# The Yang-Mills-Higgs functional on complex line bundles:
asymptotics for critical points

Giacomo Canevari, Federico Luigi Dipasquale and Giandomenico Orlandi

###### Abstract

We consider a gauge-invariant Ginzburg-Landau functional (also known as
Abelian Yang-Mills-Higgs model), on Hermitian line bundles over closed
Riemannian manifolds of dimension $n\geq 3$. Assuming a logarithmic energy
bound in the coupling parameter, we study the asymptotic behaviour of critical
points in the London limit. After a convenient choice of the gauge, we show
compactness of finite-energy critical points in Sobolev norms. Moreover,
thanks to a suitable monotonicity formula, we prove that the energy densities
of critical points, rescaled by the logarithm of the coupling parameter,
converge to the weight measure of a stationary, rectifiable varifold of
codimension 2.

## Introduction

Let $(M,\,g)$ be a smooth, compact, connected, oriented Riemannian manifold
without boundary, of dimension $n\geq 3$. Let $E\to M$ be a Hermitian line
bundle over $M$, equipped with a (smooth) reference metric connection
$\operatorname{D}_{0}$. For any $\varepsilon>0$, we consider the Ginzburg-
Landau-type functional

$G_{\varepsilon}(u,\,A):=\int_{M}\left(\frac{1}{2}\left|\operatorname{D}_{A}u\right|^{2}+\frac{1}{2}\left|F_{A}\right|^{2}+\frac{1}{4\varepsilon^{2}}\left(1-\left|u\right|^{2}\right)^{2}\right)\operatorname{vol}_{g}.$
(1)

Here $u\colon M\to E$ is a section of the bundle, $A$ is a real-valued
$1$-form on $M$, $\operatorname{D}_{A}:=\operatorname{D}_{0}-iA$ and $F_{A}$
is the curvature $2$-form of $\operatorname{D}_{A}$. We denote the integrand
of $G_{\varepsilon}$ as

$e_{\varepsilon}(u_{\varepsilon},\,A_{\varepsilon}):=\frac{1}{2}\left|\operatorname{D}_{A}u\right|^{2}+\frac{1}{2}\left|F_{A}\right|^{2}+\frac{1}{4\varepsilon^{2}}\left(1-\left|u\right|^{2}\right)^{2}\\!.$
(2)

The functional $G_{\varepsilon}$ is also known as the _Abelian Yang-Mills-
Higgs energy_. One prominent feature of $G_{\varepsilon}$ is _gauge-
invariance_ : for any $\Phi\in W^{2,2}(M,\,\mathbb{S}^{1})$ and any pair
$(u,\,A)\in W^{1,2}(M,\,E)\times W^{1,2}(M,\,\mathrm{T}^{*}M)$ with finite
energy, each term of the energy density $e_{\varepsilon}$ is invariant under
the _gauge transformation_

$(u,\,A)\mapsto\left(\Phi u,\,\Phi\cdot A\right):=\left(\Phi
u,\,A-i\Phi^{-1}\operatorname{d}\Phi\right)\\!,$ (3)

where $\Phi u$ is defined by the fibre-wise action of
$\mathbb{S}^{1}\simeq\mathrm{U}(1)$ on $E$.

For any $u$ and $A$, we define the forms (see e.g. [31])

$j(u,\,A):=\left\langle\operatorname{D}_{A}u,\,iu\right\rangle,\qquad
J(u,\,A):=\frac{1}{2}\operatorname{d}j(u,\,A)+\frac{1}{2}F_{A},$ (4)

called, respectively, the _gauge-invariant pre-Jacobian_ and the _gauge-
invariant Jacobian_ of the pair $(u,\,A)$. If $(u,\,A)$ is a pair with finite
energy, both $j(u,\,A)$ and $J(u,\,A)$ are well-defined, both in the sense of
distributions and pointwise, and invariant under gauge transformation.

In the present paper, we consider _finite-energy critical points_ of
$G_{\varepsilon}$. More precisely, we notice that the functional
$G_{\varepsilon}$ is well-defined and finite for $(u,\,A)\in\mathcal{E}$,
where

$\mathcal{E}:=(W^{1,2}\cap L^{\infty})(M,\,E)\times
W^{1,2}(M,\,\mathrm{T}^{*}M)$

and critical points $(u_{\varepsilon},\,A_{\varepsilon})$ of $G_{\varepsilon}$
in $\mathcal{E}$ satisfy the Euler-Lagrange equations

$\displaystyle\operatorname{D}_{A}^{*}\operatorname{D}_{A}u_{\varepsilon}+\frac{1}{\varepsilon^{2}}\left(\left|u_{\varepsilon}\right|^{2}-1\right)u_{\varepsilon}$
$\displaystyle=0$ (5) $\displaystyle\operatorname{d}^{*}F_{\varepsilon}$
$\displaystyle=j(u_{\varepsilon},\,A_{\varepsilon})$ (6)

where $\operatorname{D}^{*}_{A}$ is the $L^{2}$-adjoint of
$\operatorname{D}_{A}$, $F_{\varepsilon}:=F_{A_{\varepsilon}}$ and
$\operatorname{d}^{*}$ denotes the codifferential. By taking the differential
in both sides of (6), we obtain the _London equation_ (at level $\varepsilon$)

$-\Delta
F_{\varepsilon}+F_{\varepsilon}=2J(u_{\varepsilon},\,A_{\varepsilon})$ (7)

Throughout the paper, we _assume_ that
$\\{(u_{\varepsilon},\,A_{\varepsilon})\\}\subset\mathcal{E}$ is a sequence of
critical points of $G_{\varepsilon}$ satisfying the logarithmic energy bound

$G_{\varepsilon}(u_{\varepsilon},\,A_{\varepsilon})\leq\Lambda\left|\log\varepsilon\right|$
(8)

for some constant $\Lambda>0$ that does not depend on $\varepsilon$. This
assumption is natural, as the energy of minimisers of $G_{\varepsilon}$ is
precisely of order $\left|\log\varepsilon\right|$, whenever the bundle $E\to
M$ is non-trivial [13, Theorem A and Remark 2]. Under this assumption, we have
this (preliminary) convergence result:

