-K/\varepsilon_{k}}$, we deduce

$\int_{B^{n-2}_{a}\setminus
B^{n-2}_{s}}\frac{|dw|^{2}}{2}\leq\liminf_{k\to\infty}\int_{(B^{n-2}_{a}\setminus
B^{n-2}_{s})\cap\mathcal{G}^{k}}\frac{\bm{\mathrm{E}}_{z}^{(k)}}{\bm{\mathrm{E}}^{(k)}}.$

Combined with Eq. 11.1, this gives

$\int_{B^{n-2}_{a}}\bm{\mathrm{E}}_{z}^{(k)}>\bm{\mathrm{E}}^{(k)}\int_{B_{a}^{n-2}}\frac{|dw|^{2}}{2}+\frac{3\nu}{4}\bm{\mathrm{E}}^{(k)}.$

Using again Lemma 5.8 and $\bm{\mathrm{E}}^{(k)}\geq
C\varepsilon_{k}^{\beta}\geq Ce^{-K/\varepsilon_{k}}$, we obtain

(11.2) $\displaystyle\frac{1}{2\pi}\int_{B_{1}^{2}\times
B_{a}^{n-2}}e_{\varepsilon_{k}}(u_{k},\nabla_{k})>|B_{a}^{n-2}|+\bm{\mathrm{E}}^{(k)}\int_{B_{a}^{n-2}}\frac{|dw|^{2}}{2}+\frac{3\nu}{5}\bm{\mathrm{E}}^{(k)}.$

Note that, for a fixed small $\delta>0$ to be specified later, we can find $a$
and $b=a+4\delta$ in $(s,1)$ such that

(11.3) $\displaystyle\int_{B_{b}^{n-2}\setminus
B_{a}^{n-2}}[|dh_{k}|^{2}+\bm{\mathrm{E}}^{(k)}|dw|^{2}+\bm{\mathrm{E}}_{z}^{(k)}]\leq
C(n,s)\delta\bm{\mathrm{E}}^{(k)},$

along a subsequence, by the classical pigeonhole argument.

Now we take a cut-off function $\chi$ such that $\chi=1$ on $B_{a}^{n-2}$ and
$\chi=0$ outside of $B_{a+\delta}^{n-2}$, and we let

$f_{k}:=(1-\chi)h_{k}+\chi(\sqrt{\bm{\mathrm{E}}^{(k)}}w+(h_{k})_{B_{1}^{n-2}}).$

Since
$\|h_{k}-(h_{k})_{B_{1}^{n-2}}-\sqrt{\bm{\mathrm{E}}^{(k)}}w\|_{L^{2}}^{2}=o(\bm{\mathrm{E}}^{(k)})$,
the Dirichlet energy of $f_{k}$ on $B_{b}^{n-2}\setminus B_{a}^{n-2}$ is

$\int_{B_{b}^{n-2}\setminus
B_{a}^{n-2}}\left[(1-\chi)^{2}\frac{|dh_{k}|^{2}}{2}+\bm{\mathrm{E}}^{(k)}(2\chi-\chi^{2})\frac{|dw|^{2}}{2}\right]+o(\bm{\mathrm{E}}^{(k)}).$

In particular, by Eq. 11.3 we have

(11.4) $\displaystyle\int_{B_{b}^{n-2}\setminus B_{a}^{n-2}}|df_{k}|^{2}\leq
C\delta\bm{\mathrm{E}}^{(k)}.$

We apply Proposition 10.1 to obtain a new pair $(u_{f_{k}},\nabla_{f_{k}})$.

_Construction of the competitor._ We want to glue the latter to
$(u_{k},\nabla_{k})$ in a suitable annular region and obtain a new pair whose
energy in $B_{1}^{2}\times B_{b}^{n-2}$ is strictly lower than
$(u_{k},\nabla_{k})$, obtaining a contradiction to minimality. From now on, we
restrict attention to the region $B_{1}^{2}\times B_{b}^{n-2}$. We will also
drop the subscript $k$ in the sequel. Note that $f=h$ on
$B_{a+4\delta}^{n-2}\setminus B_{a+\delta}^{n-2}$.

For technical reasons, it will be convenient to glue on an annulus of width
$\sqrt{\varepsilon}$. We first select $t\in[a+2\delta,a+3\delta]$ such that

(11.5) $\displaystyle\int_{B_{t+2\sqrt{\varepsilon}}^{n-2}\setminus
B_{t-\sqrt{\varepsilon}}^{n-2}}[\bm{\mathrm{E}}_{z}+|dh|^{2}]\leq
C(n,s,\delta)\sqrt{\varepsilon}\bm{\mathrm{E}}.$

We first apply Proposition 10.2 and Corollary 10.6 to replace $(u,\nabla)$
with a gauge-equivalent pair, still denoted $(u,\nabla)$, such that
$\frac{u}{|u|}=\frac{u_{f}}{|u_{f}|}$ on $(B_{1}^{2}\setminus
B_{1/2}^{2})\times B_{b}^{n-2}$, with

