# A proof of the multiplicity one conjecture for min-max minimal surfaces in
arbitrary codimension

Alessandro Pigati  and  Tristan Rivière

###### Abstract.

Given any admissible $k$-dimensional family of immersions of a given closed
oriented surface into an arbitrary closed Riemannian manifold, we prove that
the corresponding min-max width for the area is achieved by a smooth (possibly
branched) immersed minimal surface with multiplicity one and Morse index
bounded by $k$.

###### Key words and phrases:

Minimal surfaces, min-max, viscosity method, multiplicity one conjecture

###### 2010 Mathematics Subject Classification:

49Q05, 49Q15, 49Q20, 58E20

(Pigati and Rivière) ETH Zürich, Department of Mathematics, Rämistrasse 101,
8092 Zürich, Switzerland. E-mail addresses<EMAIL_ADDRESS>and
<EMAIL_ADDRESS>
##  1\. Introduction

Recently, a new theory for the construction of branched immersed minimal
surfaces of arbitrary topology, in an assigned closed Riemannian manifold
$\mathcal{M}^{m}$, was proposed in [13]. This method is based on a
penalization of the area functional by means of the second fundamental form
$A$ of the immersion.

Namely, for a fixed parameter $\sigma>0$, one first finds an immersion
$\Phi:\Sigma\to\mathcal{M}^{m}$ which is critical for the perturbed area
functional

(1.1) $\displaystyle
A^{\sigma}(\Phi):=\int_{\\_}\Sigma\,d\operatorname{vol}_{\\_}{g_{\\_}\Phi}+\sigma^{2}\int_{\\_}\Sigma(1+\left|A\right|_{\\_}{g_{\\_}\Phi}^{2})^{2}\,d\operatorname{vol}_{\\_}{g_{\\_}\Phi},$

where $\Sigma$ is a fixed closed oriented surface and $g_{\\_}\Phi$ is the
metric induced by $\Phi$, with volume form
$\operatorname{vol}_{\\_}{g_{\\_}\Phi}$. This functional $A^{\sigma}$ enjoys a
sort of Palais–Smale condition up to diffeomorphisms.

We should mention that the idea of considering perturbed functionals goes back
to the paper [17] by Sacks–Uhlenbeck, where a perturbation of the Dirichlet
energy is used to build minimal immersed spheres. However, in order to find
minimal immersed surfaces with higher genus, one should give up working with
the Dirichlet energy and use a more tensorial functional like (1.1): among
closed orientable surfaces, only the sphere has a unique conformal structure
(up to diffeomorphisms) and, as a consequence, a harmonic map (i.e. a critical
point for the Dirichlet energy) $\Phi:\Sigma\to\mathcal{M}^{m}$ could fail to
be conformal and minimal if $\Sigma$ has positive genus. In principle, one can
overcome this issue by introducing the conformal structure as an additional
parameter in the variational problem: this program was carried out by Zhou in
[21].

Considering any sequence $\sigma_{\\_}j\downarrow 0$, one gets a sequence
$\Phi_{\\_}j:\Sigma_{\\_}j\to M$ of conformal immersions (with area bounded
above and below), where $\Sigma_{\\_}j$ denotes $\Sigma$ endowed with the
induced conformal structure. Assuming for simplicity that we are dealing with
a constant conformal structure (in general one gets a limiting Riemann surface
in the Deligne–Mumford compactification), the sequence $\Phi_{\\_}j$ is then
bounded in $W^{1,2}$ and we can consider its weak limit $\Phi_{\\_}\infty$, up
to subsequences. A priori it is not clear whether the strong
$W^{1,2}$-convergence holds, even away from a finite bubbling set. However, in
[13] the second author shows that, if the sequence $\sigma_{\\_}j$ is
carefully chosen so as to satisfy a certain _entropy condition_ , then the
surfaces $\Phi_{\\_}j(\Sigma_{\\_}j)$ converge to a _parametrized stationary
varifold_ (a notion introduced in [13, 12] and recalled in Section 3 below)
which we call $(\Sigma_{\\_}\infty,\Theta_{\\_}\infty,N_{\\_}\infty)$ in the
present paper. The limiting multiplicity $N_{\\_}\infty$ a priori could be
bigger than one.

A consequence of the main regularity result contained in [12] is that the
multiplicity $N_{\\_}\infty$ is locally constant. This result, which is
optimal for the class of parametrized stationary varifolds, leaves nonetheless
open the question whether one can have $N_{\\_}\infty>1$ on some connected
component of $\Sigma_{\\_}\infty$.

This question should be compared with the _multiplicity one conjecture_ by
Marques and Neves. In [10], the following upper bound for the Morse index of a
minimal hypersurface with locally constant multiplicity is established: if

$\displaystyle\Sigma=\sum_{\\_}{j=1}^{\ell}n_{\\_}j\Sigma_{\\_}j$

is a minimal hypersurface with locally constant multiplicity, given by a min-
max with $k$ parameters in the context of Almgren–Pitts theory, then

$\displaystyle\operatorname{index}(\operatorname{supp}\left(\Sigma\right))\leq
k,\qquad\operatorname{supp}\left(\Sigma\right):=\bigsqcup_{\\_}{j=1}^{\ell}\Sigma_{\\_}j.$

In other words, this is a bound for the Morse index of the hypersurface
obtained by replacing all the multiplicities $n_{\\_}j$ with $1$. In order for
this estimate to give more information about $\Sigma$, or at least its
unstable part, the authors make the following conjecture.

###### Conjecture 1.1 (Multiplicity one conjecture).

For generic metrics on $M^{n+1}$, with $3\leq n+1\leq 7$, two-sided unstable
components of closed minimal hypersurfaces obtained by min-max methods must
have multiplicity one.

It is natural to demand for extra information for one-sided stable components
with unstable double cover, as well. Marques and Neves were able to prove this
conjecture for one-parameter sweepouts, leaving the general case open. For
metrics with positive Ricci curvature, related results were already obtained
by Marques and Neves in [9] and later by Zhou in [20].

Further results, such as the two-sidedness of $\Sigma$ when the metric has
positive Ricci curvature, were obtained by Ketover, Marques and Neves in [7],
using the catenoid estimate.

We also mention that Ketover, Liokumovich and Song in [6, 19] started to
settle the generic, one-parameter case for the simpler and more effective
Simon–Smith variant of Almgren–Pitts theory, specially designed for
3-manifolds.

Very recently, in [1], Chodosh and Mantoulidis established the conjecture for
bumpy metrics in 3-manifolds, i.e. when $n=2$, in the setting of Allen–Cahn
level set approach. Also, half a year after this paper was written, Zhou
announced a proof for bumpy metrics in all dimensions $3\leq n+1\leq 7$, in
the context of Almgren–Pitts theory (see [22]).

The importance of this conjecture in relation to the Morse index of $\Sigma$
is twofold. First of all, there is no satisfactory definition for the Morse
index of an embedded minimal hypersurface with multiplicity bigger than one:
such $\Sigma$ could be thought as the limiting object of many qualitatively
different sequences, e.g. the elements of the sequence could realize different
covering spaces of the limit, or more pathologically they could be minimal and
have many catenoidal necks (hence $\Sigma$ would be the limit of a sequence of
highly unstable hypersurfaces).

Also, if one is able to establish a lower bound on the Morse index such as

$\displaystyle
k\leq\operatorname{index}(\operatorname{supp}\left(\Sigma\right))+\operatorname{nullity}(\operatorname{supp}\left(\Sigma\right)),$

then the multiplicity one conjecture gives infinitely many _geometrically
distinct_ minimal hypersurfaces, provided there exists at least one for every
value of $k$. This was precisely the strategy used in [1] to prove Yau’s
conjecture for generic metrics: in [1] the authors obtained the multiplicity
one result and the equality $\operatorname{index}(\Sigma)=k$ (the nullity
vanishing automatically for bumpy metrics). This was later extended to higher
ambient dimension (but in codimension one) in [22].

In this work we establish the natural counterpart of this conjecture in our
setting, namely for minimal surfaces produced by the viscous relaxation
method.

###### Theorem 1.2.

We have $N_{\\_}\infty\equiv 1$.

We stress that this result holds in _arbitrary codimension_ and _without any
genericity assumption_. This should be seen as a multiplicity one statement
from the perspective of the parametrization _domain_ , in that localization in
the domain (away from branch points) gives a genuine embedded minimal surface,
but a priori it does not exclude multiple covers of the image surface
$\Theta_{\\_}\infty(\Sigma_{\\_}\infty)$ globally. It seems to be optimal for
a min-max approach involving parametrizations, rather than e.g. approaches
involving level sets of functions, and it is sufficient to obtain an upper
bound on the Morse index. This bound, detailed in [15], relies on having a
branched immersion at our disposal, for which a good definition of Morse index
is available.

We remark that, in view of earlier work in [14], Theorem 1.2 would imply by
itself the main result of [12], for parametrized stationary varifolds arising
as a limit of stationary points for the relaxed functionals. However, the
proof of Theorem 1.2 relies substantially on the regularity result obtained in
[12], needed in several compactness arguments.

The main idea is to define a sort of _macroscopic multiplicity_ , on balls
$B_{\\_}\ell^{q}(p)$ in an ambient Euclidean space
$\mathbb{R}^{q}\supseteq\mathcal{M}^{m}$, before passing to the limit (i.e.
looking at the immersed surfaces $\Phi_{\\_}j$ rather than their limit). This
macroscopic multiplicity is roughly the closest integer to the average of a
_projected multiplicity_ , issued by the map
$\Pi\circ{\left.\kern-1.2pt\Phi_{\\_}j\vphantom{\big{|}}\right|_{B_{\\_}r^{2}(z)}}$,
where $B_{\\_}r^{2}(z)$ is a small domain ball and $\Pi$ a 2-plane close to
the image of ${\left.\kern-1.2pt\Phi_{\\_}j\vphantom{\big{|}}\right|_{\partial
B_{\\_}r^{2}(z)}}$. Then we will use a continuity argument to show that this
number stays constant as we pass from scale $1$ to scale
$\sqrt{\sigma_{\\_}j}$. At the latter scale we have a very clear understanding
of the behaviour of $\Phi_{\\_}j$ and in particular we are able to say that
here the macroscopic multiplicity equals $1$. Thus the same holds at the
original scale and this is sufficient to get $N_{\\_}\infty\equiv 1$.

Most of the work is contained in Section 5. A more detailed discussion of the
strategy, together with an informal explanation of the technical statements
contained in Section 5, is deferred to the beginning of that section.

On the other hand, Section 4 contains two auxiliary facts about harmonic maps,
the second of which is rather technical in nature, while Section 6 is more
standard and uses a simple covering argument, together with some ideas similar
to those in Section 5.

###### Corollary 1.3.

If there is no bubbling or degeneration of the underlying conformal structure,
we have strong $W^{1,2}$-convergence $\Phi_{\\_}k\to\Phi_{\\_}\infty$. In
general we have a bubble tree convergence.

Theorem 1.2 and Corollary 1.3 pave the way to obtain meaningful Morse index
bounds. Indeed, although Theorem 1.2 does not rule out the possibility of
having a surface covered multiple times by $\Phi_{\\_}\infty$, a crucial
advantage of having a parametrization at our disposal is that we have a
reasonable definition of Morse index and nullity: they are defined with
respect to the area functional and variations in
$C^{\infty}_{\\_}c(\Sigma_{\\_}\infty\setminus\left\\{z_{\\_}1,\dots,z_{\\_}s\right\\})$,
the points $z_{\\_}1,\dots,z_{\\_}s$ being the branch points of the immersion
$\Phi_{\\_}\infty$.111Although we are dealing with a weakly conformal map
$\Phi_{\\_}\infty$, for which area and energy are the same, it is important to
remark that the Morse indices for area and energy, denoted
$\operatorname{index}_{\\_}A$ and $\operatorname{index}_{\\_}E$ respectively,
should not be expected to agree. The relationship between the two is a subtle
problem: in this direction, we mention the inequality
$\operatorname{index}_{\\_}E(\Psi)\leq\operatorname{index}_{\\_}A(\Psi)\leq\operatorname{index}_{\\_}E(\Psi)+r$
established in [2], for a branched minimal immersion $\Psi$, where $r=r(g,b)$
depends on the genus and the number of branch points of $\Psi$.

The natural expected inequalities would be

$\displaystyle\operatorname{index}(\Phi_{\\_}\infty)\leq
k\leq\operatorname{index}(\Phi_{\\_}\infty)+\operatorname{nullity}(\Phi_{\\_}\infty).$

An abstract framework to show upper bounds for the Morse index, dealing with
general penalized functionals on Banach manifolds, is developed in [11].
Combining Corollary 1.3 with the general result obtained in [11] and with
[15], we reach the following conclusion (we refer the reader to [11] for the
notion of _admissible family_).

###### Corollary 1.4.

Given an admissible family
$\mathcal{A}\subseteq\mathcal{P}(\text{Imm}(\Sigma,\mathcal{M}^{m}))$ of
dimension $k$ and calling

$\displaystyle
W_{\\_}{\mathcal{A}}:=\inf_{\\_}{A\in\mathcal{A}}\sup_{\\_}{\Phi\in
A}\operatorname{area}(\Phi)$

the _width_ of $\mathcal{A}$, there exists a (possibly branched) minimal
immersion $\Phi$ of a closed surface $S$ into $\mathcal{M}^{m}$ such that

  * (i)

$\operatorname{genus}(S)\leq\operatorname{genus}(\Sigma)$,

  * (ii)

$W_{\\_}{\mathcal{A}}=\operatorname{area}(\Phi)$,

  * (iii)

$\operatorname{index}(\Phi)\leq k$.

However, proving the second inequality, namely
$k\leq\operatorname{index}(\Phi)+\operatorname{nullity}(\Phi)$, seems to
require a finer understanding of the convergence
$\Phi_{\\_}k\to\Phi_{\\_}\infty$. We hope to be able to deal with this
question elsewhere.

Also, it would be interesting to adapt Gromov’s notion of volume spectrum (and
higher codimension generalizations), used to produce infinitely many minimal
hypersurfaces in many settings, to the present situation. To this aim, a
natural topological question concerns how much _genus_ is needed to realize a
nontrivial $p$-sweepout (in the sense of Gromov–Guth), and how to realize the
sweepout within the space of immersions or in an appropriate closure of it.

##  2\. Notation

  * •

We will assume, without loss of generality, that $\mathcal{M}^{m}$ is
isometrically embedded in some Euclidean space $\mathbb{R}^{q}$. Given
$p\in\mathcal{M}^{m}$ and $\ell>0$, we set
$\mathcal{M}^{m}_{\\_}{p,\ell}:=\ell^{-1}(\mathcal{M}^{m}-p)$.

  * •

In what follows, $\Pi$ will always denote a 2-plane through the origin, which
we identify with the corresponding orthogonal projection
$\Pi:\mathbb{R}^{q}\to\Pi$. We call $\Pi^{\perp}$ the orthogonal
$(q-2)$-subspace, identified with the corresponding orthogonal projection.
Given 2-planes $\Pi,\Pi^{\prime}$, we denote by
$\operatorname{dist}(\Pi,\Pi^{\prime})$ an arbitrary distance on the
Grassmannian $\text{Gr}(2,\mathbb{R}^{q})$, e.g. the one induced by Plücker’s
embedding of $\text{Gr}(2,\mathbb{R}^{q})$ into the projectivization of
$\Lambda_{\\_}2\mathbb{R}^{q}$.

The adjoint maps, which are just the inclusions
$\Pi\hookrightarrow\mathbb{R}^{q}$ and
$\Pi^{\perp}\hookrightarrow\mathbb{R}^{q}$, are denoted $\Pi^{*}$ and
$(\Pi^{\perp})^{*}$, so that

$\displaystyle\text{id}_{\\_}{\mathbb{R}^{q}}=\Pi^{*}\Pi+(\Pi^{\perp})^{*}\Pi^{\perp}.$

Also, $\Pi_{\\_}0$ is the canonical 2-plane, so that
$\Pi_{\\_}0:\mathbb{R}^{q}\to\mathbb{R}^{2}$ is the projection onto the first
two coordinates, while $\Pi_{\\_}0^{\perp}:\mathbb{R}^{q}\to\mathbb{R}^{q-2}$
is the projection onto the remaining $q-2$.

  * •

We call $B_{\\_}r^{2}(x)$ the ball of center $x$ and radius $r$ in the plane
$\mathbb{C}=\mathbb{R}^{2}$, while $B_{\\_}s^{q}(p)$ will denote the ball of
center $p$ and radius $s$ in $\mathbb{R}^{q}$. Given $p\in\Pi$, we call
$B_{\\_}s^{\Pi}(p)$ the two-dimensional ball with center $p$ and radius $s$ in
$\Pi$, i.e. $B_{\\_}s^{\Pi}(p):=B_{\\_}s^{q}(p)\cap\Pi$. When the center is
not specified, it is always meant to be the origin.

  * •

Given a function $\Psi\in W^{1,2}(B_{\\_}r^{2}(x))$ and $0<s\leq r$, the
notation ${\left.\kern-1.2pt\Psi\vphantom{\big{|}}\right|_{\partial
B_{\\_}s^{2}(x)}}$ always refers to the trace of $\Psi$ on the circle
$\partial B_{\\_}s^{2}(x)$.

  * •

Given $K\geq 1$, we define the following set of Beltrami coefficients:

$\displaystyle\mathcal{E}_{\\_}K:=\left\\{\mu\in
L^{\infty}(\mathbb{C},\mathbb{C}):\left\|\mu\right\|_{\\_}{L^{\infty}}\leq\frac{K-1}{K+1}\right\\}.$

We let $\mathcal{D}_{\\_}K$ denote the set of $K$-quasiconformal
homeomorphisms $\varphi:\mathbb{C}\to\mathbb{C}$ such that

$\displaystyle\varphi(0)=0,\qquad\min_{\\_}{x\in\partial
B_{\\_}1^{2}}\left|\varphi(x)\right|=1.$

If $\varphi\in\mathcal{D}_{\\_}K$, we have $\varphi\in
W^{1,2}_{\\_}{loc}(\mathbb{C})$ and
$\partial_{\\_}{\overline{z}}\varphi=\mu\partial_{\\_}{z}\varphi$ for some
$\mu\in\mathcal{E}_{\\_}K$, in the weak sense; we refer the reader to [5,
Chapter 4] for the basic theory of $K$-quasiconformal homeomorphisms in the
plane. Moreover, it is immediate to check that $\varphi$ is a linear map in
$\mathcal{D}_{\\_}K$ if and only if $\varphi(e_{\\_}1)=e_{\\_}1^{\prime}$ and
$\varphi(e_{\\_}2)=\lambda e_{\\_}2^{\prime}$, for suitable orthonormal bases
$(e_{\\_}1,e_{\\_}2)$, $(e_{\\_}1^{\prime},e_{\\_}2^{\prime})$ inducing the
canonical orientation and a suitable $1\leq\lambda\leq K$.

  * •

We define

$\displaystyle
D(K):=\sup\left\\{\left|\varphi(x)\right|;x\in\overline{B}_{\\_}1^{2},\varphi\in\mathcal{D}_{\\_}K\right\\},\quad
s(K):=\inf\left\\{\left|\varphi^{-1}(y)\right|;\left|y\right|\geq\frac{1}{2},\varphi\in\mathcal{D}_{\\_}K\right\\},$

so that
$\varphi(\overline{B}_{\\_}1^{2})\subseteq\overline{B}_{\\_}{D(K)}^{2}$ and
$\varphi(\overline{B}_{\\_}{s(K)}^{2})\subseteq\overline{B}_{\\_}{1/2}^{2}$
for all $\varphi\in\mathcal{D}_{\\_}K$. The fact that $D(K)<\infty$ and
$s(K)>0$ is guaranteed by Corollary A.4. We also set

$\displaystyle\eta(K):=\frac{1}{4}\inf\left\\{\left|\varphi(x)\right|;x\in\partial
B_{\\_}{s(K)^{2}}^{2},\varphi\in\mathcal{D}_{\\_}K\right\\}>0.$

  * •

We let $\mathcal{D}_{\\_}K^{\Pi}$ denote the set of maps having the form
$\Pi^{*}\circ R\circ\varphi$, where $\varphi\in\mathcal{D}_{\\_}K$ and
$R:\mathbb{R}^{2}\to\Pi$ is a linear isometry. Given $0<\delta<1$, we call
$\mathcal{R}_{\\_}{K,\delta}^{\Pi}$ the set of maps in
$W^{1,2}(B_{\\_}1^{2},\mathbb{R}^{q})$ which are close to some
$\psi\in\mathcal{D}_{\\_}K^{\Pi}$ on the circles of radii $1,s(K),s(K)^{2}$,
namely we set

$\displaystyle\mathcal{R}_{\\_}{K,\delta}^{\Pi}:=\left\\{\Psi\in
W^{1,2}(B_{\\_}1^{2},\mathbb{R}^{q}):\min_{\\_}{\psi\in\mathcal{D}_{\\_}K^{\Pi}}\max_{\\_}{r\in\left\\{1,s(K),s(K)^{2}\right\\}}\left\|{\left.\kern-1.2pt\Psi\vphantom{\big{|}}\right|_{\partial
B_{\\_}r^{2}}}(r\cdot)-\psi(r\cdot)\right\|_{\\_}{L^{\infty}(\partial
B_{\\_}1^{2})}\leq\delta\right\\}.$

  * •

Given $\Psi\in C^{1}(\Omega,\mathbb{R}^{q})$, a ball
$B_{\\_}r^{2}(z)\subset\subset\Omega$ and a 2-plane $\Pi$, we define the
_projected multiplicity_ function

$\displaystyle
N(\Psi,B_{\\_}r^{2}(z),\Pi):\Pi\to\mathbb{N}\cup\left\\{\infty\right\\},\qquad
N(\Psi,B_{\\_}r^{2}(z),\Pi)(p):=\\#((\Pi\circ\Psi)^{-1}(p)\cap
B_{\\_}{r}^{2}(z))$

and, given $p\in\Pi$ and $t>0$, we also define the _macroscopic multiplicity_

$\displaystyle
n(\Psi,B_{\\_}r^{2}(z),B_{\\_}t^{\Pi}(p)):=\Big{\lfloor}\,\ThisStyle{\ensurestackMath{\stackinset{c}{.2\LMpt}{c}{.5\LMpt}{\SavedStyle-}{\SavedStyle\phantom{\int}}}\kern-16.56937pt}\int_{\\_}{B_{\\_}t^{\Pi}(p)}N(\Psi,B_{\\_}r^{2}(z),\Pi)+\frac{1}{2}\Big{\rfloor}\in\mathbb{N}$

The mean appearing in (• ‣ 2) is finite by the area formula and
$\left\lfloor\cdot\right\rfloor$ denotes the integer part. Note that, if the
mean is close to an integer $k$, then the macroscopic multiplicity is
precisely $k$. Note also that for any $p\in\mathbb{R}^{q}$ we have

$\displaystyle
n(\Psi,B_{\\_}r^{2}(z),B_{\\_}t^{\Pi}(\Pi(p)))=n\left(\frac{\Psi(z+r\cdot)-p}{t},B_{\\_}1^{2},B_{\\_}1^{\Pi}\right).$

##  3\. Background on parametrized stationary varifolds

Let $\mathcal{M}^{m}\subset\mathbb{R}^{q}$ be a (smooth, closed) embedded
submanifold. Assume we have a smooth conformal immersion
$\Phi:B_{\\_}1^{2}\to\mathcal{M}^{m}$, critical for the functional

(3.1)
$\displaystyle\Phi\mapsto\int_{\\_}{B_{\\_}1^{2}}\,d\operatorname{vol}_{\\_}{g_{\\_}\Phi}+\sigma^{2}\int_{\\_}{B_{\\_}1^{2}}(1+|A_{\\_}{g_{\\_}\Phi}|_{\\_}{g_{\\_}\Phi}^{2})^{2}\,d\operatorname{vol}_{\\_}{g_{\\_}\Phi}$

with respect to smooth variations $(\Phi_{\\_}t)$ such that $\Phi_{\\_}t$ is
an immersion agreeing with $\Phi$ outside some compact set $F\subset
B_{\\_}1^{2}$ (independent of $t$). Here
$g_{\\_}\Phi:=\Phi^{*}g_{\\_}{\mathbb{R}^{q}}$ and $A_{\\_}{g_{\\_}\Phi}$ is
the second fundamental form of $\Phi$. Assume that the following _entropy
condition_

(3.2)
$\displaystyle\sigma^{2}\log(\sigma^{-1})\int_{\\_}{B_{\\_}1^{2}}(1+|A|^{2})^{2}\,d\operatorname{vol}_{\\_}{g_{\\_}\Phi}\leq\varepsilon\int_{\\_}{B_{\\_}1^{2}}\,d\operatorname{vol}_{\\_}{g_{\\_}\Phi}$

holds for some $\varepsilon>0$. This condition plays a fundamental role in the
viscosity approach presented in [13], and can be enforced ultimately owing to
Struwe’s monotonicity trick (see [13, Section II] and the references therein).
Note that

$\displaystyle
g_{\\_}{\Phi}=\frac{1}{2}|\nabla\Phi|^{2}\delta,\quad\int_{\\_}{B_{\\_}1^{2}}\,d\operatorname{vol}_{\\_}{g_{\\_}\Phi}=\frac{1}{2}\int_{\\_}{B_{\\_}1^{2}}\left|\nabla\Phi\right|^{2}$

by conformality of $\Phi$.