######  Theorem 0 ([13]).

Let $\\{(u_{\varepsilon},\,A_{\varepsilon})\\}\subset\mathcal{E}$ be a
sequence of critical points of $G_{\varepsilon}$ satisfying the logarithmic
energy bound (8). Then, there exist a (non-relabelled) subsequence, $2$-forms
$J_{*}$, $F_{*}$ and a $1$-form $j_{*}$ such that the following properties
hold:

  1. (i)

$J(u_{\varepsilon},\,A_{\varepsilon})\to J_{*}$ strongly in $W^{-1,p}(M,\
\Lambda^{2}\mathrm{T}^{*}M)$ for any $p\in\left[1,\,\frac{n}{n-1}\right)$;

  2. (ii)

$F_{A_{\varepsilon}}\to F_{*}$ strongly in $W^{1,p}(M,\
\Lambda^{2}\mathrm{T}^{*}M)$ for any $p\in\left[1,\,\frac{n}{n-1}\right)$;

  3. (iii)

$j(u_{\varepsilon},\,A_{\varepsilon})\to j_{*}$ strongly in $L^{p}(M,\
\Lambda^{2}\mathrm{T}^{*}M)$ for any $p\in\left[1,\,\frac{n}{n-1}\right)$;

  4. (iv)

the Hodge dual $\star J_{*}$ is a boundary-less, integer-multiplicity
rectifiable $(n-2)$-current in $M$, whose integral homology class
$\mathcal{C}\subseteq H_{n-2}(M;\,\mathbb{Z})$ is Poincaré dual to the first
Chern class of $E\to M$;

  5. (v)

the limit curvature $F_{*}$ satisfies

$\operatorname{d}^{*}F_{*}=j_{*},\qquad-\Delta F_{*}+F_{*}=2\pi J_{*},$ (9)

in the sense of distributions.

The proof of Theorem  ‣ Theorem 0 ([13]). is contained in [13] (see, in
particular, Theorem A and Remark 3.12). Property (iii) is not mentioned
explicitly in [13], but it is an immediate consequence of (6) and (ii). Note
that Property (i) is completely independent of the Euler-Lagrange equations
(5)–(6); it holds true for any sequence
$\\{(u_{\varepsilon},\,A_{\varepsilon})\\}$ that satisfies (8).

#### Main results.

In this paper, we will address two more questions about the asymptotic
behaviour, as $\varepsilon\to 0$, of critical points satisfying (8). Our first
main result describes the concentration of the _rescaled energy densities_. We
denote by $\operatorname{spt}\mu$ the support of a Radon measure $\mu$ and by
$\mathcal{H}^{n-2}$ the $(n-2)$-dimensional Hausdorff measure.

######  Theorem 1.

Let $\\{(u_{\varepsilon},\,A_{\varepsilon})\\}\subset\mathcal{E}$ be any
sequence of critical points of $G_{\varepsilon}$ satisfying the logarithmic
energy bound (8). Then, up to extraction of a subsequence, we have

$\frac{e_{\varepsilon}(u_{\varepsilon},\,A_{\varepsilon})}{\left|\log\varepsilon\right|}\,\operatorname{vol}_{g}\rightharpoonup^{*}\mu_{*}\qquad\textrm{as
}\varepsilon\to 0$

in the sense of Radon measures in $M$. The limit measure $\mu_{*}$ satisfies
$\mathcal{H}^{n-2}(\operatorname{spt}\mu_{*})<+\infty$ and, moreover, it is
the weight measure of a stationary, rectifiable $(n-2)$-varifold in $M$.

Recall that _varifolds_ are measure-theoretic generalisations of smooth
submanifolds (see e.g. [34] and Section 5 below for details). In particular,
stationary rectifiable varifolds are generalisations of minimal surfaces.
Constructing non-trivial stationary, rectifiable varifolds in Riemannian
manifolds is a fundamental problem in Geometric Measure Theory [3, 29].

######  Remark 1.

By $\Gamma$-convergence [13, Theorem A and Remark 3.13], the limit measure
$\mu_{*}$ satisfies $\mu_{*}\geq\pi\left|J_{*}\right|$ where $J_{*}$ is the
limit Jacobian, given by Theorem  ‣ Theorem 0 ([13]).. In particular,
$\mu_{*}$ it is always nonzero, so long as the bundle $E\to M$ is nontrivial.
Moreover, if $\\{(u_{\varepsilon},\,A_{\varepsilon})\\}$ is a sequence of
minimisers, then $\mu_{*}=\pi\left|J_{*}\right|$ and the Hodge dual $\star
J_{*}$ is area-minimising in its integral homology class $\mathcal{C}$ [13,
Corollary B]. However, we do not know whether the inequality
$\mu_{*}\geq\pi\left|J_{*}\right|$ may be strict, in general. A related
question, which we do not address here, is whether or not $\mu_{*}$ is an
integral varifold (see the paragraph on open questions below).

######  Remark 2.

A more detailed description of $\mu_{*}$ can be found in Theorem 5.3 in
Section 5, of which Theorem 1 is an abridged version. Contrary to what is
observed in related problems (e.g. [9, 37]), the measure $\mu_{*}$ contains no
diffuse part; see Remark 5.4.

The proof of Theorem 1 relies on a suitable monotonicity formula (see Theorem
3 below).

In our second main result, we prove compactness for the sequence
$\\{(u_{\varepsilon},\,A_{\varepsilon})\\}$ in $W^{1,p}(M,\,E)\times
W^{2,p}(M,\,\mathrm{T}^{*}M)$, for any $p$ with $1\leq p<\frac{n}{n-1}$, so
long as $A_{\varepsilon}$ is in Coulomb gauge, that is,

$\begin{split}A_{\varepsilon}=\operatorname{d}^{*}\psi_{\varepsilon}+\zeta_{\varepsilon}\end{split}$
(10)

where $\psi_{\varepsilon}$ is an exact $2$-form, $\zeta_{\varepsilon}$ is a
harmonic $1$-form and $\left\|\zeta_{\varepsilon}\right\|_{L^{\infty}(M)}\leq
C_{M}$ for some constant $C_{M}$ that depends on $M$ only (see e.g. [13, Lemma
2.10]).