$\int_{\mathcal{A}}[\varepsilon^{-2}|u-u_{h}|^{2}+|\alpha-\alpha_{h}|^{2}]\leq
C|\log\varepsilon|^{10}\int_{P(\hat{\mathcal{A}})}\bm{\mathrm{E}}_{z}+\varepsilon^{\beta},$

where $\mathcal{A}:=B_{1}^{2}\times(B_{t+\sqrt{\varepsilon}}^{n-2}\setminus
B_{t}^{n-2})$ and
$\hat{\mathcal{A}}:=B_{1}^{2}\times(B_{t+2\sqrt{\varepsilon}}^{n-2}\setminus
B_{t-\sqrt{\varepsilon}}^{n-2})$. In particular, by Eq. 11.5 we have

(11.6)
$\displaystyle\int_{\mathcal{A}}[\varepsilon^{-2}|u-u_{h}|^{2}+|\alpha-\alpha_{h}|^{2}]\leq
C\sqrt{\varepsilon}|\log\varepsilon|^{10}\int_{P(\hat{\mathcal{A}})}\bm{\mathrm{E}}_{z}+\varepsilon^{\beta}=o(\bm{\mathrm{E}})+\varepsilon^{\beta},$

where the notation $o(\bm{\mathrm{E}})=o(\bm{\mathrm{E}}^{(k)})$ indicates a
quantity infinitesimal with respect to $\bm{\mathrm{E}}^{(k)}$, as
$k\to\infty$.

We take another cut-off function $\varphi$ with $\varphi=1$ on
$B_{1}^{2}\times B_{t}^{n-2}$ and $\varphi=0$ outside of $B_{1}^{2}\times
B_{t+\sqrt{\varepsilon}}^{n-2}$. On $B_{1}^{2}\times B_{b}^{n-2}$, we define

$\tilde{u}:=(1-\varphi)u+\varphi
u_{h},\quad\tilde{\alpha}:=(1-\varphi)\alpha+\varphi\alpha_{h}.$

We claim that

(11.7) $\displaystyle\int_{P(\mathcal{A})}\tilde{\bm{\mathrm{E}}}_{z}\leq
o(\bm{\mathrm{E}})+C\varepsilon^{\beta}.$

Once this is done, using Lemma 5.8, we obtain

$\frac{1}{2\pi}\int_{P(\mathcal{A})}e_{\varepsilon}(\tilde{u},\tilde{\nabla})\leq|P(\mathcal{A})|+o(\bm{\mathrm{E}}_{z})+C\varepsilon^{\beta},$

and hence by Proposition 10.1, together with Eq. 11.3 and Eq. 11.4, we get

$\displaystyle\frac{1}{2\pi}\int_{B_{1}^{2}\times
B_{b}^{n-2}}e_{\varepsilon}(\tilde{u},\tilde{\nabla})$
$\displaystyle\leq|B_{b}^{n-2}|+(1+O(\eta_{0}^{2}))\int_{B_{b}^{n-2}}\frac{|df|^{2}}{2}+\int_{B_{b}^{n-2}\setminus
B_{a}^{n-2}}\bm{\mathrm{E}}_{z}+o(\bm{\mathrm{E}})+C\varepsilon^{\beta}$
$\displaystyle\leq|B_{b}^{n-2}|+\bm{\mathrm{E}}\int_{B_{a}^{n-2}}\frac{|dw|^{2}}{2}+\int_{B_{b}^{n-2}\setminus
B_{a}^{n-2}}[|df|^{2}+\bm{\mathrm{E}}_{z}]+C\eta_{0}^{2}\bm{\mathrm{E}}+o(\bm{\mathrm{E}})+C\varepsilon^{\beta}$
$\displaystyle\leq|B_{b}^{n-2}|+\bm{\mathrm{E}}\int_{B_{a}^{n-2}}\frac{|dw|^{2}}{2}+\frac{\nu}{5}\bm{\mathrm{E}},$

once we take $\eta_{0}$ and $\delta$ small enough. In the same way, Eq. 11.2
gives

$\frac{1}{2\pi}\int_{B_{1}^{2}\times
B_{b}^{n-2}}e_{\varepsilon}(u,\nabla)>|B_{b}^{n-2}|+\bm{\mathrm{E}}\int_{B_{a}^{n-2}}\frac{|dw|^{2}}{2}+\frac{2\nu}{5}\bm{\mathrm{E}}.$

This gives a contradiction: near $\partial B_{1}^{2}\times
B_{t+\sqrt{\varepsilon}}^{n-2}$, using the fact that $\tilde{u}$ and $u_{f}$
have the same phase, it is easy to modify $(\tilde{u},\tilde{\nabla})$ in
order to make it agree with $(u,\nabla)$ (while this already holds on
$\partial B_{1}^{2}\times(B_{b}^{n-2}\setminus
B_{t+\sqrt{\varepsilon}}^{n-2})$), in a way which changes the energy by
$O(e^{-K/\varepsilon})\leq\frac{\nu}{5}\bm{\mathrm{E}}$: it is enough to
interpolate between the two pairs on the set

$(B_{1}^{2}\setminus B_{1/2}^{2})\times B_{b}^{n-2},$

using the fact that here the energy density is exponentially small, and hence
we can write $|\tilde{u}-u_{f}|=|(1-|\tilde{u}|)-(1-|u_{f}|)|\leq
e^{-K/\varepsilon}$ and $|\tilde{\alpha}-\alpha_{f}|\leq e^{-K/\varepsilon}$
(since, writing $\tilde{u}=e^{i\tilde{\theta}}$, we have
$|d\tilde{\theta}-\tilde{\alpha}|\leq|\tilde{u}|^{-1}|\tilde{\nabla}\tilde{u}|$
and similarly
$|d\tilde{\theta}-\alpha_{f}|=|d\theta_{f}-\alpha_{f}|\leq|u_{f}|^{-1}|\nabla_{f}u_{f}|$).