Given any $0<\ell<1$ and $p\in\mathcal{M}^{m}$, recall that
$\mathcal{M}^{m}_{\\_}{p,\ell}=\ell^{-1}(\mathcal{M}^{m}-p)$. The rescaled map

$\Psi:B_{\\_}1^{2}\to\mathcal{M}^{m}_{\\_}{p,\ell},\qquad\Psi:=\ell^{-1}(\Phi-p)$

is critical for the functional

(3.3)
$\displaystyle\int_{\\_}{B_{\\_}1^{2}}\,d\operatorname{vol}_{\\_}{g_{\\_}\Psi}+\tau^{2}\int_{\\_}{B_{\\_}1^{2}}(\ell^{2}+|A|^{2})^{2}\,d\operatorname{vol}_{\\_}{g_{\\_}\Psi},\qquad\tau:=\sigma\ell^{-2}$

and, being $\tau^{2}\log(\tau^{-1})\leq\ell^{-4}\sigma^{2}\log(\sigma^{-1})$,
it satisfies

(3.4)
$\displaystyle\tau^{2}\log(\tau^{-1})\int_{\\_}{B_{\\_}1^{2}}(\ell^{2}+|A|^{2})^{2}\,d\operatorname{vol}_{\\_}{g_{\\_}\Psi}\leq\varepsilon\int_{\\_}{B_{\\_}1^{2}}\,d\operatorname{vol}_{\\_}{g_{\\_}\Psi},$

where now $A$ denotes the second fundamental form of $\Psi$ in
$\mathcal{M}^{m}_{\\_}{p,\ell}$ and its norm is meant with respect to the
induced metric $g_{\\_}\Psi$.

In the sequel, we will establish many intermediate results on maps $\Psi$
arising in this way, by means of compactness arguments. The starting point in
these arguments is that, heuristically, if we have sequences $\Psi_{\\_}k$,
$p_{\\_}k$, $\ell_{\\_}k\to 0$, $\tau_{\\_}k\to 0$ and $\varepsilon_{\\_}k\to
0$, then by (3.3) and (3.4) $\Psi_{\\_}k$ should have a subsequential limit
$\Psi_{\\_}\infty$ (in some weak sense) which is critical for the area
functional in the tangent space $T_{\\_}{p_{\\_}\infty}\mathcal{M}^{m}$ (where
$p_{\\_}\infty$ is a subsequential limit of the sequence $p_{\\_}k$), i.e.
$\Psi_{\\_}\infty$ should be a minimal parametrization.

The kind of limiting object that we get is specified by the following
definition.

###### Definition 3.1.

A triple $(\Omega,\Phi,N)$, with $\Omega\subseteq\mathbb{C}$ open, $\Phi\in
W^{1,2}(\Omega,\mathbb{R}^{q})$ weakly conformal and $N\in
L^{\infty}(\Omega,\mathbb{N}\setminus\left\\{0\right\\})$, is a _local
parametrized stationary varifold_ if for almost every
$\omega\subset\subset\Omega$ the rectifiable $2$-varifold

$\mathbf{v}_{\\_}\omega:=(\Phi(\mathcal{G}\cap\omega),\theta_{\\_}\omega),\qquad\theta_{\\_}\omega(p):=\sum_{\\_}{x\in\mathcal{G}\cap\omega\cap\Phi^{-1}(p)}N(x)$

is stationary in the open set $\mathbb{R}^{q}\setminus\Phi(\partial\omega)$,
where $\mathcal{G}$ denotes the set of Lebesgue points for both $\Phi$ and
$d\Phi$. We also require the technical condition

$\left\|\mathbf{v}_{\\_}\Omega\right\|(B_{\\_}s^{q}(p))=\frac{1}{2}\int_{\\_}{\Phi^{-1}(B_{\\_}s^{q}(p))}N\left|\nabla\Phi\right|^{2}\,d\mathcal{L}^{2}=O(s^{2}),$

uniformly in $p\in\mathbb{R}^{q}$.

We refer the reader to [12, Definition 2.1] for the notion of _almost every
domain_ , as well as to [12, Definitions 2.2 and 2.9] for another possible
definition (in the global and local versions, respectively), whose equivalence
with Definition 3.1 is detailed in [12, Remark 2.3]. The latter formulation
will not be used here.

As already mentioned in the introduction, the main result of [12] is that
$\Phi$ is harmonic (namely, it coincides a.e. with a harmonic map) and $N$ is
(a.e.) constant on those connected components of $\Omega$ where $\Phi$ is not
itself (a.e.) constant.

Since these facts are crucially used in many intermediate steps towards the
proof of Theorem 1.2, we give a precise statement that summarizes all the
information we need to extract from the works [13] and [12].

###### Theorem 3.2 (limiting behaviour of almost flat critical immersions).

Assume that $\Psi_{\\_}k\in
C^{2}(\overline{B}_{\\_}R^{2},\mathcal{M}^{m}_{\\_}{p_{\\_}k,\ell_{\\_}k})$ is
a sequence of conformal immersions such that $\Psi_{\\_}k$ is critical for the
functional (3.3) on the interior $B_{\\_}R^{2}$ (with
$\tau_{\\_}k,\ell_{\\_}k$ in place of $\tau,\ell$) and

    * –

$\displaystyle\Psi_{\\_}k|_{\\_}{\partial B_{\\_}R^{2}}\to\gamma_{\\_}\infty$
uniformly, for some $\gamma_{\\_}\infty:\partial
B_{\\_}R^{2}\to\mathbb{R}^{q}$,

    * –

$\displaystyle\frac{1}{2}\int_{\\_}{B_{\\_}R^{2}}\left|\nabla\Psi_{\\_}k\right|^{2}\leq
E$,

    * –

$\displaystyle\tau_{\\_}k^{2}\log(\tau_{\\_}k^{-1})\int_{\\_}{B_{\\_}R^{2}}\left|A\right|^{4}\,d\operatorname{vol}_{\\_}{g_{\\_}{\Psi_{\\_}k}}\to
0$,

    * –

$\displaystyle\ell_{\\_}k,\tau_{\\_}k\to 0$.

Then, up to subsequences, $\Psi_{\\_}k\rightharpoonup\Psi_{\\_}\infty$ in
$W^{1,2}(B_{\\_}R^{2},\mathbb{R}^{q})$, for some $\Psi_{\\_}\infty$ which is
continuous (in the interior), has trace $\gamma_{\\_}\infty$, and satisfies
the convex hull property, namely

$\displaystyle\Psi_{\\_}\infty(\overline{\omega})\subseteq\operatorname{co}(\Psi_{\\_}\infty(\partial\omega))\qquad\text{for
all }\omega\subset\subset B_{\\_}R^{2}.$

The image measures
$(\Psi_{\\_}k)_{\\_}*\left(\frac{1}{2}|\nabla\Psi_{\\_}k|^{2}\right)$ in
$\mathbb{R}^{q}$ form a tight sequence.

Given $\omega\subset\subset B_{\\_}R^{2}$ with
$\Psi_{\\_}\infty(\overline{\omega})\subseteq\mathbb{R}^{q}\setminus\gamma_{\\_}\infty(\partial
B_{\\_}R^{2})$, there exists a quasiconformal homeomorphism
$\varphi_{\\_}\infty$ of $\mathbb{R}^{2}$ and a locally constant multiplicity
$N_{\\_}\infty\in L^{\infty}(\omega,\mathbb{Z}^{+})$ such that the
$2$-varifolds induced by
${\left.\kern-1.2pt\Psi_{\\_}k\vphantom{\big{|}}\right|_{\omega}}$
subsequentially converge on
$\mathbb{R}^{q}\setminus\Phi_{\\_}\infty(\partial\omega)$ to a (local)
parametrized stationary varifold

$\displaystyle(\varphi_{\\_}\infty(\omega),\Psi_{\\_}\infty\circ\varphi_{\\_}\infty^{-1},N_{\\_}\infty\circ\varphi_{\\_}\infty^{-1}).$

in the varifold sense, namely in duality with
$C_{\\_}c^{0}((\mathbb{R}^{q}\setminus\Phi_{\\_}\infty(\partial\omega))\times\operatorname{Gr}(2,\mathbb{R}^{q}))$.
Also, on $\omega$ we have the convergence of Radon measures

(3.5)
$\displaystyle\frac{1}{2}\left|\nabla\Psi_{\\_}k\right|^{2}\mathcal{L}^{2}\stackrel{{\scriptstyle*}}{{\rightharpoonup}}N_{\\_}\infty\left|\partial_{\\_}1\Psi_{\\_}\infty\wedge\partial_{\\_}2\Psi_{\\_}\infty\right|\mathcal{L}^{2}.$

Setting $\Gamma_{\\_}\infty:=\gamma_{\\_}\infty(\partial B_{\\_}R^{2})$, we
have
$N_{\\_}\infty\leq\frac{E}{\pi\operatorname{dist}(\Psi_{\\_}\infty(\overline{\omega}),\Gamma_{\\_}\infty)^{2}}$
a.e. and the distortion constant of $\varphi_{\\_}\infty$ is bounded by
$\left(\frac{E}{\pi\operatorname{dist}(\Psi_{\\_}\infty(\overline{\omega}),\Gamma_{\\_}\infty)^{2}}\right)^{2}$.

###### Proof of Theorem 3.2.

The proof is essentially already contained in [13] and [12], so we just
present the required adaptations.

Up to subsequences, we can assume that $\Psi_{\\_}k$ has a weak limit
$\Psi_{\\_}\infty$ in $W^{1,2}(B_{\\_}R^{2},\mathbb{R}^{q})$, with trace
$\gamma_{\\_}\infty$, and that the varifolds $\mathbf{v}_{\\_}k$ induced by
$\Psi_{\\_}k$ converge to a varifold $\mathbf{v}_{\\_}\infty$ in
$\mathbb{R}^{q}$.

The arguments used in [13, Section III] and in [12, Section 2] show that
$\Psi_{\\_}\infty$ has a continuous representative (on the interior
$B_{\\_}R^{2}$), satisfying the convex hull property. Also, from [13, Section
III] we have that

(3.6) $\displaystyle\mathbf{v}_{\\_}\infty\text{ is stationary in
}U:=\mathbb{R}^{q}\setminus\gamma_{\\_}\infty(\partial B_{\\_}R^{2})$

and is an integer rectifiable varifold. We claim that the measures
$\|\mathbf{v}_{\\_}k\|=(\Psi_{\\_}k)_{\\_}*\left(\frac{1}{2}|\nabla\Psi_{\\_}k|^{2}\right)$
on $\mathbb{R}^{q}$ form a tight sequence. If this were not true, up to
subsequences we could find points $q_{\\_}k\in\mathbb{R}^{q}$ with
$|q_{\\_}k|\to\infty$ and such that the argument of [13, Lemma III.3] applies
(with $q_{\\_}k$ in place of $q$), on the region
$\mathbb{R}^{q}\setminus\Psi_{\\_}k(\partial B_{\\_}R^{2})$. Hence,

$\displaystyle\liminf_{\\_}{k\to\infty}\|\mathbf{v}_{\\_}{k}\|(B_{\\_}1^{q}(q_{\\_}k))>0.$

So the varifolds $\mathbf{v}_{\\_}k-q_{\\_}k$ converge subsequentially to a
nontrivial varifold $\mathbf{v}_{\\_}\infty^{\prime}$ in $\mathbb{R}^{q}$,
stationary on $U^{\prime}:=\mathbb{R}^{q}$: indeed, the proof of (3.6) can be
repeated with $\Psi_{\\_}k-q_{\\_}k$ in place of $\Psi_{\\_}k$ (and
$U^{\prime}$ in place of $U$), using the fact that, for all $s>0$, the image
of $\Psi_{\\_}k|_{\\_}{\partial B_{\\_}R^{2}}-q_{\\_}k$ is eventually disjoint
from $B_{\\_}s^{q}$. Its total mass
$\|\mathbf{v}_{\\_}\infty^{\prime}\|(\mathbb{R}^{q})$ must be bounded by
$\liminf_{\\_}{k\to\infty}\|\mathbf{v}_{\\_}k\|(\mathbb{R}^{q})\leq E$;
however, the monotonicity formula implies that
$\|\mathbf{v}_{\\_}\infty^{\prime}\|(\mathbb{R}^{q})=\infty$, a contradiction.

We also claim that the bubbling set is empty in our setting. Indeed, by
tightness of the measures $\|\mathbf{v}_{\\_}k\|$, a bubbling point would
produce, in the limit, a nontrivial stationary varifold in $\mathbb{R}^{q}$.
Again, its mass would be bounded by $E$, contradicting the monotonicity
formula.

Now [13, Lemma III.5] gives, up to further subsequences, the limit

$\displaystyle\nu_{\\_}k:=\frac{1}{2}|\nabla\Psi_{\\_}k|^{2}\stackrel{{\scriptstyle*}}{{\rightharpoonup}}\nu_{\\_}\infty,\quad\text{with
}\nu_{\\_}\infty=m\,\mathcal{L}^{2}$

in the sense of Radon measures (i.e. in duality with
$C^{0}_{\\_}c(B_{\\_}R^{2})$). The function $m(z)\geq 0$ equals
$N_{\\_}\infty(z)|\partial_{\\_}1\Phi_{\\_}\infty\wedge\partial_{\\_}2\Phi_{\\_}\infty|(z)$
a.e., for a positive integer $N_{\\_}\infty(z)$ which is bounded by the
density of $\mathbf{v}_{\\_}\infty$ at $\Psi_{\\_}\infty(z)$ whenever
$\Psi_{\\_}\infty(z)\in U$.

Let $\omega\subset\subset B_{\\_}R^{2}$ be such that
$\Psi_{\\_}\infty(\overline{\omega})\subseteq\mathbb{R}^{q}\setminus\Gamma_{\\_}\infty$.
Defining
$s:=\operatorname{dist}(\Psi_{\\_}\infty(\overline{\omega}),\Gamma_{\\_}\infty)>0$,
note that

$\displaystyle B_{\\_}s^{q}(q)\subseteq U\qquad\text{for all
}q\in\Psi_{\\_}\infty(\overline{\omega}).$

Hence, by the monotonicity formula, the density of $\mathbf{v}_{\\_}\infty$ at
such points $q$ is bounded by $\frac{E}{\pi s^{2}}$. This gives the upper
bound for $N_{\\_}\infty$. As explained in detail in [12, Section 4], there
exists a quasiconformal homeomorphism $\varphi_{\\_}\infty$ of the plane, with
distortion constant bounded by the square of the (essential) supremum of
$N_{\\_}\infty|_{\\_}\omega$, such that
$\Psi_{\\_}\infty\circ\varphi_{\\_}\infty^{-1}$ is weakly conformal on
$\varphi_{\\_}\infty(\omega)$. Finally, it is the main outcome of [13] that
the $2$-varifolds induced by $\Phi_{\\_}k|_{\\_}\omega$ converge to the
(local) parametrized stationary varifold
$(\varphi_{\\_}\infty(\omega),\Psi_{\\_}\infty\circ\varphi_{\\_}\infty^{-1},N_{\\_}\infty\circ\varphi_{\\_}\infty^{-1})$
(whose mass measure is bounded above by $\|\mathbf{v}_{\\_}\infty\|$), in the
complement of $\Psi_{\\_}\infty(\partial\omega)$.222The convergence actually
holds on all of $\mathbb{R}^{q}$ (or, more precisely, on
$\mathbb{R}^{q}\times\operatorname{Gr}(2,\mathbb{R}^{q})$) if
$\nu_{\\_}\infty(\partial\omega)=0$. ∎

###### Theorem 3.3 (regularity of parametrized stationary varifolds).

In the situation of Theorem 3.2,
$\Psi_{\\_}\infty\circ\varphi_{\\_}\infty^{-1}:\varphi_{\\_}\infty(\omega)\to\mathbb{R}^{q}$
is harmonic. Also, if $\omega$ is connected and
$\Psi_{\\_}\infty|_{\\_}\omega$ is not constant, $N_{\\_}\infty$ equals a
constant integer (a.e.) on $\omega$ and
$\Psi_{\\_}\infty\circ\varphi_{\\_}\infty^{-1}$ is a minimal branched
immersion.

###### Proof of Theorem 3.3.

This is a special case of the main theorem in [12], namely [12, Theorem 5.7].
∎

##  4\. Two lemmas on harmonic maps

###### Lemma 4.1 (uniform convergence for Dirichlet problem with variable
domain).

Let $\gamma_{\\_}k\in C^{0}(\partial B_{\\_}1^{2},\mathbb{R}^{2})$ be a
sequence of Jordan curves converging (in $C^{0}$) to a Jordan curve
$\gamma_{\\_}\infty$ and let $f_{\\_}k\in C^{0}(\partial B_{\\_}1^{2})$ be a
sequence converging uniformly to a function $f_{\\_}\infty$. Let $D_{\\_}k$ be
the domain bounded by $\gamma_{\\_}k$, let $u_{\\_}k\in
C^{0}(\overline{D}_{\\_}k)$ be the harmonic extension of
$f_{\\_}k\circ\gamma_{\\_}k^{-1}$, and similarly define $D_{\\_}\infty$ and
$u_{\\_}\infty$. Then $u_{\\_}k\to u_{\\_}\infty$ in
$C^{0}_{\\_}{loc}(D_{\\_}\infty)$. Moreover, if $y_{\\_}k\to y_{\\_}\infty$
with $y_{\\_}k\in\overline{D}_{\\_}k$ and
$y_{\\_}\infty\in\overline{D}_{\\_}\infty$, then $u_{\\_}k(y_{\\_}k)\to
u_{\\_}\infty(y_{\\_}\infty)$.

Note that such harmonic extensions exist and are unique, since by
Carathéodory’s theorem there exist homeomorphisms
$\overline{B}_{\\_}1^{2}\to\overline{D}_{\\_}k$ restricting to biholomorphisms
$B_{\\_}1^{2}\to D_{\\_}k$ (and similarly for $D_{\\_}\infty$), allowing to
reduce matters to the well-known existence and uniqueness of the harmonic
extension on the unit disk.

###### Proof of Lemma 4.1.

Since the functions $f_{\\_}k$ are equibounded, from the maximum principle and
interior estimates it follows that the functions $u_{\\_}k$ are equibounded in
$C^{2}(\overline{\omega})$, for any $\omega\subset\subset D_{\\_}\infty$, and
hence by Ascoli–Arzelà theorem the convergence $u_{\\_}k\to u_{\\_}\infty$ in
$C^{0}_{\\_}{loc}(D_{\\_}\infty)$ follows from the second claim.

It suffices to show that the second claim holds for a subsequence: once this
is done, it can be obtained for the full sequence by a standard contradiction
argument (given a sequence $y_{\\_}k\to y_{\\_}\infty$, if
$u_{\\_}k(y_{\\_}k)$ does not converge to $u_{\\_}\infty(y_{\\_}\infty)$, we
can find a subsequence such that it converges to a different value; then we
reach a contradiction along a further subsequence where the second claim
holds).

Up to removing a finite set of indices, we can suppose that there is a point
$p$ such that $p\in D_{\\_}k$ for all
$k\in\mathbb{N}\cup\left\\{\infty\right\\}$. By Carathéodory’s theorem, we can
find homeomorphisms
$\upsilon_{\\_}k:\overline{B}_{\\_}1^{2}\to\overline{D}_{\\_}k$ restricting to
biholomorphisms from $B_{\\_}1^{2}$ to $D_{\\_}k$, so that
${\left.\kern-1.2pt\upsilon_{\\_}k\vphantom{\big{|}}\right|_{\partial
B_{\\_}1^{2}}}=\gamma_{\\_}k\circ\beta_{\\_}k$, for suitable homeomorphisms
$\beta_{\\_}k:\partial B_{\\_}1^{2}\to\partial B_{\\_}1^{2}$ (for all
$k\in\mathbb{N}$), and $\upsilon_{\\_}k(0)=p$.

Since the maps $\upsilon_{\\_}k$ and $\upsilon_{\\_}k^{-1}$ are equibounded
and harmonic, we can assume that

(4.1)
$\displaystyle\upsilon_{\\_}k\to\upsilon_{\\_}\infty,\qquad\zeta_{\\_}k:=\upsilon_{\\_}k^{-1}\to\zeta_{\\_}\infty$

in $C^{\infty}_{\\_}{loc}(B_{\\_}1^{2})$ and
$C^{\infty}_{\\_}{loc}(D_{\\_}\infty)$, respectively. Note that
$\upsilon_{\\_}\infty$ is a holomorphic map taking values into
$\overline{D}_{\\_}\infty$, while $\zeta_{\\_}\infty$ is holomorphic and takes
values into $B_{\\_}1^{2}$ (by the maximum principle, since
$\zeta_{\\_}\infty(p)=0$ and $\left|\zeta_{\\_}\infty\right|\leq 1$). So for
any $w\in D_{\\_}\infty$ the set $\left\\{\zeta_{\\_}k(w)\mid
k\in\mathbb{N}\right\\}\cup\left\\{\zeta_{\\_}\infty(w)\right\\}\subset
B_{\\_}1^{2}$ is compact and we infer

(4.2)
$\displaystyle\upsilon_{\\_}\infty\circ\zeta_{\\_}\infty(w)=\lim_{\\_}{k\to\infty}\upsilon_{\\_}k\circ\zeta_{\\_}k(w)=w.$

Hence $\upsilon_{\\_}\infty$ is surjective and thus an open map. So
$\upsilon_{\\_}\infty(B_{\\_}1^{2})=D_{\\_}\infty$ and, by [16, Theorem 10.43]
(applied with $f:=\upsilon_{\\_}\infty-w$, $g:=\upsilon_{\\_}k-w$, for a fixed
$w\in D_{\\_}\infty$ and an arbitrary circle $\partial B_{\\_}r^{2}\subseteq
B_{\\_}1^{2}$ avoiding $\upsilon_{\\_}\infty^{-1}(w)$, with $k$ large enough),
it is also injective. By Carathéodory’s theorem, it extends continuously to a
homeomorphism (still denoted $\upsilon_{\\_}\infty$) from
$\overline{B}_{\\_}1^{2}$ to $\overline{D}_{\\_}\infty$ and we have
${\left.\kern-1.2pt\upsilon_{\\_}\infty\vphantom{\big{|}}\right|_{\partial
B_{\\_}1^{2}}}=\gamma_{\\_}\infty\circ\beta_{\\_}\infty$ for a suitable
homeomorphism $\beta_{\\_}\infty:\partial B_{\\_}1^{2}\to\partial
B_{\\_}1^{2}$.