######  Theorem 2.

Let $\\{(u_{\varepsilon},\,A_{\varepsilon})\\}\subset\mathcal{E}$ be a
sequence of critical points of $G_{\varepsilon}$. Assume that
$\\{(u_{\varepsilon},\,A_{\varepsilon})\\}$ satisfies the logarithmic energy
bound (8) and that each $A_{\varepsilon}$ satisfies (10). Then, up to
extraction of a (non-relabelled) subsequence, there holds

$u_{\varepsilon}\to u_{*}\quad\textrm{strongly in }W^{1,p}(M,\,E),\qquad
A_{\varepsilon}\to A_{*}\quad\textrm{strongly in
}W^{2,p}(M,\,\mathrm{T}^{*}M)$

as $\varepsilon\to 0$, for any $p$ with $1\leq p<\frac{n}{n-1}$. Moreover,
$u_{*}$ and $A_{*}$ are smooth in $M\setminus\operatorname{spt}\mu_{*}$, they
satisfy $\left|u_{*}\right|=1$ in $M\setminus\operatorname{spt}\mu_{*}$ and

$j(u_{*},\,A_{*})=\operatorname{d}^{*}F_{A_{*}}$ (11)

in the sense of distributions in $M$.

######  Remark 3.

By continuity of the Jacobian and the differential operator, the limits
$u_{*}$ and $A_{*}$ are compatible with $J_{*}$, $F_{*}$, $j_{*}$ given by
Theorem  ‣ Theorem 0 ([13]). — that is, $F_{*}=F_{A_{*}}$,
$J_{*}=J(u_{*},\,A_{*})$, and $j_{*}=j(u_{*},\,A_{*})$. Moreover, Equation
(11) implies that $u_{*}$ is an $A_{*}$-harmonic unit section away from the
support of $\mu_{*}$, where both $u_{*}$ and $A_{*}$ are smooth. In other
words, there holds

$\operatorname{D}_{A_{*}}^{*}\operatorname{D}_{A_{*}}u_{*}=\left|\operatorname{D}_{A_{*}}u_{*}\right|^{2}u_{*}\qquad\textrm{pointwise
in }M\setminus\operatorname{spt}\mu_{*}$ (12)

(see Remark 2.3 for details). Finally, by passing to the limit in (10), it
follows that $A_{*}=\operatorname{d}^{*}\psi_{*}+\zeta_{*}$, where $\psi_{*}$
is an exact $2$-form and $\zeta_{*}$ is a harmonic $1$-form.

######  Remark 4.

Theorem 2 applies, in particular, to any sequence of _minimisers_ of
$G_{\varepsilon}$ in the class $W^{1,2}(M,\,E)\times
W^{1,2}(M,\,\mathrm{T}^{*}M)$. Indeed, by [13, Lemma 2.1], minimisers belong
to $\mathcal{E}$ and, by [13, Remark 2], they satisfy the assumption (8).

Before illustrating the ideas of the proof, let us motivate the interest
towards our results.

#### Background and motivation.

Ginzburg-Landau functionals of the form (1) were originally proposed as a
model of superconductivity. This theory gained much popularity, as it accounts
for most commonly observed effects in superconductors, and it gradually became
relevant to other areas of physics, such as particle physics and gauge
theories. The asymptotic regime as $\varepsilon\to 0$, which is known as the
_London limit_ in superconductivity theory or the _strongly repulsive limit_
in particle physics, is characterised by the emergence of _topological
singularities_ : due to topological obstructions set by the structure of the
bundle $E\to M$, critical points develop singularities, in the asymptotic
limit, which are supported on a set of dimension $n-2$. The topologically-
driven energy concentration phenomenon can be detected by the distributional
Jacobian, which both enjoys remarkable compactness properties and identifies
the topological singularities that emerge in the limit as $\varepsilon\to 0$
(see e.g. [22, 1, 2]).

The asymptotic analysis of (1) in the limit as $\varepsilon\to 0$ relies on
analogous results for the “non-magnetic” version of the functional, in which
the variable $A$ is set to zero and the energy is considered as a functional
of $u$ only. The analysis of critical points of Ginzburg-Landau-type
functionals without magnetic field was initiated by Bethuel, Brezis, and
Hélein [7] in the 2-dimensional, Euclidean setting. The analysis was then
extended to critical points on $2$-dimensional Riemann surfaces [5], higher-
dimensional Euclidean domains [23, 8] and, more recently, higher-dimensional
manifolds [14, 37, 15, 16]. In particular, our Theorems 2 and 1 generalise
results that were first obtained in [8] in the Euclidean, non-magnetic case.
In all these works, a main difficulty lies in the lack of uniform energy
bounds, as the natural energy scaling considered is the logarithmic one (8).

As for the gauge-invariant, “magnetic” version of the functional, the first
results addressed the asymptotic behaviour of minimisers in two-dimensional
Euclidean domains [10] and on Riemann surfaces [26, 30]. A detailed analysis
of the asymptotics of critical points for the gauge-invariant functional in 2
dimensions, in the Euclidean setting and in the logarithmic energy regime, has
been carried out by Sandier and Serfaty — see e.g. [32, Chapter 13] and the
references therein. Apparently, less effort has been put in studying the
asymptotic behaviour of critical points in higher dimensions. Recently, this
problem has been addressed in [27], in the context of Hermitian line bundles,
for the _self-dual_ version $G_{\varepsilon}^{\rm self}$ of $G_{\varepsilon}$,
in which the curvature term
$\int_{M}\left|F_{\varepsilon}\right|^{2}\,\operatorname{vol}_{g}$ is replaced
by
$\varepsilon^{2}\int_{M}\left|F_{\varepsilon}\right|^{2}\,\operatorname{vol}_{g}$.
The main outcome of [27] is that, for a sequence
$\\{(u_{\varepsilon},\,A_{\varepsilon})\\}$ of critical points with _bounded_
$G_{\varepsilon}^{\rm self}$-energy, the energy densities
$e_{\varepsilon}^{\rm self}(u_{\varepsilon},\,A_{\varepsilon})$ converge, up
to subsequences, to the weight measure of a limiting stationary, rectifiable,
_integral_ varifold of codimension 2 in $M$. Conversely, in [16], it is shown
that any non-degenerate minimal submanifold of codimension 2 in $M$ can be
obtained as energy concentration set of a sequences of bounded energy critical
points of $G_{\varepsilon}^{\rm self}$.

