* •

$\ell(x,r)<\varepsilon\operatorname{dist}(\Phi(x),\Phi(\partial\omega))$,

  * •

$0<\int_{\\_}{B_{\\_}{2r}^{2}(x)}\left|\nabla\Phi\right|^{2}\,d\mathcal{L}^{2}<C^{\prime}\int_{\\_}{B_{\\_}r^{2}(x)}\left|\nabla\Phi\right|^{2}\,d\mathcal{L}^{2}$,

  * •

$\frac{\left\|\mathbf{v}_{\\_}\omega\right\|(B_{\\_}s^{q}(\Phi(x)))}{\pi
s^{2}}\in(\widetilde{N}(x)-\varepsilon,\widetilde{N}(x)+\varepsilon)$ for all
$0<s<\varepsilon^{-1}\ell(x,r)$,

there exists $r^{\prime}\in\left(\varepsilon r,\frac{r}{2}\right)$ such that
$\int_{\\_}{B_{\\_}{2r^{\prime}}^{2}(x)}\left|\nabla\Phi\right|^{2}\,d\mathcal{L}^{2}<\overline{C}\int_{\\_}{B_{\\_}{r^{\prime}}^{2}(x)}\left|\nabla\Phi\right|^{2}\,d\mathcal{L}^{2}$,
for some $\overline{C}$ depending only on $\sup_{\\_}\Omega\widetilde{N}$.

###### Proof.

Assume by contradiction that the statement is false for all
$\varepsilon=2^{-k}$. We can then find a sequence of points
$x_{\\_}k\in\omega_{\\_}k$ and radii
$r_{\\_}k<\frac{1}{2}\operatorname{dist}(x_{\\_}k,\partial\omega_{\\_}k)$ such
that (• ‣ 4) and (• ‣ 4) are satisfied, as well as
$\widetilde{N}(x_{\\_}k)\to\overline{N}\in[1,\sup_{\\_}\Omega\widetilde{N}]$
(up to subsequences),

(5.2)
$\frac{\left\|\mathbf{v}_{\\_}{\omega_{\\_}k}\right\|(B_{\\_}s^{q}(\Phi(x_{\\_}k)))}{\pi
s^{2}}\in(\widetilde{N}(x_{\\_}k)-2^{-k},\widetilde{N}(x_{\\_}k)+2^{-k})\quad\text{for
}0<s<2^{k}\ell(x_{\\_}k,r_{\\_}k),$ (5.3)
$\int_{\\_}{B_{\\_}{2r^{\prime}}^{2}(x_{\\_}k)}\left|\nabla\Phi\right|^{2}\,d\mathcal{L}^{2}\geq\overline{C}\int_{\\_}{B_{\\_}{r^{\prime}}^{2}(x_{\\_}k)}\left|\nabla\Phi\right|^{2}\,d\mathcal{L}^{2}\quad\text{for
}2^{-k}r_{\\_}k<r^{\prime}<\frac{r_{\\_}k}{2}$

($\overline{C}$ will be chosen at the end of the proof). Moreover, using the
notation introduced in Section 4, the varifolds
$\mathbf{v}_{\\_}k:=(\ell_{\\_}k^{-1}(\cdot-\Phi(x_{\\_}k)))_{\\_}*\mathbf{v}_{\\_}{\omega_{\\_}k}$
have generalized mean curvature bounded by $O(\ell_{\\_}k)$ (in $L^{\infty}$)
in

$\mathbb{R}^{q}\setminus\ell_{\\_}k^{-1}(\Phi(\partial\omega_{\\_}k)-\Phi(x_{\\_}k))\supseteq
B_{\\_}{2^{k}}^{q}(0).$

Hence, up to subsequences, the varifolds $\mathbf{v}_{\\_}k$ converge to a
stationary varifold $\mathbf{v}_{\\_}\infty$ in $\mathbb{R}^{q}$. Moreover, by
(5.2), we have

$\frac{\left\|\mathbf{v}_{\\_}\infty\right\|(B_{\\_}s^{q}(0))}{\pi
s^{2}}=\frac{\left\|\mathbf{v}_{\\_}\infty\right\|(\overline{B}_{\\_}s^{q}(0))}{\pi
s^{2}}=\overline{N}.$

The varifold $\mathbf{v}_{\\_}\infty$ is rectifiable and conical, with density
in $[1,\overline{N}]$ on
$\operatorname{supp}\left(\mathbf{v}_{\\_}\infty\right)$ (see e.g. the proofs
of [44, Corollary 42.6 and Theorem 19.3]). In particular, since
$\mu_{\\_}k\leq\left\|\mathbf{v}_{\\_}k\right\|$, (• ‣ 4) follows, with
$C^{\prime\prime}:=\overline{N}$. So, up to subsequences, the conclusions of
Theorem 4.1 hold. We remark that $\Phi_{\\_}\infty$ satisfies

$\int_{\\_}{B_{\\_}1^{2}(0)}N_{\\_}\infty\left|\partial_{\\_}1\Phi_{\\_}\infty\wedge\partial_{\\_}2\Phi_{\\_}\infty\right|\,d\mathcal{L}^{2}=\nu_{\\_}\infty(B_{\\_}1^{2}(0))=\lim_{\\_}{k\to\infty}\nu_{\\_}k(B_{\\_}1^{2}(0))=\frac{1}{2}$

and in particular it is nonconstant. The varifold $\mathbf{v}_{\\_}\infty$ has
also integer multiplicity by [2, Theorem 6.4], thus it can be expressed as a
cone (with vertex $0$) over some stationary integer 1-rectifiable varifold
$\mathbf{w}$ in $S^{q-1}$ with density in $[1,\overline{N}]$
$\left\|\mathbf{w}\right\|$-a.e.

By the structure theorem in [3, Section 3], $\mathbf{w}$ is supported in a
finite union of geodesic curves. Hence, $\mathbf{v}_{\\_}\infty$ is supported
in a finite union of planes through the origin. Letting
$\Psi:=\Phi_{\\_}\infty\circ\varphi_{\\_}\infty^{-1}$, by (4.12) and Lemma 4.3
we have

$\big{\|}\mathbf{v}_{\\_}{(\Omega_{\\_}\infty,\Psi,N_{\\_}\infty\circ\varphi_{\\_}\infty^{-1})}\big{\|}=(\Phi_{\\_}\infty)_{\\_}*\nu_{\\_}\infty=\mu_{\\_}\infty\leq\left\|\mathbf{v}_{\\_}\infty\right\|.$

So, using Proposition 2.4, we deduce that
$\Psi(\Omega_{\\_}\infty)\subseteq\operatorname{supp}\left(\left\|\mathbf{v}_{\\_}\infty\right\|\right)$.
Hence, Theorem 3.7 applies: we obtain that $\Psi$ takes values in a plane and
is a holomorphic function (once this plane is suitably identified with
$\mathbb{C}$). Furthermore, by Lemma 4.5, we can assume
$\left\|N_{\\_}\infty\right\|_{\\_}\infty\leq C^{\prime\prime}=\overline{N}$.
Now, by Lemma 2.8 and Lemma 4.5,

$\frac{1}{2}\widetilde{N}(x_{\\_}k+r_{\\_}k\cdot)\left|\nabla\Phi_{\\_}k\right|^{2}\bm{1}_{\\_}{B_{\\_}2^{2}(0)}\mathcal{L}^{2}=\nu_{\\_}k\stackrel{{\scriptstyle*}}{{\rightharpoonup}}N_{\\_}\infty\left|\partial_{\\_}1\Phi_{\\_}\infty\wedge\partial_{\\_}2\Phi_{\\_}\infty\right|\mathcal{L}^{2},$

so (5.3) gives

$\int_{\\_}{B_{\\_}{2r}^{2}(0)}\left|\partial_{\\_}1\Phi_{\\_}\infty\wedge\partial_{\\_}2\Phi_{\\_}\infty\right|\,d\mathcal{L}^{2}\geq\frac{\overline{C}}{(\sup_{\\_}\Omega\widetilde{N})^{2}}\int_{\\_}{B_{\\_}r^{2}(0)}\left|\partial_{\\_}1\Phi_{\\_}\infty\wedge\partial_{\\_}2\Phi_{\\_}\infty\right|\,d\mathcal{L}^{2}\quad\text{for
}r<\frac{1}{2}.$

Let $\overline{k}\geq 1$ be the biggest integer such that
$\left|\Psi(w)\right|=O\left(\left|w\right|^{\overline{k}}\right)$. We claim
that

$\overline{k}\leq\overline{N}.$

Indeed, since $\Psi$ is nonconstant and holomorphic, we have
$K_{\\_}w=\left\\{w\right\\}$ for all $w\in\Omega_{\\_}\infty$ and the
function $\widetilde{N}$ for the local parametrized stationary varifold
$(\Omega_{\\_}\infty,\Psi,N_{\\_}\infty\circ\varphi_{\\_}\infty^{-1})$ is
everywhere finite (see Remark 2.11). We call it $N^{\prime}$ in order not to
confuse it with the same function for $(\Omega,\Phi,N)$. Since the density of
the varifold
$\mathbf{v}_{\\_}{(\Omega_{\\_}\infty,\Psi,N_{\\_}\infty\circ\varphi_{\\_}\infty^{-1})}$
is bounded everywhere by $\overline{N}$ (being this true for
$\mathbf{v}_{\\_}\infty$), the same argument used to prove Lemma 2.6 gives

$\overline{k}\leq\sum_{\\_}{j=1}^{\overline{k}}N^{\prime}(w_{\\_}j)\leq\overline{N}.$

whenever $w_{\\_}1,\dots,w_{\\_}{\overline{k}}\in\Omega_{\\_}\infty$ are
distinct points with the same image. Such points exist because $\Psi$ is a
$\overline{k}$-to-$1$ map near $0$. This establishes our claim and we deduce
that

$\lim_{\\_}{r\to
0}\frac{\int_{\\_}{B_{\\_}{2r}^{2}(0)}\left|\nabla\Psi\right|^{2}\,d\mathcal{L}^{2}}{\int_{\\_}{B_{\\_}r^{2}(0)}\left|\nabla\Psi\right|^{2}\,d\mathcal{L}^{2}}=2^{2\overline{k}}\leq
2^{2\overline{N}}.$

As was shown in the proof of Theorem 4.1, $\varphi_{\\_}\infty$ is an
$\overline{N}^{2}$-quasiconformal homeomorphism. Since
$\overline{N}\leq\sup_{\\_}\Omega\widetilde{N}$, we claim that there exists a
constant $\mathcal{K}\in\mathbb{N}$ depending only on
$\sup_{\\_}\Omega\widetilde{N}$ such that, for $r>0$ small enough, there
exists $s=s(r)>0$ with

(5.4)
$B_{\\_}s^{2}(0)\subseteq\varphi_{\\_}\infty(B_{\\_}r^{2}(0))\subseteq\varphi_{\\_}\infty(B_{\\_}{2r}^{2}(0))\subseteq
B_{\\_}{2^{\mathcal{K}}s}^{2}(0).$

Let $s:=\min_{\\_}{z\in\partial
B_{\\_}r^{2}(0)}\left|\varphi_{\\_}\infty(z)\right|$,
$s^{\prime}:=\max_{\\_}{z\in\partial
B_{\\_}{2r}^{2}(0)}\left|\varphi_{\\_}\infty(z)\right|$ and call $z_{\\_}1$
and $z_{\\_}2$ two points where the minimum and the maximum are attained,
respectively. If $r$ is small enough we have
$\overline{B}_{\\_}{s^{\prime}}^{2}(0)\subseteq\varphi_{\\_}\infty(B_{\\_}2^{2}(0))$.
Letting $A:=B_{\\_}{s^{\prime}}^{2}(0)\setminus\overline{B}_{\\_}{s}^{2}(0)$
and denoting by $M(\cdot)$ the module of a ring domain (see [28, Section
I.6.1] for the definition), by [28, Theorem I.7.1] we have

(5.5)
$\log\left(\frac{s^{\prime}}{s}\right)=M(A)\leq\overline{N}^{2}M(\varphi_{\\_}\infty^{-1}(A)).$

The ring domain $\varphi_{\\_}\infty^{-1}(A)$ separates $0$ and $z_{\\_}1$
from $z_{\\_}2$ and $\infty$ (in $\widehat{\mathbb{C}}$), so Teichmüller’s
module theorem (see [28, Section II.1.3]), together with
$\left|z_{\\_}2\right|=2\left|z_{\\_}1\right|$, implies that
$M(\varphi_{\\_}\infty^{-1}(A))$ is bounded by a constant depending only on
$\sup_{\\_}\Omega\widetilde{N}$. Since
$\varphi_{\\_}\infty(B_{\\_}r^{2}(0))\supseteq B_{\\_}{s}^{2}(0)$ and
$\varphi_{\\_}\infty(B_{\\_}{2r}^{2}(0))\subseteq B_{\\_}{s^{\prime}}^{2}(0)$,
(5.4) follows from (5.5). Thus, by the area formula,

$\frac{\overline{C}}{(\sup_{\\_}\Omega\widetilde{N})^{2}}\leq\frac{\int_{\\_}{B_{\\_}{2r}^{2}(0)}\left|\partial_{\\_}1\Phi_{\\_}\infty\wedge\partial_{\\_}2\Phi_{\\_}\infty\right|\,d\mathcal{L}^{2}}{\int_{\\_}{B_{\\_}r^{2}(0)}\left|\partial_{\\_}1\Phi_{\\_}\infty\wedge\partial_{\\_}2\Phi_{\\_}\infty\right|\,d\mathcal{L}^{2}}\leq\frac{\int_{\\_}{B_{\\_}{2^{\mathcal{K}}s(r)}^{2}(0)}\left|\nabla\Psi\right|^{2}\,d\mathcal{L}^{2}}{\int_{\\_}{B_{\\_}{s(r)}^{2}(0)}\left|\nabla\Psi\right|^{2}\,d\mathcal{L}^{2}}\to(2^{2\overline{k}})^{\mathcal{K}}\leq
2^{2\mathcal{K}\overline{N}}$

as $r\to 0$. We deduce
$\overline{C}\leq(\sup_{\\_}\Omega\widetilde{N})^{2}2^{2\mathcal{K}\sup_{\\_}\Omega\widetilde{N}}$
and this is a contradiction once we choose $\overline{C}$ so large that this
inequality fails. ∎

We need another technical result, which is again obtained by means of a
compactness argument.