Up to subsequences, applying Helly’s selection principle to suitable lifts
$\overline{\beta}_{\\_}k:\mathbb{R}\to\mathbb{R}$, we can assume that
$\beta_{\\_}k\to\widetilde{\beta}_{\\_}\infty$ everywhere, for some order-
preserving $\widetilde{\beta}_{\\_}\infty$.333The map
$\widetilde{\beta}_{\\_}\infty$ could also be order-reversing: this happens
precisely if $\beta_{\\_}k$ reverses the orientation along the subsequence.
For simplicity, we assume $\beta_{\\_}k$, $\widetilde{\beta}_{\\_}\infty$ to
be order-preserving (the other case being analogous). On the other hand, since
$\sup_{\\_}k\int_{\\_}{B_{\\_}1^{2}}\left|\upsilon_{\\_}k^{\prime}\right|^{2}=\sup_{\\_}k\mathcal{L}^{2}(D_{\\_}k)$
is finite, we have weak convergence
$\upsilon_{\\_}k\rightharpoonup\upsilon_{\\_}\infty$ in
$W^{1,2}(B_{\\_}1^{2})$ and thus weak convergence
$\gamma_{\\_}k\circ\beta_{\\_}k\rightharpoonup\gamma_{\\_}\infty\circ\beta_{\\_}\infty$
in $L^{2}(\partial B_{\\_}1^{2})$. The everywhere convergence
$\gamma_{\\_}k\circ\beta_{\\_}k\to\gamma_{\\_}\infty\circ\widetilde{\beta}_{\\_}\infty$
implies
$\gamma_{\\_}\infty\circ\beta_{\\_}\infty=\gamma_{\\_}\infty\circ\widetilde{\beta}_{\\_}\infty$
a.e. and thus $\beta_{\\_}\infty=\widetilde{\beta}_{\\_}\infty$ a.e. In
particular, $\beta_{\\_}\infty$ is also order-preserving. Since
$\beta_{\\_}\infty$ is continuous and both maps are order-preserving, we
conclude that $\beta_{\\_}\infty=\widetilde{\beta}_{\\_}\infty$ everywhere.
Using again the continuity of $\beta_{\\_}\infty$, as well as the everywhere
convergence of the order-preserving maps $\beta_{\\_}k\to\beta_{\\_}\infty$,
we also get that $\beta_{\\_}k\to\beta_{\\_}\infty$ uniformly.

Being $\upsilon_{\\_}k$ the harmonic extension of
$\gamma_{\\_}k\circ\beta_{\\_}k$ (for
$k\in\mathbb{N}\cup\left\\{\infty\right\\}$), we conclude that
$\upsilon_{\\_}k\to\upsilon_{\\_}\infty$ in $C^{0}(\overline{B}_{\\_}1^{2})$.
Let $U_{\\_}k\in C^{0}(\overline{B}_{\\_}1^{2})$ be the harmonic extension of
$f_{\\_}k\circ\beta_{\\_}k$ and note that $U_{\\_}k\to U_{\\_}\infty$ in
$C^{0}(\overline{B}_{\\_}1^{2})$. By conformal invariance,
$u_{\\_}k:=U_{\\_}k\circ\upsilon_{\\_}k^{-1}$ is the harmonic extension of
$f_{\\_}k\circ\gamma_{\\_}k^{-1}$ on $D_{\\_}k$ (for
$k\in\mathbb{N}\cup\left\\{\infty\right\\}$).

Finally, we claim that in the situation of the second claim we have
$\upsilon_{\\_}k^{-1}(y_{\\_}k)\to\upsilon_{\\_}\infty^{-1}(y_{\\_}\infty)$.
This easily follows from the injectivity of $\upsilon_{\\_}\infty$: if we had
$\left|\upsilon_{\\_}{k}^{-1}(y_{\\_}{k})-\upsilon_{\\_}\infty^{-1}(y_{\\_}\infty)\right|\geq\varepsilon$
along some subsequence (for some $\varepsilon>0$), we would have a
subsequential limit point $x_{\\_}\infty\in\overline{B}_{\\_}1^{2}$ with
$\left|x_{\\_}\infty-\upsilon_{\\_}\infty^{-1}(y_{\\_}\infty)\right|\geq\varepsilon$
and
$\upsilon_{\\_}\infty(x_{\\_}\infty)=\lim_{\\_}{k\to\infty}y_{\\_}k=y_{\\_}\infty$,
which is a contradiction. Hence,

(4.3) $\displaystyle
u_{\\_}k(y_{\\_}k)=U_{\\_}k(\upsilon_{\\_}k^{-1}(y_{\\_}k))\to
U_{\\_}\infty(\upsilon_{\\_}\infty^{-1}(y_{\\_}\infty))=u_{\\_}\infty(y_{\\_}\infty),$

as desired. ∎

###### Lemma 4.2 (injectivity under a boundary constraint).

Given $K\geq 1$ and $s,\varepsilon>0$, there exists a constant
$0<\delta_{\\_}0<\varepsilon$, depending only on $q,K,s,\varepsilon$, with the
following property: whenever

    * –

$\displaystyle\Psi\in W^{1,2}\cap
C^{0}(\overline{B}_{\\_}1^{2},\mathbb{R}^{q})$ has
$\left\|{\left.\kern-1.2pt\Psi\vphantom{\big{|}}\right|_{\partial
B_{\\_}1^{2}}}-{\left.\kern-1.2pt\psi(s\cdot)\vphantom{\big{|}}\right|_{\partial
B_{\\_}1^{2}}}\right\|_{\\_}{C^{0}(\partial B_{\\_}1^{2})}\leq\delta_{\\_}0$
for some $\psi\in\mathcal{D}_{\\_}K^{\Pi}$,

    * –

$\displaystyle\Psi\circ\varphi^{-1}$ is harmonic and weakly conformal on
$\varphi(B_{\\_}1^{2})$, where $\varphi:\mathbb{R}^{2}\to\mathbb{R}^{2}$ is a
$K$-quasiconformal homeomorphism,444The maps $\psi$ and $\varphi$ are not
necessarily related to each other.

then $\Pi\circ\Psi\circ\varphi^{-1}$ is a diffeomorphism from
$\varphi(\overline{B}_{\\_}{1/2}^{2})$ onto its image and

(4.4)
$\displaystyle\operatorname{dist}(\Pi,\Pi(x))<\varepsilon,\qquad\Pi(x):=\text{2-plane
spanned by }\nabla(\Psi\circ\varphi^{-1})(x),$

for all $x\in\varphi(\overline{B}_{\\_}{1/2}^{2})$. In particular,
$\Pi\circ\Psi$ is injective on $\overline{B}_{\\_}{1/2}^{2}$.

###### Proof of Lemma 4.2.

Assume by contradiction that, for a sequence $\delta_{\\_}k\downarrow 0$,
there exist maps $\Psi_{\\_}k:B_{\\_}1^{2}\to\mathbb{R}^{q}$, planes
$\Pi_{\\_}k$, homeomorphisms $\varphi_{\\_}k:\mathbb{R}^{2}\to\mathbb{R}^{2}$
and coefficients $\mu_{\\_}k$ such that the claim fails with
$\delta_{\\_}0=\delta_{\\_}k$. By Corollary A.4, up to subsequences we have
$\Pi_{\\_}k\to\Pi_{\\_}\infty$ and
${\left.\kern-1.2pt\Psi_{\\_}k\vphantom{\big{|}}\right|_{\partial
B_{\\_}1^{2}}}\to\Gamma$, where $\Gamma:\partial
B_{\\_}1^{2}\to\mathbb{R}^{q}$ is the restriction of a map in
$\mathcal{D}_{\\_}K^{\Pi_{\\_}\infty}$.

We can assume that $\varphi_{\\_}k\in\mathcal{D}_{\\_}K$ (replacing
$\varphi_{\\_}k$ with
$\frac{\varphi_{\\_}k-\varphi_{\\_}k(0)}{\min_{\\_}{\partial
B_{\\_}1^{2}}|\varphi_{\\_}k-\varphi_{\\_}k(0)|}$). By Corollary A.4, we can
assume that $\varphi_{\\_}k\to\varphi_{\\_}\infty$ and
$\varphi_{\\_}k^{-1}\to\varphi_{\\_}\infty^{-1}$ in
$C^{0}_{\\_}{loc}(\mathbb{R}^{2})$, for some homeomorphism
$\varphi_{\\_}\infty:\mathbb{R}^{2}\to\mathbb{R}^{2}$.

By harmonicity, up to subsequences we get
$\Theta_{\\_}k:=\Psi_{\\_}k\circ\varphi_{\\_}k^{-1}\to\Theta_{\\_}\infty$ in
$C^{2}_{\\_}{loc}(\varphi_{\\_}\infty(B_{\\_}1^{2}))$, for some
$\Theta_{\\_}\infty:\varphi_{\\_}\infty(B_{\\_}1^{2})\to\mathbb{R}^{q}$, so
that $\Theta_{\\_}\infty$ is weakly conformal and harmonic.

On the other hand, by Lemma 4.1 applied to the sequence of harmonic maps
$\Theta_{\\_}k$ on the Jordan domains $\varphi_{\\_}k(B_{\\_}1^{2})$,
$\Theta_{\\_}\infty$ is the harmonic extension of
$\Gamma\circ\varphi_{\\_}\infty^{-1}$ and
$\Psi_{\\_}k\to\Theta_{\\_}\infty\circ\varphi_{\\_}\infty=:\Psi_{\\_}\infty$
in $C^{0}(\overline{B}_{\\_}1^{2})$. By the maximum principle we have
$\Pi_{\\_}\infty^{\perp}\circ\Theta_{\\_}\infty=0$ and thus
$\Pi_{\\_}\infty\circ\Theta_{\\_}\infty$ is either holomorphic or
antiholomorphic on $\varphi_{\\_}\infty(B_{\\_}1^{2})$ (once $\Pi_{\\_}\infty$
is identified with $\mathbb{C}$).

Now, given two Jordan domains $U,V\subset\mathbb{C}$, if a holomorphic map
$h:U\to\mathbb{C}$ extends to a continuous map $h:\overline{U}\to\mathbb{C}$
mapping $\partial U$ onto $\partial V$ homeomorphically, then $h$ maps $U$
diffeomorphically onto $V$.555Indeed,
${\left.\kern-1.2pth\vphantom{\big{|}}\right|_{U}}$ must be an open map, hence
$h(\overline{U})\setminus\partial V$ is closed and open in
$\mathbb{C}\setminus\partial V$ and it follows that $h(U)=V$. We can find
biholomorphisms $u:B_{\\_}1^{2}\to U$ and $v:B_{\\_}1^{2}\to V$ extending to
homeomorphisms of the closures. The map $g:=v^{-1}\circ h\circ u$ satisfies
$g(B_{\\_}1^{2})\subseteq B_{\\_}1^{2}$ and maps $\partial B_{\\_}1^{2}$ to
itself homeomorphically. Given $w\in B_{\\_}1^{2}$, for $r<1$ close enough to
$1$ the loop $g(re^{i\theta})-w$ is homotopic to $g(e^{i\theta})$ in
$\mathbb{C}\setminus\left\\{0\right\\}$, so the classical argument principle
gives $\\#g^{-1}(w)=1$. Being
${\left.\kern-1.2pt\Pi_{\\_}\infty\circ\Theta_{\\_}\infty\vphantom{\big{|}}\right|_{\partial\varphi_{\\_}\infty(B_{\\_}1^{2})}}=\Pi_{\\_}\infty\circ\Gamma\circ\varphi_{\\_}\infty^{-1}$
a Jordan curve, we deduce that $\Pi_{\\_}\infty\circ\Theta_{\\_}\infty$ is a
diffeomorphism from $\varphi_{\\_}\infty(B_{\\_}1^{2})$ onto its image.

Fix now a compact neighborhood $F$ of
$\varphi_{\\_}\infty(\overline{B}_{\\_}{1/2}^{2})$ in
$\varphi_{\\_}\infty(B_{\\_}1^{2})$, with smooth boundary. Since
$\Theta_{\\_}k\to\Theta_{\\_}\infty$ in
$C^{1}_{\\_}{loc}(\varphi_{\\_}\infty(B_{\\_}1^{2}))$, we obtain that
eventually $\Pi_{\\_}k\circ\Theta_{\\_}k$ is a diffeomorphism of $F$ onto its
image, with

$\displaystyle\operatorname{dist}(\Pi_{\\_}k,\Pi_{\\_}k(x))<\varepsilon,\qquad
x\in F.$

The fact that eventually $\varphi_{\\_}k(\overline{B}_{\\_}{1/2}^{2})\subseteq
F$ yields the desired contradiction. ∎

##  5\. Technical iteration lemmas

###  5.1. Informal discussion of the results

Since the intermediate results contained in this section have rather involved
statements, with several different constants and thresholds appearing along
the way, we find it helpful to provide an informal explanation of the meaning
of these statements and constants, as well as a rough sketch of the underlying
ideas in the proofs.

This section contains four important intermediate results, namely Lemmas 5.2,
5.3, 5.5 and 5.6, which all invoke Theorem 3.2 (except for Lemma 5.6) by means
of a compactness-and-contradiction argument. All statements are about a
conformal immersion
$\Psi:\overline{B}_{\\_}r^{2}(z)\to\mathcal{M}^{m}_{\\_}{p,\ell}$, critical
for the functional (3.3) (on the interior). For simplicity, in this discussion
we assume $z=0$ and $r=1$. The first three statements require the following:

    * (i)

a control of the shape of the images of three circles, dictated by a
distortion constant $K$; namely we require that

$\displaystyle\Psi\in\mathcal{R}_{\\_}{K,\delta_{\\_}0}^{\Pi}$

for some $2$-plane $\Pi$ and some small $\delta_{\\_}0$; recall from Section 2
that this means that (up to rotations of $\mathbb{R}^{q}$) $\Psi$ is
$C^{0}$-close to a $K$-quasiconformal homeomorphism
$\varphi\in\mathcal{D}_{\\_}K:\mathbb{R}^{2}\to\mathbb{R}^{2}\subseteq\mathbb{R}^{q}$
on the three circles $\partial B_{\\_}1^{2}$, $\partial B_{\\_}{s(K)}^{2}$,
$\partial B_{\\_}{s(K)^{2}}^{2}$ (it would be far too restrictive to ask for
$C^{0}$-closeness on all of $B_{\\_}1^{2}$);

    * (ii)

an upper bound $E$ on the Dirichlet energy
$\frac{1}{2}\int_{\\_}{B_{\\_}1^{2}}|\nabla\Psi|^{2}$;

    * (iii)

an upper bound $V$ on the area (divided by $\pi$) of the immersed surface
$\Psi(B_{\\_}1^{2})\cap B_{\\_}1^{q}$, taking into account multiplicity;
namely,

$\displaystyle\int_{\\_}{\Psi^{-1}(B_{\\_}1^{q})}\,d\operatorname{vol}_{\\_}{g_{\\_}\Psi}=\frac{1}{2}\int_{\\_}{\Psi^{-1}(B_{\\_}1^{q})}|\nabla\Psi|^{2}\leq
V\pi,$

where $g_{\\_}{\Psi}$ is the pullback of the Euclidean metric, which equals
$\frac{1}{2}|\nabla\Psi|^{2}\delta$ by conformality; this upper bound will
give a crucial improvement on the last conclusions of Theorem 3.2, as
discussed below.

Also, in the same spirit as Theorem 3.2, these lemmas assume $\tau,\ell\ll 1$
and

$\displaystyle\tau^{2}\log(\tau^{-1})\int_{\\_}{B_{\\_}1^{2}}|A|^{4}\,d\operatorname{vol}_{\\_}{g_{\\_}\Psi}\ll\int_{\\_}{B_{\\_}1^{2}}\,d\operatorname{vol}_{\\_}{g_{\\_}\Psi}.$

In Lemmas 5.3 and 5.5, the closeness in (i) is measured by a threshold
$\delta_{\\_}0$ (which will be specified according to Lemma 4.2), while other
closeness or smallness constraints will be measured by thresholds
$\varepsilon_{\\_}0,\varepsilon_{\\_}0^{\prime},\varepsilon_{\\_}0^{\prime\prime}$
in Lemmas 5.2, 5.3, 5.5, respectively.

Observe that the hypotheses guarantee that $\Pi\circ\Psi$ maps
$B_{\\_}{s(K)}^{2}$ to a subset of $B_{\\_}{1/2}^{\Pi}$ and $\partial
B_{\\_}1^{2}$ to a subset of $\Pi\setminus B_{\\_}1^{\Pi}$ (approximately),
hence $\Psi(B_{\\_}{s(K)}^{2})$ is far away from $\Psi(\partial
B_{\\_}1^{2})$. Hence, when arguing by contradiction, we can apply the last
part of Theorem 3.2 and obtain in the limit a parametrized stationary varifold
close to $\Psi|_{\\_}\omega$ (we will choose either
$\omega:=B_{\\_}{s(K)}^{2}$ or the smaller domain
$\omega:=B_{\\_}{s(K)^{2}}^{2}$). The reason to impose the geometric control
on three circles, rather than two, is merely technical and is convenient for
the proofs.

Lemma 5.2 says that the _projected multiplicity_
$N(\Psi,B_{\\_}{s(K)^{2}}^{2},\Pi)$ (introduced in Section 2) issued by $\Psi$
from the ball $B_{\\_}{s(K)^{2}}^{2}$ has an average close to a positive
integer $k$, on the ball $B_{\\_}{\eta(K)}^{\Pi}$. It also asserts that this
holds for $2$-planes $\Pi^{\prime}$ close enough to $\Pi$. As a consequence,
the corresponding _macroscopic multiplicity_ will be precisely $k$.

Observe that the hypotheses guarantee that $\Pi\circ\Psi$ maps
$B_{\\_}{s(K)^{2}}^{2}$ approximately to a superset of
$B_{\\_}{\eta(K)}^{\Pi}$ (see Section 2 for the definition of these
geometrical constants). Hence, arguing by contradiction, we obtain in the
limit a (parametrized) stationary varifold which is close, in the varifold
sense, to $\Psi(B_{\\_}{s(K)^{2}}^{2})$. The constraints on $\Psi$ force this
limiting varifold to lie on a $2$-plane, so by the constancy theorem it has
constant integer multiplicity on $B_{\\_}{\eta(K)}^{\Pi}$, giving a
contradiction. Note that the volume constraint $V$ is not used here.

As already mentioned in the introduction, we would now like to find a
decreasing sequence of radii $r_{\\_}0:=1,\dots,r_{\\_}k\approx\sqrt{\tau}$,
with $r_{\\_}j$ comparable to $r_{\\_}{j+1}$, such that the maps
$\Psi(r_{\\_}j\cdot)$ satisfy the same assumptions (with different scales
$\ell_{\\_}0:=\ell,\dots,\ell_{\\_}k$ in the target). The strategy to get
Theorem 1.2 is then to show that the corresponding macroscopic multiplicities
$n_{\\_}j$ _do not change_ from one scale to the next one:
$n_{\\_}0=n_{\\_}1=\cdots=n_{\\_}k$. At the smallest scale, we will be able to
say that the immersed surface $\ell_{\\_}k^{-1}\Psi(B_{\\_}{r_{\\_}k}^{2})$
has small second fundamental form in $L^{4}$, implying a strong graphical
control that allows to conclude $n_{\\_}k=1$ and thus $n_{\\_}0=1$. In the
situation where we will apply this strategy (namely, in Section 6), upon
careful selection of the center $z$, it will be easy to impose the “maximal”
bounds

$\displaystyle(\ell^{\prime})^{-2}\int_{\\_}{\Psi^{-1}(B_{\\_}{\ell^{\prime}}^{q})}\,d\operatorname{vol}_{\\_}{g_{\\_}\Psi}\leq
V\pi,$
$\displaystyle\tau^{2}\log(\tau^{-1})\int_{\\_}{B_{\\_}{r^{\prime}}^{2}}|A|^{4}\,d\operatorname{vol}_{\\_}{g_{\\_}\Psi}\ll\int_{\\_}{B_{\\_}{r^{\prime}}^{2}}\,d\operatorname{vol}_{\\_}{g_{\\_}\Psi},$

for _all_ $0<\ell^{\prime}<1$ and $0<r^{\prime}<1$, by means of covering
arguments. However, we _cannot_ a priori impose similar bounds on the
Dirichlet energy and on the shape of the images of small circles (items (ii)
and (i)). Note that if $(\ell^{\prime})^{-1}\Psi(r^{\prime}\cdot)$ satisfies
(i), then we can bound the Dirichlet energy of this rescaled map on the ball
$B_{\\_}{s(K)}^{2}$, in terms of $V$, as $\Psi$ maps
$B_{\\_}{s(K)r^{\prime}}^{2}$ into $B_{\\_}{\ell^{\prime}}^{q}$
(approximately). So (i) would give (ii) for free (on a smaller domain ball),
with a uniform bound (depending on $K$, $V$) in place of $E$.

Lemma 5.3 is the main technical workhorse, and essentially says that we can
circumvent this difficulty: namely, the hypotheses (i)–(iii) are still
satisfied for a smaller radius $r^{\prime}\leq\frac{r}{2}$ in the domain, with
a smaller scale $\ell^{\prime}\ell\leq\frac{\ell}{2}$ in the codomain. Note
that the reference point $p$ also changes; this will in principle destroy the
maximal volume bound, but we can still recover (iii) in the new situation,
exploiting the fact that the multiplicity is quantized in the limit (see the
proof of Lemma 5.3 and Definition 5.4 for the details).

The idea of the proof of Lemma 5.3 is that, up to a quasiconformal
homeomorphism $\varphi$, $\Psi$ is close (in the weak $W^{1,2}$-topology) to a
conformal harmonic map with small oscillation with respect to $\Pi$. Hence, by
Lemma 4.2, it will be arbitrarily close to an affine injective conformal map
$L$ on smaller and smaller balls $B_{\\_}{r^{\prime}}^{2}$. If $\varphi$ were
the identity, given a (finite) collection of circles centered at $0$ we would
get $C^{0}$-closeness of $\Psi(r^{\prime}\cdot)$ to $L$ on all these circles,
for some $r^{\prime}$ small, and we would be done.

The important observation now is that the distortion constant of
$\varphi_{\\_}\infty$ can be bounded solely in terms of $V$: indeed, as in the
proof of Theorem 3.2, $\Psi(B_{\\_}1^{2})\cap B_{\\_}1^{q}$ is close to a
stationary varifold $\mathbf{v}$ (in $B_{\\_}1^{q}$), whose density on
$B_{\\_}{1/2}^{q}$ is bounded in terms of $V$. Since
$\Psi(B_{\\_}{s(K)}^{2})\subseteq B_{\\_}{1/2}^{q}$ (approximately), the upper
bound on the distortion constant given by Theorem 3.2 can be improved to a
constant $K^{\prime}(V)$ depending _only_ on $V$. Hence, we get (i) also for a
smaller radius $r^{\prime}$ (with $K^{\prime}(V)$ replacing $K$) and, as
already said, this gives also (ii) with a bound $E^{\prime}(V)$ in place of
$E$. Our sequence of radii is now obtained by iterated application of Lemma
5.3 with parameters $K^{\prime}(V)$, $E^{\prime}(V)$, $V$.