#### Proofs of the main results: a sketch.

In the paper, we first address the proof of Theorem 2, which ultimately relies
on the $L^{p}$-compactness for the prejacobians
$j(u_{\varepsilon},\,A_{\varepsilon})$ given by Theorem  ‣ Theorem 0 ([13])..
Instead, the proof of Theorem 1 combines ideas from [8] in order to deal with
the logarithmic energy regime and from [26, 27] to deal with the Hermitian
line bundle setting. The most difficult part is to identify $\mu_{*}$ as the
weight measure of a stationary, rectifiable $(n-2)$-varifold. The key point is
showing that, if $\mu_{*}(\mathcal{B}_{R}(x_{0}))$ is smaller than
$\eta_{0}\,R^{n-2}$, where $\eta_{0}$ is suitable uniform “small” constant,
then $\mu_{*}(\mathcal{B}_{R/2}(x_{0}))=0$. This is shown in Lemma 5.6. The
proof goes along the lines of that of [8, Proposition VIII.1] and it is based
on a _clearing-out_ result (see the analysis in Section 4, in particular
Proposition 4.1 and Proposition 4.6).

A crucial rôle is played by the following (almost) $(n-2)$-_monotonicity
inequality_. Let $\operatorname{inj}(M)$ be the injectivity radius of $M$. For
$x_{0}\in M$ and $r\in(0,\operatorname{inj}(M))$, we denote by
$\mathcal{B}_{r}(x_{0})$ the geodesic ball of radius $r$ centered at $x_{0}$.
Given $(u_{\varepsilon},\,A_{\varepsilon})\in\mathcal{E}$, we define

$E_{\varepsilon}(x_{0},\,\rho):=\int_{\mathcal{B}_{\rho}(x_{0})}e_{\varepsilon}(u_{\varepsilon},\,A_{\varepsilon})\,\operatorname{vol}_{g}.$
(13)

and

$X_{\varepsilon}(x_{0},\,\rho):=\int_{\partial\mathcal{B}_{\rho}(x_{0})}\left(\left|\operatorname{D}_{A_{\varepsilon},\nu}u_{\varepsilon}\right|^{2}+\left|{\rm
i}_{\nu}F_{\varepsilon}\right|^{2}\right)\operatorname{vol}_{\hat{g}}+\frac{1}{2\varepsilon^{2}\,\rho}\int_{\mathcal{B}_{\rho}(x_{0})}\left(1-\left|u_{\varepsilon}\right|^{2}\right)^{2}\operatorname{vol}_{g}$
(14)

where $\nu$ is the exterior unit normal to $\partial B_{\rho}(x_{0})$, ${\rm
i}_{\nu}$ denotes the contraction against $\nu$ (see (17) below),
$F_{\varepsilon}:=F_{A_{\varepsilon}}$ and $\hat{g}$ is the metric induced by
$g$ on $\partial\mathcal{B}_{\rho}(x_{0})$.

######  Theorem 3.

There exist $\alpha\in(0,\,2)$, $R_{0}\in(0,\,\operatorname{inj}(M))$, and
$C>0$, all depending only on $M$ and $g$, such that the following holds. If
$(u_{\varepsilon},\,A_{\varepsilon})\in\mathcal{E}$ is any critical point of
$G_{\varepsilon}$ and $x_{0}$ is any point of $M$, then the inequality

$\begin{split}\frac{e^{Cr^{2}}}{r^{n-2}}E_{\varepsilon}(x_{0},\,r)&+\int_{r}^{R}\frac{e^{C\rho^{2}}}{\rho^{n-2}}X_{\varepsilon}(x_{0},\,\rho)\,\operatorname{d}\rho\\\
&\leq\frac{e^{CR^{2}}}{R^{n-2}}E_{\varepsilon}(x_{0},\,R)+\frac{Ce^{CR^{2}}(R^{\alpha}-r^{\alpha})}{R^{n-2+\alpha}}E_{\varepsilon}(x_{0},\,R)\end{split}$
(15)

holds for any $0<r<R<R_{0}$.

The proof of Theorem 3 is delicate. A classical argument (see e.g. Proposition
3.3 below), based on the Pohozaev identity, proves that

$\frac{\operatorname{d}}{\operatorname{d}\rho}\left(\frac{e^{C\rho^{2}}E_{\varepsilon}(x_{0},\,\rho)}{\rho^{n-4}}\right)\geq
0$ (16)

for any $x_{0}\in M$, $\rho\in(0,\,\operatorname{inj}(M))$ and some uniform
constant $C>0$. The exponent of $n-4$ at the denominator appears for scaling
reasons (heuristically, the curvature $F_{\varepsilon}$ scales as a second
derivative, so the integral of $\left|F_{\varepsilon}\right|^{2}$ scales as a
length to the power of $n-4$). The inequality (16) is known to be sharp in the
context of non-Abelian Yang-Mills-Higgs theories [35]. However, in our Abelian
setting, Theorem 3 gives an (almost) monotonicity formula for the energy
rescaled by $\rho^{n-2}$. A similar result is obtained in [27, Section 4], for
the Ginzburg-Landau functional in the self-dual scaling. However, the
arguments of [27] fall apart in our setting, because we work with a different
scaling of the energy. Instead, we exploit the properties of the Euler-
Lagrange equations. In particular, our arguments rely on Equation (6), which
can be reformulated as a _uniformly elliptic_ equation for the variable
$A_{\varepsilon}$. By elliptic regularity estimates, we can control the
curvature terms and improve on the monotonicity, obtaining in the end Theorem
3. As a byproduct of our arguments, we obtain a bound for the Morrey-Campanato
$L^{q,2}$-norm of $F_{\varepsilon}$, in terms of the rescaled energy (13), for
any $q<n$ (see Proposition 3.7 below).

Back on the proof of Theorem 1, the clearing-out property combined with the
energy bound (8) implies a two-sided estimate for the $(n-2)$-density of
$\mu_{*}$, see Lemma 5.7. Following ideas in [8, pp. 498–499] (extended first
in the gauge-invariant setting in [32] and to the context of Hermitian line
bundles in [27, Section 6]), the conclusion is achieved once we characterise
$\mu_{*}$ in terms of the _stress-energy tensor fields_ $T_{\varepsilon}$,
defined in (5.3) below. Indeed, by criticality of
$(u_{\varepsilon},\,A_{\varepsilon})$, each tensor field $T_{\varepsilon}$ is
stationary (i.e., divergence-free). Since stationarity is a linear condition,
their weak limit $T_{*}$ is still stationary. On the other hand, $T_{*}$ is
naturally identified with a $(n-2)$-generalised varifold in $M$ whose weight
measure $\left|T_{*}\right|$ is absolutely continuous with respect to
$\mu_{*}$. Then, by Lemma 5.7, we can apply a suitable version of the
Ambrosio-Soner rectifiability criterion [4, Theorem 3.1] to deduce that
$T_{*}$ is also rectifiable and that $\mu_{*}=\left|T_{*}\right|$, whence the
conclusion of Theorem 1 follows.