_Bounding the energy on the interpolation annulus._ It remains to check the
previous claim. We first write

$\displaystyle 2\pi\tilde{\bm{\mathrm{E}}}_{z}$
$\displaystyle=\int_{B^{2}_{1}\times\\{z\\}}\sum_{j\geq
3}|\tilde{\nabla}_{e_{j}}\tilde{u}|^{2}+\sum_{(j,j^{\prime})\neq(1,2)}\varepsilon^{2}|d\tilde{\alpha}(e_{j},e_{j^{\prime}})|^{2}+|\tilde{\nabla}_{e_{1}}\tilde{u}+i\tilde{\nabla}_{e_{2}}\tilde{u}|^{2}+\left|\varepsilon
d\tilde{\alpha}(e_{1},e_{2})-\frac{1-|\tilde{u}|^{2}}{2\varepsilon}\right|^{2}$
$\displaystyle=:\bm{\mathrm{I}}+\bm{\mathrm{II}}+\bm{\mathrm{III}}+\bm{\mathrm{IV}}.$

We start by bounding $\bm{\mathrm{II}}$: we have

$d\tilde{\alpha}=(1-\varphi)d\alpha+\varphi
d\alpha_{h}+d\varphi\wedge(\alpha_{h}-\alpha).$

Hence, using the fact that $|d\varphi|\leq C\varepsilon^{-1/2}$, we have

$\bm{\mathrm{II}}\leq C\bm{\mathrm{E}}_{z}+C|dh|^{2}(z)+C\varepsilon^{2}\cdot
C\varepsilon^{-1}\int_{B^{2}_{1}\times\\{z\\}}|\alpha_{h}-\alpha|^{2}.$

The last term is bounded by the left-hand side of Eq. 11.6; together with Eq.
11.5, this gives the desired bound.

As for $\bm{\mathrm{IV}}$, we note that

$\frac{1-|\tilde{u}|^{2}}{2\varepsilon}=(1-\varphi)\frac{1-|u|^{2}}{2\varepsilon}+\varphi\frac{1-|u_{h}|^{2}}{2\varepsilon}+O(\varepsilon^{-1}|u-u_{h}|),$

and hence $\bm{\mathrm{IV}}$ is the squared norm of

$(1-\varphi)\left[\varepsilon
d\alpha(e_{1},e_{2})-\frac{1-|u|^{2}}{2\varepsilon}\right]+\varphi\left[\varepsilon
d\alpha_{h}(e_{1},e_{2})-\frac{1-|u_{h}|^{2}}{2\varepsilon}\right]+O(\sqrt{\varepsilon}|\alpha-\alpha_{h}|)+O(\varepsilon^{-1}|u-u_{h}|).$

Thus, we have

$\bm{\mathrm{IV}}\leq
C\bm{\mathrm{E}}_{z}+C\varepsilon\int_{B^{2}_{1}\times\\{z\\}}|\alpha-\alpha_{h}|^{2}+C\varepsilon^{-2}\int_{B^{2}_{1}\times\\{z\\}}|u-u_{h}|^{2},$

again bounded by Eq. 11.5 and Eq. 11.6.

We finally turn to $\bm{\mathrm{I}}$; the bound for $\bm{\mathrm{III}}$ is
obtained in the same way and hence will be skipped. We note that

$\displaystyle\tilde{\nabla}\tilde{u}$ $\displaystyle=d[(1-\varphi)u+\varphi
u_{h}]-i[(1-\varphi)u+\varphi u_{h}][(1-\varphi)\alpha+\varphi\alpha_{h}]$
$\displaystyle=(1-\varphi)\nabla
u+\varphi\nabla_{h}u_{h}+(u_{h}-u)d\varphi+O(|u-u_{h}||\alpha-\alpha_{h}|).$

Hence,

$\int_{B^{2}_{1}\times\\{z\\}}|\tilde{\nabla}_{e_{j}}\tilde{u}|^{2}\leq
C\bm{\mathrm{E}}_{z}+C|dh|^{2}(z)+C\varepsilon^{-1}\int_{B^{2}_{1}\times\\{z\\}}|u-u_{h}|^{2}+C\int_{B^{2}_{1}\times\\{z\\}}|\alpha-\alpha_{h}|^{2}.$

Again, the last two terms are bounded by the left-hand side of Eq. 11.6. This
completes the proof of Eq. 11.7, and hence the proof of the proposition. ∎