Given constants $K^{\prime\prime}$, $E^{\prime\prime}$ and $V$, which will be
chosen when applying these results in Section 6, we then fix
$K_{\\_}0:=\max\\{K^{\prime}(V),K^{\prime\prime}\\}$ and
$E_{\\_}0:=\max\\{E^{\prime}(V),E^{\prime\prime}\\}$, so that all the
statements apply for all radii $r_{\\_}0,r_{\\_}1,\dots,r_{\\_}k$.

Lemma 5.5 says that the macroscopic multiplicity does not change after
applying Lemma 5.3, namely when replacing the domain and codomain scales $r$,
$\ell$ with $r^{\prime}$, $\ell\ell^{\prime}$ (and $p,\Pi$ with
$p^{\prime},\Pi^{\prime}$). Its proof uses Lemma 4.2 to claim that $\Psi$ is
approximately a graph over $\Pi$, and then applies the constancy theorem (in
the limiting situation).

Finally, as it will be clear along the proof of Theorem 6.1, Lemma 5.6
concerns the behaviour of a conformal immersion
$\Phi:B_{\\_}1^{2}\to\mathcal{M}^{m}$ at a scale (comparable to)
$\ell:=\sqrt{\sigma}$ in the codomain, when $\Phi$ is critical for (3.1).
Assume that $\Phi(B_{\\_}r^{2})$ has diameter approximately $\ell^{2}$, and
assume the smallness

(5.1)
$\displaystyle\sigma^{2}\int_{\\_}{B_{\\_}r^{2}}|A|^{4}\,d\operatorname{vol}_{\\_}{g_{\\_}\Phi}\ll\int_{\\_}{B_{\\_}r^{2}}\,d\operatorname{vol}_{\\_}{g_{\\_}\Phi}$

and the bound
$\int_{\\_}{B_{\\_}r^{2}}\,d\operatorname{vol}_{\\_}{g_{\\_}\Phi}\leq
C\ell^{2}$. When dilating the codomain by a factor $\ell^{-1}$, (5.1) becomes
$\int_{\\_}{B_{\\_}r^{2}}|A_{\\_}\Psi|^{4}\,d\operatorname{vol}_{\\_}{g_{\\_}\Psi}\ll
1$, for $\Psi:=\ell^{-1}(\Phi-\Phi(0))$. As $\Psi$ is conformal, we have
$\Delta\Psi=2H_{\\_}\Psi e^{2\lambda}$ (where $e^{\lambda}$ is the conformal
factor). Thus, we get that $\Delta\Psi$ is small in $L^{4}$ provided we can
obtain an upper bound on $\lambda$; once this is done, by Sobolev’s embedding
we obtain a $C^{1}$-control on $\Psi$, which implies that the macroscopic
multiplicity is $1$ at this scale.

In order to bound $\lambda$, we use a result by Hélein (belonging to a broad
class of phenomena of integrability by compensation, whose study dates back to
the discovery of Wente’s inequality), guaranteeing the existence of an
orthonormal frame $\left\\{e_{\\_}1(z),e_{\\_}2(z)\right\\}$ for the tangent
space of the immersed surface $\Psi$, with a bound on $\left\|\nabla
e_{\\_}i\right\|_{\\_}{L^{2}}$ depending only the $L^{2}$-norm of the second
fundamental form. Then we show that

$\displaystyle-\Delta\lambda=\partial_{\\_}1e_{\\_}1\cdot\partial_{\\_}2e_{\\_}2-\partial_{\\_}2e_{\\_}1\cdot\partial_{\\_}1e_{\\_}2$

and we compare $\lambda$ with the solution $\mu$ to the same equation, with
zero boundary conditions on a ball. A pointwise bound for $\mu$ now follows
from Wente’s inequality, from which one easily deduces the desired upper bound
for $\lambda$. Although not necessary, we will also show how to obtain a
pointwise lower bound on $\lambda$ in this situation.

While reading Sections 5 and 6, it can be useful to refer to the following
diagrams, illustrating how the constants depend on each other:

${K}$${V}$${E}$${\varepsilon_{\\_}0}$${\varepsilon_{\\_}0^{\prime}}$${\delta_{\\_}0}$${V}$${K^{\prime\prime}}$${K^{\prime}(V)}$${E^{\prime}(V)}$${E^{\prime\prime}}$${K_{\\_}0}$${E_{\\_}0}$${\varepsilon_{\\_}0^{\prime}}$${\varepsilon_{\\_}0^{\prime\prime}}$

(an arrow $A\to B$ means that $B$ depends on $A$).

###  5.2. Rigorous statements and proofs

We now make the above discussion rigorous. In a first reading, it can be
helpful to pretend that all quasiconformal homeomorphisms appearing in the
proofs coincide with the identity.

###### Definition 5.1.

Given $V>0$ with $V=\left\lfloor V\right\rfloor+\frac{1}{2}$, we define the
constants

$\displaystyle K^{\prime}(V):=(4V)^{2},\qquad E^{\prime}(V):=2\pi
K^{\prime}(V)D(K^{\prime}(V))^{2}.$

In the sequel, it will be convenient to assume always that
$V\in\mathbb{N}+\frac{1}{2}$, so that $V=\left\lfloor
V\right\rfloor+\frac{1}{2}$.

###### Lemma 5.2 (almost integrality of averaged projected multiplicity).

There exists $0<\varepsilon_{\\_}0<\eta(K)$, depending on $E,V>0$, $K\geq 1$
and $\mathcal{M}^{m}$, such that whenever $\Psi\in
C^{2}(\overline{B}_{\\_}{r}^{2}(z),\mathcal{M}^{m}_{\\_}{p,\ell})$ is a
conformal immersion, critical for the functional (3.3) on $B_{\\_}{r}^{2}(z)$,
and $\Pi,\Pi^{\prime}$ are 2-planes satisfying

    * –

$\displaystyle\Psi(z+r\cdot)\in\mathcal{R}_{\\_}{K,\varepsilon_{\\_}0}^{\Pi}$,

    * –

$\displaystyle\frac{1}{2}\int_{\\_}{B_{\\_}{r}^{2}(z)}\left|\nabla\Psi\right|^{2}\leq
E$,

    * –

$\displaystyle\int_{\\_}{\Psi^{-1}(B_{\\_}1^{q})}\,d\operatorname{vol}_{\\_}{g_{\\_}\Psi}=\frac{1}{2}\int_{\\_}{\Psi^{-1}(B_{\\_}1^{q})}\left|\nabla\Psi\right|^{2}\leq
V\pi$,

    * –

$\displaystyle\tau^{2}\log(\tau^{-1})\int_{\\_}{B_{\\_}{r}^{2}(z)}\left|A\right|^{4}\,d\operatorname{vol}_{\\_}{g_{\\_}\Psi}\leq\varepsilon_{\\_}0$
for some $0<\tau\leq\varepsilon_{\\_}0$,

    * –

$\displaystyle\operatorname{dist}(\Pi,\Pi^{\prime})\leq\varepsilon_{\\_}0$ and
$0<\ell\leq\varepsilon_{\\_}0$,

then the projected multiplicity $N(\Psi,B_{\\_}{s(K)^{2}r}^{2}(z),\Pi)$
satisfies

(5.2)
$\displaystyle\operatorname{dist}\Big{(}\,\ThisStyle{\ensurestackMath{\stackinset{c}{.2\LMpt}{c}{.5\LMpt}{\SavedStyle-}{\SavedStyle\phantom{\int}}}\kern-16.56937pt}\int_{\\_}{B_{\\_}{\eta(K)}^{\Pi}}N(\Psi,B_{\\_}{s(K)^{2}r}^{2}(z),\Pi),\,\mathbb{Z}^{+}\Big{)}<\frac{1}{8},$
(5.3)
$\displaystyle\Bigg{|}\,\ThisStyle{\ensurestackMath{\stackinset{c}{.2\LMpt}{c}{.5\LMpt}{\SavedStyle-}{\SavedStyle\phantom{\int}}}\kern-16.56937pt}\int_{\\_}{B_{\\_}{\eta(K)}^{\Pi}}N(\Psi,B_{\\_}{s(K)^{2}r}^{2}(z),\Pi)-\,\ThisStyle{\ensurestackMath{\stackinset{c}{.2\LMpt}{c}{.5\LMpt}{\SavedStyle-}{\SavedStyle\phantom{\int}}}\kern-16.56937pt}\int_{\\_}{B_{\\_}{\eta(K)}^{\Pi^{\prime}}}N(\Psi,B_{\\_}{s(K)^{2}r}^{2}(z),\Pi^{\prime})\Bigg{|}<\frac{1}{8},$

where $\mathbb{Z}^{+}$ is the set of positive integers.

###### Proof of Lemma 5.2.

We can assume $z=0$ and $r=1$. Suppose by contradiction that there exist
sequences $\varepsilon_{\\_}k\downarrow 0$, $\tau_{\\_}k$, $\ell_{\\_}k$,
points $p_{\\_}k$, maps $\Psi_{\\_}k$ and planes
$\Pi_{\\_}k,\Pi_{\\_}k^{\prime}$ making the claim false for
$\varepsilon_{\\_}0=\varepsilon_{\\_}k$. Up to subsequences, we can assume
that $\Pi_{\\_}k,\Pi_{\\_}k^{\prime}\to\Pi_{\\_}\infty$, that $\Psi_{\\_}k$
has a weak limit $\Psi_{\\_}\infty$ in $W^{1,2}(B_{\\_}1^{2},\mathbb{R}^{q})$,
with traces
${\left.\kern-1.2pt\Psi_{\\_}\infty\vphantom{\big{|}}\right|_{\partial
B_{\\_}s^{2}}}(s\cdot)=\psi(s\cdot)$ for some
$\psi\in\mathcal{D}_{\\_}K^{\Pi_{\\_}\infty}$ and all
$s\in\left\\{1,s(K),s(K)^{2}\right\\}$ (thanks to Corollary A.4), and that the
varifolds $\mathbf{v}_{\\_}k$ induced by $\Psi_{\\_}k$ converge to a varifold
$\mathbf{v}_{\\_}\infty$ in $\mathbb{R}^{q}$.

We now invoke Theorem 3.2. Recalling the definition of
$\mathcal{D}_{\\_}K^{\Pi_{\\_}\infty}$ and $s(K)$ from Section 2, the convex
hull property satisfied by $\Psi_{\\_}\infty$ gives

(5.4)
$\displaystyle\Psi_{\\_}\infty(\overline{B}_{\\_}{s(K)}^{2})\subseteq\operatorname{co}\left(\Psi_{\\_}\infty(\partial
B_{\\_}{s(K)}^{2})\right)\subseteq\overline{B}_{\\_}{1/2}^{q},$

so that, being $\Gamma_{\\_}\infty=\Psi_{\\_}\infty(\partial B_{\\_}1^{2})$
disjoint from $B_{\\_}1^{q}$,

(5.5)
$\displaystyle\operatorname{dist}(\Psi_{\\_}\infty(x),\Gamma_{\\_}\infty)\geq\frac{1}{2}\quad\text{for
}x\in\overline{B}_{\\_}{s(K)}^{2}.$

Theorem 3.2 gives the varifold convergence
$\mathbf{v}_{\\_}k^{\prime}\stackrel{{\scriptstyle*}}{{\rightharpoonup}}\mathbf{v}_{\\_}\infty^{\prime}$
and
$\mathbf{v}_{\\_}k^{\prime\prime}\stackrel{{\scriptstyle*}}{{\rightharpoonup}}\mathbf{v}_{\\_}\infty^{\prime\prime}$
as $k\to\infty$, as well as the tightness of the sequences of mass measures
$\left\|\mathbf{v}_{\\_}k^{\prime}\right\|$ and
$\left\|\mathbf{v}_{\\_}k^{\prime\prime}\right\|$, where
$\mathbf{v}_{\\_}k^{\prime}$ and $\mathbf{v}_{\\_}k^{\prime\prime}$ are the
varifolds issued by
${\left.\kern-1.2pt\Psi_{\\_}k\vphantom{\big{|}}\right|_{B_{\\_}{s(K)}^{2}}}$
and
${\left.\kern-1.2pt\Psi_{\\_}k\vphantom{\big{|}}\right|_{B_{\\_}{s(K)^{2}}^{2}}}$
respectively, while $\mathbf{v}_{\\_}\infty^{\prime}$ and
$\mathbf{v}_{\\_}\infty^{\prime\prime}$ are the ones issued by
$(\varphi_{\\_}\infty(B_{\\_}{s(K)}^{2}),\Psi_{\\_}\infty\circ\varphi_{\\_}\infty^{-1},N_{\\_}\infty\circ\varphi_{\\_}\infty^{-1})$
and
$(\varphi_{\\_}\infty(B_{\\_}{s(K)^{2}}^{2}),\Psi_{\\_}\infty\circ\varphi_{\\_}\infty^{-1},N_{\\_}\infty\circ\varphi_{\\_}\infty^{-1})$.666The
fact that one can choose the same multiplicity $N_{\\_}\infty$ and the same
quasiconformal homeomorphism
$\varphi_{\\_}\infty:\mathbb{R}^{2}\to\mathbb{R}^{2}$ for both domains is
evident from the proof of Theorem 3.2.

Although not needed in the present proof, let us remark the following
improvement on the last statement in Theorem 3.2: we have

$\displaystyle
N_{\\_}\infty\leq\frac{V\pi}{\pi\Big{(}\frac{1}{2}\Big{)}^{2}}=4V$

and the distortion constant of $\varphi_{\\_}\infty$ is bounded by
$K^{\prime}(V)=(4V)^{2}$. Indeed, since $\mathbf{v}_{\\_}\infty$ is stationary
in $B_{\\_}1^{q}$ and $\|\mathbf{v}_{\\_}\infty\|(B_{\\_}1^{q})\leq V\pi$, by
the monotonicity formula its density is bounded by
$\frac{V\pi}{\pi(1-|p|)^{2}}$ at any $p\in B_{\\_}1^{q}$. In particular, (5.4)
gives an upper bound $4V$ at points in $\Psi_{\\_}\infty(B_{\\_}{s(K)}^{2})$,
which implies our claim.

The support of $\mathbf{v}_{\\_}\infty^{\prime\prime}$ is contained in the
plane $\Pi_{\\_}\infty$, by the convex hull property enjoyed by
$\Psi_{\\_}\infty$ and the fact that $\Psi_{\\_}\infty$ maps $\partial
B_{\\_}{s(K)^{2}}^{2}$ to $\Pi_{\\_}\infty$. Since $\Psi_{\\_}\infty(\partial
B_{\\_}{s(K)^{2}}^{2})$ does not intersect
$B_{\\_}{\eta(K)}^{\Pi_{\\_}\infty}$, the varifold
$\mathbf{v}_{\\_}\infty^{\prime\prime}$ is stationary here and thus, by the
constancy theorem [18, Theorem 41.1], it has a constant density
$\nu\in\mathbb{N}$. We must have $\nu>0$, since
$\Psi_{\\_}\infty(B_{\\_}{s(K)^{2}}^{2})$ is a superset of
$B_{\\_}{\eta(K)}^{\Pi_{\\_}\infty}$ by Lemma A.1 (applied to
$\eta(K)^{-1}\Psi_{\\_}\infty(s(K)^{2}\cdot)$). The area formula and the
tightness of $\left\|\mathbf{v}_{\\_}k^{\prime\prime}\right\|$ then give

$\displaystyle\,\ThisStyle{\ensurestackMath{\stackinset{c}{.2\LMpt}{c}{.5\LMpt}{\SavedStyle-}{\SavedStyle\phantom{\int}}}\kern-16.56937pt}\int_{\\_}{B_{\\_}{\eta(K)}^{\Pi_{\\_}k}}N(\Psi_{\\_}k,B_{\\_}{s(K)^{2}}^{2},\Pi_{\\_}k)=\frac{\left\|(\Pi_{\\_}k)_{\\_}*\mathbf{v}_{\\_}k^{\prime\prime}\right\|(B_{\\_}{\eta(K)}^{\Pi_{\\_}k})}{\pi\eta(K)^{2}}\to\frac{\left\|(\Pi_{\\_}\infty)_{\\_}*\mathbf{v}_{\\_}\infty^{\prime\prime}\right\|(B_{\\_}{\eta(K)}^{\Pi_{\\_}\infty})}{\pi\eta(K)^{2}}=\nu.$

Similarly,
$\,\ThisStyle{\ensurestackMath{\stackinset{c}{.2\LMpt}{c}{.5\LMpt}{\SavedStyle-}{\SavedStyle\phantom{\int}}}\kern-16.56937pt}\int_{\\_}{B_{\\_}{\eta(K)}^{\Pi_{\\_}k^{\prime}}}N(\Psi_{\\_}k,B_{\\_}{s(K)^{2}}^{2},\Pi_{\\_}k^{\prime})\to\nu$
as $k\to\infty$. Hence the claim is eventually true, yielding the desired
contradiction. ∎

We now specify $\delta_{\\_}0$ so that Lemma 4.2 applies, with
$\varepsilon:=\varepsilon_{\\_}0$ and $s:=s(K)$. Note that
$\delta_{\\_}0<\varepsilon_{\\_}0<\eta(K)$ and that $\varepsilon_{\\_}0$ and
$\delta_{\\_}0$ still depend on $V$, $K$ and $E$.

###### Lemma 5.3 (existence of a smaller good scale).

Given $E>0$ and $K\geq 1$ there exists a constant
$0<\varepsilon_{\\_}0^{\prime}<\varepsilon_{\\_}0$ (depending on
$E,V,K,\mathcal{M}^{m}$) with the following property: if a conformal immersion
$\Psi\in C^{2}(\overline{B}_{\\_}{r}^{2}(z),\mathcal{M}^{m}_{\\_}{p,\ell})$ is
critical for the functional (3.3) (on the interior) and satisfies

    * –

$\displaystyle\Psi(z+r\cdot)\in\mathcal{R}_{\\_}{K,\delta_{\\_}0}^{\Pi}$,

    * –

$\displaystyle\frac{1}{2}\int_{\\_}{B_{\\_}{r}^{2}(z)}\left|\nabla\Psi\right|^{2}\leq
E$,

    * –

$\displaystyle\frac{1}{\pi}\int_{\\_}{\Psi^{-1}(B_{\\_}1^{q})}\,d\operatorname{vol}_{\\_}{g_{\\_}\Psi},\,\frac{1}{\pi\eta(K)^{2}}\int_{\\_}{\Psi^{-1}(B_{\\_}{\eta(K)}^{q})}\,d\operatorname{vol}_{\\_}{g_{\\_}\Psi}\leq
V$,

    * –

$\displaystyle\tau^{2}\log(\tau^{-1})\int_{\\_}{B_{\\_}{r}^{2}(z)}\left|A\right|^{4}\,d\operatorname{vol}_{\\_}{g_{\\_}\Psi}\leq\varepsilon_{\\_}0^{\prime}$
for some $0<\tau\leq\varepsilon_{\\_}0^{\prime}$,

    * –

$\displaystyle 0<\ell\leq\varepsilon_{\\_}0^{\prime}$,

then there exist a new point $p^{\prime}\in\mathcal{M}^{m}_{\\_}{p,\ell}$, new
scales $r^{\prime},\ell^{\prime}$ and a new 2-plane $\Pi^{\prime}$ with

    * –

$\displaystyle\varepsilon_{\\_}0^{\prime}r<r^{\prime}<s(K)r$,

    * –

$\displaystyle\varepsilon_{\\_}0^{\prime}<\ell^{\prime}<\frac{1}{2}$,

    * –

$\displaystyle\operatorname{dist}(\Pi,\Pi^{\prime})<\varepsilon_{\\_}0$,

    * –

$\displaystyle(\ell^{\prime})^{-1}(\Psi(z+r^{\prime}\cdot)-p^{\prime})\in\mathcal{R}_{\\_}{K^{\prime}(V),\delta_{\\_}0}^{\Pi^{\prime}}$,

    * –

$\displaystyle\frac{1}{2}\int_{\\_}{B_{\\_}{r^{\prime}}^{2}(z)}\left|\nabla\Psi^{\prime}\right|^{2}<E^{\prime}(V)$,
for $\Psi^{\prime}:=(\ell^{\prime})^{-1}(\Psi-p^{\prime})$ (defined on
$B_{\\_}{r^{\prime}}^{2}(z)$),

    * –

$\displaystyle\frac{1}{\pi}\int_{\\_}{(\Psi^{\prime})^{-1}(B_{\\_}1^{q})}\,d\operatorname{vol}_{\\_}{g_{\\_}{\Psi^{\prime}}},\,\frac{1}{\pi\eta(K)^{2}}\int_{\\_}{(\Psi^{\prime})^{-1}(B_{\\_}{\eta(K)}^{q})}\,d\operatorname{vol}_{\\_}{g_{\\_}{\Psi^{\prime}}}<\Big{\lfloor}\left(\frac{\eta(K)}{\eta(K)-\varepsilon_{\\_}0}\right)^{2}V\Big{\rfloor}+\frac{1}{2}$.

###### Proof of Lemma 5.3.

We can assume $z=0$ and $r=1$. By contradiction, suppose that there is a
sequence $\varepsilon_{\\_}k\downarrow 0$ such that the claim fails (with
$\varepsilon_{\\_}0^{\prime}=\varepsilon_{\\_}k$) for all radii
$\varepsilon_{\\_}k<r^{\prime}<s(K)$, for some $\Psi_{\\_}k$ and $\Pi_{\\_}k$
satisfying all the hypotheses. As observed in the proof of Lemma 5.2, up to
subsequences we get a limiting local parametrized stationary varifold
$(\Omega_{\\_}\infty,\Theta_{\\_}\infty,N_{\\_}\infty\circ\varphi_{\\_}\infty^{-1})$
in $\mathbb{R}^{q}$, where
$\Theta_{\\_}\infty=\Psi_{\\_}\infty\circ\varphi_{\\_}\infty^{-1}$ and
$\Omega_{\\_}\infty=\varphi_{\\_}\infty(B_{\\_}{s(K)}^{2})$ for a suitable
$K^{\prime}(V)$-quasiconformal homeomorphism $\varphi_{\\_}\infty$ of the
plane. Moreover, assuming also that $\Pi_{\\_}k\to\Pi_{\\_}\infty$ and
$p_{\\_}k\to p_{\\_}\infty$, by weak convergence of traces and Corollary A.4
we still have
$\Psi_{\\_}\infty\in\mathcal{R}_{\\_}{K,\delta_{\\_}0}^{\Pi_{\\_}\infty}$. By
Theorem 3.3, $\Theta_{\\_}\infty$ is harmonic. Also, it takes values in the
tangent space $T$ at $p_{\\_}\infty$ (translated to the origin).

We can assume that $\varphi_{\\_}\infty(0)=0$. By definition of
$\delta_{\\_}0$ and Lemma 4.2, applied to $\Psi_{\\_}\infty(s(K)\cdot)$ and
$\varphi_{\\_}\infty(s(K)\cdot)$, $\Theta_{\\_}\infty$ is a diffeomorphism
from $\varphi_{\\_}\infty(\overline{B}_{\\_}{s(K)/2}^{2})$ onto its image and
the differential $\nabla\Theta_{\\_}\infty(0)$ is a conformal linear map of
full rank, spanning a plane $\Pi^{\prime}$ with
$\operatorname{dist}(\Pi_{\\_}\infty,\Pi^{\prime})<\varepsilon_{\\_}0$.