#### Further comments and open questions.

We did not address specifically the problem of the existence, in full
generality, of non-minimising critical points of $G_{\varepsilon}$. We believe
that they could be constructed by adapting the general framework developed in
[21].

Another interesting question that we have not addressed in this paper is
whether the varifold $\frac{1}{\pi}\mu_{*}$ is _integral_ or not. For the
magnetic Ginzburg-Landau functional in the self-dual scaling, it is known that
the measure $\frac{1}{\pi}\mu_{*}$ is integral [27]. By contrast, for critical
points of the non-magnetic Ginzburg-Landau functional (5.4) in Euclidean
domains, the situation is more involved [28] — that is, the density of the
limiting varifold $\frac{1}{\pi}\mu_{*}$ at a generic point is either $1$ or
possibly a non integer number larger or equal than 2. Unfortunately, the
arguments in [27] are really tailored to the bounded energy regime and do not
carry over to our setting.

Finally, it would be interesting to study, along the lines of [16], whether
any non-degenerate minimal codimension 2 submanifold of $M$ can be obtained as
energy concentration set of critical points of $G_{\varepsilon}$, for an
appropriate choice of the total space $E$ of the bundle $E\to M$. We leave all
these questions to future investigation.

#### Organisation of the paper.

The paper is organised as follows. In Section 1 we derive several _a priori_
estimates for gauge-invariant quantities, such as
$\left|u_{\varepsilon}\right|$,
$\left|\operatorname{D}_{A_{\varepsilon}}u_{\varepsilon}\right|$, and
$\left|F_{\varepsilon}\right|$, that are used throughout the whole paper. In
Section 2 we prove Theorem 2. In Section 3 we prove the almost
$(n-2)$-monotonicity inequality, i.e., Theorem 3. In Section 4, the longest
and by far the most technical of the paper, we prove the clearing-out property
with its consequences (Propositions 4.1 and 4.6). In the last section, Section
5, we complete the proof of Theorem 1.

The paper is completed by two appendices containing technical results
concerning Poincaré and trace-type inequalities for differential forms
(Appendix A) and elliptic estimates for the Hodge Laplacian (Appendix B).
These results are certainly known to experts but we provide full proofs
because they are crucial to our arguments and we could not find the exact
statements we needed in the available literature.

#### Notation.

In this paper, we use the symbol $\\{X_{\varepsilon}\\}$ to denote a family of
objects indexed by the parameter $\varepsilon>0$. Although in some instances
$\\{X_{\varepsilon}\\}$ may also be considered as a continuous family, we
always look at it as a sequence. In other words,
$\left\\{X_{\varepsilon}\right\\}$ is a shorthand for the sequence
$\left\\{X_{\varepsilon_{k}}\right\\}_{k}$, where $\varepsilon_{k}\to 0$ as
$k\to+\infty$. Usually, for the sake of a lighter notation, we do not relabel
subsequences.

In inequalities like $A\lesssim B$, the symbol $\lesssim$ means that there
exists a constant $C$, independent of $A$ and $B$, such that $A\leq CB$. In
particular, dealing with sequences indexed by $\varepsilon$, we use $\lesssim$
to denote inequality up to a constant independent of $\varepsilon$. Whenever
it is relevant, we keep track of the dependences of the implicit constants. In
most cases, they will depend on the manifold $(M,\,g)$ and on the constant
$\Lambda$ in the energy bound (8). Since we only consider Riemannian manifolds
$(M,\,g)$ as base spaces of $E\to M$ and the metric $g$ on $M$ is fixed from
the very beginning, we will sometimes write $M$ as a shorthand for $(M,\,g)$.

We denote $\Lambda^{k}\mathrm{T}^{*}M$ the bundle of $k$-forms on $M$. Sobolev
spaces of sections of a bundle $E\to M$ are denoted $W^{k,p}(M,\,E)$. In
particular, $W^{1,2}(M,\,\mathrm{T}^{*}M)$ is the Sobolev space of one-forms
on $M$ of class $W^{1,2}$. More details on such spaces can be found, for
instance, in Appendix A of [13]. If $\omega\in\Lambda^{k}\mathrm{T}^{*}M$ is a
$k$-form on $M$ and $X$ is a vector field, we denote by ${\rm i}_{X}\omega$
the _contraction_ of $\omega$ with $X$, defined by setting

${\rm i}_{X}\omega(X_{1},\dots,X_{k-1}):=\omega(X,X_{1},\dots,X_{k-1})$ (17)

for any vector fields $X_{1}$, …,$X_{k-1}$ on $M$.

Finally, as mentioned above, we will always write $\operatorname{inj}(M)$ for
the injectivity radius of $M$ and $\mathcal{B}_{r}(x_{0})$ for the geodesic
ball in $M$ of centre $x_{0}\in M$ and $r>0$.

##  1 Preliminary estimates for gauge-invariant quantities

We recall from [13] the following decomposition of $\operatorname{D}_{A}u$
which we will use several times.

######  Lemma 1.1 ([13, Lemma 2.2]).

For any $u\in W^{1,2}(M,\,E)$ and any $A\in W^{1,2}(M,\,\mathrm{T}^{*}M)$,
there holds

$\operatorname{D}_{A}u=\frac{\operatorname{d}(\left|u\right|)}{\left|u\right|}u+\frac{j(u,\,A)}{\left|u\right|^{2}}iu$
(1.1)

a.e. on the set $\\{u\neq 0\\}$.

Below, we collect some useful consequences of the equations (5), (6), and (7)
satisfied by critical points of the functionals $G_{\varepsilon}$. All the
results in this section are gauge-invariant and will be used repeatedly in the
rest of the paper.