###  11.2. Proof of 1.10

In this section we finish the proof of the stronger decay of excess for
minimizers. We rescale $B_{1}^{n}(0)$ to $B_{2}^{n}(0)$ and apply Proposition
11.1, with some $s\in(0,\frac{1}{2})$ and $\nu>0$ to be chosen later and with
$\beta+1$ in place of $\beta$. We obtain that either
$\bm{\mathrm{E}}=\bm{\mathrm{E}}(u,\nabla,B_{2}^{n}(0),\mathbb{R}^{n-2})\leq
C\varepsilon^{\beta+1}$ or the conclusions of Proposition 11.1 hold true
(provided that the picture is rotated in such a way that $\bm{\mathrm{E}}$ is
small enough).

In the first situation, we clearly have
$\min_{S}\bm{\mathrm{E}}(u,\nabla,B_{2}^{n},S)\leq\varepsilon^{\beta}$ for
$\varepsilon$ small enough and we are done. Hence, in the sequel, we can
assume that we are in the second situation.

We will assume for simplicity that $\mathbb{R}^{n-2}$ minimizes
$\bm{\mathrm{E}}(u,\nabla,B_{2}^{n},\cdot)$ and that

(11.8) $|dw(0)|\leq\delta$

with $\delta>0$ to be chosen momentarily.

Since $|dw(z)-dw(0)|\leq s\sup_{B_{s}^{n-2}}|D^{2}w|$ on $B_{s}^{n-2}$, we
have

$\int_{B_{s}^{n-2}}|dw|^{2}\leq
C(n)s^{n-2}\delta^{2}+C(n)s^{n}\sup_{B_{s}^{n-2}}|D^{2}w|^{2}\leq
C(n)s^{n-2}(\delta^{2}+s^{2}),$

where the last inequality comes from the bound $\|dw\|_{L^{2}}\leq C(n)$ and
standard elliptic estimates.

By item (iii) from Proposition 11.1 we then have

$\int_{B_{s}^{n-2}}\bm{\mathrm{E}}_{z}\leq\bm{\mathrm{E}}\int_{B_{s}^{n-2}}\frac{|dw|^{2}}{2}+\nu\bm{\mathrm{E}}\leq
C(n)s^{n-2}(\delta^{2}+s^{2}+s^{2-n}\nu)\bm{\mathrm{E}}.$

This immediately gives

$\bm{\mathrm{E}}(u,\nabla,B_{s}^{n},\mathbb{R}^{n-2})\leq
C(n)(\delta^{2}+s^{2}+s^{2-n}\nu)\bm{\mathrm{E}}.$

The theorem follows under the assumption (11.8) by taking $\delta,s$ and
_subsequently_ $\nu$ small enough. The general case can be reduced to this one
by the very same argument of Section 8.2; the only differences here are that
we use item (ii) from Proposition 11.1 in order to bound

$\|h-(h)_{B_{1}^{n-2}}-\sqrt{\bm{\mathrm{E}}}w\|_{L^{2}}^{2}\leq\nu\bm{\mathrm{E}}$

and that $\bm{\mathrm{E}}_{1}$ is replaced by $\bm{\mathrm{E}}$ throughout
that argument.

###  11.3. Proof of 1.9: the case of minimizers

We prove the following theorem, which contains the second part of 1.9.

######  Theorem 11.2.

For $n\geq 2$, there exists $\tau_{0}(n)>0$ such that the following holds. If
$(u,\nabla)$ is an entire, local minimizer for the energy $E_{1}$, with
$u(0)=0$ and the energy bound

$\displaystyle\frac{1}{|B^{n-2}_{R}|}\int_{B_{R}^{n}}\left[|\nabla
u|^{2}+|F_{\nabla}|^{2}+\frac{1}{4}(1-|u|^{2})^{2}\right]\leq 2\pi+\tau_{0},$

then $(u,\nabla)$ is two-dimensional. More precisely, we have
$(u,\nabla)=P^{*}(u_{0},\nabla_{0})$ up to a change of gauge, where $P$ is the
orthogonal projection onto a two-dimensional subspace and $(u_{0},\nabla_{0})$
is the standard degree-one solution of Taubes [51] (or its conjugate),
centered at the origin.

###### Proof.

We can assume $n\geq 3.$ We proceed exactly as in the proof of 1.9: letting
$\beta:=n-2>0$, it is enough to prove that

$\lim_{R\to\infty}R^{\beta}\min_{S}\bm{\mathrm{E}}_{1}(u,\nabla,B_{R}^{n},S)=0.$

This follows from the stronger excess decay statement for minimizers, using
the same iteration argument employed in the proof of 1.9. ∎

##  Appendix A Barycenter and variance of good slices

We show two lemmas which give a more refined control of a critical pair on a
_good slice_ $B_{1}^{2}\times\\{z\\}$, with $z\in\mathcal{G}^{\eta}$, the good
set defined in Eq. 6.2. We assume that $(u,\nabla)$ is a critical pair for
$E_{\varepsilon}$, defined on $B_{1}^{2}\times B_{1}^{n-2}$, with
$\varepsilon\leq\varepsilon_{0}$ and