###### Lemma 5.5 (absence of energy jumps).

For every $\delta\in(0,1)$ there exists
$\varepsilon^{\prime}=\varepsilon^{\prime}(\Omega,\Phi,N,\delta)$ with the
following property: whenever

  * •

$x\in\omega\subset\subset\Omega$ and
$0<r<\frac{1}{2}\operatorname{dist}(x,\partial\omega)$,

  * •

$\ell(x,r)<\varepsilon^{\prime}\operatorname{dist}(\Phi(x),\Phi(\partial\omega))$,

  * •

$0<\int_{\\_}{B_{\\_}{2r}^{2}(x)}\left|\nabla\Phi\right|^{2}\,d\mathcal{L}^{2}<\overline{C}\int_{\\_}{B_{\\_}r^{2}(x)}\left|\nabla\Phi\right|^{2}\,d\mathcal{L}^{2}$,
with $\overline{C}$ given by Lemma 5.4,

  * •

$\frac{\left\|\mathbf{v}_{\\_}\omega\right\|(B_{\\_}s^{q}(\Phi(x)))}{\pi
s^{2}}\in(\widetilde{N}(x)-\varepsilon^{\prime},\widetilde{N}(x)+\varepsilon^{\prime})$
for all $0<s<(\varepsilon^{\prime})^{-1}\ell(x,r)$,

we have $\int_{\\_}{B_{\\_}{\delta
r}^{2}(0)}\left|\nabla\Phi\right|^{2}\,d\mathcal{L}^{2}\geq\varepsilon^{\prime}\int_{\\_}{B_{\\_}r^{2}(0)}\left|\nabla\Phi\right|^{2}\,d\mathcal{L}^{2}$.

###### Proof.

Arguing by contradiction as in the proof of Lemma 5.4, we would get a local
parametrized stationary varifold
$(\Omega_{\\_}\infty,\Phi_{\\_}\infty\circ\varphi_{\\_}\infty^{-1},N_{\\_}\infty\circ\varphi_{\\_}\infty^{-1})$
with

$\int_{\\_}{B_{\\_}\delta^{2}(0)}\left|\nabla\Phi_{\\_}\infty\right|^{2}\,d\mathcal{L}^{2}=0.$

But then $\Psi:=\Phi_{\\_}\infty\circ\varphi_{\\_}\infty^{-1}$ could be
identified with a nonconstant holomorphic function, on the connected domain
$\Omega_{\\_}\infty=\varphi_{\\_}\infty(B_{\\_}2^{2}(0))$, with uncountably
many zeroes. This is a contradiction. ∎

###### Corollary 5.6 (slow energy decay at admissible points).

If $x$ lies in the admissible set $\mathcal{A}$, then there exists an
arbitrarily small $r>0$ such that
$\int_{\\_}{B_{\\_}{2r}^{2}(x)}\left|\nabla\Phi\right|^{2}\,d\mathcal{L}^{2}<\overline{C}\int_{\\_}{B_{\\_}r^{2}(0)}\left|\nabla\Phi\right|^{2}\,d\mathcal{L}^{2}$.
Moreover,

$\limsup_{\\_}{r\to
0}\frac{\int_{\\_}{B_{\\_}{2r}^{2}(0)}\left|\nabla\Phi\right|^{2}\,d\mathcal{L}^{2}}{\int_{\\_}{B_{\\_}r^{2}(0)}\left|\nabla\Phi\right|^{2}\,d\mathcal{L}^{2}}<\infty.$

###### Proof.

Since $x\in\mathcal{A}$, we have $\liminf_{\\_}{r\to
0}\frac{\int_{\\_}{B_{\\_}{2r}^{2}(0)}\left|\nabla\Phi\right|^{2}\,d\mathcal{L}^{2}}{\int_{\\_}{B_{\\_}r^{2}(0)}\left|\nabla\Phi\right|^{2}\,d\mathcal{L}^{2}}<C^{\prime}$
for some finite $C^{\prime}$. Let
$K_{\\_}x\subseteq\omega\subset\subset\Omega$ with $\overline{\omega}$
disjoint from $\Phi^{-1}(\Phi(x))\setminus K_{\\_}x$ and choose a radius $r$
such that

$\int_{\\_}{B_{\\_}{2r}^{2}(0)}\left|\nabla\Phi\right|^{2}\,d\mathcal{L}^{2}<C^{\prime}\int_{\\_}{B_{\\_}r^{2}(0)}\left|\nabla\Phi\right|^{2}\,d\mathcal{L}^{2}$

and $r$ so small that it satisfies the other hypotheses of Lemma 5.4 with both
$\varepsilon=\varepsilon(\Omega,\Phi,N,C^{\prime})$ and
$\varepsilon=\varepsilon(\Omega,\Phi,N,\overline{C})$ (the density assumptions
for $\left\|\mathbf{v}_{\\_}\omega\right\|$ are eventually satisfied by
definition of $\widetilde{N}$). Then, by Lemma 5.4, we can find

$\varepsilon(\Omega,\Phi,N,C^{\prime})r<r_{\\_}1<\frac{r}{2}$

such that
$\int_{\\_}{B_{\\_}{2r_{\\_}1}^{2}(0)}\left|\nabla\Phi\right|^{2}\,d\mathcal{L}^{2}<\overline{C}\int_{\\_}{B_{\\_}{r_{\\_}1}^{2}(0)}\left|\nabla\Phi\right|^{2}\,d\mathcal{L}^{2}$.
This new radius $r_{\\_}1$ satisfies the hypotheses of Lemma 5.4 with
$\varepsilon=\varepsilon(\Omega,\Phi,N,\overline{C})$, so there exists

$\varepsilon(\Omega,\Phi,N,\overline{C})r_{\\_}1<r_{\\_}2<\frac{r_{\\_}1}{2}$

such that
$\int_{\\_}{B_{\\_}{2r_{\\_}2}^{2}(0)}\left|\nabla\Phi\right|^{2}\,d\mathcal{L}^{2}<\overline{C}\int_{\\_}{B_{\\_}{r_{\\_}2}^{2}(0)}\left|\nabla\Phi\right|^{2}\,d\mathcal{L}^{2}$.
Again, $r_{\\_}2$ satisfies the hypotheses of Lemma 5.4 with
$\varepsilon=\varepsilon(\Omega,\Phi,N,\overline{C})$, so we can find
$\varepsilon(\Omega,\Phi,N,\overline{C})r_{\\_}2<r_{\\_}3<\frac{r_{\\_}2}{2}$
and so on. Eventually $r_{\\_}k$ satisfies the hypotheses of Lemma 5.5 with
$\varepsilon^{\prime}=\varepsilon^{\prime}(\Omega,\Phi,N,\varepsilon(\Omega,\Phi,N,\overline{C}))$.
Thus,

$\int_{\\_}{B_{\\_}{r_{\\_}{k+1}}^{2}(0)}\left|\nabla\Phi\right|^{2}\,d\mathcal{L}^{2}\geq\varepsilon^{\prime}\int_{\\_}{B_{\\_}{r_{\\_}k}^{2}(0)}\left|\nabla\Phi\right|^{2}\,d\mathcal{L}^{2}.$

Any radius $s>0$ small enough lies in some interval $[r_{\\_}{k+1},r_{\\_}k]$
and $2s\leq r_{\\_}{k-1}$. The result follows. ∎

We are finally ready to show the full regularity result for parametrized
stationary varifolds.

###### Theorem 5.7 (regularity in the general case).

Let $(\Omega,\Phi,N)$ be a local parametrized stationary varifold in
${\mathcal{M}^{m}}$. Then $\Phi$ solves
$-\Delta\Phi=A(\Phi)(\nabla\Phi,\nabla\Phi)$ and, on each connected component
where $\Phi$ is nonconstant, $\Phi$ is a $C^{\infty}$-smooth branched
immersion and $N$ is a.e. constant.

For the definition of branched immersion, see e.g. [21, Definitions 1.2 and
1.6].

###### Proof.

Assume that $\Omega^{\prime}\subset\subset\Omega$ satisfies
$\Phi(\Omega^{\prime})\cap\Phi(\partial\Omega^{\prime})=\emptyset$ and
$\sup_{\\_}{\Omega^{\prime}}\widetilde{N}<\infty$. We show that the partial
differential equation holds in $\Omega^{\prime}$ (the full result will be
obtained at the end of the proof). More precisely, letting

$\gamma:=\frac{1}{3}\varepsilon\left(\Omega^{\prime},{\left.\kern-1.2pt\Phi\vphantom{\big{|}}\right|_{\Omega^{\prime}}},{\left.\kern-1.2ptN\vphantom{\big{|}}\right|_{\Omega^{\prime}}},\overline{C}\right)<1$

(where $\overline{C}$ is the constant given by Lemma 5.4, depending only on
$\sup_{\\_}{\Omega^{\prime}}\widetilde{N}$), we will show that the equation
holds on the open set
$\Omega_{\\_}k:=\Omega^{\prime}\cap\\{\gamma^{-1}\widetilde{N}<k+1\\}$, by
induction on $k$. The base case $k=0$ is trivial, since
$\\{\widetilde{N}<\gamma\\}=\emptyset$. Assume that the equation holds for
$k-1$.

We call $\mathcal{C}_{\\_}k$ the set of accumulation points of
$\Omega_{\\_}k\setminus\Omega_{\\_}{k-1}$ in $\Omega_{\\_}k$, i.e. its derived
set in $\Omega_{\\_}k$. Notice that
$\mathcal{C}_{\\_}k\subseteq\Omega_{\\_}k\setminus\Omega_{\\_}{k-1}$ is closed
in $\Omega_{\\_}k$. We also set
$\mathcal{A}_{\\_}k:=\mathcal{C}_{\\_}k\cap\mathcal{A}$ and
$\mathcal{B}_{\\_}k:=\mathcal{C}_{\\_}k\setminus\mathcal{A}_{\\_}k$. Notice
that, by Lemma 5.1, the equation holds in the open set
$\Omega_{\\_}k\setminus\mathcal{C}_{\\_}k$, since here the points with
$k\gamma\leq\widetilde{N}(\cdot)<(k+1)\gamma$ form a discrete set.

_Step 1._ We first show that $\mathcal{A}_{\\_}k$ is relatively open in
$\mathcal{C}_{\\_}k$, so that $\mathcal{B}_{\\_}k$ is closed in
$\Omega_{\\_}k$. Let $x_{\\_}0\in\mathcal{A}_{\\_}k$. First of all, by Remark
2.12, we can find
$K_{\\_}{x_{\\_}0}\subseteq\omega\subset\subset\Omega^{\prime}$ with
$\overline{\omega}$ disjoint from $\Phi^{-1}(\Phi(x_{\\_}0))\setminus
K_{\\_}{x_{\\_}0}$, $\Phi(\omega)\cap\Phi(\partial\omega)=\emptyset$ and
$\omega\cap\Phi^{-1}(\Phi(y))=K_{\\_}y$ whenever $y\in\omega$ has
$k\gamma\leq\widetilde{N}(y)<(k+1)\gamma$ (thanks to the fact that
$\gamma<1$).

Let
$\varepsilon:=\varepsilon\left(\Omega^{\prime},{\left.\kern-1.2pt\Phi\vphantom{\big{|}}\right|_{\Omega^{\prime}}},{\left.\kern-1.2ptN\vphantom{\big{|}}\right|_{\Omega^{\prime}}},\overline{C}\right)$
and assume that $x_{\\_}j\to x_{\\_}0$, with
$x_{\\_}j\in\omega\cap\mathcal{C}_{\\_}k$. In particular, by definition of
$\widetilde{N}$, the density of $\mathbf{v}_{\\_}\omega$ at $\Phi(x_{\\_}j)$
coincides with $\widetilde{N}(x_{\\_}j)$. Using Corollary 5.6, we choose a
radius $0<r<\frac{1}{2}\operatorname{dist}(x_{\\_}0,\partial\omega)$ with

$\int_{\\_}{B_{\\_}{2r}^{2}(x_{\\_}0)}\left|\nabla\Phi\right|^{2}\,d\mathcal{L}^{2}<\overline{C}\int_{\\_}{B_{\\_}r^{2}(x_{\\_}0)}\left|\nabla\Phi\right|^{2}\,d\mathcal{L}^{2},\quad\ell(x_{\\_}0,r)<\varepsilon\operatorname{dist}(\Phi(x_{\\_}0),\Phi(\partial\omega)),$
$\frac{\left\|\mathbf{v}_{\\_}\omega\right\|(\overline{B}_{\\_}{\varepsilon^{-1}\ell(x_{\\_}0,r)}^{q}(\Phi(x_{\\_}0)))}{\pi(\varepsilon^{-1}\ell(x_{\\_}0,r))^{2}}<\widetilde{N}(x_{\\_}0)+\gamma.$

Eventually all the assumptions of Lemma 5.4 are satisfied by
$x_{\\_}j\in\omega$ (with $\Omega^{\prime}$ and $C^{\prime}:=\overline{C}$),
provided $\ell(x_{\\_}0,r)$ is small enough: eventually we have
$B_{\\_}{\varepsilon^{-1}\ell(x_{\\_}j,r)}^{q}(\Phi(x_{\\_}j))\cap\Phi(\partial\omega)=\emptyset$,
so by the monotonicity formula we get, for
$0<s<\varepsilon^{-1}\ell(x_{\\_}j,r)$,