####  $L^{\infty}$-bounds.

Let $(u_{\varepsilon},\,A_{\varepsilon})\in\mathcal{E}$ be a critical point of
$G_{\varepsilon}$. Elliptic regularity theory implies that, for each
$\varepsilon>0$, the pair $(u_{\varepsilon},\,A_{\varepsilon})$ is smooth up
to a suitable gauge transformation (see e.g. [20] or [27, Appendix]). In
particular, gauge-invariant quantities such as $\left|u_{\varepsilon}\right|$
or $\left|\operatorname{D}_{A_{\varepsilon}}u_{\varepsilon}\right|$ are
continuous (and their squares are smooth). Below, we prove several a priori
estimates on $u_{\varepsilon}$,
$\operatorname{D}_{A_{\varepsilon}}u_{\varepsilon}$, and
$F_{A_{\varepsilon}}$. We will apply these estimates to obtain compactness
results. For the reader’s convenience, we provide self-contained proofs of the
estimates we need. We recall that $\mathcal{E}:=(W^{1,2}\cap
L^{\infty})(M,\,E)\times W^{1,2}(M,\,\mathrm{T}^{*}M)$. First, we prove that

######  Proposition 1.2.

Let $\varepsilon>0$ and let
$(u_{\varepsilon},\,A_{\varepsilon})\in(W^{1,2}\cap L^{\infty})(M,\,E)\times
W^{1,2}(M,\,\mathrm{T}^{*}M)$ be any critical point of $G_{\varepsilon}$.
Then, $\left|u_{\varepsilon}\right|$ satisfies

$-\frac{1}{2}\Delta(\left|u_{\varepsilon}\right|^{2})+\frac{1}{\varepsilon^{2}}\left(\left|u_{\varepsilon}\right|^{2}-1\right)\left|u_{\varepsilon}\right|^{2}+\left|\operatorname{D}_{A_{\varepsilon}}u_{\varepsilon}\right|^{2}=0$
(1.2)

in the sense of distributions on $M$. As a consequence,

$\left|u_{\varepsilon}\right|\leq 1\qquad\textrm{a.e. in }M.$ (1.3)

######  Remark 1.3.

The assumption $u_{\varepsilon}\in(W^{1,2}\cap L^{\infty})(M,\,E)$ implies
that $\left|u_{\varepsilon}\right|^{2}\in W^{1,2}(M)$. Moreover, since
$G_{\varepsilon}(u_{\varepsilon},\,A_{\varepsilon})<+\infty$, we have
$\left|u_{\varepsilon}\right|\in L^{4}(M)$ and
$\left|\operatorname{D}_{A_{\varepsilon}}u_{\varepsilon}\right|^{2}\in
L^{1}(M)$. In particular, it makes sense to test (1.2) against test functions
in $(L^{\infty}\cap W^{1,2})(M)$.

In order to prove (1.3), we write the equation for
$\left|u_{\varepsilon}\right|$, as in [26, Proposition 2.4], then apply the
maximum principle. To this purpose, we first establish the following auxiliary
result.

######  Lemma 1.4.

For any section $u\in L^{2}(M,\,E)$ and any 1-form $A$ on $M$, such that
$\operatorname{D}_{A}u\in L^{2}(M,\,\mathrm{T}^{*}M\otimes E)$, there holds

$\begin{split}-\frac{1}{2}\Delta(\left|u\right|^{2})=\left\langle\operatorname{D}^{*}_{A}\operatorname{D}_{A}u,\,u\right\rangle-\left|\operatorname{D}_{A}u\right|^{2}\end{split}$
(1.4)

in the sense of distributions on $M$.

###### Proof.

Let $u\in L^{2}(M,\,E)$ be a section and let $A$ be $1$-form on $M$, such that
$\operatorname{D}_{A}u\in L^{2}(M,\,\mathrm{T}^{*}M\otimes E)$. For any smooth
function $\varphi\colon M\to\mathbb{R}$, we have

$\begin{split}\int_{M}\left\langle\operatorname{D}^{*}_{A}\operatorname{D}_{A}u,\,u\right\rangle\,\varphi\,\operatorname{vol}_{g}-\int_{M}\left|\operatorname{D}_{A}u\right|^{2}\,\varphi\,\operatorname{vol}_{g}&=\int_{M}\left\langle\operatorname{D}_{A}u,\,\operatorname{D}_{A}(u\varphi)\right\rangle\,\operatorname{vol}_{g}-\int_{M}\left|\operatorname{D}_{A}u\right|^{2}\,\varphi\,\operatorname{vol}_{g}\\\
&=\int_{M}\left\langle\operatorname{D}_{A}u,\,\operatorname{d}\varphi\otimes
u\right\rangle\,\operatorname{vol}_{g}\end{split}$

Let $x\in M$, and let $\\{\tau_{1},\,\ldots,\,\tau_{n}\\}$ be an orthonormal
basis of $\mathrm{T}_{x}M$. The scalar product on $\mathrm{T}^{*}M\otimes E$
is the one induced by the Hermitian form on $E$ and the Riemannian metric on
$M$; therefore, at the given point $x$, we have

$\left\langle\operatorname{D}_{A}u,\,\operatorname{d}\varphi\otimes
u\right\rangle=\sum_{j=1}^{n}\left\langle\operatorname{D}_{A\,\tau_{j}}u,\,u\right\rangle\,\frac{\partial\varphi}{\partial\tau_{j}}$
(1.5)

However, the right-hand side of (1.5) is also the scalar product between the
(real-valued) $1$-forms $\left\langle\operatorname{D}_{A}u,\,u\right\rangle$
and $\operatorname{d}\varphi$, evaluated at $x$. As a consequence, we have

$\begin{split}\int_{M}\left(\left\langle\operatorname{D}^{*}_{A}\operatorname{D}_{A}u,\,u\right\rangle-\left|\operatorname{D}_{A}u\right|^{2}\right)\varphi\,\operatorname{vol}_{g}=\int_{M}\left\langle\left\langle\operatorname{D}_{A}u,\,u\right\rangle\\!,\,\operatorname{d}\varphi\right\rangle\,\operatorname{vol}_{g}&=\int_{M}\varphi\,\operatorname{d}^{*}\left\langle\operatorname{D}_{A}u,\,u\right\rangle\,\operatorname{vol}_{g}\end{split}$