$E_{\varepsilon}(u,\nabla)\leq|B_{1}^{n-2}|(2\pi+\tau_{0})$

(as well as Eq. 4.1–Eq. 4.2). Under this assumption, we have

$\int_{(B_{3/4}^{2}\setminus
B_{1/2}^{2})\times\\{z\\}}e_{\varepsilon}(u,\nabla)\leq e^{-K(n)/\varepsilon}$

for $z\in B_{3/4}^{n-2}$, since this part of the slice is far from the
vorticity set. Recall that the barycenter

$h(z)=\Phi_{\chi(x_{1},x_{2})}(z)$

was defined using a cut-off function $\chi$ supported in $B_{3/4}^{2}$, with
$\chi=1$ on $B_{1/2}^{2}$ (the notation in the subscript means
$\chi\cdot(x_{1},x_{2})$).

######  Lemma A.1 (Barycenter of a good slice).

For $\varepsilon_{0},\tau_{0},\eta_{0}>0$ small enough, if $\eta\leq\eta_{0}$
and $z\in\mathcal{G}^{\eta}$, then we have the following estimate (for a
possibly different $K=K(n)$):

$\displaystyle|h(z)-h_{0}(z)|\leq
C(n)\varepsilon|\log(\bm{\mathrm{E}}_{2})_{z}|(\bm{\mathrm{E}}_{2})_{z}^{1/2}+e^{-K/\varepsilon},$

where $h_{0}$ is the map from Proposition 6.6 giving the zero set on good
slices.

In other words, the barycenter of the good slice is close to the actual zero
of $u$ here (unique in $B_{1/2}^{2}\times\\{z\\}$).

###### Proof.

Recall that, by definition, we have

$\displaystyle
h(z)=\Phi_{\chi(x_{1},x_{2})}(z)=\frac{1}{2\pi}\int_{B^{2}_{1/2}\times\\{z\\}}(x_{1},x_{2})J(u,\nabla)(e_{1},e_{2})+Ce^{-K/\varepsilon}.$

Since the integral of the Jacobian on $B_{1/2}^{2}\times\\{z\\}$ is
$2\pi+O(e^{-K/\varepsilon})$ (see, e.g., the proof of [42, Lemma 6.11]), we
get

$h(z)-h_{0}(z)=\frac{1}{2\pi}\int_{B^{2}_{1/2}\times\\{z\\}}[(x_{1},x_{2})-h_{0}(z)]J(u,\nabla)(e_{1},e_{2})+Ce^{-K/\varepsilon}.$

On the other hand, using the notation from Proposition 10.1, we have

$\left|\frac{1}{2\pi}\int_{B_{1/2}^{2}\times\\{z\\}}[(x_{1},x_{2})-h_{0}(z)]J(u_{h_{0}},\nabla_{h_{0}})(e_{1},e_{2})\right|\leq
Ce^{-K/\varepsilon},$

by symmetry of the standard planar solution.

Moreover, $u(\cdot,z)$ vanishes linearly at $h_{0}(z)$, as observed in Lemma
5.7. We can then apply a rescaling of [31, Theorem C.1], which gives

(A.1)
$\displaystyle\int_{B^{2}_{1/2}\times\\{z\\}}|J(u,\nabla)(e_{1},e_{2})-J(u_{h_{0}},\nabla_{h_{0}})(e_{1},e_{2})|\leq
C\sqrt{(\bm{\mathrm{E}}_{2})_{z}}+Ce^{-{K}/{\varepsilon}}.$

Selecting a radius $C(n)\varepsilon\leq r\leq\frac{1}{4}$, we have

$e_{\varepsilon}(u,\nabla)(y,z)\leq
C(n)\varepsilon^{-2}e^{-K|y-h_{0}(z)|/\varepsilon}\quad\text{on
}[B_{1/2}^{2}\setminus B_{r}^{2}(h_{0}(z))]\times\\{z\\}$

for a possibly different $K$, since as observed in Lemma 5.7 the distance from
the vorticity set $Z$ is comparable to the distance from

$Z\cap(B_{3/4}^{2}\times\\{z\\})\subseteq
B_{C(n)\varepsilon}^{2}(h_{0}(z))\times\\{z\\},$

on good slices. Hence,

$\displaystyle\int_{B^{2}_{1/2}\times\\{z\\}}|(x_{1},x_{2})-h_{0}(z)||J(u,\nabla)(e_{1},e_{2})-J(u_{h_{0}},\nabla_{h_{0}})(e_{1},e_{2})|$
$\displaystyle\leq
r\int_{B_{r}^{2}(h_{0}(z))\times\\{z\\}}|J(u,\nabla)(e_{1},e_{2})-J(u_{h_{0}},\nabla_{h_{0}})(e_{1},e_{2})|$
$\displaystyle\quad+C\varepsilon^{-2}\int_{B_{1/2}^{2}\setminus
B_{r}^{2}(h_{0}(z))}|y-h_{0}(z)|e^{-K|y-h_{0}(z)|/\varepsilon}\,dy$
$\displaystyle\leq Cr\sqrt{\bm{\mathrm{E}}_{z}}+C\varepsilon
e^{-{Kr}/{\varepsilon}}$

for a possibly different $K$. Taking
$r:=M\varepsilon|\log\bm{\mathrm{E}}_{z}|$ for big enough $M$, we get