Since $s(K)^{2}\leq\frac{s(K)}{2}$, the varifolds $\mathbf{v}_{\\_}k$ induced
by
${\left.\kern-1.2pt\Psi_{\\_}k\vphantom{\big{|}}\right|_{B_{\\_}{s(K)^{2}}^{2}}}$
converge to $\mathbf{v}_{\\_}\infty$, induced by
$(\varphi_{\\_}\infty(B_{\\_}{s(K)^{2}}^{2}),\Theta_{\\_}\infty,N_{\\_}\infty\circ\varphi_{\\_}\infty^{-1})$.
Using Lemma A.1, applied to
$\eta(K)^{-1}\Pi_{\\_}\infty\circ\Psi_{\\_}\infty(s(K)^{2}\cdot)$, and the
fact that $\delta_{\\_}0<\eta(K)$, we deduce the existence of a point $y\in
B_{\\_}{s(K)^{2}}^{2}$ such that $\Pi_{\\_}\infty\circ\Psi_{\\_}\infty(y)=0$.
By the convex hull property enjoyed by $\Psi_{\\_}\infty$, it follows that

$\displaystyle\left|\Psi_{\\_}\infty(y)\right|=\left|\Pi_{\\_}\infty^{\perp}\circ\Psi_{\\_}\infty(y)\right|\leq\delta_{\\_}0,$

as $\Psi_{\\_}\infty(\partial
B_{\\_}{s(K)^{2}}^{2})\subseteq\left\\{p:\left|\Pi_{\\_}\infty^{\perp}(p)\right|\leq\delta_{\\_}0\right\\}$.
Since $\left\|\mathbf{v}_{\\_}\infty\right\|(B_{\\_}{\eta(K)}^{q})\leq
V\pi\eta(K)^{2}$, the stationarity of $\mathbf{v}_{\\_}\infty$ on
$B_{\\_}{\eta(K)}^{q}$ implies that its density at $\Psi_{\\_}\infty(y)$ is at
most

(5.6)
$\displaystyle\frac{V\pi\eta(K)^{2}}{\pi(\eta(K)-\delta_{\\_}0)^{2}}\leq\left(\frac{\eta(K)}{\eta(K)-\varepsilon_{\\_}0}\right)^{2}V.$

Being $\mathbf{v}_{\\_}\infty$ stationary in the embedded surface
$\Theta_{\\_}\infty(\varphi_{\\_}\infty(B_{\\_}{s(K)^{2}}^{2}))$, the
constancy theorem gives that its density $\theta$ is a constant integer here.
Thus we have

(5.7)
$\displaystyle\left\|\mathbf{v}_{\\_}\infty\right\|(\overline{B}_{\\_}t^{q}(p_{\\_}\infty^{\prime}))<\Bigg{(}\Big{\lfloor}\left(\frac{\eta(K)}{\eta(K)-\varepsilon_{\\_}0}\right)^{2}V\Big{\rfloor}+\frac{1}{2}\Bigg{)}\pi
t^{2},\qquad p_{\\_}\infty^{\prime}:=\Theta_{\\_}\infty(0)\in T,$

for all $t>0$ small enough. Fix now any $r^{\prime}<s(K)$ such that we have
the strong convergence
$\Psi_{\\_}k(r^{\prime}\cdot)\to\Psi_{\\_}\infty(r^{\prime}\cdot)$ in
$C^{0}(\partial B_{\\_}1^{2}\cup\partial B_{\\_}{s(K)}^{2}\cup\partial
B_{\\_}{s(K)^{2}}^{2})$ along a subsequence.777This can be obtained by
applying e.g. [12, Lemma A.5] to the weakly converging
$\mathbb{R}^{3q}$-valued maps
$\displaystyle(\Psi_{\\_}k,\Psi_{\\_}k(s(K)\cdot),\Psi_{\\_}k(s(K)^{2}\cdot))\rightharpoonup(\Psi_{\\_}\infty,\Psi_{\\_}\infty(s(K)\cdot),\Psi_{\\_}\infty(s(K)^{2}\cdot)).$
Note that
$\lambda^{-1}\varphi_{\\_}\infty(r^{\prime}\cdot)\in\mathcal{D}_{\\_}{K^{\prime}(V)}$,
where
$\lambda:=\min_{\\_}{\left|x\right|=r^{\prime}}\left|\varphi_{\\_}\infty(x)\right|$.
Also, the fact that
$\Psi_{\\_}\infty=\Theta_{\\_}\infty\circ\varphi_{\\_}\infty$ and the
smoothness of $\Theta_{\\_}\infty$ give

(5.8)
$\displaystyle\left|\Psi_{\\_}\infty(r^{\prime}x)-\Psi_{\\_}\infty(0)-\left\langle\nabla\Theta_{\\_}\infty(0),\varphi_{\\_}\infty(r^{\prime}x)\right\rangle\right|<\frac{\delta_{\\_}0\left|\nabla\Theta_{\\_}\infty(0)\right|}{\sqrt{2}D(K^{\prime}(V))}\left|\varphi_{\\_}\infty(r^{\prime}x)\right|\leq\delta_{\\_}0\ell^{\prime}$

if $r^{\prime}$ is chosen small enough, where
$\ell^{\prime}:=\frac{\left|\nabla\Theta_{\\_}\infty(0)\right|}{\sqrt{2}}\lambda$
and $x\in\overline{B}_{\\_}1^{2}$. This implies

$\displaystyle(\ell^{\prime})^{-1}(\Psi_{\\_}k(r^{\prime}\cdot)-p_{\\_}\infty^{\prime})\in\mathcal{R}_{\\_}{K^{\prime}(V),\delta_{\\_}0}^{\Pi^{\prime}}$

by conformality of $\nabla\Theta_{\\_}\infty(0)$. Shrinking $r^{\prime}$, we
can also ensure that $\ell^{\prime}<\frac{1}{2}$, as well as

(5.9) $\displaystyle\begin{aligned}
\int_{\\_}{B_{\\_}{r^{\prime}}^{2}}N_{\\_}\infty|\partial_{\\_}1\Psi_{\\_}\infty\wedge\partial_{\\_}2\Psi_{\\_}\infty|&\leq\frac{K^{\prime}(V)}{2}\int_{\\_}{B_{\\_}{D(K^{\prime}(V))\lambda}^{2}}\left|\nabla\Theta_{\\_}\infty\right|^{2}\\\
&<K^{\prime}(V)(D(K^{\prime}(V))\lambda)^{2}\pi\left|\nabla\Theta_{\\_}\infty(0)\right|^{2}.\end{aligned}$

Calling $\mathbf{v}_{\\_}\infty^{\prime}$ the varifold induced by
$(\varphi_{\\_}\infty(B_{\\_}{r^{\prime}}^{2}),(\ell^{\prime})^{-1}(\Theta_{\\_}\infty-
p_{\\_}\infty^{\prime}),N_{\\_}\infty\circ\varphi_{\\_}\infty^{-1})$, in view
of (5.7) we can even guarantee that

$\displaystyle\frac{\left\|\mathbf{v}_{\\_}\infty^{\prime}\right\|(\overline{B}_{\\_}1^{q})}{\pi},\,\frac{\left\|\mathbf{v}_{\\_}\infty^{\prime}\right\|(\overline{B}_{\\_}{\eta(K)}^{q})}{\pi\eta(K)^{2}}<\Big{\lfloor}\left(\frac{\eta(K)}{\eta(K)-\varepsilon_{\\_}0}\right)^{2}V\Big{\rfloor}+\frac{1}{2}.$

Calling $p_{\\_}k^{\prime}$ the closest point to $p_{\\_}\infty^{\prime}$ in
$\mathcal{M}^{m}_{\\_}{p_{\\_}k,\ell_{\\_}k}$ (eventually defined and
converging to $p_{\\_}\infty^{\prime}$, since
$\mathcal{M}^{m}_{\\_}{p_{\\_}k,\ell_{\\_}k}\to T$), thanks to (5.8) and
$\lambda^{-1}\varphi_{\\_}\infty(r^{\prime}\cdot)\in\mathcal{D}_{\\_}{K^{\prime}(V)}$,
eventually we have

$\displaystyle(\ell^{\prime})^{-1}(\Psi_{\\_}k(r^{\prime}\cdot)-p_{\\_}k^{\prime})\in\mathcal{R}_{\\_}{K^{\prime}(V),\delta_{\\_}0}^{\Pi^{\prime}}.$

Moreover, (5.9) and (3.5) give

$\displaystyle\frac{1}{2}\int_{\\_}{B_{\\_}{r^{\prime}}^{2}(z)}\left|\nabla\Psi_{\\_}k\right|^{2}\to\int_{\\_}{B_{\\_}{r^{\prime}}^{2}(z)}N_{\\_}\infty\left|\partial_{\\_}1\Psi_{\\_}\infty\wedge\partial_{\\_}2\Psi_{\\_}\infty\right|<(\ell^{\prime})^{2}E^{\prime}(V).$

From the convergence of the varifolds $\mathbf{v}_{\\_}k^{\prime}$ induced by
${\left.\kern-1.2pt(\ell^{\prime})^{-1}(\Psi_{\\_}k-p_{\\_}k^{\prime})\vphantom{\big{|}}\right|_{B_{\\_}{r^{\prime}}^{2}}}$
to $\mathbf{v}_{\\_}\infty^{\prime}$ we get

$\displaystyle\limsup_{\\_}{k\to\infty}\frac{\left\|\mathbf{v}_{\\_}k^{\prime}\right\|(B_{\\_}1^{q})}{\pi},\,\limsup_{\\_}{k\to\infty}\frac{\left\|\mathbf{v}_{\\_}k^{\prime}\right\|(B_{\\_}{\eta(K)}^{q})}{\pi\eta(K)^{2}}<\Big{\lfloor}\left(\frac{\eta(K)}{\eta(K)-\varepsilon_{\\_}0}\right)^{2}V\Big{\rfloor}+\frac{1}{2}.$

So eventually
$(\ell^{\prime})^{-1}(\Psi_{\\_}k(r^{\prime}\cdot)-p_{\\_}k^{\prime})$
satisfies all the conclusions. This yields the desired contradiction. ∎

###### Definition 5.4.

Given constants $K^{\prime\prime}\geq 1$ and $E^{\prime\prime}>0$, we define
$K_{\\_}0:=\max\left\\{K^{\prime}(V),K^{\prime\prime}\right\\}$ and
$E_{\\_}0:=\max\left\\{E^{\prime}(V),E^{\prime\prime}\right\\}$. We also let
$s_{\\_}0:=s(K_{\\_}0)$ and $\eta_{\\_}0:=\eta(K_{\\_}0)$.

We fix $\varepsilon_{\\_}0$ (and thus $\delta_{\\_}0$) and
$\varepsilon_{\\_}0^{\prime}$ so that Lemmas 5.2 and 5.3 apply with
$K:=K_{\\_}0$, $E:=E_{\\_}0$. Since $\varepsilon_{\\_}0$ depends on $V$, we
can assume that it is chosen so small that

(5.10)
$\displaystyle\Big{\lfloor}\left(\frac{\eta_{\\_}0}{\eta_{\\_}0-\varepsilon_{\\_}0}\right)^{2}V\Big{\rfloor}+\frac{1}{2}=\left\lfloor
V\right\rfloor+\frac{1}{2}=V.$

This makes the last conclusion of Lemma 5.3 match one of the hypotheses,
making it possible to iterate that result. On the other hand, the constants
$V$, $K^{\prime\prime}$, $E^{\prime\prime}$ (upon which all the aforementioned
constants depend) will be fixed only in Section 6.

###### Lemma 5.5 ($\bm{n(\cdot)}$ does not change from one scale to the next
one).

There exists a constant
$0<\varepsilon_{\\_}0^{\prime\prime}\leq\frac{(\varepsilon_{\\_}0^{\prime})^{2}}{2}<\varepsilon_{\\_}0^{\prime}$
with the following property: if a conformal immersion $\Psi\in
C^{2}(\overline{B}_{\\_}{r}^{2}(z),\mathcal{M}^{m}_{\\_}{p,\ell})$ satisfies
the hypotheses of the previous lemma (with
$\varepsilon_{\\_}0^{\prime\prime},E_{\\_}0,K_{\\_}0$ in place of
$\varepsilon_{\\_}0^{\prime},E,K$), then the new point $p^{\prime}$ and the
new radius $r^{\prime}$ provided by Lemma 5.3 satisfy

(5.11) $\displaystyle
n(\Psi,B_{\\_}{s_{\\_}0^{2}r}^{2}(z),B_{\\_}{\eta_{\\_}0}^{\Pi})=n(\Psi-p^{\prime},B_{\\_}{s_{\\_}0^{2}r^{\prime}}^{2}(z),B_{\\_}{\eta_{\\_}0\ell^{\prime}}^{\Pi})=n(\Psi-p^{\prime},B_{\\_}{s_{\\_}0^{2}r^{\prime}}^{2}(z),B_{\\_}{\eta_{\\_}0\ell^{\prime}}^{\Pi^{\prime}}).$

###### Proof of Lemma 5.5.

The second equality in (5.11) follows immediately from Lemma 5.2 (applied with
$(\ell^{\prime})^{-1}(\Psi-p^{\prime})$ on $B_{\\_}{r^{\prime}}^{2}$), which
gives

$\displaystyle
n(\Psi-p^{\prime},B_{\\_}{s_{\\_}0^{2}r^{\prime}}^{2},B_{\\_}{\eta_{\\_}0\ell^{\prime}}^{\Pi})=n(\Psi-p^{\prime},B_{\\_}{s_{\\_}0^{2}r^{\prime}}^{2},B_{\\_}{\eta_{\\_}0\ell^{\prime}}^{\Pi^{\prime}})$

since $\operatorname{dist}(\Pi^{\prime},\Pi)<\varepsilon_{\\_}0$.

Assume again $z=0$, $r=1$ and, by contradiction, that the first equality in
(5.11) fails, so that we have again two sequences
$\varepsilon_{\\_}k\downarrow 0$ and $\Psi_{\\_}k$. We can assume that
$\Pi_{\\_}k\to\Pi_{\\_}\infty$, $p_{\\_}k^{\prime}\to p_{\\_}\infty^{\prime}$,
$\ell_{\\_}k^{\prime}\to\ell_{\\_}\infty^{\prime}$ and $r_{\\_}k^{\prime}\to
r_{\\_}\infty^{\prime}$, with $p_{\\_}\infty^{\prime}\in\mathcal{M}^{m}$,
$\varepsilon_{\\_}0^{\prime}\leq\ell_{\\_}\infty^{\prime}\leq\frac{1}{2}$ and
$\varepsilon_{\\_}0^{\prime}\leq r_{\\_}\infty^{\prime}\leq s_{\\_}0$.
Moreover, as in the proof of Lemma 5.3, up to further subsequences we get a
limiting local parametrized stationary varifold
$(\Omega_{\\_}\infty,\Theta_{\\_}\infty,N_{\\_}\infty\circ\varphi_{\\_}\infty^{-1})$
in $\mathbb{R}^{q}$, with
$\Omega_{\\_}\infty=\varphi_{\\_}\infty(B_{\\_}{s_{\\_}0}^{2})$. From Theorem
3.3 we know that $\Theta_{\\_}\infty$ is harmonic and $N_{\\_}\infty=\nu$ is
constant, so Lemma 4.2 gives that $\Pi_{\\_}\infty\circ\Theta_{\\_}\infty$ is
a diffeomorphism from
$\varphi_{\\_}\infty(\overline{B}_{\\_}{s_{\\_}0/2}^{2})$ onto its image.

Calling $\mathbf{v}_{\\_}k$ the varifold issued by
${\left.\kern-1.2pt\Psi_{\\_}k\vphantom{\big{|}}\right|_{B_{\\_}{s_{\\_}0^{2}}^{2}}}$
and $\mathbf{v}_{\\_}\infty$ the one issued by
$(\varphi_{\\_}\infty(B_{\\_}{s_{\\_}0^{2}}^{2}),\Theta_{\\_}\infty,\nu)$, we
have the varifold convergence
$\mathbf{v}_{\\_}k\stackrel{{\scriptstyle*}}{{\rightharpoonup}}\mathbf{v}_{\\_}\infty$
as $k\to\infty$. The area formula gives

$\displaystyle\,\ThisStyle{\ensurestackMath{\stackinset{c}{.2\LMpt}{c}{.5\LMpt}{\SavedStyle-}{\SavedStyle\phantom{\int}}}\kern-16.56937pt}\int_{\\_}{B_{\\_}{\eta_{\\_}0}^{\Pi_{\\_}k}}N(\Psi_{\\_}k,B_{\\_}{s_{\\_}0^{2}}^{2},\Pi_{\\_}k)=\frac{\left\|(\Pi_{\\_}k)_{\\_}*\mathbf{v}_{\\_}k\right\|(B_{\\_}{\eta_{\\_}0}^{\Pi_{\\_}k})}{\pi\eta_{\\_}0^{2}}\to\frac{\left\|(\Pi_{\\_}\infty)_{\\_}*\mathbf{v}_{\\_}\infty\right\|(B_{\\_}{\eta_{\\_}0}^{\Pi_{\\_}\infty})}{\pi\eta_{\\_}0^{2}}=\nu,$

since $(\Pi_{\\_}\infty)_{\\_}*\mathbf{v}_{\\_}\infty$ equals an open superset
of $B_{\\_}{\eta_{\\_}0}^{\Pi_{\\_}\infty}$ in $\Pi_{\\_}\infty$ (by Lemma
A.1), equipped with the constant integer multiplicity $\nu$. Hence,
$n(\Psi_{\\_}k,B_{\\_}{s_{\\_}0^{2}}^{2},B_{\\_}{\eta_{\\_}0}^{\Pi_{\\_}k})=\nu$
eventually.

Similarly, calling $\mathbf{v}_{\\_}k^{\prime}$ the varifold induced by
${\left.\kern-1.2pt\Psi_{\\_}k\vphantom{\big{|}}\right|_{B_{\\_}{s_{\\_}0^{2}r_{\\_}k^{\prime}}^{2}}}$
and $\mathbf{v}_{\\_}\infty^{\prime}$ the varifold induced by
$(\varphi_{\\_}\infty(B_{\\_}{s_{\\_}0^{2}r_{\\_}\infty^{\prime}}^{2}),\Theta_{\\_}\infty,\nu)$,
we have
$\mathbf{v}_{\\_}k^{\prime}\stackrel{{\scriptstyle*}}{{\rightharpoonup}}\mathbf{v}_{\\_}\infty^{\prime}$
as $k\to\infty$, as is readily seen by approximating with domains which do not
vary along the sequence. Since
$(\ell_{\\_}\infty^{\prime})^{-1}(\Psi_{\\_}\infty(r_{\\_}\infty^{\prime}\cdot)-p_{\\_}\infty^{\prime})\in\mathcal{R}_{\\_}{K_{\\_}0,\delta_{\\_}0}^{\Pi_{\\_}\infty}$,
again $(\Pi_{\\_}\infty)_{\\_}*\mathbf{v}_{\\_}\infty^{\prime}$ equals a
superset of $B_{\\_}{\eta_{\\_}0\ell_{\\_}\infty^{\prime}}^{\Pi_{\\_}\infty}$
in $\Pi_{\\_}\infty$, with constant density $\nu$. This gives again

$\displaystyle\,\ThisStyle{\ensurestackMath{\stackinset{c}{.2\LMpt}{c}{.5\LMpt}{\SavedStyle-}{\SavedStyle\phantom{\int}}}\kern-16.56937pt}\int_{\\_}{B_{\\_}{\eta_{\\_}0\ell_{\\_}k^{\prime}}^{\Pi_{\\_}k}(q_{\\_}k)}N(\Psi_{\\_}k,B_{\\_}{s_{\\_}0^{2}r_{\\_}k^{\prime}}^{2},\Pi_{\\_}k)=\frac{\left\|(\Pi_{\\_}k)_{\\_}*\mathbf{v}_{\\_}k^{\prime}\right\|(B_{\\_}{\eta_{\\_}0\ell_{\\_}k^{\prime}}^{\Pi_{\\_}k}(q_{\\_}k))}{\pi\eta_{\\_}0^{2}(\ell_{\\_}k^{\prime})^{2}}\to\frac{\left\|(\Pi_{\\_}\infty)_{\\_}*\mathbf{v}_{\\_}\infty^{\prime}\right\|(B_{\\_}{\eta_{\\_}0\ell_{\\_}\infty^{\prime}}^{\Pi_{\\_}\infty}(q_{\\_}\infty))}{\pi\eta_{\\_}0^{2}(\ell_{\\_}\infty^{\prime})^{2}}=\nu,$

where $q_{\\_}k:=\Pi_{\\_}k(p_{\\_}k^{\prime})$ for
$k\in\mathbb{N}\cup\left\\{\infty\right\\}$. Hence,
$n(\Psi_{\\_}k-p_{\\_}k^{\prime},B_{\\_}{s_{\\_}0^{2}r_{\\_}k^{\prime}}^{2},B_{\\_}{\eta_{\\_}0\ell_{\\_}k^{\prime}}^{\Pi_{\\_}k})=\nu$
eventually. So the first equality in (5.11) holds eventually, giving the
desired contradiction. ∎

###### Lemma 5.6 ($\bm{n(\cdot)=1}$ when $\bm{\tau=1}$).

Assume that $\Psi\in
C^{\infty}(\overline{B}_{\\_}{r}^{2}(z),\mathcal{M}^{m}_{\\_}{p,\ell})$ is a
conformal immersion and $\Pi$ is a 2-plane with
$\Psi(z+r\cdot)\in\mathcal{R}_{\\_}{K_{\\_}0,\delta_{\\_}0}^{\Pi}$ and
$\frac{1}{2}\int_{\\_}{B_{\\_}r^{2}(z)}\left|\nabla\Psi\right|^{2}\leq E$. If
$\int_{\\_}{B_{\\_}1^{2}}\left|A\right|^{4}\,d\operatorname{vol}_{\\_}{g_{\\_}\Psi}$
and $\ell$ are sufficiently small, then $\Pi\circ\Psi$ is a diffeomorphism
from $\overline{B}_{\\_}{s_{\\_}0^{2}r}^{2}(z)$ onto its image.