Since $\varphi$ is arbitrary, we deduce
$\left\langle\operatorname{D}^{*}_{A}\operatorname{D}_{A}u,\,u\right\rangle-\left|\operatorname{D}_{A}u\right|^{2}=\operatorname{d}^{*}\left\langle\operatorname{D}_{A}u,\,u\right\rangle$.
Finally, as $\left\langle\operatorname{D}_{A}u,\,u\right\rangle\in
L^{1}(M,\,\mathrm{T}^{*}M)$ and the connection $\operatorname{D}_{A}$ is
compatible with the Hermitian form on $E$, we have
$\left\langle\operatorname{D}_{A}u,\,u\right\rangle=\operatorname{d}(\left|u\right|^{2}/2)$
in the sense of distributions and hence,

$\left\langle\operatorname{D}^{*}_{A}\operatorname{D}_{A}u,\,u\right\rangle-\left|\operatorname{D}_{A}u\right|^{2}=\frac{1}{2}\operatorname{d}^{*}\operatorname{d}\left(\left|u\right|^{2}\right)=-\frac{1}{2}\Delta\left(\left|u\right|^{2}\right)$

in the sense of distributions on $M$. The conclusion follows. ∎

###### Proof of Proposition 1.2.

Since $G_{\varepsilon}(u_{\varepsilon},\,A_{\varepsilon})<+\infty$, it follows
that $\operatorname{D}_{A_{\varepsilon}}u_{\varepsilon}\in
L^{2}(M,\,\mathrm{T}^{*}M\otimes E)$. By Lemma 1.4,
$(u_{\varepsilon},\,A_{\varepsilon})$ satisfies (1.4) in the sense of
distributions on $M$. On the other hand, being a critical point of
$G_{\varepsilon}$, $(u_{\varepsilon},\,A_{\varepsilon})$ satisfies (5), too.
Combining (1.4) with (5), we immediately obtain (1.2). Then, taking Remark 1.3
into account, (1.3) follows by testing (1.2) against
$(\left|u_{\varepsilon}\right|^{2}-1)^{+}:=\max\\{\left|u_{\varepsilon}\right|^{2}-1,\,0\\}$.
∎

Next, we prove a $L^{\infty}$-bound for
$\operatorname{D}_{A_{\varepsilon}}u_{\varepsilon}$. Precisely,

######  Lemma 1.5.

Let $\varepsilon>0$ and let
$(u_{\varepsilon},\,A_{\varepsilon})\in(W^{1,2}\cap L^{\infty})(M,\,E)\times
W^{1,2}(M,\,\mathrm{T}^{*}M)$ be a critical point of $G_{\varepsilon}$. Then,

$\left\|\operatorname{D}_{A_{\varepsilon}}u_{\varepsilon}\right\|_{L^{\infty}(M)}\lesssim\varepsilon^{-1}.$
(1.6)

###### Proof.

Let $\varepsilon>0$ be arbitrary and let $(u_{\varepsilon},\,A_{\varepsilon})$
be any critical point of $G_{\varepsilon}$. In particular,
$\operatorname{D}_{A_{\varepsilon}}u_{\varepsilon}\in
L^{2}(M,\,\mathrm{T}^{*}M\otimes E)$, whence almost every $x_{0}\in M$ is a
Lebesgue point for
$\left|\operatorname{D}_{A_{\varepsilon}}u_{\varepsilon}\right|^{2}$. Testing
(5) against any $v\in W^{1,2}(M,\,E)$, we get

$\int_{M}\left\langle\operatorname{D}_{A_{\varepsilon}}u_{\varepsilon},\,\operatorname{D}_{A_{\varepsilon}}v\right\rangle\,\operatorname{vol}_{g}=\frac{1}{\varepsilon^{2}}\int_{M}\left\langle
v,\,u_{\varepsilon}\right\rangle(1-\left|u_{\varepsilon}\right|^{2})\,\operatorname{vol}_{g}.$
(1.7)

Let $x_{0}\in M$ be a Lebesgue point for
$\left|\operatorname{D}_{A_{\varepsilon}}u_{\varepsilon}\right|^{2}$. Let
$\mathcal{B}_{r}(x_{0})$ be the geodesic ball of center $x_{0}$ and radius
$r>0$, with $r$ smaller than the injectivity radius of $M$. For any $s>0$ such
that $r+s$ is still smaller than the injectivity radius of $M$, we pick
$\varphi_{s}\in C^{\infty}(M)$ such that

$\operatorname{spt}(\varphi_{s})\subset\mathcal{B}_{r+s}(x_{0}),\quad
0\leq\varphi_{s}\leq 1,\quad\varphi_{s}\equiv 1\mbox{ on
}\mathcal{B}_{r}(x_{0}),\quad\left|\operatorname{d}\varphi_{s}\right|\lesssim\frac{1}{s}.$
(1.8)

Since $v:=u_{\varepsilon}\varphi_{s}$ belongs to $W^{1,2}(M,\,E)$, it is
admissible to test (5). By (1.7), we get

$\int_{\mathcal{B}_{r}(x_{0})}\left|\operatorname{D}_{A_{\varepsilon}}u_{\varepsilon}\right|^{2}\,\operatorname{vol}_{g}-\frac{1}{\varepsilon^{2}}\int_{\mathcal{B}_{s}(x_{0})}\varphi_{s}\left|u_{\varepsilon}\right|^{2}(1-\left|u_{\varepsilon}\right|^{2})\,\operatorname{vol}_{g}\\\
=-\int_{\mathcal{B}_{r+s}(x_{0})\setminus\mathcal{B}_{r}(x_{0})}\left\langle\operatorname{D}_{A_{\varepsilon}}u_{\varepsilon},\,\operatorname{d}\varphi_{s}\otimes
u_{\varepsilon}+\varphi_{s}\operatorname{D}_{A_{\varepsilon}}u_{\varepsilon}\right\rangle\,\operatorname{vol}_{g}.$
(1.9)

Since $\left\|u_{\varepsilon}\right\|_{L^{\infty}(M)}\leq 1$ by Proposition
1.2, (1.8) and Schwarz’ inequality imply that the last term at right hand side
of (1.9) tends to zero as $s\to 0$. Hence, from (1.9) we obtain