$r\sqrt{\bm{\mathrm{E}}_{z}}+\varepsilon e^{-{Kr}/{\varepsilon}}\leq
M\varepsilon\sqrt{\bm{\mathrm{E}}_{z}}|\log\bm{\mathrm{E}}_{z}|+\varepsilon\sqrt{\bm{\mathrm{E}}_{z}}\leq
C(n)\varepsilon\sqrt{\bm{\mathrm{E}}_{z}}|\log\bm{\mathrm{E}}_{z}|$

(recall that $\bm{\mathrm{E}}_{z}\leq\frac{1}{2}$, by definition of good set),
unless $r<C(n)\varepsilon$ or $r>\frac{1}{4}$. The situation
$r<C(n)\varepsilon$ cannot happen, once $M$ is taken large enough, while in
the last case we obtain $\bm{\mathrm{E}}_{z}\leq e^{-K^{\prime}/\varepsilon}$
and thus we are done again, by taking $r:=\frac{1}{4}$ above. ∎

Next we show that on a good slice the variance is close to
$\varepsilon^{2}v_{0}$, where $v_{0}$ is the variance of the standard degree-
one planar solution.

######  Lemma A.2 (Variance of a good slice).

For any $\sigma\in(\varepsilon,1)$ such that $|h_{0}(z)|\leq\sigma$, we have
the following estimate on good slices, for any $c\in\mathbb{R}^{2}$ with
$|c|\leq\sigma$:

$\displaystyle\left|\frac{1}{2\pi}\int_{B_{1/2}^{2}\times\\{z\\}}[(x_{1}-c_{1})^{2}+(x_{2}-c_{2})^{2}]e_{\varepsilon}(u,\nabla)-\varepsilon^{2}v_{0}\right|$
$\displaystyle\leq
C(n)\varepsilon^{2}|\log(\bm{\mathrm{E}}_{2})_{z}|^{2}\sqrt{(\bm{\mathrm{E}}_{2})_{z}}+C(n)\sigma^{2}(\bm{\mathrm{E}}_{1})_{z}+C(n)|h(z)-c|^{2}\sqrt{(\bm{\mathrm{E}}_{2})_{z}}+C(n)e^{-{K\sigma}/{\varepsilon}},$

for a possibly different $K=K(n)$.

###### Proof.

First of all, since the integrand in the definition of excess
$\bm{\mathrm{E}}=\bm{\mathrm{E}}_{1}+\bm{\mathrm{E}}_{2}$ upper bounds
$e_{\varepsilon}(u,\nabla)-J(u,\nabla)$, we can replace
$e_{\varepsilon}(u,\nabla)$ with $J(u,\nabla)$, up to an error bounded as
follows: for $\bm{\mathrm{E}}_{1}$, we bound separately the contribution of
$B_{2\sigma}^{2}$ and the complement (where we use exponential decay)
obtaining the error

$9\sigma^{2}(\bm{\mathrm{E}}_{1})_{z}+C(n)e^{-K\sigma/\varepsilon};$

as for $\bm{\mathrm{E}}_{2}$, we argue as in the previous proof, obtaining the
error

$C(n)|h_{0}(z)-c|^{2}(\bm{\mathrm{E}}_{2})_{z}+C(n)\varepsilon^{2}|\log(\bm{\mathrm{E}}_{2})_{z}|^{2}(\bm{\mathrm{E}}_{2})_{z}+e^{-K/\varepsilon},$

where the first term comes from replacing $c$ with the actual location
$h_{0}(z)$ of the zero. Moreover, by definition of $v_{0}$, we have

$\displaystyle\frac{1}{2\pi}\int_{B^{2}_{1/2}\times\\{z\\}}|(x_{1},x_{2})-h_{0}(z)|^{2}J(u_{h_{0}},\nabla_{h_{0}})(e_{1},e_{2})=\varepsilon^{2}v_{0}+O(e^{-{K}/{\varepsilon}});$

since
$|(x_{1},x_{2})-h_{0}(z)|^{2}-|(x_{1},x_{2})-c|^{2}=2\langle(x_{1},x_{2})-h_{0}(z),c-h_{0}(z)\rangle-|c-h_{0}(z)|^{2}$
and

$\int_{B_{1/2}^{2}\times\\{z\\}}[(x_{1},x_{2})-h_{0}(z)]J(u_{h_{0}},\nabla_{h_{0}})=O(e^{-K/\varepsilon}),$

we obtain

$\frac{1}{2\pi}\int_{B^{2}_{1/2}\times\\{z\\}}|(x_{1},x_{2})-c|^{2}J(u_{h_{0}},\nabla_{h_{0}})(e_{1},e_{2})=|h_{0}(z)-c|^{2}+\varepsilon^{2}v_{0}+O(e^{-{K}/{\varepsilon}}).$