$\begin{split}&\widetilde{N}(x_{\\_}j)\leq
e^{(\sqrt{2}\left\|A\right\|_{\\_}\infty)s}\frac{\left\|\mathbf{v}_{\\_}\omega\right\|(B_{\\_}s^{q}(\Phi(x_{\\_}j)))}{\pi
s^{2}}\leq
e^{(\sqrt{2}\left\|A\right\|_{\\_}\infty)\varepsilon^{-1}\ell(x_{\\_}j,r)}\frac{\left\|\mathbf{v}_{\\_}\omega\right\|(B_{\\_}{\varepsilon^{-1}\ell(x_{\\_}j,r)}^{q}(\Phi(x_{\\_}j)))}{\pi(\varepsilon^{-1}\ell(x_{\\_}j,r))^{2}}\\\
&\leq
e^{(\sqrt{2}\left\|A\right\|_{\\_}\infty)\varepsilon^{-1}\ell(x_{\\_}j,r)}(\widetilde{N}(x_{\\_}0)+\gamma)\leq(e^{(\sqrt{2}\left\|A\right\|_{\\_}\infty)\varepsilon^{-1}\ell(x_{\\_}0,r)}+o_{\\_}j(1))\frac{\widetilde{N}(x_{\\_}0)+\gamma}{\widetilde{N}(x_{\\_}0)+2\gamma}(\widetilde{N}(x_{\\_}j)+\varepsilon)\end{split}$

eventually and, since
$\frac{\widetilde{N}(x_{\\_}j)}{\widetilde{N}(x_{\\_}j)-\varepsilon}\geq\frac{\widetilde{N}(x_{\\_}0)-\gamma}{\widetilde{N}(x_{\\_}0)-2\gamma}$,
it suffices to impose additionally that

$e^{(\sqrt{2}\left\|A\right\|_{\\_}\infty)\varepsilon^{-1}\ell(x_{\\_}0,r)}<\min\left\\{\frac{\widetilde{N}(x_{\\_}0)-\gamma}{\widetilde{N}(x_{\\_}0)-2\gamma},\frac{\widetilde{N}(x_{\\_}0)+2\gamma}{\widetilde{N}(x_{\\_}0)+\gamma}\right\\}.$

By Lemma 5.4 applied to the parametrized varifold
$\left(\Omega^{\prime},{\left.\kern-1.2pt\Phi\vphantom{\big{|}}\right|_{\Omega^{\prime}}},{\left.\kern-1.2ptN\vphantom{\big{|}}\right|_{\Omega^{\prime}}}\right)$
and the point $x_{\\_}j\in\omega$, for $j$ big enough there exists
$r^{\prime}<\frac{r}{2}$ (depending on $j$) such that
$\int_{\\_}{B_{\\_}{2r^{\prime}}^{2}(x_{\\_}j)}\left|\nabla\Phi\right|^{2}\,d\mathcal{L}^{2}<\overline{C}\int_{\\_}{B_{\\_}{r^{\prime}}^{2}(x_{\\_}j)}\left|\nabla\Phi\right|^{2}\,d\mathcal{L}^{2}$.
Since $r^{\prime}$ satisfies again all the hypotheses of Lemma 5.4, we can
iterate and deduce that $x_{\\_}j\in\mathcal{A}$. Hence,
$x_{\\_}j\in\mathcal{A}_{\\_}k$ eventually.

_Step 2._ We now claim that $\widetilde{N}(x)$ is an integer for any
admissible point $x\in\Omega^{\prime}\cap\mathcal{A}$. Indeed, as in the proof
of Lemma 5.4, we can apply Theorem 4.1 with $x_{\\_}k:=x$ and a suitable
sequence of radii $r_{\\_}k\to 0$: whenever
$K_{\\_}x\subseteq\widetilde{\omega}\subset\subset\Omega^{\prime}$ has its
closure disjoint from $\Phi^{-1}(\Phi(x))\setminus K_{\\_}x$, the varifolds
$(\ell_{\\_}k^{-1}(\cdot-\Phi(x_{\\_}k)))_{\\_}*\mathbf{v}_{\\_}{\widetilde{\omega}}$
converge to a stationary cone $\mathbf{v}_{\\_}\infty$ having density at most
$\widetilde{N}(x)$ at $0$, so we have
$\left\|\mathbf{v}_{\\_}\infty\right\|(B_{\\_}s^{q}(p))\leq\widetilde{N}(x)\pi
s^{2}$ (see the proof of [44, Theorem 42.4]) and thus (• ‣ 4) holds with
$C^{\prime\prime}:=\widetilde{N}(x)$.

We obtain a local parametrized stationary varifold
$(\Omega_{\\_}\infty,\Phi_{\\_}\infty\circ\varphi_{\\_}\infty^{-1},N_{\\_}\infty\circ\varphi_{\\_}\infty^{-1})$
with $\Psi:=\Phi_{\\_}\infty\circ\varphi_{\\_}\infty^{-1}$ nonconstant and
holomorphic (again by Theorem 3.7, since the mass measure of this parametrized
varifold is bounded by the mass measure of $\mathbf{v}_{\\_}\infty$, which is
an integer rectifiable stationary cone).

Let $\omega^{\prime}\subset\subset B_{\\_}2^{2}(0)$ be a smooth neighborhood
of $0$ with $0\not\in\Phi_{\\_}\infty(\partial\omega^{\prime})$ and
$\Phi_{\\_}\infty^{-1}(0)\cap\omega^{\prime}=\left\\{0\right\\}$. Using the
notation of Section 4, from the locally uniform convergence
$\Phi_{\\_}k\to\Phi_{\\_}\infty$ we infer that eventually
$\Phi(x)\not\in\Phi(x+r_{\\_}k\partial\omega^{\prime})$ and, for all
$0<s<\operatorname{dist}(0,\Phi_{\\_}\infty(\partial\omega^{\prime}))$,

$\begin{split}\widetilde{N}(x)&\leq\lim_{\\_}{k\to\infty}\frac{\left\|\mathbf{v}_{\\_}{x+r_{\\_}k\omega^{\prime}}\right\|(B_{\\_}{\ell_{\\_}ks}^{q}(\Phi(x)))}{\pi(\ell_{\\_}ks)^{2}}=\lim_{\\_}{k\to\infty}\frac{(\Phi_{\\_}k)_{\\_}*(\bm{1}_{\\_}{\omega^{\prime}}\nu_{\\_}k)(B_{\\_}s^{q}(0))}{\pi
s^{2}}\\\
&=\frac{(\Phi_{\\_}\infty)_{\\_}*(\bm{1}_{\\_}{\omega^{\prime}}\nu_{\\_}\infty)(B_{\\_}s^{q}(0))}{\pi
s^{2}},\end{split}$

by the definition of $\widetilde{N}$, the monotonicity formula and Lemma 4.3.
We deduce that, with the same notation as in the proof of Lemma 5.4,
$N^{\prime}(0)\geq\widetilde{N}(x)$. We also have the converse inequality
$N^{\prime}(0)\leq\widetilde{N}(x)$, since the density of $\mu_{\\_}\infty$ is
everywhere at most $C^{\prime\prime}=\widetilde{N}(x)$. This argument also
shows that $\Psi^{-1}(0)=0$ and
$\sum_{\\_}{i=1}^{r}N^{\prime}(z_{\\_}i)\leq\widetilde{N}(x_{\\_}0)$ whenever
$z_{\\_}i\in\Omega_{\\_}\infty$ are distinct points in a fiber $\Psi^{-1}(p)$
(since $z_{\\_}i$ contributes by $N^{\prime}(z_{\\_}i)$ to the density of
$\mu_{\\_}\infty$ at $p$).

Finally, arguing as in the proof of Theorem 3.3, we conclude that
$N^{\prime}(0)$ is integer since it equals the constant density of a suitable
localization of
$(\Omega_{\\_}\infty,\Psi,N_{\\_}\infty\circ\varphi_{\\_}\infty^{-1})$ (in an
open subset of $\Psi(\Omega_{\\_}\infty)$). This establishes our claim.

_Step 3._ We show by contradiction that there cannot be any
$x_{\\_}0\in\partial\mathcal{A}_{\\_}k\cap\mathcal{A}_{\\_}k\cap\Omega_{\\_}k$.
Indeed, if this happens, then we can find
$x_{\\_}1\in\Omega_{\\_}k\setminus\mathcal{A}_{\\_}k$ with

$\left|x_{\\_}1-x_{\\_}0\right|<\frac{1}{2}\min\left\\{\operatorname{dist}(x_{\\_}0,\mathcal{B}_{\\_}k),\operatorname{dist}(x_{\\_}0,\partial\Omega_{\\_}k)\right\\},$

thanks to the fact that the latter is positive, as $\mathcal{B}_{\\_}k$ is
closed in $\Omega_{\\_}k$. We infer that

$r:=\operatorname{dist}(x_{\\_}1,\mathcal{C}_{\\_}k)<\frac{1}{2}\min\left\\{\operatorname{dist}(x_{\\_}0,\mathcal{B}_{\\_}k),\operatorname{dist}(x_{\\_}0,\partial\Omega_{\\_}k)\right\\},$

there exists $y_{\\_}0\in\mathcal{C}_{\\_}k$ with
$\left|y_{\\_}0-x_{\\_}1\right|=r$ and necessarily we have

$y_{\\_}0\in\mathcal{A}_{\\_}k,\quad
B_{\\_}r^{2}(x_{\\_}1)\subseteq\Omega_{\\_}k\setminus\mathcal{C}_{\\_}k.$

Let $H\subset\mathbb{C}$ be the unique open half-plane with $0\in\partial H$
and $B_{\\_}r^{2}(x_{\\_}1)\subseteq y_{\\_}0+H$. By definition of
$\mathcal{C}_{\\_}k$, we can find a sequence $y_{\\_}j\to y_{\\_}0$ with
$y_{\\_}j\in\Omega_{\\_}k\setminus\Omega_{\\_}{k-1}$ and $y_{\\_}j\neq
y_{\\_}0$. We can assume that
$\frac{y_{\\_}j-y_{\\_}0}{\left|y_{\\_}j-y_{\\_}0\right|}\to\overline{y}$. Let
$r_{\\_}j:=\left|y_{\\_}j-y_{\\_}0\right|$ and set
$\ell_{\\_}j^{2}:=\int_{\\_}{B_{\\_}{r_{\\_}j}^{2}(y_{\\_}0)}\left|\nabla\Phi\right|^{2}\,d\mathcal{L}^{2}$,
$\Phi_{\\_}j:=\ell_{\\_}j^{-1}(\Phi(y_{\\_}0+r_{\\_}j\cdot)-\Phi(y_{\\_}0))$,
$N_{\\_}j:=N(y_{\\_}0+r_{\\_}j\cdot)$. Thanks to Corollary 5.6, we can apply
Theorem 4.6 and obtain, up to subsequences, a limiting local parametrized
stationary varifold
$(\mathbb{C},\Phi_{\\_}\infty\circ\varphi_{\\_}\infty^{-1},N_{\\_}\infty\circ\varphi_{\\_}\infty^{-1})$.

\begin{overpic}[width=312.9803pt]{touching.png}
\put(19.0,47.0){$\Omega_{\\_}k$} \put(51.0,36.5){$\mathcal{A}_{\\_}k$}
\put(55.0,59.5){$\mathcal{B}_{\\_}k$} \put(36.0,27.5){$\mathcal{B}_{\\_}k$}
\put(64.0,40.0){$B_{\\_}r^{2}(x_{\\_}1)$} \put(65.0,75.0){$y_{\\_}0+H$}
\put(53.0,28.0){$x_{\\_}0$} \put(70.0,32.5){$x_{\\_}1$}
\put(60.5,34.0){$y_{\\_}0$} \end{overpic}

With the same notation and the same argument used in Step 2, we get
$N^{\prime}(0)=\widetilde{N}(y_{\\_}0)$. Actually, we also have
$N^{\prime}(\varphi_{\\_}\infty(\overline{y}))\geq k\gamma$: letting
$\omega^{\prime\prime}$ be a smooth neighborhood of $\overline{y}$ with
$\Phi_{\\_}\infty(\overline{y})\not\in\Phi_{\\_}\infty(\partial\omega^{\prime\prime})$
and
$\Phi_{\\_}\infty^{-1}(\Phi_{\\_}\infty(\overline{y}))\cap\omega^{\prime\prime}=\left\\{\overline{y}\right\\}$,
as soon as
$\Phi_{\\_}j(r_{\\_}j^{-1}(y_{\\_}j-y_{\\_}0))\not\in\Phi_{\\_}j(\partial\omega^{\prime\prime})$
and $r_{\\_}j^{-1}(y_{\\_}j-y_{\\_}0)\in\omega^{\prime\prime}$ the varifold

$\mathbf{v}_{\\_}{(\omega^{\prime\prime},\Phi_{\\_}j,N_{\\_}j)}=(\ell_{\\_}j^{-1}(\cdot-\Phi(y_{\\_}0)))_{\\_}*\mathbf{v}_{\\_}{(y_{\\_}0+r_{\\_}j\omega^{\prime\prime},\Phi,N)}$

has density at least $\widetilde{N}(y_{\\_}j)\geq k\gamma$ at
$\ell_{\\_}j^{-1}(\Phi(y_{\\_}j)-\Phi(y_{\\_}0))=\Phi_{\\_}j(r_{\\_}j^{-1}(y_{\\_}j-y_{\\_}0))\to\Phi_{\\_}\infty(\overline{y})$,
has infinitesimal mean curvature in
$\mathbb{R}^{q}\setminus\Phi_{\\_}j(\partial\omega^{\prime\prime})$ and its
mass measure converges to
$(\Phi_{\\_}\infty)_{\\_}*(\bm{1}_{\\_}{\omega^{\prime\prime}}\nu_{\\_}\infty)$,
by Lemma 4.3. Thus, by the monotonicity formula,
$N^{\prime}(\varphi_{\\_}\infty(\overline{y}))$ (i.e. the density of this last
measure at $\Phi_{\\_}\infty(\overline{y})$) is at least $k\gamma$.