###### Proof of Lemma 5.6.

We can suppose $z=0$, $r=1$. Assume by contradiction that the claim does not
hold, for a sequence of 2-planes $\Pi_{\\_}k\to\Pi_{\\_}\infty$ and immersions
$\Psi_{\\_}k:\overline{B}_{\\_}1^{2}\to\mathcal{M}^{m}_{\\_}{p_{\\_}k,\ell_{\\_}k}$
with $\ell_{\\_}k\to 0$ and second fundamental forms $A_{\\_}k$ satisfying

(5.12)
$\displaystyle\int_{\\_}{B_{\\_}1^{2}}\left|A_{\\_}k\right|^{4}\,d\operatorname{vol}_{\\_}{g_{\\_}{\Psi_{\\_}k}}\to
0.$

Let $\lambda_{\\_}k\in C^{\infty}(\overline{B}_{\\_}1^{2})$ be defined by
$\left|\partial_{\\_}1\Psi_{\\_}k\right|=\left|\partial_{\\_}2\Psi_{\\_}k\right|=:e^{\lambda_{\\_}k}$
and let $A_{\\_}{p,\ell}$ and $\widetilde{A}_{\\_}k$ denote the second
fundamental forms of $\mathcal{M}^{m}_{\\_}{p,\ell}\subseteq\mathbb{R}^{q}$
and of the immersion $\Psi_{\\_}k$ in $\mathbb{R}^{q}$ respectively, so that
$\widetilde{A}_{\\_}k=A_{\\_}{p_{\\_}k,\ell_{\\_}k}+A_{\\_}k$. Note that

(5.13)
$\displaystyle\left\|A_{\\_}{p_{\\_}k,\ell_{\\_}k}\right\|_{\\_}{L^{\infty}}\leq
C(\mathcal{M}^{m})\ell_{\\_}k\to 0,$

so that

(5.14)
$\displaystyle\int_{\\_}{B_{\\_}1^{2}}\left|\widetilde{A}_{\\_}k\right|^{4}\,d\operatorname{vol}_{\\_}{g_{\\_}{\Psi_{\\_}k}}\to
0.$

With a slight abuse of notation, let us drop the dependence on $k$ in the
subsequent computations. We define the orthonormal frame

(5.15)
$\displaystyle\widetilde{e}_{\\_}1:=e^{-\lambda}\partial_{\\_}1\Psi,\qquad\widetilde{e}_{\\_}2:=e^{-\lambda}\partial_{\\_}2\Psi$

for the tangent space of the immersed surface $\Psi$. It is straightforward to
check that the map
$\widetilde{e}_{\\_}1\wedge\widetilde{e}_{\\_}2:\overline{B}_{\\_}1^{2}\to\Lambda_{\\_}2\mathbb{R}^{q}$
has
$\left|\nabla(\widetilde{e}_{\\_}1\wedge\widetilde{e}_{\\_}2)\right|=e^{\lambda}\left|\widetilde{A}\right|$,
so

(5.16)
$\displaystyle\int_{\\_}{B_{\\_}1^{2}}\left|\nabla(\widetilde{e}_{\\_}1\wedge\widetilde{e}_{\\_}2)\right|^{2}\,d\mathcal{L}^{2}=\int_{\\_}{B_{\\_}1^{2}}e^{2\lambda}\left|\widetilde{A}\right|^{2}\,d\mathcal{L}^{2}=\int_{\\_}{B_{\\_}1^{2}}\left|\widetilde{A}\right|^{2}\,d\operatorname{vol}_{\\_}{g_{\\_}\Psi}\to
0$

by Hölder’s inequality, since
$\int_{\\_}{B_{\\_}1^{2}}\,d\operatorname{vol}_{\\_}{g_{\\_}\Psi}\leq E$. We
identify the Grassmannian $\text{Gr}(2,\mathbb{R}^{q})$ of 2-planes in
$\mathbb{R}^{q}$ with a submanifold of the projectivization of
$\Lambda_{\\_}2\mathbb{R}^{q}$, by means of Plücker’s embedding. For $k$ large
enough [3, Lemma 5.1.4] applies and provides a rotated frame
$(e_{\\_}1,e_{\\_}2)$, given by

(5.17) $\displaystyle
e_{\\_}\mathbb{C}:=e_{\\_}1+ie_{\\_}2=e^{i\theta}\widetilde{e}_{\\_}\mathbb{C},\qquad\widetilde{e}_{\\_}\mathbb{C}:=\widetilde{e}_{\\_}1+i\widetilde{e}_{\\_}2,$

for a suitable real function $\theta\in W^{1,2}(B_{\\_}1^{2})$ minimizing
$\int_{\\_}{B_{\\_}1^{2}}\left|\nabla\theta+\widetilde{e}_{\\_}1\cdot\nabla\widetilde{e}_{\\_}2\right|^{2}$
(in particular, $\theta$ and $e_{\\_}\mathbb{C}$ are smooth functions on
$\overline{B}_{\\_}1^{2}$) and with $\left\|\nabla
e_{\\_}\mathbb{C}\right\|_{\\_}{L^{2}}^{2}$ becoming arbitrarily small as
$k\to\infty$. We will assume in the sequel that $\left\|\nabla
e_{\\_}\mathbb{C}\right\|_{\\_}{L^{2}}^{2}\leq 1$. Observe that, whenever
$\alpha,\beta\in C^{1}(\overline{B}_{\\_}1^{2})$,

$\displaystyle\partial_{\\_}1\alpha\partial_{\\_}2\beta-\partial_{\\_}2\alpha\partial_{\\_}1\beta$
$\displaystyle=\frac{1}{4}(\partial_{\\_}1\alpha+\partial_{\\_}2\beta)^{2}+\frac{1}{4}(\partial_{\\_}2\alpha-\partial_{\\_}1\beta)^{2}-\frac{1}{4}(\partial_{\\_}1\alpha-\partial_{\\_}2\beta)^{2}-\frac{1}{4}(\partial_{\\_}2\alpha+\partial_{\\_}1\beta)^{2}$
$\displaystyle=\left|\partial_{\\_}z(\alpha+i\beta)\right|^{2}-\left|\partial_{\\_}{\overline{z}}(\alpha+i\beta)\right|^{2}.$

Hence, being
$\widetilde{e}_{\\_}1+i\widetilde{e}_{\\_}2=2e^{-\lambda}\partial_{\\_}{\overline{z}}\Psi$
and
$\partial_{\\_}z\Psi\cdot\partial_{\\_}z\Psi=\partial_{\\_}{\overline{z}}\Psi\cdot\partial_{\\_}{\overline{z}}\Psi=0$
by conformality, we get

$\displaystyle-(\partial_{\\_}1\widetilde{e}_{\\_}1\cdot\partial_{\\_}2\widetilde{e}_{\\_}2-\partial_{\\_}2\widetilde{e}_{\\_}1\cdot\partial_{\\_}1\widetilde{e}_{\\_}2)$
$\displaystyle=4\partial_{\\_}{\overline{z}}(e^{-\lambda}\partial_{\\_}{\overline{z}}\Psi)\cdot\partial_{\\_}z(e^{-\lambda}\partial_{\\_}z\Psi)-4\partial_{\\_}{\overline{z}}(e^{-\lambda}\partial_{\\_}z\Psi)\cdot\partial_{\\_}z(e^{-\lambda}\partial_{\\_}{\overline{z}}\Psi)$
$\displaystyle=4e^{-2\lambda}(\partial_{\\_}{\overline{z}\overline{z}}^{2}\Psi\cdot\partial_{\\_}{zz}^{2}\Psi-\partial_{\\_}{\overline{z}z}^{2}\Psi\cdot\partial_{\\_}{\overline{z}z}^{2}\Psi-\partial_{\\_}{\overline{z}}\lambda\partial_{\\_}{\overline{z}}\Psi\cdot\partial_{\\_}{zz}^{2}\Psi-\partial_{\\_}z\lambda\partial_{\\_}z\Psi\cdot\partial_{\\_}{\overline{z}\overline{z}}^{2}\Psi)$
$\displaystyle\quad+2e^{-2\lambda}\partial_{\\_}{\overline{z}}\lambda\partial_{\\_}{\overline{z}}(\partial_{\\_}z\Psi\cdot\partial_{\\_}z\Psi)+2e^{-2\lambda}\partial_{\\_}z\lambda\partial_{\\_}z(\partial_{\\_}{\overline{z}}\Psi\cdot\partial_{\\_}{\overline{z}}\Psi)$
$\displaystyle=4e^{-2\lambda}(\partial_{\\_}{\overline{z}\overline{z}}^{2}\Psi\cdot\partial_{\\_}{zz}^{2}\Psi-\partial_{\\_}{\overline{z}z}^{2}\Psi\cdot\partial_{\\_}{\overline{z}z}^{2}\Psi-\partial_{\\_}{\overline{z}}\lambda\partial_{\\_}{\overline{z}}\Psi\cdot\partial_{\\_}{zz}^{2}\Psi-\partial_{\\_}z\lambda\partial_{\\_}z\Psi\cdot\partial_{\\_}{\overline{z}\overline{z}}^{2}\Psi).$

On the other hand we have

$\displaystyle
2e^{2\lambda}\partial_{\\_}z\lambda=\partial_{\\_}z(e^{2\lambda})=\partial_{\\_}z(2\partial_{\\_}{\overline{z}}\Psi\cdot\partial_{\\_}z\Psi)=\partial_{\\_}{\overline{z}}(\partial_{\\_}z\Psi\cdot\partial_{\\_}z\Psi)+2\partial_{\\_}{\overline{z}}\Psi\cdot\partial_{\\_}{zz}^{2}\Psi=2\partial_{\\_}{\overline{z}}\Psi\cdot\partial_{\\_}{zz}^{2}\Psi,$
$\displaystyle\Delta(e^{2\lambda})$
$\displaystyle=4\partial_{\\_}{\overline{z}z}^{2}(2\partial_{\\_}{\overline{z}}\Psi\cdot\partial_{\\_}z\Psi)=8\partial_{\\_}{\overline{z}}(\partial_{\\_}{\overline{z}}\Psi\cdot\partial_{\\_}{zz}^{2}\Psi)+4\partial_{\\_}{\overline{z}\overline{z}}^{2}(\partial_{\\_}z\Psi\cdot\partial_{\\_}z\Psi)$
$\displaystyle=8(\partial_{\\_}{\overline{z}\overline{z}}^{2}\Psi\cdot\partial_{\\_}{zz}^{2}\Psi-\partial_{\\_}{\overline{z}z}^{2}\Psi\cdot\partial_{\\_}{\overline{z}z}^{2}\Psi)+4\partial_{\\_}{zz}^{2}(\partial_{\\_}{\overline{z}}\Psi\cdot\partial_{\\_}{\overline{z}}\Psi)$
$\displaystyle=8(\partial_{\\_}{\overline{z}\overline{z}}^{2}\Psi\cdot\partial_{\\_}{zz}^{2}\Psi-\partial_{\\_}{\overline{z}z}^{2}\Psi\cdot\partial_{\\_}{\overline{z}z}^{2}\Psi),$

so we arrive at

(5.18)
$\displaystyle\partial_{\\_}1\widetilde{e}_{\\_}1\cdot\partial_{\\_}2\widetilde{e}_{\\_}2-\partial_{\\_}2\widetilde{e}_{\\_}1\cdot\partial_{\\_}1\widetilde{e}_{\\_}2=-\frac{\Delta(e^{2\lambda})}{2e^{2\lambda}}+8\partial_{\\_}{\overline{z}}\lambda\partial_{\\_}z\lambda=-\Delta\lambda.$

Alternatively, since the projections of $\partial_{\\_}j\widetilde{e}_{\\_}1$
and $\partial_{\\_}k\widetilde{e}_{\\_}2$ onto the tangent space of the
immersion $\Psi$ are orthogonal (being the projection of
$\partial_{\\_}j\widetilde{e}_{\\_}1$ a multiple of $\widetilde{e}_{\\_}2$ and
the projection of $\partial_{\\_}k\widetilde{e}_{\\_}2$ a multiple of
$\widetilde{e}_{\\_}1$), we have

$\displaystyle\partial_{\\_}1\widetilde{e}_{\\_}1\cdot\partial_{\\_}2\widetilde{e}_{\\_}2-\partial_{\\_}2\widetilde{e}_{\\_}1\cdot\partial_{\\_}1\widetilde{e}_{\\_}2=e^{2\lambda}(\widetilde{A}(\widetilde{e}_{\\_}1,\widetilde{e}_{\\_}1)\cdot\widetilde{A}(\widetilde{e}_{\\_}2,\widetilde{e}_{\\_}2)-\widetilde{A}(\widetilde{e}_{\\_}1,\widetilde{e}_{\\_}2)\cdot\widetilde{A}(\widetilde{e}_{\\_}1,\widetilde{e}_{\\_}2))=e^{2\lambda}K,$

by Gauss’ formula, $K$ denoting the Gaussian curvature of the immersed
surface. But, by the well-known formula for the curvature of a conformal
metric, we have $K=-e^{-2\lambda}\Delta\lambda$, which gives again (5.18).
Moreover,

$\displaystyle\partial_{\\_}1e_{\\_}1\cdot\partial_{\\_}2e_{\\_}2-\partial_{\\_}2e_{\\_}1\cdot\partial_{\\_}1e_{\\_}2$
$\displaystyle=\Im\left\langle\nabla\overline{e_{\\_}\mathbb{C}};\nabla
e_{\\_}\mathbb{C}\right\rangle=\Im\left\langle\nabla\overline{\widetilde{e}_{\\_}\mathbb{C}}-i\overline{\widetilde{e}_{\\_}\mathbb{C}}\otimes\nabla\theta;\nabla\widetilde{e}_{\\_}\mathbb{C}+i\widetilde{e}_{\\_}\mathbb{C}\otimes\nabla\theta\right\rangle$
$\displaystyle=\Im\left\langle\nabla\overline{\widetilde{e}_{\\_}\mathbb{C}};\nabla\widetilde{e}_{\\_}\mathbb{C}\right\rangle=\partial_{\\_}1\widetilde{e}_{\\_}1\cdot\partial_{\\_}2\widetilde{e}_{\\_}2-\partial_{\\_}2\widetilde{e}_{\\_}1\cdot\partial_{\\_}1\widetilde{e}_{\\_}2,$

since
$\left\langle\overline{\widetilde{e}_{\\_}\mathbb{C}}\otimes\nabla\theta;\widetilde{e}_{\\_}\mathbb{C}\otimes\nabla\theta\right\rangle$
is real and
$\left\langle-i\overline{\widetilde{e}_{\\_}\mathbb{C}}\otimes\nabla\theta;\nabla\widetilde{e}_{\\_}\mathbb{C}\right\rangle=\overline{\left\langle\nabla\overline{\widetilde{e}_{\\_}\mathbb{C}};i\widetilde{e}_{\\_}\mathbb{C}\otimes\nabla\theta\right\rangle}$.
Thus, calling $\mu\in C^{\infty}(\overline{B}_{\\_}1^{2})$ the solution to

$\displaystyle\begin{cases}-\Delta\mu=\partial_{\\_}1e_{\\_}1\cdot\partial_{\\_}2e_{\\_}2-\partial_{\\_}2e_{\\_}1\cdot\partial_{\\_}1e_{\\_}2&\text{on
}B_{\\_}1^{2}\\\ \ \mu=0&\text{on }\partial B_{\\_}1^{2},\end{cases}$

we obtain that $\lambda-\mu$ is harmonic and, by Wente’s inequality,

(5.19) $\displaystyle\left\|\mu\right\|_{\\_}{L^{\infty}}\leq
C(q)\left(\left\|\nabla e_{\\_}1\right\|_{\\_}{L^{2}}^{2}+\left\|\nabla
e_{\\_}2\right\|_{\\_}{L^{2}}^{2}\right)\leq C(q).$

Since $\lambda<e^{2\lambda}$, for all $x\in\overline{B}_{\\_}{3/4}^{2}$ we get

(5.20)
$\displaystyle(\lambda-\mu)(x)=\,\ThisStyle{\ensurestackMath{\stackinset{c}{.2\LMpt}{c}{.5\LMpt}{\SavedStyle-}{\SavedStyle\phantom{\int}}}\kern-16.56937pt}\int_{\\_}{B_{\\_}{1/4}^{2}(x)}(\lambda-\mu)\leq\,\ThisStyle{\ensurestackMath{\stackinset{c}{.2\LMpt}{c}{.5\LMpt}{\SavedStyle-}{\SavedStyle\phantom{\int}}}\kern-16.56937pt}\int_{\\_}{B_{\\_}{1/4}^{2}(x)}e^{2\lambda}+\left\|\mu\right\|_{\\_}{L^{\infty}}\leq\frac{E}{\mathcal{L}^{2}(B_{\\_}{1/4}^{2})}+C(q).$

Together with (5.19), this gives an upper bound for $\lambda$ on
$B_{\\_}{3/4}^{2}$, depending only on $E,q$. Although this is sufficient for
the present purposes, one can also get a lower bound for $\lambda$ on
$B_{\\_}{s_{\\_}0}^{2}$. Indeed, calling $M$ the right-hand side of (5.20), we
obtain that $M-(\lambda-\mu)$ is a nonnegative harmonic function on
$B_{\\_}{3/4}^{2}$. Moreover, the length of the curve
${\left.\kern-1.2pt\Psi\vphantom{\big{|}}\right|_{\partial
B_{\\_}{s_{\\_}0}^{2}}}$ is

(5.21) $\displaystyle\int_{\\_}{\partial B_{\\_}{s_{\\_}0}^{2}}e^{\lambda}\geq
2\pi\eta_{\\_}0,$

since the composition of
$\Pi\circ{\left.\kern-1.2pt\Psi\vphantom{\big{|}}\right|_{\partial
B_{\\_}{s_{\\_}0}^{2}}}$ with the radial projection onto $\partial
B_{\\_}{\eta_{\\_}0}^{\Pi}$ (which does not increase the length) is surjective
(being a generator of the fundamental group of $\partial
B_{\\_}{\eta_{\\_}0}^{\Pi}$). Hence, there exists some $x\in\partial
B_{\\_}{s_{\\_}0}^{2}$ such that
$\lambda(x)\geq\log\left(s_{\\_}0^{-1}\eta_{\\_}0\right)$. We deduce that

(5.22) $\displaystyle\inf_{\\_}{B_{\\_}{s_{\\_}0}^{2}}(M-(\lambda-\mu))\leq
M+C(q)-\log(s_{\\_}0^{-1}\eta_{\\_}0)$

and so, by Harnack’s inequality, the supremum of $M-(\lambda-\mu)$ on
$B_{\\_}{s_{\\_}0}^{2}$ is bounded by a constant depending only on
$E,s_{\\_}0,\eta_{\\_}0,q$. This, together with (5.20) and (5.19), gives

(5.23)
$\displaystyle\left\|\lambda\right\|_{\\_}{L^{\infty}(B_{\\_}{s_{\\_}0}^{2})}\leq
C(E,s_{\\_}0,\eta_{\\_}0,q).$

The mean curvature of the immersion $\Psi$ is
$\widetilde{H}=\frac{1}{2e^{2\lambda}}(\widetilde{A}(\partial_{\\_}1\Psi,\partial_{\\_}1\Psi)+\widetilde{A}(\partial_{\\_}2\Psi,\partial_{\\_}2\Psi))=-\frac{\Delta\Psi}{2e^{2\lambda}}$
(note that $\Delta\Psi$ is already orthogonal to the tangent space of the
immersion, since
$\partial_{\\_}z\Psi\cdot\Delta\Psi=4\partial_{\\_}z\Psi\cdot\partial_{\\_}{\overline{z}z}^{2}\Psi=2\partial_{\\_}{\overline{z}}(\partial_{\\_}z\Psi\cdot\partial_{\\_}z\Psi)=0$).
So we get

(5.24) $\displaystyle\begin{aligned}
&\int_{\\_}{B_{\\_}{3/4}^{2}}\left|\Delta\Psi_{\\_}k\right|^{4}\,d\mathcal{L}^{2}=16\int_{\\_}{B_{\\_}{3/4}^{2}}\left|\widetilde{H}_{\\_}k\right|^{4}e^{6\lambda_{\\_}k}\,d\operatorname{vol}_{\\_}{g_{\\_}{\Psi_{\\_}k}}\\\
&\leq
C(E,q)\int_{\\_}{B_{\\_}{3/4}^{2}}\left|\widetilde{A}_{\\_}k\right|^{4}\,d\operatorname{vol}_{\\_}{g_{\\_}{\Psi_{\\_}k}}\to
0.\end{aligned}$

Since $s_{\\_}0\leq\frac{1}{2}$, this implies that $(\Psi_{\\_}k)$ is a
bounded sequence in $W^{2,4}(B_{\\_}{s_{\\_}0}^{2})$ (by Lemma A.2 applied to
$\Psi_{\\_}k(\frac{3}{4}\cdot)$), so by the compact embedding
$W^{2,4}(B_{\\_}{s_{\\_}0}^{2})\hookrightarrow
C^{1}(\overline{B}_{\\_}{s_{\\_}0}^{2})$ we obtain a strong limit
$\Psi_{\\_}\infty$ in $C^{1}(\overline{B}_{\\_}{s_{\\_}0}^{2})$, up to
subsequences. Thus $\Psi_{\\_}\infty$ is weakly conformal and, by (5.24), it
is also harmonic. Lemma 4.2 applies (with $\Psi_{\\_}\infty(s_{\\_}0\cdot)$
and $\text{id}_{\\_}{\mathbb{R}^{2}}$ in place of $\Psi$ and $\varphi$) and
gives that $\Pi_{\\_}\infty\circ\Psi_{\\_}\infty$ is a diffeomorphism from
$\overline{B}_{\\_}{s_{\\_}0/2}^{2}\supseteq\overline{B}_{\\_}{s_{\\_}0^{2}}^{2}$
onto its image, hence the same is eventually true for
$\Pi_{\\_}k\circ\Psi_{\\_}k$, giving the desired contradiction. ∎

##  6\. Multiplicity one in the limit

###### Theorem 6.1 ($\bm{n(\cdot)=1}$ for small
$\bm{\sigma^{2}\log(\sigma^{-1})\int|A|^{4}}$).

Assume $\Phi\in C^{\infty}(\overline{B}_{\\_}{r}^{2}(z),\mathcal{M}^{m})$ is a
conformal immersion, critical for (3.1) on $B_{\\_}{r}^{2}(z)$ and satisfying

    * –

$\displaystyle\ell^{-1}(\Phi(z+r\cdot)-p)\in\mathcal{R}_{\\_}{K_{\\_}0,\delta_{\\_}0}^{\Pi}$
for some $\sqrt{\sigma/\varepsilon_{\\_}0^{\prime\prime}}<\ell<1$ and
$p\in\mathcal{M}^{m}$,

    * –

$\displaystyle\frac{1}{2}\int_{\\_}{B_{\\_}r^{2}(z)}\left|\nabla\Phi\right|^{2}\leq
E_{\\_}0\ell^{2}$,

    * –

$\displaystyle\int_{\\_}{\Phi^{-1}(B_{\\_}\ell^{q}(p))}\,d\operatorname{vol}_{\\_}{g_{\\_}\Phi}\leq
V\pi\ell^{2}$ and
$\displaystyle\int_{\\_}{\Phi^{-1}(B_{\\_}{\eta_{\\_}0\ell}^{q}(p))}\,d\operatorname{vol}_{\\_}{g_{\\_}\Phi}\leq
V\pi(\eta_{\\_}0\ell)^{2}$,

    * –

$\displaystyle\sigma^{2}\log(\sigma^{-1})\int_{\\_}{B_{\\_}s^{2}}\left|A\right|^{4}\,d\operatorname{vol}_{\\_}{g_{\\_}\Phi}\leq\frac{\varepsilon_{\\_}0^{\prime\prime}}{E_{\\_}0}\int_{\\_}{B_{\\_}s^{2}}\,d\operatorname{vol}_{\\_}{g_{\\_}\Phi}$
for all $0<s\leq r$.