As in the previous proof, we can replace $J(u_{0},\nabla_{0})$ with
$J(u,\nabla)$ here, up to an error of the form
$C(n)(|h_{0}(z)-c|^{2}+\varepsilon^{2}|\log\bm{\mathrm{E}}_{z}|^{2})\sqrt{(\bm{\mathrm{E}}_{2})_{z}}$
(using Eq. 4.15 from Proposition 4.8). Finally, we can bound

$|h_{0}(z)-c|^{2}\leq 2|h_{0}(z)-h(z)|^{2}+2|h(z)-c|^{2}\leq
2|h(z)-c|^{2}+C(n)\varepsilon^{2}|\log(\bm{\mathrm{E}}_{2})_{z}|^{2}(\bm{\mathrm{E}}_{2})_{z}$

using the previous proposition, and the claim follows. ∎

######  Remark A.3.

Since $t\mapsto t|\log t|^{2}$ is concave for $t>0$ small enough, we have

$\int_{S}\bm{\mathrm{E}}_{z}|\log\bm{\mathrm{E}}_{z}|^{2}\leq\left(\int_{S}\bm{\mathrm{E}}_{z}\right)\left|\log\left(\fint_{S}\bm{\mathrm{E}}_{z}\right)\right|^{2}\leq
C\left(\int_{S}\bm{\mathrm{E}}_{z}\right)\left|\log\left(\fint_{S}\bm{\mathrm{E}}_{z}\right)\right|^{2}$

for sets $S\subseteq\mathcal{G}^{\eta}$ of measure comparable with $1$.

##  Appendix B Poincaré–Gaffney-type inequalities

In the construction of the interpolation gauge in Proposition 10.2 we make
frequent use of Poincaré-type inequalities for functions and differential
forms. These inequalities are well known; we present the special cases used in
this paper for the convenience of the reader.

The following lemma is a consequence of results first appeared in the original
paper of Gaffney [27] (see also [35] for a systematic treatment on manifolds
with boundary), but for our application we need it to hold uniformly for
cylinders of the form $B^{2}_{1}\times B^{n-2}_{r}$ of arbitrarily small width
$r>0$.

######  Lemma B.1 (Poincaré–Gaffney-type inequality for thin cylinders).

Given a $1$-form
$\alpha\in\Omega^{1}(\overline{B}^{2}_{1}\times\overline{B}_{r}^{n-2})$ with
$r\leq 1$ and the Neumann boundary condition $\alpha(\nu)=0$ on
$\partial(B^{2}_{1}\times B^{n-2}_{r})$, the following inequality holds:

$\displaystyle\int_{B^{2}_{1}\times B_{r}^{n-2}}|\alpha|^{2}\leq
C(n)\int_{B^{2}_{1}\times B^{n-2}_{r}}[|d\alpha|^{2}+|d^{*}\alpha|^{2}].$

###### Proof.

Since $B^{2}_{1}\times B^{n-2}_{r}$ is a convex domain and
$\iota_{\nu}\alpha=0$ at its boundary, we can apply [15, Remark 9] to see that

$\displaystyle\int_{B^{2}_{1}\times
B^{n-2}_{r}}|\nabla\alpha|^{2}\leq\int_{B^{2}_{1}\times
B^{n-2}_{r}}[|d\alpha|^{2}+|d^{*}\alpha|^{2}].$

Now we rescale the domain with the map $\phi:B^{2}_{1}\times
B^{n-2}_{1}\rightarrow B^{2}_{1}\times B_{r}^{n-2}$ given by

$\phi(x_{1},\dots,x_{n}):=(x_{1},x_{2},rx_{3},\dots,rx_{n}),$

and define $\tilde{\alpha}(x):=\alpha(\phi(x))$ (notice that this is different
from the pullback $\phi^{*}(\alpha)$). Then we claim that there exists a
constant $C(n)>0$ such that

(B.1) $\displaystyle\int_{B^{2}_{1}\times B^{n-2}_{1}}|\tilde{\alpha}|^{2}\leq
C(n)\int_{B^{2}_{1}\times B^{n-2}_{1}}|\nabla\tilde{\alpha}|^{2}\,.$

We prove this by compactness and contradiction. By homogeneity, suppose there
exists a sequence $\tilde{\alpha}_{k}$ with $\iota_{\nu}\tilde{\alpha}_{k}=0$
on $\partial(B_{1}^{2}\times B^{n-2}_{1})$ and

$\displaystyle\int_{B^{2}_{1}\times
B^{n-2}_{1}}|\tilde{\alpha}_{k}|^{2}=1,\quad\lim_{k\rightarrow\infty}\int_{B^{2}_{1}\times
B^{n-2}_{1}}|\nabla\tilde{\alpha}_{k}|^{2}=0.$

Note that by the display above we have the bound
$\|\tilde{\alpha}_{k}\|_{W^{1,2}(B^{2}_{1}\times B_{1}^{n-2})}\leq 2$ for all
large $k\geq 0$. Up to extracting a subsequence, we can assume that
$\tilde{\alpha}_{k}$ converges weakly to $\tilde{\alpha}_{\infty}$ in
$W^{1,2}$. By Rellich–Kondrachov, the convergence is strong in $L^{2}$. Thus,