Since $k\gamma+1>\widetilde{N}(y_{\\_}0)$, as observed in Step 2 we must have
$\Phi_{\\_}\infty^{-1}(0)=\left\\{0\right\\}$ and
$\Phi_{\\_}\infty^{-1}(\Phi_{\\_}\infty(\overline{y}))=\left\\{\overline{y}\right\\}$.
Recall that, as in Step 2, the map
$\Psi:=\Phi_{\\_}\infty\circ\varphi_{\\_}\infty^{-1}$ takes values in a plane
and is an entire holomorphic function, up to suitable identification of this
plane with $\mathbb{C}$. Since it has two values having only one preimage, by
Picard’s great theorem it does not have an essential singularity at $\infty$
and is thus a polynomial. Actually, Picard’s great theorem can be easily
avoided: by Corollary 5.6 the Dirichlet energy
$\int_{\\_}{B_{\\_}R^{2}(0)}\left|\nabla\Phi_{\\_}\infty\right|^{2}\,d\mathcal{L}^{2}$
grows at most polynomially in $R$ and, by inspecting the proof of [24, Theorem
4.30] (as well as inequalities (4.21) and (4.24) in [24]), we see that
$\sup_{\\_}{z\in B_{\\_}R^{2}(0)}\left|\varphi_{\\_}\infty^{-1}\right|(z)$
also grows at most polynomially, hence the same is true for
$\int_{\\_}{B_{\\_}R^{2}(0)}\left|\nabla\Psi\right|^{2}\,d\mathcal{L}^{2}$ and
thus (by the mean value property for harmonic functions) for $\sup_{\\_}{z\in
B_{\\_}R^{2}(0)}\left|\nabla\Psi\right|(z)$, i.e. $\Psi$ is a polynomial.
Since $\Psi^{-1}(0)=\left\\{0\right\\}$, it must have the form

$\Psi(z)=cz^{\overline{k}}$

for some $\overline{k}$ and finally the fact that
$\Psi^{-1}(\Phi_{\\_}\infty(\overline{y}))$ is a singleton gives
$\overline{k}=1$. We deduce that $\Psi^{\prime}(0)\neq 0$. Arguing as in the
proof of Corollary 3.4, we get
$N_{\\_}\infty\circ\varphi_{\\_}\infty^{-1}=N^{\prime}(0)\geq k\gamma$ a.e.
near $0$.

We finally show that $\Phi_{\\_}j\to\Phi_{\\_}\infty$ in
$W^{1,2}_{\\_}{loc}(H,\mathbb{R}^{q})$. In particular, for any small ball
$B\subset\subset H$ close enough to $0$, this will contradict the estimate

$\begin{split}&\kappa\gamma\int_{\\_}B\left|\partial_{\\_}1\Phi_{\\_}\infty\wedge\partial_{\\_}2\Phi_{\\_}\infty\right|\,d\mathcal{L}^{2}\leq\nu_{\\_}\infty(B)=\lim_{\\_}{j\to\infty}\nu_{\\_}j(B)\\\
&=\frac{1}{2}\lim_{\\_}{j\to\infty}\int_{\\_}BN_{\\_}j\left|\nabla\Phi_{\\_}j\right|^{2}\,d\mathcal{L}^{2}\leq\alpha\lim_{\\_}{j\to\infty}\int_{\\_}B\left|\partial_{\\_}1\Phi_{\\_}j\wedge\partial_{\\_}2\Phi_{\\_}j\right|\,d\mathcal{L}^{2},\end{split}$

where $\alpha$ is the biggest integer smaller than $\kappa\gamma$ (the last
inequality comes from the fact that $N_{\\_}j\in\mathbb{N}$ and
$N_{\\_}j=\widetilde{N}(y_{\\_}0+r_{\\_}j\cdot)$ a.e. on
$\left\\{\nabla\Phi_{\\_}j\neq 0\right\\}$, together with the fact that
eventually $y_{\\_}0+r_{\\_}jB\subseteq
B_{\\_}r^{2}(x_{\\_}1)\subseteq\Omega_{\\_}k\setminus\mathcal{C}_{\\_}k$).

Fix any $U\subset\subset H$. Since eventually
$-\Delta\Phi_{\\_}j=\ell_{\\_}jA(\Phi(y_{\\_}0+r_{\\_}j\cdot))(\nabla\Phi_{\\_}j,\nabla\Phi_{\\_}j)$
on $U$ and the right-hand side converges to $0$ in $L^{1}(U,\mathbb{R}^{q})$,
we get

$-\Delta\Phi_{\\_}\infty=0$

on $U$ and hence (since $U$ was arbitrary) on $H$. Fix any nonnegative
$\rho\in C^{\infty}_{\\_}c(H)$ with $\rho=1$ on $U$. Setting
$\Psi_{\\_}j:=\Phi_{\\_}j-\Phi_{\\_}\infty$, we have
$-\Delta\Psi_{\\_}j=\ell_{\\_}jA(\Phi(y_{\\_}0+r_{\\_}j\cdot))(\nabla\Phi_{\\_}j,\nabla\Phi_{\\_}j)$
and thus

$\begin{split}&\int_{\\_}H\rho\left|\nabla\Psi_{\\_}j\right|^{2}\,d\mathcal{L}^{2}+\int_{\\_}H\Psi_{\\_}j\cdot\left\langle\nabla\rho,\nabla\Psi_{\\_}j\right\rangle\,d\mathcal{L}^{2}=\int_{\\_}H\left\langle\nabla(\rho\Psi_{\\_}j);\nabla\Psi_{\\_}j\right\rangle\,d\mathcal{L}^{2}\\\
&=\ell_{\\_}j\int_{\\_}H\rho\Psi_{\\_}j\cdot
A(\Phi(y_{\\_}0+r_{\\_}j\cdot))(\nabla\Phi_{\\_}j,\nabla\Phi_{\\_}j)\,d\mathcal{L}^{2}.\end{split}$

But both the right-hand side and the second term in the left-hand side
converge to $0$. Hence,

$\int_{\\_}U\left|\nabla\Psi_{\\_}j\right|^{2}\,d\mathcal{L}^{2}\leq\int_{\\_}H\rho\left|\nabla\Psi_{\\_}j\right|^{2}\,d\mathcal{L}^{2}\to
0.$

_Step 4._ From the previous step we have
$\partial\mathcal{A}_{\\_}k\cap\Omega_{\\_}k\subseteq\mathcal{B}_{\\_}k$,
hence $\mathcal{A}_{\\_}k$ is open. Since $\gamma<1$ and $\widetilde{N}$ is
integer-valued on $\mathcal{A}_{\\_}k$ (by Step 2), $\widetilde{N}$ takes
exactly a single value here. We can then apply Theorem 2.13 (as replacing $N$
with $\widetilde{N}$ does not affect the stationarity) and obtain that the
partial differential equation holds on $\mathcal{A}_{\\_}k$.

_Step 5._ From the two previous steps, it follows that
$-\Delta\Phi=A(\Phi)(\nabla\Phi,\nabla\Phi)$ on the open set
$\Omega_{\\_}k\setminus\mathcal{B}_{\\_}k$. Using Lemma 5.3 and Lemma 5.1, we
deduce that the partial differential equation holds on the whole
$\Omega_{\\_}k$ (and $\Phi$ is $C^{\infty}$-smooth on $\Omega_{\\_}k$). This
completes the induction.

_Step 6._ The extra assumptions made at the beginning of the proof can be
dropped by arguing as in the proof of Theorem 3.7 and using the upper
semicontinuity of $\widetilde{N}$. For the fact that $\Phi$ is a branched
immersion on a connected component $\Omega^{\prime\prime}$ where it is
nonconstant, we refer the reader to the proofs of [22, Theorems 1 and 2] and
[21, Lemmas 2.1 and 2.2]. Finally, let $D\subset\Omega^{\prime\prime}$ denote
the discrete subset where $\nabla\Phi=0$. Whenever a connected
$\omega\subset\subset\Omega^{\prime\prime}\setminus D$ is such that
$\Phi(\omega)\cap\Phi(\partial\omega)=\emptyset$ and
${\left.\kern-1.2pt\Phi\vphantom{\big{|}}\right|_{\omega}}$ is an embedding,
the constancy theorem (see [44, Theorem 41.1]) implies that $\widetilde{N}$ is
constant on $\omega$. Since $\Omega^{\prime\prime}\setminus D$ is connected,
Proposition 2.8 gives that $N$ is a.e. constant on
$\Omega^{\prime\prime}\setminus D$, hence on $\Omega^{\prime\prime}$. ∎

###### Corollary 5.8 (global regularity).

If $(\Sigma,\Phi,N)$ is a parametrized stationary varifold, then $\Phi$ solves
$-\Delta\Phi=A(\Phi)(\nabla\Phi,\nabla\Phi)$ in local conformal coordinates,
$\Phi$ is a $C^{\infty}$-smooth branched immersion and $N$ is a.e. constant.

###### Remark 5.9 (converse statement).

We notice that the converse statement holds as well: namely, if
$\Phi:\Sigma\to{\mathcal{M}^{m}}$ is a nonconstant weakly conformal, weakly
harmonic map and $N$ is a positive integer, then $(\Sigma,\Phi,N)$ is a
parametrized stationary varifold. Indeed, for almost every
$\omega\subseteq\Sigma$, the continuous representative of
${\left.\kern-1.2pt\Phi\vphantom{\big{|}}\right|_{\partial\omega}}$ coincides
with the trace (by [15, Theorem 5.7] this holds whenever
$\mathcal{H}^{1}$-a.e. point of $\partial\omega$ is a Lebesgue point for
$\Phi$). Thus, for any smooth $F\in
C^{\infty}_{\\_}c(\mathbb{R}^{q}\setminus\Phi(\partial\omega),\mathbb{R}^{q})$,
${\left.\kern-1.2ptF(\Phi)\vphantom{\big{|}}\right|_{\omega}}$ has zero trace
on $\partial\omega$ and (2.1) follows. Notice that we did not need Hélein’s
regularity result to show this assertion: on the contrary, we can immediately
deduce the continuity of $\Phi$ (and hence the smoothness) from Proposition
2.4. ∎

##  6\. An application to the conductivity equation

In this section we illustrate an application of Theorem 5.7 to the regularity
theory for the conductivity equation

$-\operatorname{div}(N\nabla\Phi)=0\quad\text{on }B_{\\_}1^{2}(0).$

This partial differential equation was already investigated by many authors:
see e.g. [6, 16, 29]. We show below that, assuming $\Phi\in
W^{1,2}(B_{\\_}1^{2}(0),\mathbb{R}^{q})$ weakly conformal and $N\in
L^{\infty}(\mathbb{N}\setminus\left\\{0\right\\})$, $\Phi$ is necessarily
harmonic and $N$ is a.e. constant, unless $\Phi$ is itself constant.

This statement initially originated as a possible intermediate step in order
to achieve Theorem 5.7, but as a matter of fact we are able to prove the
former only as a consequence of the latter. It would be interesting to find an
independent, purely PDE-theoretic proof.

We can do this in the case $q=2$, where the following slightly stronger result
holds.

###### Theorem 6.1 (regularity for $\bm{q=2}$).

Assume $\Phi\in W^{1,2}(B_{\\_}1^{2}(0),\mathbb{R}^{2})$ is weakly conformal,
$N\in L^{\infty}(B_{\\_}1^{2}(0))$ is bounded below by a positive constant and

(6.1) $-\operatorname{div}(N\nabla\Phi)=0\quad\text{in
}\mathcal{D}^{\prime}(B_{\\_}1^{2}(0),\mathbb{R}^{2}).$

Then $\Phi_{\\_}1+i\,\Phi_{\\_}2$ is holomorphic or antiholomorphic and, if
$\Phi$ is nonconstant, $N$ is a.e. constant.

###### Proof.

First of all, $\Phi$ is continuous (see e.g. [19, Section 4.4]). We can assume
that $\Phi$ is nonconstant, so that the set ${\mathcal{G}^{f}}\neq\emptyset$
of Lebesgue points $z$ for $\nabla\Phi$ with $\nabla\Phi(z)\neq 0$ is
nonempty. Notice that $\Phi_{\\_}1$ and $\Phi_{\\_}2$ are both nonconstant:
otherwise e.g. $\nabla\Phi_{\\_}1$ would be a.e. $0$ and in particular we
would have $\nabla\Phi_{\\_}1=0$ on ${\mathcal{G}^{f}}$, contradicting the
weak conformality. From (6.1) and standard Hodge theory, we can find real
functions $\Psi_{\\_}k\in W^{1,2}(B_{\\_}1^{2}(0))$ with

$N\nabla\Phi_{\\_}k=-\nabla^{\perp}\Psi_{\\_}k,\quad\nabla^{\perp}:=(-\partial_{\\_}2,\partial_{\\_}1),$

for $k=1,2$. This equation can be equivalently rewritten as

(6.2)
$N\partial_{\\_}z\Phi_{\\_}k=i\partial_{\\_}z\Psi_{\\_}k\quad\text{or}\quad
N\partial_{\\_}{\overline{z}}\Phi_{\\_}k=-i\partial_{\\_}{\overline{z}}\Psi_{\\_}k.$

Let $f_{\\_}k:=\Phi_{\\_}k+i\,\Psi_{\\_}k$. We have

$\partial_{\\_}{\overline{z}}f_{\\_}k=(1-N)\partial_{\\_}{\overline{z}}\Phi_{\\_}k,\quad\partial_{\\_}{{z}}f_{\\_}k=(1+N)\partial_{\\_}{{z}}\Phi_{\\_}k.$

We define the Beltrami coefficient $\mu_{\\_}k$ on $\mathbb{C}$ as

$\mu_{\\_}k:=\frac{(1-N)\partial_{\\_}{\overline{z}}\Phi_{\\_}k}{(1+N)\partial_{\\_}{{z}}\Phi_{\\_}k}\bm{1}_{\\_}{B_{\\_}1^{2}(0)\cap{\mathcal{G}^{f}}}.$

Notice that, by weak conformality, $\partial_{\\_}z\Phi_{\\_}k\neq 0$ on
${\mathcal{G}^{f}}$. Our hypotheses on $N$ clearly imply

$\|\mu_{\\_}k\|_{\\_}{L^{\infty}}<1$

and $f_{\\_}k$ satisfies the Beltrami equation

$\partial_{\\_}{\overline{z}}f_{\\_}k=\mu_{\\_}k\partial_{\\_}zf_{\\_}k$

on $B_{\\_}1^{2}(0)$. Let $\varphi_{\\_}k$ be the normal solution of

$\partial_{\\_}{\overline{z}}\varphi_{\\_}k=\mu_{\\_}k\partial_{\\_}z\varphi_{\\_}k$

(see [24, Theorem 4.24]). As already pointed out in the proof of Theorem 4.1,
$\varphi_{\\_}k,\varphi_{\\_}k^{-1}\in
W^{1,2}_{\\_}{loc}(\mathbb{C},\mathbb{C})$ are homeomorphisms of $\mathbb{C}$
mapping negligible sets to negligible sets and

$\partial_{\\_}{\overline{w}}\varphi_{\\_}k^{-1}=-(\mu_{\\_}k\circ\varphi_{\\_}k^{-1})\overline{\partial_{\\_}w\varphi_{\\_}k^{-1}}.$

By the chain rule (see [28, Lemma III.6.4]),
$h_{\\_}k:=f_{\\_}k\circ\varphi_{\\_}k^{-1}$ (defined on
$\varphi_{\\_}k(B_{\\_}1^{2}(0))$) satisfies

$\partial_{\\_}{\overline{w}}h_{\\_}k=(\partial_{\\_}zf_{\\_}k\circ\varphi_{\\_}k^{-1})\partial_{\\_}{\overline{w}}\varphi_{\\_}k^{-1}+(\partial_{\\_}{\overline{z}}f_{\\_}k\circ\varphi_{\\_}k^{-1})\overline{\partial_{\\_}w\varphi_{\\_}k^{-1}}=0$

and is thus a nonconstant holomorphic function. Pick now any $z_{\\_}0\in
B_{\\_}1^{2}(0)$ such that the points $\varphi_{\\_}1(z_{\\_}0)$ and
$w_{\\_}0:=\varphi_{\\_}2(z_{\\_}0)$ satisfy
$h_{\\_}1^{\prime}(\varphi_{\\_}1(z_{\\_}0))\neq 0$ and
$h_{\\_}2^{\prime}(w_{\\_}0)\neq 0$: by holomorphicity of $h_{\\_}1$ and
$h_{\\_}2$, this holds true for all $z_{\\_}0$ outside a discrete, relatively
closed subset $D\subset B_{\\_}1^{2}(0)$.