Then, if $\sigma$ and $\ell$ are small enough (independently of each other),
we have

$\displaystyle
n(\Phi-p,B_{\\_}{s_{\\_}0^{2}r}^{2}(z),B_{\\_}{\eta_{\\_}0\ell}^{\Pi})=1.$

###### Proof of Theorem 6.1.

Let $r_{\\_}0:=r$, $p_{\\_}0:=p$, $\ell_{\\_}0:=\ell$,
$\tau_{\\_}0:=\sigma\ell_{\\_}0^{-2}$ and $\Pi_{\\_}0:=\Pi$. Note that

$\displaystyle\Psi_{\\_}0:=\ell^{-1}(\Phi-p)=\ell_{\\_}0^{-1}(\Phi-p_{\\_}0)$

is critical for (3.3), with
$\tau:=\tau_{\\_}0\leq\varepsilon_{\\_}0^{\prime\prime}$. Thus Lemma 5.3
applies to $\Psi_{\\_}0$ (if $\ell$ is small enough), giving a new radius
$\varepsilon_{\\_}0^{\prime}r_{\\_}0<r_{\\_}1<s_{\\_}0r_{\\_}0$, a new point
$p^{\prime}\in\mathcal{M}^{m}_{\\_}{p,\ell}$, a new scale
$\varepsilon_{\\_}0^{\prime}<\ell^{\prime}<\frac{1}{2}$ and a new 2-plane
$\Pi^{\prime}$. Setting $r_{\\_}1:=r^{\prime}$,
$p_{\\_}1:=p_{\\_}0+\ell_{\\_}0p^{\prime}$,
$\ell_{\\_}1:=\ell^{\prime}\ell_{\\_}0$,
$\tau_{\\_}1:=\sigma\ell_{\\_}1^{-2}$, $\Pi_{\\_}1:=\Pi^{\prime}$ and
recalling (5.10), the map

$\displaystyle\Psi_{\\_}1:=(\ell^{\prime})^{-1}(\Psi_{\\_}0-p^{\prime})=\ell_{\\_}1^{-1}(\Phi-
p_{\\_}1)$

still satisfies the hypotheses of Lemma 5.3, with the parameters
$r_{\\_}1,\tau_{\\_}1,p_{\\_}1,\ell_{\\_}1,\Pi_{\\_}1$, provided that
$\tau_{\\_}1\leq\varepsilon_{\\_}0^{\prime}$: indeed, note that (assuming
$\tau_{\\_}1\leq\varepsilon_{\\_}0^{\prime}<1$)

$\displaystyle\tau_{\\_}1^{2}\log(\tau_{\\_}1^{-1})\int_{\\_}{B_{\\_}{r_{\\_}1}^{2}(z)}\left|A_{\\_}{\Psi_{\\_}1}\right|^{4}\,d\operatorname{vol}_{\\_}{g_{\\_}{\Psi_{\\_}1}}\leq\tau_{\\_}1^{2}\log(\sigma^{-1})\int_{\\_}{B_{\\_}{r_{\\_}1}^{2}(z)}\left|A_{\\_}{\Psi_{\\_}1}\right|^{4}\,d\operatorname{vol}_{\\_}{g_{\\_}{\Psi_{\\_}1}}$
$\displaystyle=\ell_{\\_}1^{-2}\sigma^{2}\log(\sigma^{-1})\int_{\\_}{B_{\\_}{r_{\\_}1}^{2}(z)}\left|A_{\\_}\Phi\right|^{4}\,d\operatorname{vol}_{\\_}{g_{\\_}\Phi}\leq\frac{\varepsilon_{\\_}0^{\prime\prime}\ell_{\\_}1^{-2}}{E_{\\_}0}\int_{\\_}{B_{\\_}{r_{\\_}1}^{2}(z)}\,d\operatorname{vol}_{\\_}{g_{\\_}\Phi}=\frac{\varepsilon_{\\_}0^{\prime\prime}}{2E_{\\_}0}\int_{\\_}{B_{\\_}{r_{\\_}1}^{2}(z)}\left|\nabla\Psi_{\\_}1\right|^{2}\leq\varepsilon_{\\_}0^{\prime\prime}.$

Hence, we can iterate and define
$r_{\\_}j,\tau_{\\_}j,p_{\\_}j,\ell_{\\_}j,\Pi_{\\_}j$, for $j=0,1,\dots$, up
to a maximum index $k\geq 1$ such that
$\tau_{\\_}j\leq\varepsilon_{\\_}0^{\prime\prime}\leq\varepsilon_{\\_}0^{\prime}$
for $1\leq j<k$ and $\tau_{\\_}k>\varepsilon_{\\_}0^{\prime\prime}$: such $k$
exists since $\tau_{\\_}j=\ell_{\\_}j^{-2}\sigma\geq 4^{j}\tau_{\\_}0$. With
the same computation as above, this implies

(6.1)
$\displaystyle\int_{\\_}{B_{\\_}{r_{\\_}k}^{2}(z)}\left|A\right|^{4}\,d\operatorname{vol}_{\\_}{g_{\\_}{\Psi_{\\_}k}}\leq\frac{\varepsilon_{\\_}0^{\prime\prime}}{\tau_{\\_}k^{2}\log(\sigma^{-1})}\leq\frac{1}{\varepsilon_{\\_}0^{\prime\prime}\log(\sigma^{-1})}.$

If $\sigma$ and $\ell$ are small enough, Lemma 5.6 applies to the map
$\Psi_{\\_}k:=\ell_{\\_}k^{-1}(\Psi-p_{\\_}k)$, on the ball
$B_{\\_}{r_{\\_}k}^{2}(z)$: indeed, note that $\ell_{\\_}k\leq\ell$ and
$\int_{\\_}{B_{\\_}{r_{\\_}k}^{2}(z)}|A|^{4}\,d\operatorname{vol}_{\\_}{g_{\\_}{\Psi_{\\_}k}}$
can be assumed arbitrarily small (by taking $\sigma$ and $\ell$ small enough),
by virtue of (6.1). This, together with Lemma A.1, gives

$\displaystyle
n(\Psi_{\\_}k,B_{\\_}{s_{\\_}0^{2}r_{\\_}k}^{2}(z),B_{\\_}{\eta_{\\_}0}^{\Pi_{\\_}k})=1.$

Also, Lemma 5.5 applies for all $j=0,\dots,k-1$, giving

$\displaystyle
n(\Phi-p,B_{\\_}{s_{\\_}0^{2}r}^{2}(z),B_{\\_}{\eta_{\\_}0\ell}^{\Pi})$
$\displaystyle=n(\Psi_{\\_}0,B_{\\_}{s_{\\_}0^{2}r_{\\_}0}^{2}(z),B_{\\_}{\eta_{\\_}0}^{\Pi_{\\_}0})$
$\displaystyle=n(\Psi_{\\_}1,B_{\\_}{s_{\\_}0^{2}r_{\\_}1}^{2}(z),B_{\\_}{\eta_{\\_}0}^{\Pi_{\\_}1})$
$\displaystyle=\cdots$
$\displaystyle=n(\Psi_{\\_}k,B_{\\_}{s_{\\_}0^{2}r_{\\_}k}^{2}(z),B_{\\_}{\eta_{\\_}0}^{\Pi_{\\_}k})$
$\displaystyle=1.\qed$

As in Section 3, assume now that $\Phi_{\\_}k:\Sigma\to\mathcal{M}^{m}$ is a
sequence of critical points for

(6.2)
$\displaystyle\int_{\\_}\Sigma\,d\operatorname{vol}_{\\_}{g_{\\_}{\Phi_{\\_}k}}+\sigma_{\\_}k^{2}\int_{\\_}\Sigma(1+|A|^{2})^{2}\,d\operatorname{vol}_{\\_}{g_{\\_}{\Phi_{\\_}k}}$

with controlled area, namely

$\displaystyle\lambda\leq\int_{\\_}\Sigma\,d\operatorname{vol}_{\\_}{g_{\\_}{\Phi_{\\_}k}}\leq\Lambda,$

and with

$\displaystyle\sigma_{\\_}k\to
0,\qquad\sigma_{\\_}k^{2}\log(\sigma_{\\_}k^{-1})\int_{\\_}\Sigma(1+|A|^{2})^{2}\,d\operatorname{vol}_{\\_}{g_{\\_}{\Phi_{\\_}k}}\to
0.$

By the main result of [13], up to subsequences the varifolds
$\mathbf{v}_{\\_}k$ induced by $\Phi_{\\_}k$ converge to a parametrized
stationary varifold (see e.g. [12, Definition 2.2] and [12, Remark 2.3] for
two equivalent definitions of this notion).

As explained in [13], there could be bubbling points, and also the conformal
structures induced by $\Phi_{\\_}k$ could degenerate (in the space of
conformal structures up to diffeomorphisms). The latter cannot happen if
$\Sigma$ is a sphere, while it can yield a cylinder $\mathbb{C}/\mathbb{Z}$
(or, equivalently,
$\mathbb{C}\setminus\\{0\\}=\widehat{\mathbb{C}}\setminus\\{0,\infty\\}$) in
the limit when $\Sigma$ is a torus. While both of these statements are easy
consequences of the uniformization theorem for Riemann surfaces, if the genus
is at least two then the general picture of degenerating conformal structures
is more complicated; we refer the reader to [4, Section IV.5] for its
description.888Given a sequence of closed hyperbolic surfaces, we can assume
that they have been decomposed into marked pairs of pants, with a constant
combinatorial configuration (see e.g. [4, Section IV.3]). We can then cut them
open along the marked geodesics whose length converges to zero and apply [4,
Proposition IV.5.1] to the resulting surfaces with boundary.

In the remainder of the paper, we will assume for simplicity that there is no
bubbling and no degeneration of the conformal structure. Note that the
arguments will apply also to the general case, working on appropriate domains
different from $\Sigma$.

Up to precomposing $\Phi_{\\_}k$ with suitable diffeomorphisms of $\Sigma$, we
can thus assume that there exist metrics $g_{\\_}k$ of constant curvature
($1$, $0$ or $-1$, depending on the genus of $\Sigma$) such that
$\Phi_{\\_}k:(\Sigma,g_{\\_}k)\to\mathcal{M}^{m}$ is conformal, and such that
$g_{\\_}k$ converges smoothly to a limiting Riemannian metric $g_{\\_}\infty$.
The limiting varifold $\mathbf{v}_{\\_}\infty$ is a parametrized stationary
varifold, of the form $(\Sigma_{\\_}\infty,\Theta_{\\_}\infty,N_{\\_}\infty)$,
where $\Theta_{\\_}\infty:\Sigma_{\\_}\infty\to\mathcal{M}^{m}$ is a smooth
branched minimal immersion. Also, calling $\Phi_{\\_}\infty\in
W^{1,2}(\Sigma,\mathcal{M}^{m})$ the weak limit of $\Phi_{\\_}k$ (up to
subsequences),
$\Theta_{\\_}\infty=\Phi_{\\_}\infty\circ\varphi_{\\_}\infty^{-1}$ for some
(locally) quasiconformal homeomorphism
$\varphi_{\\_}\infty:\Sigma\to\Sigma_{\\_}\infty$.999Here $\Sigma_{\\_}\infty$
is a Riemann surface homeomorphic (and thus diffeomorphic) to $\Sigma$. In
particular, $\Phi_{\\_}\infty$ is continuous.

In local conformal coordinates for $(\Sigma,g_{\\_}\infty)$, as in (3.5) we
have

(6.3)
$\displaystyle\operatorname{vol}_{\\_}{g_{\\_}{\Phi_{\\_}k}}\stackrel{{\scriptstyle*}}{{\rightharpoonup}}N_{\\_}\infty\left|\partial_{\\_}1\Phi_{\\_}\infty\wedge\partial_{\\_}2\Phi_{\\_}\infty\right|\,\mathcal{L}^{2}\geq\frac{1}{2}|\nabla\Phi_{\\_}\infty|^{2}\,\mathcal{L}^{2}.$

By the regularity result of [12], which was already exploited in Section 5,
$N_{\\_}\infty$ is locally a.e. constant and thus a.e. constant (being
$\Sigma_{\\_}\infty$ connected).

Setting $\nu_{\\_}k:=\operatorname{vol}_{\\_}{g_{\\_}{\Phi_{\\_}k}}$ and
$\nu_{\\_}\infty:=N_{\\_}\infty|\partial_{\\_}1\Phi_{\\_}\infty\wedge\partial_{\\_}2\Phi_{\\_}\infty|\,\mathcal{L}^{2}$
(in local conformal coordinates for $\Sigma$), by (6.3) we have
$\nu_{\\_}k\stackrel{{\scriptstyle*}}{{\rightharpoonup}}\nu_{\\_}\infty$. We
can find a conformal disk $U\subset(\Sigma,g_{\\_}\infty)$, which we identify
with $B_{\\_}1^{2}\subset\mathbb{C}$ and fix in the sequel, such that
$\nu_{\\_}\infty(B_{\\_}{1/2}^{2})>0$.

###### Definition 6.2.

We denote by $\nu$ the constant value of $N_{\\_}\infty$. Also, we call $T$
the set of bad points $z\in B_{\\_}1^{2}$ which are not Lebesgue for
$\nabla\Phi_{\\_}\infty$, or such that $\nabla\Phi_{\\_}\infty(z)$ does not
have full rank, or such that

(6.4)
$\displaystyle\max_{\\_}{\left|x\right|=1}\left|\left\langle\nabla\Phi_{\\_}\infty(z),x\right\rangle\right|>2\nu\min_{\\_}{\left|x\right|=1}\left|\left\langle\nabla\Phi_{\\_}\infty(z),x\right\rangle\right|.$

We have $\mathcal{L}^{2}(T)=0$, since $\nabla\Theta_{\\_}\infty$ has full rank
a.e. by conformality (hence the same holds for $\Phi_{\\_}\infty$ by the chain
rule101010See e.g. [5, Lemma 4.12] and [8, Lemma III.6.4].) and since (6.4)
implies
$\nu|\partial_{\\_}1\Phi_{\\_}\infty\wedge\partial_{\\_}2\Phi_{\\_}\infty|(z)<\frac{1}{2}|\nabla\Phi_{\\_}\infty|^{2}(z)$
(as it can be immediately verified using a singular value decomposition for
$\nabla\Phi_{\\_}\infty(z)$).

###### Definition 6.3.

We now specify $K^{\prime\prime}:=2\nu$ and we set
$E^{\prime\prime}:=\pi\nu((K^{\prime\prime})^{2}+1)$. Finally, we choose $V>0$
such that $V=\left\lfloor V\right\rfloor+\frac{1}{2}$ and

(6.5)
$\displaystyle\left\|\mathbf{v}_{\\_}\infty\right\|(\overline{B}_{\\_}\ell^{q}(p))<V\pi\ell^{2}$

for all $\ell>0$ and all $p\in\mathcal{M}^{m}$. Such $V$ exists by the
monotonicity formula satisfied by the stationary varifold
$\mathbf{v}_{\\_}\infty$. Note that now also the constants $K_{\\_}0$,
$E_{\\_}0$, $s_{\\_}0$, $\eta_{\\_}0$, as well as $\varepsilon_{\\_}0$,
$\delta_{\\_}0$, $\varepsilon_{\\_}0^{\prime}$ and
$\varepsilon_{\\_}0^{\prime\prime}$, are determined.

###### Theorem 6.4 (multiplicity one).

We have $N_{\\_}\infty=1$ a.e., or equivalently $\nu=1$.

###### Proof of Theorem 6.4.

Let $\mathcal{B}_{\\_}k$ be the Borel set of points $z\in B_{\\_}{1/2}^{2}$
such that

$\displaystyle\sigma_{\\_}k^{2}\log(\sigma_{\\_}k^{-1})\int_{\\_}{B_{\\_}{r}^{2}(z)}\left|A\right|^{4}\,d\operatorname{vol}_{\\_}{g_{\\_}{\Phi_{\\_}k}}\geq\frac{\varepsilon_{\\_}0^{\prime\prime}}{E_{\\_}0}\int_{\\_}{B_{\\_}{r}^{2}(z)}\,d\operatorname{vol}_{\\_}{g_{\\_}{\Phi_{\\_}k}}$

for some radius $0<r<\frac{1}{2}$. By Besicovitch’s covering lemma, we get a
collection of points $z_{\\_}i\in\mathcal{B}_{\\_}k$ and radii
$0<r_{\\_}i<\frac{1}{2}$ such that

$\displaystyle\sigma_{\\_}k^{2}\log(\sigma_{\\_}k^{-1})\int_{\\_}{B_{\\_}{r_{\\_}i}^{2}(z_{\\_}i)}\left|A\right|^{4}\,d\operatorname{vol}_{\\_}{g_{\\_}{\Phi_{\\_}k}}\geq\frac{\varepsilon_{\\_}0^{\prime\prime}}{E_{\\_}0}\int_{\\_}{B_{\\_}{r_{\\_}i}^{2}(z_{\\_}i)}\,d\operatorname{vol}_{\\_}{g_{\\_}{\Phi_{\\_}k}},\qquad\bm{1}_{\\_}{\mathcal{B}_{\\_}k}\leq\sum_{\\_}i\bm{1}_{\\_}{B_{\\_}{r_{\\_}i}^{2}(z_{\\_}i)}\leq\mathfrak{N},$

for some universal constant $\mathfrak{N}$. Thus we get

$\displaystyle\nu_{\\_}k(\mathcal{B}_{\\_}k)$
$\displaystyle\leq\sum_{\\_}i\operatorname{vol}_{\\_}{g_{\\_}{\Phi_{\\_}k}}(B_{\\_}{r_{\\_}i}^{2}(z_{\\_}i))\leq\frac{E_{\\_}0}{\varepsilon_{\\_}0^{\prime\prime}}\sigma_{\\_}k^{2}\log(\sigma_{\\_}k^{-1})\sum_{\\_}i\int_{\\_}{B_{\\_}{r_{\\_}i}^{2}(z_{\\_}i)}\left|A\right|^{4}\,d\operatorname{vol}_{\\_}{g_{\\_}{\Phi_{\\_}k}}$
$\displaystyle\leq\frac{E_{\\_}0\mathfrak{N}}{\varepsilon_{\\_}0^{\prime\prime}}\sigma_{\\_}k^{2}\log(\sigma_{\\_}k^{-1})\int_{\\_}\Sigma\left|A\right|^{4}\,d\operatorname{vol}_{\\_}{g_{\\_}{\Phi_{\\_}k}}\to
0.$

Up to subsequences, we can assume that
$\overline{B_{\\_}{1/2}^{2}\setminus\mathcal{B}_{\\_}k}$ converges in the
Hausdorff topology to some compact set
$S\subseteq\overline{B}_{\\_}{1/2}^{2}$. We remark that
$\nu_{\\_}\infty(S)>0$: indeed, for any compact neighborhood $F$ of $S$ in
$B_{\\_}1^{2}$, we have $B_{\\_}{1/2}^{2}\setminus\mathcal{B}_{\\_}k\subseteq
F$ eventually and so

$\displaystyle\nu_{\\_}\infty(F)\geq\limsup_{\\_}{k\to\infty}\nu_{\\_}k(F)\geq\limsup_{\\_}{k\to\infty}(\nu_{\\_}k(B_{\\_}{1/2}^{2})-\nu_{\\_}k(\mathcal{B}_{\\_}k))=\limsup_{\\_}{k\to\infty}\nu_{\\_}k(B_{\\_}{1/2}^{2})\geq\nu_{\\_}\infty(B_{\\_}{1/2}^{2})>0.$

It follows from (6.3) that $\mathcal{L}^{2}(S)>0$.

We now show that $N_{\\_}\infty=1$ a.e. on $S\setminus T$, which has positive
Lebesgue measure. This will show that $\nu=1$, as desired. Fix any $z\in
S\setminus T$ and take a sequence $z_{\\_}k\in
B_{\\_}{1/2}^{2}\setminus\mathcal{B}_{\\_}k$ with $z_{\\_}k\to z$. Locally we
can find conformal reparametrizations $\widetilde{\Phi}_{\\_}k$ of
$\Phi_{\\_}k(z_{\\_}k+\cdot)$, by means of diffeomorphisms converging smoothly
to the identity on a small neighborhood of $0$.111111For instance, one can
isometrically identify a neighborhood of $z_{\\_}k$ in $(\Sigma,g_{\\_}k)$
with a neighborhood of $z$ in $(\Sigma,g_{\\_}\infty)$, by means of the
exponential map. By weak convergence
$\widetilde{\Phi}_{\\_}k\rightharpoonup\Phi_{\\_}\infty(z+\cdot)$ in
$W^{1,2}$, for a.e. radius $r>0$ we have

(6.6)
$\displaystyle\quad\widetilde{\Phi}_{\\_}k(r\cdot)\to\Phi_{\\_}\infty(z+r\cdot)\quad\text{in
}C^{0}(\partial B_{\\_}1^{2}\cup\partial B_{\\_}{s_{\\_}0}^{2}\cup\partial
B_{\\_}{s_{\\_}0^{2}}^{2})$

up to further subsequences.121212This can be obtained by applying [12, Lemma
A.5] to the weakly converging $\mathbb{R}^{3q}$-valued maps
$\displaystyle(\widetilde{\Phi}_{\\_}k,\widetilde{\Phi}_{\\_}k(s_{\\_}0\cdot),\widetilde{\Phi}_{\\_}k(s_{\\_}0^{2}\cdot))\rightharpoonup(\Phi_{\\_}\infty(z+\cdot),\Phi_{\\_}\infty(z+s_{\\_}0\cdot),\Phi_{\\_}\infty(z+s_{\\_}0^{2}\cdot)).$
Using [12, Lemma A.4] and the fact that $z\not\in T$, we can assume that $r$
satisfies

(6.7)
$\displaystyle\left|\Phi_{\\_}\infty(z+rx)-\Phi_{\\_}\infty(z)-\left\langle\nabla\Phi_{\\_}\infty(z),rx\right\rangle\right|<\delta_{\\_}0\ell\quad\text{for
}x\in\partial B_{\\_}1^{2}\cup\partial B_{\\_}{s_{\\_}0}^{2}\cup\partial
B_{\\_}{s_{\\_}0^{2}}^{2},$ (6.8)
$\displaystyle\frac{1}{2}\int_{\\_}{B_{\\_}{r}^{2}(z)}\left|\nabla\Phi_{\\_}\infty\right|^{2}<(\pi
r^{2})\left|\nabla\Phi_{\\_}\infty(z)\right|^{2}\leq\ell^{2}\pi((K^{\prime\prime})^{2}+1),$

with
$\ell:=r\min_{\\_}{\left|x\right|=1}\left|\left\langle\nabla\Phi_{\\_}\infty(0),x\right\rangle\right|$.
Setting $p:=\Phi_{\\_}\infty(z)$, note that (6.5) gives

$\displaystyle\left\|\mathbf{v}_{\\_}k\right\|(B_{\\_}\ell^{q}(p))<V\pi\ell^{2},\quad\left\|\mathbf{v}_{\\_}k\right\|(B_{\\_}{\eta_{\\_}0\ell}^{q}(p))<V\pi(\eta_{\\_}0\ell)^{2}$

eventually, which trivially implies

(6.9)
$\displaystyle\int_{\\_}{\widetilde{\Phi}_{\\_}k^{-1}(B_{\\_}\ell^{q}(p))}\,d\operatorname{vol}_{\\_}{g_{\\_}{\widetilde{\Phi}_{\\_}k}}<V\pi\ell^{2},\quad\int_{\\_}{\widetilde{\Phi}_{\\_}k^{-1}(B_{\\_}{\eta_{\\_}0\ell}^{q}(p))}\,d\operatorname{vol}_{\\_}{g_{\\_}{\widetilde{\Phi}_{\\_}k}}<V\pi(\eta_{\\_}0\ell)^{2}.$

Also, (6.3) and (6.8) give

$\displaystyle\lim_{\\_}{k\to\infty}\frac{1}{2}\int_{\\_}{B_{\\_}{r}^{2}(z)}\left|\nabla\widetilde{\Phi}_{\\_}k\right|^{2}=\lim_{\\_}{k\to\infty}\nu_{\\_}k(B_{\\_}r^{2}(z))\leq\frac{\nu}{2}\int_{\\_}{B_{\\_}r^{2}(z)}\left|\nabla\Phi_{\\_}\infty\right|^{2}<E^{\prime\prime}\ell^{2}.$

Thanks to the fact that $z_{\\_}k\not\in\mathcal{B}_{\\_}k$ and the above
inequalities, eventually $\widetilde{\Phi}_{\\_}k$ satisfies the hypotheses of
Theorem 6.1 on the ball $B_{\\_}r^{2}$, provided that $r$ (and thus $\ell$) is
chosen small enough. Setting
$\Psi_{\\_}k:=\ell^{-1}(\widetilde{\Phi}_{\\_}k-p)$, we infer that

(6.10) $\displaystyle
n(\Psi_{\\_}k,B_{\\_}{s_{\\_}0^{2}r}^{2},B_{\\_}{\eta_{\\_}0}^{\Pi})=1,$

where $\Pi$ is the 2-plane spanned by $\nabla\Phi_{\\_}\infty(0)$.