$\int_{B^{2}_{1}\times
B^{n-2}_{1}}|\tilde{\alpha}_{\infty}|^{2}=1,\quad\nabla\tilde{\alpha}_{\infty}=0.$

Hence, $\tilde{\alpha}_{\infty}=v$ is a constant covector. The boundary
condition passes to the limit, giving that $v(\nu)=0$ on
$\partial(B^{2}_{1}\times B^{n-2}_{1})$; since the normal vectors to the
boundary of this domain span all of $\mathbb{R}^{n}$, we get that $v=0$, a
contradiction establishing Eq. B.1. Then we compute that

$\displaystyle\int_{B^{2}_{1}\times B^{n-2}_{r}}|\alpha|^{2}$
$\displaystyle=r^{n-2}\int_{B^{2}_{1}\times B^{n-2}_{1}}|\tilde{\alpha}|^{2}$
$\displaystyle\leq C(n)r^{n-2}\int_{B^{2}_{1}\times
B^{n-2}_{1}}|\nabla\tilde{\alpha}|^{2}$ $\displaystyle\leq
C(n)\int_{B^{2}_{1}\times B^{n-2}_{r}}|\nabla\alpha|^{2}$ $\displaystyle\leq
C(n)\int_{B^{2}_{1}\times B^{n-2}_{r}}[|d\alpha|^{2}+|d^{*}\alpha|^{2}],$

as desired. ∎

The next lemma is a weighted Poincaré estimate for functions in two
dimensions.

######  Lemma B.2.

There exists a constant $C>0$ such that for any compactly supported function
$f\in C^{1}_{c}(B^{2}_{R})$ the following weighted Poincare type estimate
holds:

$\displaystyle\int_{B^{2}_{R}}|x|^{2}|f|^{2}(x)\leq
CR^{3/2}\int_{B^{2}_{R}}|x|^{5/2}|df|^{2}(x).$

###### Proof.

By scaling the domain, we can assume that $R=1$. Then by [13, eq. (1.4)] (for
the choice of constants $\alpha:=5/4$, $a:=1$, $p=q=r:=2$,
$\gamma,\sigma:=1/4$) we can see that

$\displaystyle\int_{B^{2}_{1}}|x|^{2}|f|^{2}(x)\leq\int_{B^{2}_{1}}|x|^{1/2}|f|^{2}(x)\leq
C\int_{B^{2}_{1}}|x|^{5/2}|df|^{2}(x).$

This is indeed the desired conclusion. ∎

######  Lemma B.3 (Poincaré inequality on a thin annulus).

Given $a>b\geq c\geq 0$, there exists a constant $C(n,a,b,c)>0$ with the
following property. Let $f$ be a function in $W^{1,2}((B^{2}_{a}\setminus
B^{2}_{c})\times\Omega)$, where $\Omega\subseteq\mathbb{R}^{n-2}$ is a convex
bounded domain, such that

$\displaystyle\int_{(B^{2}_{a}\setminus B^{2}_{b})\times\Omega}f=0.$

Then the following Poincaré inequality holds:

$\displaystyle\int_{(B^{2}_{a}\setminus B^{2}_{c})\times\Omega}|f|^{2}\leq
C(n,a,b,c)\operatorname{diam}(\Omega)^{2}\int_{(B^{2}_{a}\setminus
B^{2}_{c})\times\Omega}|df|^{2}.$

###### Proof.

First we apply the standard Poincaré inequality on each two dimensional slice
$(B^{2}_{a}\setminus B^{2}_{c})\times\\{z\\}$ for any $z\in\Omega$:

$\displaystyle\int_{\Omega}\left[\int_{(B^{2}_{a}\setminus
B^{2}_{c})\times\\{z\\}}|f|^{2}\right]\,dz$ $\displaystyle\leq
C(a,b,c)\int_{(B^{2}_{a}\setminus
B^{2}_{c})\times\Omega}|df|^{2}+\int_{\Omega}\left|\int_{(B^{2}_{c}\setminus
B^{2}_{b})\times\\{z\\}}f\right|^{2}\,dz.$

Notice that the function $g(z):=\int_{(B^{2}_{a}\setminus
B^{2}_{b})\times\\{z\\}}f$ has zero average on $\Omega$. Hence we can apply
the Poincaré inequality on $\Omega$ to see that

$\int_{\Omega}|g|^{2}\leq
C(n)\operatorname{diam}(\Omega)^{2}\int_{\Omega}|dg|^{2}.$

Indeed, it is well-known that the Poincaré inequality on a convex domain holds
with a constant depending only on its diameter and $n$. This yields the
desired conclusion. ∎

######  Remark B.4.

The same conclusion holds if we assume that

$\int_{(B^{2}_{a}\setminus B^{2}_{b})\times\Omega^{\prime}}f=0$

for some $\Omega^{\prime}$ with $|\Omega^{\prime}|\geq\alpha|\Omega|$ (the
constant depending also on $\alpha$).

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