By the Cauchy–Riemann equations, the harmonic map
$\Phi_{\\_}2\circ\varphi_{\\_}2^{-1}=\Re h_{\\_}2$ has nonzero differential at
$w_{\\_}0$. By the inverse function theorem, there exists a local chart $\psi$
centered at $w_{\\_}0$, with some ball $B_{\\_}\delta^{2}(0)$ as its image,
such that

(6.3)
$\Phi_{\\_}2\circ\varphi_{\\_}2^{-1}\circ\psi^{-1}(y)-\Phi_{\\_}2(z_{\\_}0)=y_{\\_}2\quad\text{on
}B_{\\_}\delta^{2}(0).$

Since $\Phi$ is weakly conformal, we have
$(\partial_{\\_}z\Phi_{\\_}1)^{2}+(\partial_{\\_}z\Phi_{\\_}2)^{2}=0$ a.e.,
hence there exists a measurable function $\varepsilon\in
L^{\infty}(B_{\\_}1^{2}(0),\left\\{-1,1\right\\})$ such that

(6.4)
$\partial_{\\_}z\Phi_{\\_}1(z)=i\varepsilon(z)\partial_{\\_}z\Phi_{\\_}2(z)\quad\text{a.e.
on }B_{\\_}1^{2}(0).$

Combining (6.2) and (6.4) we obtain

$\nabla\Psi_{\\_}1=\varepsilon N\nabla\Phi_{\\_}2$

a.e. Using (6.3) and the chain rule again, we get

$\partial_{\\_}1(\Psi_{\\_}1\circ\varphi_{\\_}2^{-1}\circ\psi^{-1})=0,\quad\partial_{\\_}2(\Psi_{\\_}1\circ\varphi_{\\_}2^{-1}\circ\psi^{-1})=(\varepsilon
N)\circ\varphi_{\\_}2^{-1}\circ\psi^{-1}$

a.e. and, since
$\partial_{\\_}{12}^{2}(\Psi_{\\_}1\circ\varphi_{\\_}2^{-1}\circ\psi^{-1})=0$
distributionally, we deduce that

$(\varepsilon
N)\circ\varphi_{\\_}2^{-1}\circ\psi^{-1}(y)=\partial_{\\_}2(\Psi_{\\_}1\circ\varphi_{\\_}2^{-1}\circ\psi^{-1})(y)=g(y_{\\_}2)$

a.e. on $B_{\\_}\delta^{2}(0)$, for a suitable $g\in
L^{\infty}((-\delta,\delta))$, as is immediately verified e.g. by
mollification.

Let $G$ be a Lipschitz primitive of $g$ on $(-\delta,\delta)$. We have

$\nabla(\Psi_{\\_}1\circ\varphi_{\\_}2^{-1}\circ\psi^{-1})=\nabla(G(y_{\\_}2))\quad\text{on
}B_{\\_}\delta^{2}(0),$

so up to subtracting a constant from $G$ we obtain

$\Psi_{\\_}1\circ\varphi_{\\_}2^{-1}\circ\psi^{-1}(y)=G(y_{\\_}2)$

and finally, using (6.3),

(6.5) $\Psi_{\\_}1=G\circ(\Phi_{\\_}2-\Phi_{\\_}2(z_{\\_}0))\quad\text{on
}\varphi_{\\_}2^{-1}\circ\psi^{-1}(B_{\\_}\delta^{2}(0))\ni z_{\\_}0.$

Since $f_{\\_}1=\Phi_{\\_}1+i\Psi_{\\_}1$ is injective in a neighborhood of
$z_{\\_}0$ (being $h_{\\_}1^{\prime}(\varphi_{\\_}1(z_{\\_}0))\neq 0$), from
(6.5) we deduce that $\Phi$ is injective on some neighborhood
$B_{\\_}\eta^{2}(z_{\\_}0)$.

We claim that, as a consequence, $\det(\nabla\Phi)$ has a constant sign on
$B_{\\_}\eta^{2}(z_{\\_}0)\cap{\mathcal{G}^{f}}$: indeed, the induced map

$\Phi_{\\_}*:H_{\\_}1(\partial B_{\\_}r^{2}(z))\to
H_{\\_}1(\mathbb{C}\setminus\left\\{\Phi(z)\right\\})$

is clearly independent of $z\in B_{\\_}\eta^{2}(z_{\\_}0)$ and
$0<r<\operatorname{dist}(z,\partial B_{\\_}\eta^{2}(z_{\\_}0))$, once the two
groups are canonically identified with $\mathbb{Z}$. But, for any $z\in
B_{\\_}\eta^{2}(z_{\\_}0)\cap{\mathcal{G}^{f}}$, applying Lemma A.4 to
$\Phi-\Phi(z)-\left\langle\nabla\Phi(z),\cdot-z\right\rangle$ we can find a
small radius $r$ such that
${\left.\kern-1.2pt\Phi\vphantom{\big{|}}\right|_{\partial B_{\\_}r^{2}(z)}}$
is homotopic to $\Phi(z)+\left\langle\nabla\Phi(z),\cdot-z\right\rangle$ (as a
map $\partial B_{\\_}r^{2}(z)\to\mathbb{C}\setminus\left\\{\Phi(z)\right\\}$).
Thus the above map $\Phi_{\\_}*$ coincides with the multiplication by
$\operatorname{sgn}\det(\nabla\Phi(z))$ and our claim follows.

From this fact and the weak conformality assumption, on
$B_{\\_}\eta^{2}(z_{\\_}0)$ either $(\Phi_{\\_}1,\Phi_{\\_}2)$ or
$(\Phi_{\\_}1,-\Phi_{\\_}2)$ satisfy the Cauchy–Riemann equations. In
particular, $\Phi$ is real analytic on the connected set
$B_{\\_}1^{2}(0)\setminus D$. Since locally we have either
$\partial_{\\_}{\overline{z}}(\Phi_{\\_}1+i\Phi_{\\_}2)=0$ or
$\partial_{\\_}{z}(\Phi_{\\_}1+i\Phi_{\\_}2)=0$, by analyticity
$\Phi_{\\_}1+i\Phi_{\\_}2$ is globally holomorphic or antiholomorphic on
$B_{\\_}1^{2}(0)\setminus D$, hence also on $B_{\\_}1^{2}(0)$. Finally, (6.1)
gives

$\partial_{\\_}1\Phi_{\\_}k\partial_{\\_}1N+\partial_{\\_}2\Phi_{\\_}k\partial_{\\_}2N=0\quad\text{in
}\mathcal{D}^{\prime}(B_{\\_}1^{2}(0))$

for $k=1,2$ (as $\Delta\Phi_{\\_}k=0$). Since $\nabla\Phi_{\\_}1$ and
$\nabla\Phi_{\\_}2$ are smooth and linearly independent outside a closed
discrete set, we infer $\nabla N=0$ here and thus, by connectedness, $N$ is
a.e. constant. ∎

We now prove the result for arbitrary $q$.

###### Theorem 6.2 (regularity for $\bm{q>2}$).

Assume $\Phi\in W^{1,2}(B_{\\_}1^{2}(0),\mathbb{R}^{q})$ is weakly conformal,
$N\in L^{\infty}(B_{\\_}1^{2}(0),\mathbb{N}\setminus\left\\{0\right\\})$ and

$-\operatorname{div}(N\nabla\Phi)=0\quad\text{in
}\mathcal{D}^{\prime}(B_{\\_}1^{2}(0),\mathbb{R}^{q}).$

Then $\Delta\Phi=0$ and, if $\Phi$ is nonconstant, $N$ is a.e. constant.

###### Proof.

As in the previous proof, we notice that $\Phi$ is continuous. We can assume
that $\Phi$ is not constant. The triple $(B_{\\_}1^{2}(0),\Phi,N)$ satisfies
Definition 2.9 (with ${\mathcal{M}^{m}}=\mathbb{R}^{q}$), except possibly for
the technical condition (2.6): indeed, for any
$\omega\subset\subset\mathcal{B}_{\\_}1^{2}(0)$ and any $F\in
C^{\infty}_{\\_}c(\mathbb{R}^{q}\setminus\Phi(\partial\omega),\mathbb{R}^{q})$,
$F(\Phi)\bm{1}_{\\_}\omega$ lies in
$W^{1,2}_{\\_}0(B_{\\_}1^{2}(0),\mathbb{R}^{q})$ and thus $\int_{\\_}\omega
N\left\langle\nabla(F(\Phi));\nabla\Phi\right\rangle\,d\mathcal{L}^{2}=0$.

Assume, without loss of generality, that $\Phi_{\\_}1$ is not constant and let
$\Psi_{\\_}1$, $f_{\\_}1$, $\varphi_{\\_}1$ and $h_{\\_}1$ be the functions
constructed as in the preceding proof. Let
$D\subset\varphi_{\\_}1(B_{\\_}1^{2}(0))$ be the discrete set of points where
$h_{\\_}1^{\prime}=0$, or equivalently (by the Cauchy–Riemann equations) where
$\nabla(\Phi_{\\_}1\circ\varphi_{\\_}1^{-1})=0$. From the chain rule and the
fact that $\varphi_{\\_}1$ and $\varphi_{\\_}1^{-1}$ map negligible sets to
negligible sets (see [28, Lemma III.6.4] and [24, Lemma 4.12]) we deduce that,
for a.e. $x\in B_{\\_}1^{2}(0)$, $\nabla\varphi_{\\_}1(x)$ is invertible,
$\nabla(\Phi_{\\_}1\circ\varphi_{\\_}1^{-1})(\varphi_{\\_}1(x))\neq 0$ and
$\nabla\Phi_{\\_}1(x)$ is given by the composition of these differentials.
Hence, $\nabla\Phi\neq 0$ a.e. and thus has full rank a.e. (by weak
conformality).

Let $S\subseteq B_{\\_}1^{2}(0)$ denote the complement of the biggest open
subset where $\Delta\Phi=0$. We remark that, given $x\in B_{\\_}1^{2}(0)$, if
there exists a neighborhood $\omega\subset\subset B_{\\_}1^{2}(0)$ with
$\Phi(x)\cap\Phi(\partial\omega)=\emptyset$ then $x\not\in S$: indeed,
$\mathbf{v}_{\\_}\omega$ is stationary in
$\mathbb{R}^{q}\setminus\Phi(\partial\omega)$, so by the monotonicity formula
$\mathbf{v}_{\\_}\omega$ satisfies (2.6) for $p\in B_{\\_}\rho^{q}(\Phi(x))$
and $s<\rho$, where
$\rho:=\frac{1}{2}\operatorname{dist}(\Phi(x),\Phi(\partial\omega))$. Thus,
replacing $\omega$ with $\omega\cap\Phi^{-1}(B_{\\_}{\rho/2}^{2}(\Phi(x)))$,
the triple $(\omega,\Phi,N)$ is a local parametrized stationary varifold and
Theorem 5.7 gives $\Delta\Phi=0$ near $x$. In particular, arguing as in the
proof of Theorem 3.7, we infer that $x\not\in S$ for any Lebesgue point $x$
for $\nabla\Phi$ with $\nabla\Phi(x)\neq 0$. Being $\nabla\Phi\neq 0$ a.e., we
get that $\mathcal{L}^{2}(S)=0$ and that $\Phi$ is nonconstant on any ball
outside $S$.

Moreover, $N$ is a.e. constant on every connected component $U$ of
$B_{\\_}1^{2}(0)\setminus S$, since on $U$ we have
$\partial_{\\_}1\Phi_{\\_}k\partial_{\\_}1N+\partial_{\\_}2\Phi_{\\_}k\partial_{\\_}2N=0$
for all $k=1,\dots,q$ and the (classical) differential $\nabla\Phi$ has full
rank except for a discrete set (being $\Phi$ a nonconstant harmonic function
on $U$), which does not disconnect $U$.