Since $r$ can be chosen arbitrarily small (possibly changing the subsequence
guaranteeing (6.6)), the argument used in the proof of [13, Lemma III.10]
shows that $N_{\\_}\infty(z)=1$.

Alternatively, (6.10) gives

$\displaystyle\left|\frac{\left\|\Pi_{\\_}*\mathbf{v}_{\\_}k^{\prime}\right\|(B_{\\_}{\eta_{\\_}0}^{\Pi})}{\pi\eta_{\\_}0^{2}}-1\right|<\frac{1}{8},$

where the varifold $\mathbf{v}_{\\_}k^{\prime}$ is induced by
${\left.\kern-1.2pt\Psi_{\\_}k\vphantom{\big{|}}\right|_{B_{\\_}{s_{\\_}0^{2}r}^{2}}}$
and converges to the varifold $\mathbf{v}_{\\_}\infty^{\prime}$ induced by
$(\varphi_{\\_}\infty(B_{\\_}{s_{\\_}0^{2}r}^{2}(z)),\ell^{-1}(\Theta_{\\_}\infty-p),\nu)$.
Assuming without loss of generality that
$\nabla\Theta_{\\_}\infty(\varphi_{\\_}\infty(z))\neq 0$,
$\Pi\circ\Theta_{\\_}\infty$ is a diffeomorphism from
$\varphi_{\\_}\infty(B_{\\_}{s_{\\_}0^{2}r}^{2}(z))$ onto its image (for $r$
small enough). Hence, at a.e. point of $\Pi$ the varifold
$\Pi_{\\_}*\mathbf{v}_{\\_}\infty$ has density either $0$ or $\nu$. Since
$\Pi\circ\Phi_{\\_}\infty(B_{\\_}{s_{\\_}0^{2}r}^{2}(z))$ is a superset of
$B_{\\_}{\eta_{\\_}0\ell}^{\Pi}(\Pi(p))$ (by Lemma A.1), it follows that

$\displaystyle\frac{\left\|\Pi_{\\_}*\mathbf{v}_{\\_}\infty^{\prime}\right\|(B_{\\_}{\eta_{\\_}0}^{\Pi})}{\pi\eta_{\\_}0^{2}}=\nu.$

The convergence
$\frac{\left\|\Pi_{\\_}*\mathbf{v}_{\\_}k^{\prime}\right\|(B_{\\_}{\eta_{\\_}0}^{\Pi})}{\pi\eta_{\\_}0^{2}}\to\frac{\left\|\Pi_{\\_}*\mathbf{v}_{\\_}\infty^{\prime}\right\|(B_{\\_}{\eta_{\\_}0}^{\Pi})}{\pi\eta_{\\_}0^{2}}$
thus gives $\left|\nu-1\right|\leq\frac{1}{8}$, and again we conclude that
$\nu=1$. ∎

## Appendix.

###### Lemma A.1 (big image under a boundary constraint).

Assume that $F\in C^{0}(\overline{B}_{\\_}1^{2},\mathbb{R}^{2})$ satisfies

(A.1) $\displaystyle\left|F(x)-\varphi(x)\right|\leq\delta\qquad\text{for all
}x\in\partial B_{\\_}1^{2}$

for some $0<\delta<1$ and some homeomorphism
$\varphi:\mathbb{R}^{2}\to\mathbb{R}^{2}$, with $\varphi(0)=0$ and
$\min_{\\_}{\left|x\right|=1}\left|\varphi(x)\right|\geq 1$. Then

(A.2) $\displaystyle F(B_{\\_}1^{2})\supseteq B_{\\_}{1-\delta}^{2}.$

###### Proof of Lemma A.1.

It suffices to show that, for a fixed $y\in B_{\\_}{1-\delta}^{2}$, the closed
curve $\Gamma^{\prime}:={\left.\kern-1.2ptF\vphantom{\big{|}}\right|_{\partial
B_{\\_}1^{2}}}$ is not contractible in
$\mathbb{R}^{2}\setminus\left\\{y\right\\}$: once this is done, if we had
$y\not\in F(B_{\\_}1^{2})$, i.e. $y\not\in F(\overline{B}_{\\_}1^{2})$, then
$F$ would provide a homotopy from $\Gamma^{\prime}$ to the constant curve
$F(0)$ in $\mathbb{R}^{2}\setminus\left\\{y\right\\}$, yielding a
contradiction.

Letting $\Gamma:={\left.\kern-1.2pt\varphi\vphantom{\big{|}}\right|_{\partial
B_{\\_}1^{2}}}$ and $\gamma:=\Gamma^{\prime}-\Gamma$, we have
$\left|\gamma(x)\right|\leq\delta$ for all $x\in\partial B_{\\_}1^{2}$. Hence,
$\Gamma$ is homotopic to $\Gamma^{\prime}$ in $\mathbb{R}^{2}\setminus
B_{\\_}{1-\delta}^{2}\subseteq\mathbb{R}^{2}\setminus\left\\{y\right\\}$ by
means of the homotopy

$\displaystyle\Gamma+t\gamma,\qquad 0\leq t\leq 1.$

So we are left to show that $\Gamma$ is not contractible in
$\mathbb{R}^{2}\setminus\left\\{y\right\\}$, i.e. that $\Gamma-y$ is not
contractible in $\mathbb{R}^{2}\setminus\left\\{0\right\\}$. The curve
$\Gamma-y$ is homotopic to $\Gamma$ in
$\mathbb{R}^{2}\setminus\left\\{0\right\\}$, by means of the homotopy

$\displaystyle\Gamma-ty,\qquad 0\leq t\leq 1,$

which avoids the origin since $\left|y\right|<1$. Finally, $\Gamma$ is not
contractible in $\mathbb{R}^{2}\setminus\left\\{0\right\\}$, since $\varphi$
(once restricted to a homeomorphism of
$\mathbb{R}^{2}\setminus\left\\{0\right\\}$) induces an automorphism of
$\pi_{\\_}1(\mathbb{R}^{2}\setminus\left\\{0\right\\})$ sending the class of
the generator $\text{id}_{\\_}{\partial B_{\\_}1^{2}}$ to the class of
$\Gamma$. Hence, $\Gamma-y$ is not contractible in
$\mathbb{R}^{2}\setminus\left\\{0\right\\}$, too, as desired. ∎

###### Lemma A.2 (elliptic $\bm{W^{2,4}}$-estimate).

For a function $\Psi\in C^{\infty}(\overline{B}_{\\_}1)$ and a $0<\tau<1$ we
have

$\displaystyle\left\|\Psi\right\|_{\\_}{W^{2,4}(B_{\\_}{\tau}^{2})}\leq
C(\tau)(\left\|\Delta\Psi\right\|_{\\_}{L^{4}(B_{\\_}1^{2})}+\left\|\nabla\Psi\right\|_{\\_}{L^{2}(B_{\\_}1^{2})}+\left\|\Psi\right\|_{\\_}{L^{2}(B_{\\_}1^{2})}).$

###### Proof of Lemma A.2.

Given two radii $0<r<s\leq 1$, let us choose a cut-off function $\rho\in
C^{\infty}_{\\_}c(B_{\\_}s^{2})$ with $\rho=1$ on $B_{\\_}r^{2}$. Since
$\rho\Psi\in C^{\infty}_{\\_}c(\mathbb{R}^{2})$, standard Calderón–Zygmund
estimates give

(A.3) $\displaystyle\begin{aligned}
\left\|\nabla^{2}\Psi\right\|_{\\_}{L^{p}(B_{\\_}r^{2})}&\leq\left\|\nabla^{2}(\rho\Psi)\right\|_{\\_}{L^{p}(\mathbb{R}^{2})}\leq
C(p)\left\|\Delta(\rho\Psi)\right\|_{\\_}{L^{p}(\mathbb{R}^{2})}\\\ &\leq
C(p,r,s)(\left\|\Delta\Psi\right\|_{\\_}{L^{p}(B_{\\_}s^{2})}+\left\|\nabla\Psi\right\|_{\\_}{L^{p}(B_{\\_}s^{2})}+\left\|\Psi\right\|_{\\_}{L^{p}(B_{\\_}s^{2})})\end{aligned}$

for all $1<p<\infty$. Setting $t:=\frac{1+\tau}{2}$ and applying (A.3) with
$p:=2$, $r:=t$ and $s:=1$ we get

$\displaystyle\left\|\nabla^{2}\Psi\right\|_{\\_}{L^{2}(B_{\\_}t^{2})}\leq
C(\tau)(\left\|\Delta\Psi\right\|_{\\_}{L^{2}(B_{\\_}1^{2})}+\left\|\nabla\Psi\right\|_{\\_}{L^{2}(B_{\\_}1^{2})}+\left\|\Psi\right\|_{\\_}{L^{2}(B_{\\_}1^{2})}),$

hence $\left\|\Psi\right\|_{\\_}{W^{2,2}(B_{\\_}t^{2})}$ is bounded by the
desired quantity. Using Sobolev’s embedding
$W^{2,2}(B_{\\_}t^{2})\hookrightarrow W^{1,4}(B_{\\_}t^{2})$ and (A.3) with
$p:=4$, $r:=\tau$ and $s:=t$, we obtain

$\displaystyle\left\|\Psi\right\|_{\\_}{W^{2,4}(B_{\\_}{\tau}^{2})}$
$\displaystyle\leq
C(\tau)(\left\|\Delta\Psi\right\|_{\\_}{L^{4}(B_{\\_}{t}^{2})}+\left\|\Psi\right\|_{\\_}{W^{2,2}(B_{\\_}{t}^{2})})$
$\displaystyle\leq
C(\tau)(\left\|\Delta\Psi\right\|_{\\_}{L^{4}(B_{\\_}1^{2})}+\left\|\nabla\Psi\right\|_{\\_}{L^{2}(B_{\\_}1^{2})}+\left\|\Psi\right\|_{\\_}{L^{2}(B_{\\_}1^{2})}).\qed$

###### Lemma A.3 (compactness of normal solutions to Beltrami equation).

Given a sequence $\psi_{\\_}k:\mathbb{C}\to\mathbb{C}$ of $K$-quasiconformal
homeomorphisms with the normalization conditions

$\displaystyle\psi_{\\_}k(0)=0,\qquad\psi_{\\_}k(1)=1,$

there exists a $K$-quasiconformal homeomorphism
$\psi_{\\_}\infty:\mathbb{C}\to\mathbb{C}$ satisfying the same normalization
condition and such that, up to subsequences, $\psi_{\\_}k\to\psi_{\\_}\infty$
and $\psi_{\\_}k^{-1}\to\psi_{\\_}\infty^{-1}$ in
$C^{0}_{\\_}{loc}(\mathbb{C})$.

###### Proof of Lemma A.3.

Let $\mu_{\\_}k\in\mathcal{E}_{\\_}K$ be defined by
$\partial_{\\_}{\overline{z}}\psi_{\\_}k=\mu_{\\_}k\partial_{\\_}{z}\psi_{\\_}k$.131313Actually,
the coefficient $\mu_{\\_}k$ is uniquely determined a.e., as
$\partial_{\\_}z\psi_{\\_}k\neq 0$ a.e. (this follows from
$\operatorname{id}_{\\_}\mathbb{C}=\psi_{\\_}k^{-1}\circ\psi_{\\_}k$ and the
chain rule [8, Lemma III.6.4], together with
$\left|\partial_{\\_}{\overline{z}}\psi_{\\_}k\right|\leq\left|\partial_{\\_}z\psi_{\\_}k\right|$).
Existence and uniqueness of a $K$-quasiconformal homeomorphism satisfying this
equation and the normalization conditions is shown in [5, Theorem 4.30].

Given $M>0$, we consider the set
$\mathcal{E}_{\\_}K^{M}:=\left\\{\mu\in\mathcal{E}_{\\_}K:\mu=0\text{ a.e. on
}\mathbb{C}\setminus B_{\\_}M^{2}\right\\}$. If $F^{\mu}$ denotes the normal
solution to the equation
$\partial_{\\_}{\overline{z}}F^{\mu}=\mu\partial_{\\_}zF^{\mu}$ (in the sense
of [5, Theorem 4.24]), then $F^{\mu}$ satisfies estimates (4.21) and (4.24) in
[5]. Applying them with the points $0$ and $1$, we infer that also the map
$f^{\mu}:=F^{\mu}(1)^{-1}F^{\mu}$ satisfies estimates of the form

(A.4) $\displaystyle\left|f^{\mu}(z_{\\_}1)-f^{\mu}(z_{\\_}2)\right|\leq
C\left|z_{\\_}1-z_{\\_}2\right|^{\alpha}+C\left|z_{\\_}1-z_{\\_}2\right|,$
(A.5) $\displaystyle\left|z_{\\_}1-z_{\\_}2\right|\leq
C\left|f^{\mu}(z_{\\_}1)-f^{\mu}(z_{\\_}2)\right|^{\alpha}+C\left|f^{\mu}(z_{\\_}1)-f^{\mu}(z_{\\_}2)\right|,$

with $C$ and $0<\alpha<1$ depending only on $K$ and $M$. Given a sequence of
homeomorphisms $f_{\\_}k:\mathbb{C}\to\mathbb{C}$ satisfying these estimates,
Ascoli–Arzelà theorem applies to $f_{\\_}k$ and $f_{\\_}k^{-1}$ and so we can
extract a subsequence (not relabeled) such that

$\displaystyle f_{\\_}k\to f_{\\_}\infty,\quad
f_{\\_}k^{-1}\to\widetilde{f}_{\\_}\infty\quad\text{in
}C^{0}_{\\_}{loc}(\mathbb{C}).$

From $f_{\\_}k^{-1}\circ f_{\\_}k=f_{\\_}k\circ
f_{\\_}k^{-1}=\operatorname{id}_{\\_}{\mathbb{C}}$ we get
$\widetilde{f}_{\\_}\infty\circ
f_{\\_}\infty=f_{\\_}\infty\circ\widetilde{f}_{\\_}\infty=\operatorname{id}_{\\_}{\mathbb{C}}$
and thus $f_{\\_}\infty:\mathbb{C}\to\mathbb{C}$ is a homeomorphism, with
$\widetilde{f}_{\\_}\infty=f_{\\_}\infty^{-1}$. Also, since
$f_{\\_}k(z),f_{\\_}k^{-1}(z)\to\infty$ uniformly as $z\to\infty$, we deduce
that the canonical extensions
$\widehat{f}_{\\_}k:\widehat{\mathbb{C}}\to\widehat{\mathbb{C}}$ converge
uniformly to $\widehat{f}_{\\_}\infty$ and that the same holds for
$\widehat{f}_{\\_}k^{-1}$.

We now closely examine the proof of [5, Theorem 4.30]: let
$\widetilde{\mu}_{\\_}k\in\mathcal{E}_{\\_}K^{1}$ be given by equation (4.25)
in [5], with $\mu_{\\_}k\bm{1}_{\\_}{\mathbb{C}\setminus B_{\\_}1^{2}}$ in
place of $\mu$, and

$\displaystyle g_{\\_}k:\widehat{\mathbb{C}}\to\widehat{\mathbb{C}},\qquad
g_{\\_}k(z):=\widehat{f^{\widetilde{\mu}_{\\_}k}}(z^{-1})^{-1}.$

This map corresponds to the map $f^{\mu_{\\_}1}$ in the aforementioned proof
(with $\mu_{\\_}k$ in place of $\mu$). The lower bound (A.5), applied with
$f^{\widetilde{\mu}_{\\_}k}$ and
$z_{\\_}1:=f^{\widetilde{\mu}_{\\_}k}(z^{-1})$, $z_{\\_}2:=0$, shows that
$\left|g_{\\_}k(z)\right|$ is bounded above by some $M$, for all $k$ and all
$z\in\overline{B}_{\\_}1^{2}$. Hence, defining $\mu_{\\_}{k,2}$ as in equation
(4.27) in [5] (with $\mu_{\\_}k$ in place of $\mu$), we get
$\mu_{\\_}{k,2}\in\mathcal{E}_{\\_}{K}^{M}$. Calling
$h_{\\_}k:\widehat{\mathbb{C}}\to\widehat{\mathbb{C}}$ the associated
quasiconformal homeomorphism, normalized so that $h_{\\_}k(0)=0$ and
$h_{\\_}k(1)=1$, by the above argument we obtain the uniform convergence

$\displaystyle g_{\\_}k\to g_{\\_}\infty,\quad g_{\\_}k^{-1}\to
g_{\\_}\infty^{-1},\quad h_{\\_}k\to h_{\\_}\infty,\quad h_{\\_}k^{-1}\to
h_{\\_}\infty^{-1}$

up to subsequences, for suitable homeomorphisms $g_{\\_}\infty$ and
$h_{\\_}\infty$ of the Riemann sphere $\widehat{\mathbb{C}}$. Setting
$\psi_{\\_}\infty:={\left.\kern-1.2pth_{\\_}\infty\circ
g_{\\_}\infty\vphantom{\big{|}}\right|_{\mathbb{C}}}$ and observing that
$\psi_{\\_}k={\left.\kern-1.2pth_{\\_}k\circ
g_{\\_}k\vphantom{\big{|}}\right|_{\mathbb{C}}}$, we get the desired
convergence $\psi_{\\_}k\to\psi_{\\_}\infty$ and
$\psi_{\\_}k^{-1}\to\psi_{\\_}\infty^{-1}$ in $C^{0}_{\\_}{loc}(\mathbb{C})$.

Finally, we show that $\psi_{\\_}\infty$ is a $K$-quasiconformal
homeomorphism. Given an open rectangle $R\subset\subset\mathbb{C}$, [5, Lemma
4.12] gives

$\displaystyle\mathcal{L}^{2}(\psi_{\\_}k(R))=\int_{\\_}R(\left|\partial_{\\_}z\psi_{\\_}k\right|^{2}-\left|\partial_{\\_}{\overline{z}}\psi_{\\_}k\right|^{2})\geq\int_{\\_}R(1-k^{2})\left|\partial_{\\_}z\psi_{\\_}k\right|^{2}\geq(1-k^{2})k^{-2}\int_{\\_}R\left|\partial_{\\_}{\overline{z}}\psi_{\\_}k\right|^{2},$

where $k:=\frac{K-1}{K+1}$. Since
$\mathcal{L}^{2}(\psi_{\\_}k(R))\to\mathcal{L}^{2}(\psi_{\\_}\infty(R))$, we
deduce that $\psi_{\\_}k$ is bounded in $W^{1,2}(R)$, thus $\psi_{\\_}\infty$
is the limit of $\psi_{\\_}k$ in the weak
$W^{1,2}_{\\_}{loc}(\mathbb{C})$-topology. Given $\rho,\psi^{1},\psi^{2}\in
C^{\infty}_{\\_}c(\mathbb{C})$, integration by parts shows that

(A.6)
$\displaystyle\int_{\\_}\mathbb{C}\rho(\partial_{\\_}1\psi^{1}\partial_{\\_}2\psi^{2}-\partial_{\\_}2\psi^{1}\partial_{\\_}1\psi^{2})=-\int_{\\_}\mathbb{C}(\partial_{\\_}1\rho\psi^{1}\partial_{\\_}2\psi^{2}-\partial_{\\_}2\rho\psi^{1}\partial_{\\_}1\psi^{2}).$

Writing $\psi_{\\_}k=\varphi_{\\_}k^{1}+i\psi_{\\_}k^{2}$, a standard density
argument shows that (A.6) still holds with $\psi^{1},\psi^{2}$ replaced by
$\psi_{\\_}k^{1},\psi_{\\_}k^{2}$, for
$k\in\mathbb{N}\cup\left\\{\infty\right\\}$. Hence, observing that
$\left|\partial_{\\_}z\psi_{\\_}k\right|^{2}-\left|\partial_{\\_}{\overline{z}}\psi_{\\_}k\right|^{2}=(\partial_{\\_}1\psi_{\\_}k^{1}\partial_{\\_}2\psi_{\\_}k^{2}-\partial_{\\_}2\psi_{\\_}k^{1}\partial_{\\_}1\psi_{\\_}k^{2})$,
we get

(A.7)
$\displaystyle\int_{\\_}\mathbb{C}\rho(\left|\partial_{\\_}z\psi_{\\_}k\right|^{2}-\left|\partial_{\\_}{\overline{z}}\psi_{\\_}k\right|^{2})\to\int_{\\_}\mathbb{C}\rho(\left|\partial_{\\_}z\psi_{\\_}\infty\right|^{2}-\left|\partial_{\\_}{\overline{z}}\psi_{\\_}\infty\right|^{2}).$

Defining the positive measures
$\nu_{\\_}k:=(\left|\partial_{\\_}z\psi_{\\_}k\right|^{2}-\left|\partial_{\\_}{\overline{z}}\psi_{\\_}k\right|^{2})\mathcal{L}^{2}$,
up to further subsequences we can assume that
$\nu_{\\_}k\stackrel{{\scriptstyle*}}{{\rightharpoonup}}\nu_{\\_}\infty$ as
Radon measures. For any rectangle $R$ such that $\nu_{\\_}\infty(\partial
R)=0$, approximating $\bm{1}_{\\_}R$ from above and below with smooth
functions and applying (A.7) we get

$\displaystyle\int_{\\_}R(\left|\partial_{\\_}z\psi_{\\_}k\right|^{2}-\left|\partial_{\\_}{\overline{z}}\psi_{\\_}k\right|^{2})\to\int_{\\_}R(\left|\partial_{\\_}z\psi_{\\_}\infty\right|^{2}-\left|\partial_{\\_}{\overline{z}}\psi_{\\_}\infty\right|^{2}).$

By monotonicity of both sides, this actually holds for every rectangle $R$. On
the other hand, by lower semicontinuity of the $L^{2}$-norm,

$\displaystyle\int_{\\_}R(1-k^{2})\left|\partial_{\\_}z\psi_{\\_}\infty\right|^{2}$
$\displaystyle\leq\liminf_{\\_}{k\to\infty}\int_{\\_}R(1-k^{2})\left|\partial_{\\_}z\psi_{\\_}k\right|^{2}\leq\lim_{\\_}{k\to\infty}\int_{\\_}R(\left|\partial_{\\_}z\psi_{\\_}k\right|^{2}-\left|\partial_{\\_}{\overline{z}}\psi_{\\_}k\right|^{2})$
$\displaystyle=\int_{\\_}R(\left|\partial_{\\_}z\psi_{\\_}\infty\right|^{2}-\left|\partial_{\\_}{\overline{z}}\psi_{\\_}\infty\right|^{2}).$

Since $R$ is arbitrary, we get
$\left|\partial_{\\_}{\overline{z}}\psi_{\\_}\infty\right|\leq
k\left|\partial_{\\_}z\psi_{\\_}\infty\right|$ a.e., as desired. ∎

###### Corollary A.4 (compactness of $\bm{\mathcal{D}_{\\_}K}$).

Given a sequence $\varphi_{\\_}k\in\mathcal{D}_{\\_}K$, there exists
$\varphi_{\\_}\infty\in\mathcal{D}_{\\_}K$ such that, up to subsequences,
$\varphi_{\\_}k\to\varphi_{\\_}\infty$ and
$\varphi_{\\_}k^{-1}\to\varphi_{\\_}\infty^{-1}$ in
$C^{0}_{\\_}{loc}(\mathbb{C})$.

###### Proof of Corollary A.4.

Let $\mu_{\\_}k\in\mathcal{E}_{\\_}K$ be defined by