It suffices to show that $S\subseteq\varphi_{\\_}1^{-1}(D)$, since then any
point of $S$ is a removable singularity and thus $S=\emptyset$. Assume by
contradiction that there exists a point $x_{\\_}0\in
S\setminus\varphi_{\\_}1^{-1}(D)$. Let $\psi:V\to(-1,1)^{2}$ be a local chart
centered at $\varphi_{\\_}1(x_{\\_}0)$ and such that

(6.6)
$\Phi_{\\_}1\circ\varphi_{\\_}1^{-1}\circ\psi^{-1}(y)=\Phi_{\\_}1(x_{\\_}0)+y_{\\_}2\quad\text{on
}(-1,1)^{2}.$

Let $\Xi:=\Phi\circ\varphi_{\\_}1^{-1}\circ\psi^{-1}$. We claim that the
negligible set $S^{\prime}:=\psi(V\cap\varphi_{\\_}1(S))$, relatively closed
in $(-1,1)^{2}$, has the following property: if $y=(y_{\\_}1,y_{\\_}2)\in
S^{\prime}$, then $\Xi^{-1}(\Xi(y))$ contains either
$\left(-1,y_{\\_}1\right]\times\left\\{y_{\\_}2\right\\}$ or
$\left[y_{\\_}1,1\right)\times\left\\{y_{\\_}2\right\\}$. If this were not
true, we could find $a\in(-1,y_{\\_}1)$ and $b\in(y_{\\_}1,1)$ with
$\Xi(a,y_{\\_}2),\Xi(b,y_{\\_}2)\neq\Xi(y)$. Given any
$\varepsilon<1-\left|y_{\\_}2\right|$, by (6.6) we would have
$\Xi(y)\not\in\Xi(\partial([a,b]\times[y_{\\_}2-\varepsilon,y_{\\_}2+\varepsilon]))$.
But, as remarked earlier, this would imply $y\not\in S^{\prime}$.

This horizontal segment, i.e. either
$\left(-1,y_{\\_}1\right]\times\left\\{y_{\\_}2\right\\}$ or
$\left[y_{\\_}1,1\right)\times\left\\{y_{\\_}2\right\\}$, has to be contained
in $S^{\prime}$ (being $\Xi$ locally injective at points outside $S^{\prime}$,
with at most countably many exceptions). As a consequence we have
$\mathcal{L}^{1}(\left\\{t:(t,y_{\\_}2)\in S^{\prime}\right\\})>0$ and thus,
by Fubini’s theorem and $\mathcal{L}^{2}(S^{\prime})=0$, we infer

(6.7) $((-1,1)\times\left\\{t\right\\})\cap S^{\prime}=\emptyset\quad\text{for
a.e. }t\in(-1,1).$

Pick now any $\mu>0$ with $((-1,1)\times\left\\{\mu\right\\})\cap
S^{\prime}=\emptyset$. Recall that $(0,0)\in S^{\prime}$ and let
$\lambda:=\max\left\\{t<\mu:(0,t)\in S^{\prime}\right\\}\geq 0$. From (6.7) it
follows that the open set $((-1,1)\times(\lambda,\mu))\setminus S^{\prime}$ is
connected. We infer that $N$ is a.e. constant on
$\varphi_{\\_}1^{-1}\circ\psi^{-1}((-1,1)\times(\lambda,\mu))$, hence
$\Delta\Phi=0$ on this open set. However, this contradicts the weak
conformality assumption: let

$\omega:=\varphi_{\\_}1^{-1}\circ\psi^{-1}\left(\left(-\frac{1}{2},\frac{1}{2}\right)\times\left(\lambda,\frac{\lambda+\mu}{2}\right)\right),$

on which $\Delta\Phi=0$, and take any homeomorphism
$\upsilon:\overline{\omega}\to\overline{B}_{\\_}1^{2}(0)$ biholomorphic on
$\omega$ (using Riemann’s mapping theorem and [18, Theorem I.3.1]). We have
$\Phi_{\\_}1\circ\upsilon^{-1}>\lambda$ on $B_{\\_}1^{2}(0)$, as well as
$\Phi_{\\_}1\circ\upsilon^{-1}=\lambda$ and $\Phi\circ\upsilon^{-1}$ constant
on some open arc $A$ in $\partial B_{\\_}1^{2}(0)$. Since
$\Delta(\Phi\circ\upsilon^{-1})=0$ on $B_{\\_}1^{2}(0)$, by standard
regularity theory $\Phi\circ\upsilon^{-1}$ is smooth in a neighborhood of $A$
in $\overline{B}_{\\_}1^{2}(0)$. But, by Hopf’s maximum principle, the radial
derivative of $\Phi_{\\_}1\circ\upsilon^{-1}$ is nonzero on $A$. So, by weak
conformality of $\Phi\circ\upsilon^{-1}$, the tangential derivative of
$\Phi\circ\upsilon^{-1}$ does not vanish on $A$. This contradicts the fact
that $\Phi\circ\upsilon^{-1}$ is constant on $A$. ∎

## Appendix

This appendix collects some useful general facts about Sobolev functions on
the plane. Most of them are standard results, with the possible exception of
Lemmas A.7 and A.8; however, we preferred to include the proofs since a
precise reference for these statements is not immediately available in the
literature.

###### Lemma A.1 (essential image of a Sobolev map).

Let $\Omega\subseteq\mathbb{C}$ be an open connected set. Let $\Psi\in
W^{1,1}_{\\_}{loc}(\Omega,\mathbb{R}^{q})$ and assume that any point in a
nonempty measurable set $E\subseteq\Omega$ is a Lebesgue point for $\Psi$, as
well as $\nabla\Psi=0$ a.e. on $\Omega\setminus E$. Then the essential image
of $\Psi$ equals $\overline{\Psi(E)}$.

We recall that the essential image of $\Psi$ is the (closed) set of values
$p\in\mathbb{R}^{q}$ such that $\mathcal{L}^{2}(\Psi^{-1}(B_{\\_}s^{q}(p)))>0$
for all $s>0$, or equivalently it is the support of the positive measure
$\Psi_{\\_}*(\bm{1}_{\\_}\Omega\mathcal{L}^{2})$.

###### Proof.

Fix $x\in E$, $s>0$ and let $p:=\Psi(x)$. By Chebyshev’s inequality

$\mathcal{L}^{2}(B_{\\_}r^{2}(x)\setminus\Psi^{-1}(B_{\\_}s^{q}(p)))\leq
s^{-1}\int_{\\_}{B_{\\_}r^{2}(x)}\left|\Psi-\Psi(x)\right|\,d\mathcal{L}^{2}=o(r^{2}),$

so $p$ belongs to the essential image. We deduce that $\overline{\Psi(E)}$ is
included in the essential image. Conversely, assume that $B_{\\_}s^{q}(p)$ is
disjoint from $\overline{\Psi(E)}$. We can find a function $\rho\in
C^{\infty}(\mathbb{R}^{q},\mathbb{R}^{q})$ with $\rho=\operatorname{id}$ on
$\mathbb{R}^{q}\setminus B_{\\_}s^{q}(p)$ and
$\rho=\operatorname{id}+e_{\\_}1$ on $B_{\\_}{s/2}^{q}(p)$. By the chain rule,

$\nabla(\rho\circ\Psi)=(\nabla\rho\circ\Psi)\nabla\Psi=\nabla\Psi$

a.e. on $\Omega$, since $\nabla\Psi=0$ a.e. on
$\left\\{\nabla\rho\circ\Psi\neq\operatorname{id}\right\\}$. Thus,
$\rho\circ\Psi-\Psi$ is constant a.e. But this function vanishes on $E$, hence
$0$ belongs to its essential image (by the same argument used above). Thus
$\rho\circ\Psi=\Psi$ a.e. and we infer that
$\Psi^{-1}(B_{\\_}{s/2}^{q}(p))\subseteq\left\\{\rho\circ\Psi\neq\Psi\right\\}$
is negligible. This shows that $p$ does not belong to the essential image. ∎

###### Lemma A.2 (rectifiability of the image).

Let $\Omega\subseteq\mathbb{C}$ be open. If $\Psi\in
W^{1,1}_{\\_}{loc}(\Omega,\mathbb{R}^{q})$ and $\mathcal{G}$ denotes the set
of Lebesgue points for both $\Psi$ and $\nabla\Psi$, then there exist Lebesgue
measurable sets $E_{\\_}1,E_{\\_}2,\dots$ such that
$\mathcal{G}=\bigcup_{\\_}iE_{\\_}i$ and
${\left.\kern-1.2pt\Psi\vphantom{\big{|}}\right|_{E_{\\_}i}}$ is Lipschitz.

###### Proof.

We set
$F_{\\_}j:=\big{\\{}x\in\mathcal{G}\cap\Omega_{\\_}{1/j}:\,\ThisStyle{\ensurestackMath{\stackinset{c}{.2\LMpt}{c}{.5\LMpt}{\SavedStyle-}{\SavedStyle\phantom{\int}}}\kern-16.56937pt}\int_{\\_}{B_{\\_}r^{2}(x)}\left|\Psi(y)-\Psi(x)\right|\,dy\leq
jr\text{ for }0<r\leq j^{-1}\big{\\}}$ (where
$\Omega_{\\_}\delta:=\\{x\in\Omega:\operatorname{dist}(x,\partial\Omega)>\delta\\}$),
which is a Lebesgue measurable set. By [15, Theorem 6.1] (and its proof), for
all $x\in\mathcal{G}$

$\begin{split}\,\ThisStyle{\ensurestackMath{\stackinset{c}{.2\LMpt}{c}{.5\LMpt}{\SavedStyle-}{\SavedStyle\phantom{\int}}}\kern-16.56937pt}\int_{\\_}{B_{\\_}r^{2}(x)}\left|\Psi(y)-\Psi(x)\right|\,dy&\leq
r\left|\nabla\Psi(x)\right|+\,\ThisStyle{\ensurestackMath{\stackinset{c}{.2\LMpt}{c}{.5\LMpt}{\SavedStyle-}{\SavedStyle\phantom{\int}}}\kern-16.56937pt}\int_{\\_}{B_{\\_}r^{2}(x)}\left|\Psi(y)-\Psi(x)-\left\langle\nabla\Psi(x),y-x\right\rangle\right|\,dy\\\
&=r\left|\nabla\Psi(x)\right|+o(r).\end{split}$

We infer that $\mathcal{G}=\bigcup_{\\_}{j\geq 1}F_{\\_}j$. Assume now that
$x,x^{\prime}\in F_{\\_}j$ are distinct and
$\left|x-x^{\prime}\right|\leq\frac{1}{2j}$. Let
$r:=\left|x-x^{\prime}\right|$ and notice that $B_{\\_}r^{2}(x)\subseteq
B_{\\_}{2r}^{2}(x^{\prime})$. We can estimate

$\begin{split}\left|\Psi(x)-\Psi(x^{\prime})\right|&\leq\,\ThisStyle{\ensurestackMath{\stackinset{c}{.2\LMpt}{c}{.5\LMpt}{\SavedStyle-}{\SavedStyle\phantom{\int}}}\kern-16.56937pt}\int_{\\_}{B_{\\_}r^{2}(x)}\left|\Psi(y)-\Psi(x)\right|\,dy+\,\ThisStyle{\ensurestackMath{\stackinset{c}{.2\LMpt}{c}{.5\LMpt}{\SavedStyle-}{\SavedStyle\phantom{\int}}}\kern-16.56937pt}\int_{\\_}{B_{\\_}r^{2}(x)}\left|\Psi(y)-\Psi(x^{\prime})\right|\,dy\\\
&\leq\,\ThisStyle{\ensurestackMath{\stackinset{c}{.2\LMpt}{c}{.5\LMpt}{\SavedStyle-}{\SavedStyle\phantom{\int}}}\kern-16.56937pt}\int_{\\_}{B_{\\_}r^{2}(x)}\left|\Psi(y)-\Psi(x)\right|\,dy+4\,\ThisStyle{\ensurestackMath{\stackinset{c}{.2\LMpt}{c}{.5\LMpt}{\SavedStyle-}{\SavedStyle\phantom{\int}}}\kern-16.56937pt}\int_{\\_}{B_{\\_}{2r}^{2}(x^{\prime})}\left|\Psi(y)-\Psi(x)\right|\,dy\\\
&\leq jr+8jr=9j\left|x-x^{\prime}\right|\end{split}$

(since $2r\leq j^{-1}$). The result follows once we split each $F_{\\_}j$ into
countably many subsets of diameter at most $\frac{1}{2j}$. ∎

###### Lemma A.3 (small diameter for suitable slices).

Assume $\Psi\in W^{1,2}(B_{\\_}{(1+\tau)r}^{2}(0),\mathbb{R}^{q})$, with
$r,\tau>0$. Then there is a measurable subset $E\subseteq(r,(1+\tau)r)$ of
positive measure such that, for all $r^{\prime}\in E$,
${\left.\kern-1.2pt\Psi\vphantom{\big{|}}\right|_{\partial
B_{\\_}{r^{\prime}}^{2}(0)}}$ has an absolutely continuous representative
whose image satisfies

$\operatorname{diam}\Psi(\partial
B_{\\_}{r^{\prime}}^{2}(0))\leq\bigg{(}\frac{2\pi(1+\tau)}{\tau}\int_{\\_}{B_{\\_}{(1+\tau)r}^{2}(0)}\left|\nabla\Psi\right|^{2}\,d\mathcal{L}^{2}\bigg{)}^{1/2}.$

###### Proof.

For a.e. $r^{\prime}\in(r,(1+\tau)r)$ the function
$\theta\mapsto\Psi(r^{\prime}\cos\theta,r^{\prime}\sin\theta)$ has an
absolutely continuous representative, whose weak derivative is
$r^{\prime}\left\langle\nabla\Psi(r^{\prime}\cos\theta,r^{\prime}\sin\theta),(-\sin\theta,\cos\theta)\right\rangle$
(see [15, Theorem 4.21]). Moreover,

$\,\ThisStyle{\ensurestackMath{\stackinset{c}{.2\LMpt}{c}{.5\LMpt}{\SavedStyle-}{\SavedStyle\phantom{\int}}}\kern-16.56937pt}\int_{\\_}r^{r+\tau
r}\int_{\\_}{\partial
B_{\\_}t^{2}(0)}\left|\nabla\Psi\right|^{2}\,d\mathcal{H}^{1}\,dt\leq\frac{1}{\tau
r}\int_{\\_}{B_{\\_}{(1+\tau)r}^{2}(0)}\left|\nabla\Psi\right|^{2}\,d\mathcal{L}^{2}.$

Hence, for a set of radii $r^{\prime}$ of positive measure, the inner integral
is less or equal than the right-hand side, giving

$\bigg{(}\int_{\\_}{\partial
B_{\\_}{r^{\prime}}^{2}(0)}\left|\nabla\Psi\right|\,d\mathcal{H}^{1}\bigg{)}^{2}\leq
2\pi r^{\prime}\int_{\\_}{\partial
B_{\\_}{r^{\prime}}^{2}(0)}\left|\nabla\Psi\right|^{2}\,d\mathcal{H}^{1}\leq\frac{2\pi(1+\tau)}{\tau}\int_{\\_}{B_{\\_}{(1+\tau)r}^{2}(0)}\left|\nabla\Psi\right|^{2}\,d\mathcal{L}^{2}.\qed$

###### Lemma A.4 (vanishing first order on suitable slices).

Assume that $0$ is a Lebesgue point for $\Psi\in
W^{1,2}(B_{\\_}R^{2}(0),\mathbb{R}^{q})$ and $\nabla\Phi$, as well as
$\Psi(0)=\nabla\Psi(0)=0$. Then, for any $\tau,\varepsilon>0$ and any
$0<r\leq\overline{r}$, we can select $r^{\prime}\in(r,(1+\tau)r)$ such that
${\left.\kern-1.2pt\Psi\vphantom{\big{|}}\right|_{\partial
B_{\\_}{r^{\prime}}^{2}(0)}}$ has an absolutely continuous representative with

$\max_{\\_}{y\in S^{1}}\left|\Psi(r^{\prime}y)\right|\leq\varepsilon r,$

for a suitable $\overline{r}=\overline{r}(\Psi,\tau,\varepsilon)$.

###### Proof.

We can assume $\varepsilon<1$. By [15, Theorem 6.1] and its proof we have

(A.1) $\int_{\\_}{B_{\\_}r^{2}(0)}|\Psi|^{2}\,d\mathcal{L}^{2}=o(r^{4})$

(since $1^{*}=2$) and, as a consequence,

$\begin{split}&\,\ThisStyle{\ensurestackMath{\stackinset{c}{.2\LMpt}{c}{.5\LMpt}{\SavedStyle-}{\SavedStyle\phantom{\int}}}\kern-16.56937pt}\int_{\\_}r^{r+\tau
r}\int_{\\_}{\partial B_{\\_}t^{2}(0)}((2\pi
t)^{-1}|\Psi|^{2}+r|\nabla\Psi|)\,d\mathcal{H}^{1}\,dt\\\
&\leq\frac{1}{2\pi\tau
r^{2}}\int_{\\_}{B_{\\_}{(1+\tau)r}^{2}(0)}|\Psi|^{2}\,d\mathcal{L}^{2}+\frac{1}{\tau}\int_{\\_}{B_{\\_}{(1+\tau)r}^{2}(0)}|\nabla\Psi|\,d\mathcal{L}^{2}=o(r^{2}).\end{split}$

Hence, if $r$ is small enough, there exists some $r^{\prime}\in(r,r+\tau r)$
such that

$(2\pi r^{\prime})^{-1}\left\|\Psi\right\|_{\\_}{L^{2}(\partial
B_{\\_}{r^{\prime}}^{2}(0))}^{2}+r\left\|\nabla\Psi\right\|_{\\_}{L^{1}(\partial
B_{\\_}{r^{\prime}}^{2}(0))}\leq\frac{\varepsilon^{2}r^{2}}{4}$

and ${\left.\kern-1.2pt\Psi\vphantom{\big{|}}\right|_{\partial
B_{\\_}{r^{\prime}}^{2}(0)}}$ is absolutely continuous (a.e.), with derivative
given by the chain rule. The elementary inequality

$\left\|\Psi\right\|_{\\_}{L^{\infty}(\partial
B_{\\_}{r^{\prime}}^{2}(0))}\leq\bigg{|}\,\ThisStyle{\ensurestackMath{\stackinset{c}{.2\LMpt}{c}{.5\LMpt}{\SavedStyle-}{\SavedStyle\phantom{\int}}}\kern-16.56937pt}\int_{\\_}{\partial
B_{\\_}{r^{\prime}}^{2}(0)}\Psi\,d\mathcal{H}^{1}\bigg{|}+\int_{\\_}{\partial
B_{\\_}{r^{\prime}}^{2}(0)}\left|\nabla\Psi\right|\,d\mathcal{H}^{1}\leq\frac{\left\|\Psi\right\|_{\\_}{L^{2}(\partial
B_{\\_}{r^{\prime}}^{2}(0))}}{\sqrt{2\pi
r^{\prime}}}+\left\|\nabla\Psi\right\|_{\\_}{L^{1}(\partial
B_{\\_}{r^{\prime}}^{2}(0))}$

gives the desired result. ∎

###### Lemma A.5 (strong subconvergence on a.e. slice).

If $\Psi_{\\_}k\rightharpoonup\Psi_{\\_}\infty$ in
$W^{1,2}(B_{\\_}R^{2}(0),\mathbb{R}^{q})$, then for a.e. $r^{\prime}\in(0,R)$
there exists a subsequence $(k_{\\_}i)$ such that
${\left.\kern-1.2pt\Psi_{\\_}\infty\vphantom{\big{|}}\right|_{\partial
B_{\\_}{r^{\prime}}^{2}(0)}}$ and
${\left.\kern-1.2pt\Psi_{\\_}{k_{\\_}i}\vphantom{\big{|}}\right|_{\partial
B_{\\_}{r^{\prime}}^{2}(0)}}$ have absolutely continuous representatives and

${\left.\kern-1.2pt\Psi_{\\_}{k_{\\_}i}\vphantom{\big{|}}\right|_{\partial
B_{\\_}{r^{\prime}}^{2}(0)}}\to{\left.\kern-1.2pt\Psi_{\\_}\infty\vphantom{\big{|}}\right|_{\partial
B_{\\_}{r^{\prime}}^{2}(0)}}\quad\text{in }L^{\infty}.$

###### Proof.

We observe that
${\left.\kern-1.2pt\Psi_{\\_}\infty\vphantom{\big{|}}\right|_{\partial
B_{\\_}{r^{\prime}}^{2}(0)}}$ coincides with the trace of $\Psi_{\\_}\infty$
on $\partial B_{\\_}{r^{\prime}}^{2}(0)$ for a.e. $r^{\prime}\in(0,R)$:
actually, this happens whenever $\mathcal{H}^{1}$-a.e. point on $\partial
B_{\\_}{r^{\prime}}^{2}(0)$ is a Lebesgue point of $\Psi_{\\_}\infty$ (see
[15, Theorem 5.7]); the same holds for $\Psi_{\\_}k$. Now, by Fatou’s lemma,

$\begin{split}&\int_{\\_}0^{R}\liminf_{\\_}{k\to\infty}\int_{\\_}{\partial
B_{\\_}{r^{\prime}}^{2}(0)}(\left|\Psi_{\\_}k\right|^{2}+\left|\nabla\Psi_{\\_}k\right|^{2})\,d\mathcal{H}^{1}\,dr^{\prime}\\\
&\leq\liminf_{\\_}{k\to\infty}\int_{\\_}0^{R}\int_{\\_}{\partial
B_{\\_}{r^{\prime}}^{2}(0)}(\left|\Psi_{\\_}k\right|^{2}+\left|\nabla\Psi_{\\_}k\right|^{2})\,d\mathcal{H}^{1}\,dr^{\prime}\\\
&=\liminf_{\\_}{k\to\infty}\left\|\Psi_{\\_}k\right\|_{\\_}{W^{1,2}}^{2}<\infty.\end{split}$

Hence, for a.e. $r^{\prime}\in(0,R)$,

$\liminf_{\\_}{k\to\infty}\int_{\\_}{\partial
B_{\\_}{r^{\prime}}^{2}(0)}(\left|\Psi_{\\_}k\right|^{2}+\left|\nabla\Psi_{\\_}k\right|^{2})\,d\mathcal{H}^{1}<\infty,$

which means that there exists a subsequence $(k_{\\_}i)$ such that

$\sup_{\\_}i\int_{\\_}{\partial
B_{\\_}{r^{\prime}}^{2}(0)}(\left|\Psi_{\\_}{k_{\\_}i}\right|^{2}+\left|\nabla\Psi_{\\_}{k_{\\_}i}\right|^{2})\,d\mathcal{H}^{1}<\infty.$

We can also assume that
${\left.\kern-1.2pt\Psi_{\\_}\infty\vphantom{\big{|}}\right|_{\partial
B_{\\_}{r^{\prime}}^{2}(0)}}$ equals the trace, that it has a $W^{1,2}$
representative with weak derivative given by the chain rule and that the
analogous statements hold also for $\Psi_{\\_}{k_{\\_}i}$.

In particular, the sequence
$\left({\left.\kern-1.2pt\Psi_{\\_}{k_{\\_}i}\vphantom{\big{|}}\right|_{\partial
B_{\\_}{r^{\prime}}^{2}(0)}}\right)$ is bounded in $W^{1,2}(\partial
B_{\\_}{r^{\prime}}^{2}(0),\mathbb{R}^{q})$: by the compact embedding into
$C^{0}(\partial B_{\\_}{r^{\prime}}^{2}(0),\mathbb{R}^{q})$ we deduce that (up
to subsequences)
${\left.\kern-1.2pt\Psi_{\\_}{k_{\\_}i}\vphantom{\big{|}}\right|_{\partial
B_{\\_}{r^{\prime}}^{2}(0)}}\to f$ in $L^{\infty}$, for some $f\in
C^{0}(\partial B_{\\_}{r^{\prime}}^{2}(0),\mathbb{R}^{q})$. By the weak
continuity of the trace, we have
${\left.\kern-1.2pt\Psi_{\\_}{k_{\\_}i}\vphantom{\big{|}}\right|_{\partial
B_{\\_}{r^{\prime}}^{2}(0)}}\rightharpoonup{\left.\kern-1.2pt\Psi_{\\_}\infty\vphantom{\big{|}}\right|_{\partial
B_{\\_}{r^{\prime}}^{2}(0)}}$ in $L^{2}(\partial
B_{\\_}{r^{\prime}}^{2}(0),\mathbb{R}^{q})$, hence
$f={\left.\kern-1.2pt\Psi_{\\_}\infty\vphantom{\big{|}}\right|_{\partial
B_{\\_}{r^{\prime}}^{2}(0)}}$ $\mathcal{H}^{1}$-a.e. ∎

###### Lemma A.6 (almost planar conformal matrices).

Let $\mathcal{C}\subseteq\mathbb{R}^{q\times 2}$ denote the set of matrices
$M$ with
$\sum_{\\_}{i=1}^{q}M_{\\_}{ij}M_{\\_}{ik}=\frac{|M|^{2}}{2}\delta_{\\_}{jk}$,
where $|M|$ is the Hilbert–Schmidt norm of $M$ ($\mathcal{C}$ can be
identified with the set of linear weakly conformal maps
$\mathbb{R}^{2}\to\mathbb{R}^{q}$). For any $\tau>0$ there exists
$C=C(\tau,q)$ such that

$|M_{\\_}{11}M_{\\_}{21}+M_{\\_}{12}M_{\\_}{22}|+|M_{\\_}{11}^{2}+M_{\\_}{12}^{2}-J(M)|+|M_{\\_}{21}^{2}+M_{\\_}{22}^{2}-J(M)|\leq\tau|M|^{2}+C\sum_{\\_}{i=3}^{q}\sum_{\\_}{j=1}^{2}M_{\\_}{ij}^{2}$

for all $M\in\mathcal{C}$, where
$J(M):=\left|M_{\\_}{11}M_{\\_}{22}-M_{\\_}{12}M_{\\_}{21}\right|$.

###### Proof.

Assume by contradiction that, for a sequence $(M^{k})\subseteq\mathcal{C}$, we
have

$\begin{split}&|M_{\\_}{11}^{k}M_{\\_}{21}^{k}+M_{\\_}{12}^{k}M_{\\_}{22}^{k}|+|(M_{\\_}{11}^{k})^{2}+(M_{\\_}{12}^{k})^{2}-J(M^{k})|+|(M_{\\_}{21}^{k})^{2}+(M_{\\_}{22}^{k})^{2}-J(M^{k})|\\\
&>\tau|M^{k}|^{2}+k\sum_{\\_}{i=3}^{q}\sum_{\\_}{j=1}^{2}(M_{\\_}{ij}^{k})^{2}.\end{split}$

By homogeneity we can assume $|M^{k}|=1$ for all $k$. As a consequence,
$\sum_{\\_}{i=3}^{q}\sum_{\\_}{j=1}^{2}(M_{\\_}{ij}^{k})^{2}\to 0$. So the
sequence has a limit point $M^{\infty}\in\mathbb{R}^{q\times 2}$ satisfying

(A.2) $M^{\infty}\in\mathcal{C},\qquad M_{\\_}{ij}^{\infty}=0\quad\text{for
}i=3,\dots,q\text{ and }j=1,2,$
$|M_{\\_}{11}^{\infty}M_{\\_}{21}^{\infty}+M_{\\_}{12}^{\infty}M_{\\_}{22}^{\infty}|+|(M_{\\_}{11}^{\infty})^{2}+(M_{\\_}{12}^{\infty})^{2}-J(M^{\infty})|+|(M_{\\_}{21}^{\infty})^{2}+(M_{\\_}{22}^{\infty})^{2}-J(M^{\infty})|\geq\tau.$

But conditions (A.2) force
$\begin{pmatrix}M_{\\_}{11}^{\infty}&M_{\\_}{12}^{\infty}\\\
M_{\\_}{21}^{\infty}&M_{\\_}{22}^{\infty}\end{pmatrix}=\begin{pmatrix}a&\mp
b\\\ b&\pm a\end{pmatrix}$ for some $a,b\in\mathbb{R}$, so the left-hand side
of the last inequality vanishes, yielding the desired contradiction. ∎

###### Lemma A.7 ($\bm{L^{1,\infty}}$-variance estimate).

For any $\varepsilon>0$ there exists a $C=C(\varepsilon)$ such that

$\begin{split}&\big{\|}f-(f)_{\\_}{B_{\\_}{r/2}^{2}(0)}\big{\|}_{\\_}{L^{1,\infty}(B_{\\_}{r/2}^{2}(0))}\leq\varepsilon\big{\|}f-(f)_{\\_}{B_{\\_}{3r/4}^{2}(0)}\big{\|}_{\\_}{L^{1}(B_{\\_}{3r/4}^{2}(0))}\\\
&+C\sup\Big{\\{}\Big{|}\int_{\\_}{B_{\\_}r^{2}(0)}f\nabla\phi\,d\mathcal{L}^{2}\Big{|}\,;\
\phi\in
C^{\infty}_{\\_}c(B_{\\_}r^{2}(0)),\left\|\nabla\phi\right\|_{\\_}{L^{\infty}}\leq
1\Big{\\}},\end{split}$

for all $f\in L^{1}(B_{\\_}r^{2}(0))$, where
$(f)_{\\_}{B_{\\_}s^{2}(0)}:=\,\ThisStyle{\ensurestackMath{\stackinset{c}{.2\LMpt}{c}{.5\LMpt}{\SavedStyle-}{\SavedStyle\phantom{\int}}}\kern-16.56937pt}\int_{\\_}{B_{\\_}s^{2}(0)}f\,d\mathcal{L}^{2}$.

###### Proof.

By scaling-invariance and homogeneity, we can assume that $r=1$ and that the
supremum in the right-hand side equals $1$. Thanks to Hahn–Banach theorem
applied to the functional

$(\partial_{\\_}1\phi,\partial_{\\_}2\phi)\mapsto\int_{\\_}{B_{\\_}1^{2}(0)}f\partial_{\\_}i\phi\,d\mathcal{L}^{2},\quad\text{on
}\left\\{(\partial_{\\_}1\phi,\partial_{\\_}2\phi)\mid\phi\in
C_{\\_}c^{\infty}(B_{\\_}1^{2}(0))\right\\}\subseteq
C_{\\_}c^{0}(B_{\\_}1^{2}(0))^{2},$

we can find finite Borel measures $g_{\\_}{ij}\in\mathcal{M}(B_{\\_}1^{2}(0))$
(for $i,j=1,2$) with

(A.3) $\displaystyle\left\|g_{\\_}{ij}\right\|(B_{\\_}1^{2}(0))\leq
1,\qquad\partial_{\\_}if=\sum_{\\_}j\partial_{\\_}jg_{\\_}{ij}\quad\text{in
}\mathcal{D}^{\prime}(B_{\\_}1^{2}(0)).$

By convolution with a mollifier $\rho_{\\_}\delta$, we can assume
$f,g_{\\_}{ij}\in C^{\infty}(B_{\\_}{3/4}^{2}(0))$ and that (A.3) holds in the
ball $B_{\\_}{3/4}^{2}(0)$: once we show that (A.3), together with the
requirement that
$\big{|}\int_{\\_}{B_{\\_}{3/4}^{2}(0)}f\nabla\phi\,d\mathcal{L}^{2}\big{|}\leq
1$ whenever $\phi\in C^{\infty}_{\\_}c(B_{\\_}{3/4}^{2}(0))$ and
$\left\|\nabla\phi\right\|_{\\_}{L^{\infty}}\leq 1$, imply

(A.4)
$\displaystyle\big{\|}f-(f)_{\\_}{B_{\\_}{1/2}^{2}(0)}\big{\|}_{\\_}{L^{1,\infty}(B_{\\_}{1/2}^{2}(0))}\leq\varepsilon\big{\|}f-(f)_{\\_}{B_{\\_}{3/4}^{2}(0)}\big{\|}_{\\_}{L^{1}(B_{\\_}{3/4}^{2}(0))}+C$

for the mollified $f$, the result follows for the original function by letting
$\delta\to 0$ (notice that the above requirement holds if $\delta$ is small
enough). We choose a cut-off function $\chi\in
C^{\infty}_{\\_}c(B_{\\_}{3/4}^{2}(0))$ such that $\chi=1$ on a neighborhood
of $\overline{B}_{\\_}{1/2}^{2}(0)$. Notice that

$\partial_{\\_}i(\chi f)=\sum_{\\_}j\partial_{\\_}j(\chi
g_{\\_}{ij})-\sum_{\\_}j\partial_{\\_}j\chi g_{\\_}{ij}+\partial_{\\_}i\chi
f,$

so we get the following distributional identity in $\mathbb{R}^{2}$:

$\begin{split}(1-\Delta)(\chi f)&=(1-\Delta)\chi
f-\sum_{\\_}{i,j}\partial_{\\_}{ij}^{2}(\chi
g_{\\_}{ij})+\sum_{\\_}{i,j}\partial_{\\_}{ij}^{2}\chi
g_{\\_}{ij}+\sum_{\\_}{i,j}\partial_{\\_}j\chi\partial_{\\_}ig_{\\_}{ij}-\sum_{\\_}i\partial_{\\_}i\chi\partial_{\\_}if\\\
&=(1-\Delta)\chi f-\sum_{\\_}{i,j}\partial_{\\_}{ij}^{2}(\chi
g_{\\_}{ij})+\sum_{\\_}{i,j}\partial_{\\_}{ij}^{2}\chi
g_{\\_}{ij}+\sum_{\\_}{i,j}\partial_{\\_}j\chi\partial_{\\_}ig_{\\_}{ij}-\sum_{\\_}{i,j}\partial_{\\_}i\chi\partial_{\\_}jg_{\\_}{ij}\\\
&=(1-\Delta)\chi f-\sum_{\\_}{i,j}\partial_{\\_}{ij}^{2}(\chi
g_{\\_}{ij})+\sum_{\\_}{i,j}\partial_{\\_}{ij}^{2}\chi
g_{\\_}{ij}+\sum_{\\_}{i,j}\partial_{\\_}i(\partial_{\\_}j\chi
g_{\\_}{ij})-\sum_{\\_}{i,j}\partial_{\\_}j(\partial_{\\_}i\chi
g_{\\_}{ij}).\end{split}$

Applying $(1-\Delta)^{-1}$ to both sides and observing that the operators
$(1-\Delta)^{-1}$, $(1-\Delta)^{-1}\nabla$ and $(1-\Delta)^{-1}\nabla^{2}$
correspond to Hörmander–Mikhlin Fourier multipliers, we obtain

$\chi f=(1-\Delta)^{-1}((1-\Delta)\chi f)+r,$

where the Schwartz function $r$ satisfies
$\left\|r\right\|_{\\_}{L^{1,\infty}(\mathbb{R}^{2})}\leq C^{\prime}$ for some
absolute constant $C^{\prime}$. The desired estimate (A.4) now follows from
Lemma A.8 below. ∎

###### Lemma A.8 ($\bm{L^{1}}$-estimate for a commutator).

For any $\varepsilon>0$ there exists a $C=C(\varepsilon)$ such that

$\begin{split}&\big{\|}(1-\Delta)^{-1}((1-\Delta)\chi
f)-(f)_{\\_}{B_{\\_}{1/2}^{2}(0)}\big{\|}_{\\_}{L^{1}(B_{\\_}{1/2}^{2}(0))}\leq\varepsilon\big{\|}f-(f)_{\\_}{B_{\\_}{3/4}^{2}(0)}\big{\|}_{\\_}{L^{1}(B_{\\_}{3/4}^{2}(0))}\\\
&+C\sup\Big{\\{}\Big{|}\int_{\\_}{B_{\\_}{3/4}^{2}(0)}f\nabla\phi\,d\mathcal{L}^{2}\Big{|}\,;\
\phi\in
C^{\infty}_{\\_}c(B_{\\_}{3/4}^{2}(0)),\left\|\nabla\phi\right\|_{\\_}{L^{\infty}}\leq
1\Big{\\}},\end{split}$

for all $f\in C^{\infty}(B_{\\_}{3/4}^{2}(0))\cap L^{1}(B_{\\_}{3/4}^{2}(0))$.

###### Proof.

We can assume $(f)_{\\_}{B_{\\_}{3/4}^{2}(0)}=0$ and
$\left\|f\right\|_{\\_}{L^{1}(B_{\\_}{3/4}^{2}(0))}=1$. If the statement is
false, we can find a sequence $(f_{\\_}n)$ of such functions with

(A.5) $\begin{split}&\big{\|}(1-\Delta)^{-1}((1-\Delta)\chi
f_{\\_}n)-(f_{\\_}n)_{\\_}{B_{\\_}{1/2}^{2}(0)}\big{\|}_{\\_}{L^{1}(B_{\\_}{1/2}^{2}(0))}\\\
&\geq\varepsilon+n\sup\Big{\\{}\Big{|}\int_{\\_}{B_{\\_}{3/4}^{2}(0)}f_{\\_}n\nabla\phi\,d\mathcal{L}^{2}\Big{|}\,;\
\phi\in
C^{\infty}_{\\_}c(B_{\\_}{3/4}^{2}(0)),\left\|\nabla\phi\right\|_{\\_}{L^{\infty}}\leq
1\Big{\\}}.\end{split}$

We remark that, for any $h\in C^{\infty}_{\\_}c(B_{\\_}{3/4}^{2}(0))$,

(A.6) $(1-\Delta)^{-1}h=K*h\quad\text{on }B_{\\_}{1/2}^{2}(0),\qquad
K:=1_{\\_}{B_{\\_}2^{2}(0)}\mathcal{F}^{-1}((1+4\pi^{2}|\xi|^{2})^{-1})\in
L^{1}(\mathbb{R}^{2}).$

In particular, thanks to (A.6), the left-hand side of (A.5) is bounded by a
constant and we deduce
$\int_{\\_}{B_{\\_}{3/4}^{2}(0)}f_{\\_}n\nabla\phi\,d\mathcal{L}^{2}\to 0$ for
all $\phi\in C^{\infty}_{\\_}c(B_{\\_}{3/4}^{2}(0))$. There exists a
subsequence $(f_{\\_}{n_{\\_}k})$ such that
$f_{\\_}{n_{\\_}k}\mathcal{L}^{2}\stackrel{{\scriptstyle*}}{{\rightharpoonup}}\mu$,
for some $\mu\in\mathcal{M}(B_{\\_}{3/4}^{2}(0))$. From the identity
$\int_{\\_}{B_{\\_}{3/4}^{2}(0)}\nabla\phi\,d\mu=0$ for all $\phi\in
C^{\infty}_{\\_}c(B_{\\_}{3/4}^{2}(0))$ we infer
$\mu=c\bm{1}_{\\_}{B_{\\_}{3/4}^{2}(0)}\mathcal{L}^{2}$ (with $c$ a real
constant). Hence, letting

$h_{\\_}k:=(1-\Delta)\chi f_{\\_}{n_{\\_}k},\quad
h_{\\_}\infty:=c(1-\Delta)\chi,$

we have
$h_{\\_}k\mathcal{L}^{2}\stackrel{{\scriptstyle*}}{{\rightharpoonup}}h_{\\_}\infty\mathcal{L}^{2}$
in $\mathcal{M}(\mathbb{R}^{2})$ and, since $\mu(\partial
B_{\\_}{1/2}^{2}(0))=0$, we deduce that
$(f_{\\_}{n_{\\_}k})_{\\_}{B_{\\_}{1/2}^{2}(0)}\to c$. We also have

(A.7) $\psi*h_{\\_}k\to\psi*h_{\\_}\infty\quad\text{in
}C^{0}(\mathbb{R}^{2}),$

for any $\psi\in C^{0}_{\\_}c(B_{\\_}2^{2}(0))$: if this were not true, up to
further subsequences we could find $\delta>0$ and $(x_{\\_}k)\subseteq
B_{\\_}3^{2}(0)$ such that

(A.8)
$\left|\psi*h_{\\_}k(x_{\\_}k)-\psi*h_{\\_}\infty(x_{\\_}k)\right|\geq\delta$

and we could also assume that $x_{\\_}k\to
x_{\\_}\infty\in\overline{B}_{\\_}3^{2}(0)$; but
$\psi(x_{\\_}k-\cdot)\to\psi(x_{\\_}\infty-\cdot)$ uniformly, so

$\psi*h_{\\_}k(x_{\\_}k)=\int_{\\_}{\mathbb{R}^{2}}\psi(x_{\\_}k-\cdot)\,d(h_{\\_}k\mathcal{L}^{2})\to\int_{\\_}{\mathbb{R}^{2}}\psi(x_{\\_}\infty-\cdot)\,d(h_{\\_}\infty\mathcal{L}^{2})=\psi*h_{\\_}\infty(x_{\\_}\infty),$

while trivially
$\psi*h_{\\_}\infty(x_{\\_}k)\to\psi*h_{\\_}\infty(x_{\\_}\infty)$,
contradicting (A.8) for $k$ large enough.

Moreover, $K*h_{\\_}k\to K*h_{\\_}\infty$ in $L^{1}(\mathbb{R}^{2})$, as is
readily seen from (A.7) by approximating $K$ with functions $\psi\in
C^{0}_{\\_}c(B_{\\_}2^{2}(0))$ in the $L^{1}$-topology. Using (A.6), it
follows that

$1_{\\_}{B_{\\_}{1/2}^{2}(0)}(1-\Delta)^{-1}h_{\\_}k\to
1_{\\_}{B_{\\_}{1/2}^{2}(0)}(1-\Delta)^{-1}h_{\\_}\infty=1_{\\_}{B_{\\_}{1/2}^{2}(0)}(1-\Delta)^{-1}(c(1-\Delta)\chi)=c1_{\\_}{B_{\\_}{1/2}^{2}(0)}$

in $L^{1}(\mathbb{R}^{2})$. However, for $k$ large enough, this contradicts
the inequality

$\big{\|}(1-\Delta)^{-1}((1-\Delta)\chi
f_{\\_}{n_{\\_}k})-(f_{\\_}{n_{\\_}k})_{\\_}{B_{\\_}{1/2}^{2}(0)}\big{\|}_{\\_}{L^{1}(B_{\\_}{1/2}^{2}(0))}\geq\varepsilon.\qed$

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