$\begin{split}\int_{\mathcal{B}_{r}(x_{0})}&\left|\operatorname{D}_{A_{\varepsilon}}u_{\varepsilon}\right|^{2}\,\operatorname{vol}_{g}-\frac{1}{\varepsilon^{2}}\int_{\mathcal{B}_{r}(x_{0})}\left|u_{\varepsilon}\right|^{2}(1-\left|u_{\varepsilon}\right|^{2})\,\operatorname{vol}_{g}\\\
&\leq\liminf_{s\to
0}\frac{C}{s}\int_{\mathcal{B}_{r+s}(x_{0})\setminus\mathcal{B}_{r}(x_{0})}\left|\operatorname{D}_{A_{\varepsilon}}u_{\varepsilon}\right|\,\operatorname{vol}_{g}\\\
&=\liminf_{s\to
0}\frac{C}{s}\int_{r}^{r+s}\left(\int_{\partial\mathcal{B}_{r}(x_{0})}\left|\operatorname{D}_{A_{\varepsilon}}u_{\varepsilon}\right|\,\operatorname{d}\mathcal{H}^{n-1}\right)\,\operatorname{d}t\\\
&=C\int_{\partial\mathcal{B}_{r}(x_{0})}\left|\operatorname{D}_{A_{\varepsilon}}u_{\varepsilon}\right|\,\operatorname{d}\mathcal{H}^{n-1}\\\
&\leq
C(r^{n-1})^{1/2}\left(\int_{\partial\mathcal{B}_{r}(x_{0})}\left|\operatorname{D}_{A_{\varepsilon}}u_{\varepsilon}\right|^{2}\,\operatorname{d}\mathcal{H}^{n-1}\right)^{1/2}\end{split}$

Using once again $\left\|u_{\varepsilon}\right\|_{L^{\infty}(M)}\leq 1$, we
estimate

$\int_{\mathcal{B}_{r}(x_{0})}\left|\operatorname{D}_{A_{\varepsilon}}u_{\varepsilon}\right|^{2}\,\operatorname{vol}_{g}\leq
Cr^{n}\varepsilon^{-2}+Cr^{n/2-1/2}\left(\int_{\partial\mathcal{B}_{r}(x_{0})}\left|\operatorname{D}_{A_{\varepsilon}}u\right|^{2}\,\operatorname{d}\mathcal{H}^{n-1}\right)^{1/2}$
(1.10)

Now, we define

$y(r)\equiv
y(x_{0},r):=\int_{\mathcal{B}_{r}(x_{0})}\left|\operatorname{D}_{A_{\varepsilon}}u_{\varepsilon}\right|^{2}\,\operatorname{vol}_{g}.$

Recasting (1.10) in terms of $y$ and rearranging, we see that have shown

$C_{*}y(r)\leq r^{n}\varepsilon^{-2}+r^{n/2-1/2}(y^{\prime}(r))^{1/2},$

for some constant $C_{*}>0$, whence

$y^{\prime}(r)\geq\frac{1}{r^{n-1}}\left(\left(C_{*}y(r)-r^{n}\varepsilon^{-2}\right)^{+}\right)^{2}.$
(1.11)

Suppose now

$C_{*}y(r)-r^{n}\varepsilon^{-2}\geq\frac{C_{*}}{2}y(r)\quad\mbox{for any
}r>0\mbox{ small enough}.$ (1.12)

Then, for $r>0$ small enough we obtain

$y^{\prime}(r)\geq\frac{C}{r^{n-1}}y(r)^{2},$

which implies

$\frac{1}{y(r)}\geq\frac{1}{y(R)}+C\left(\frac{1}{r^{n-2}}-\frac{1}{R^{n-2}}\right)\geq\frac{C}{r^{n-2}}$

for any $0<r<R$, with $R>0$ small enough (and, perhaps, a different constant
$C>0$). In particular,

$y(r)\leq C_{1}r^{n-2}\quad\mbox{for any }r>0\mbox{ small enough}$ (1.13)

for some constant $C_{1}>0$; but (1.12) implies

$y(r)\geq C_{2}r^{n}\varepsilon^{-2}\quad\mbox{for any }r>0\mbox{ small
enough}.$ (1.14)

Inequality (1.14) contradicts (1.13) for $0<r\ll\varepsilon$. Therefore,
(1.12) does not hold; i.e., there is a sequence $r_{k}\to 0$ such that

$C_{*}y(r_{k})-r^{n}_{k}\varepsilon^{-2}<\frac{C_{*}}{2}y(r_{k}).$

Hence, by definition of $y(r)$,

$\frac{C_{*}}{2}\int_{\mathcal{B}_{r_{k}}(x_{0})}\left|\operatorname{D}_{A_{\varepsilon}}u\right|^{2}\leq
r^{n}_{k}\varepsilon^{-2},\quad\mbox{i.e.,}\quad\frac{1}{r^{n}}\int_{\mathcal{B}_{r_{k}}(x_{0})}\left|\operatorname{D}_{A_{\varepsilon}}u\right|^{2}\lesssim\varepsilon^{-2}.$

Taking the limit as $k\to+\infty$, we obtain

$\left|\operatorname{D}_{A_{\varepsilon}}u_{\varepsilon}(x_{0})\right|^{2}\lesssim\varepsilon^{-2},$

Since almost every $x_{0}\in M$ is a Lebesgue point for
$\left|\operatorname{D}_{A_{\varepsilon}}u_{\varepsilon}\right|^{2}$, the
conclusion follows. ∎

Now, we provide a bound on the $L^{\infty}(M)$-norm of
$\\{F_{\varepsilon}\\}$.

######  Lemma 1.6.

Let $\varepsilon>0$ and let
$(u_{\varepsilon},\,A_{\varepsilon})\in(W^{1,2}\cap L^{\infty})(M,\,E)\times
W^{1,2}(M,\,\mathrm{T}^{*}M)$ be a critical point of $G_{\varepsilon}$. Then,

$\left\|F_{\varepsilon}\right\|_{L^{\infty}(M)}\lesssim\varepsilon^{-1}$

###### Proof.

The Euler-Lagrange equation (6), together with Proposition 1.2 and Lemma 1.5,
implies

$\left\|\operatorname{d}^{*}F_{\varepsilon}\right\|_{L^{\infty}(M)}=\left\|j(u_{\varepsilon},\,A_{\varepsilon})\right\|_{L^{\infty}(M)}\leq\left\|u_{\varepsilon}\right\|_{L^{\infty}(M)}\left\|\operatorname{D}_{A_{\varepsilon}}u_{\varepsilon}\right\|_{L^{\infty}(M)}\lesssim\varepsilon^{-1}$
(1.15)

Let $F_{0}$ be the curvature of the reference connection
$\operatorname{D}_{0}$. The difference
$F_{\varepsilon}-F_{0}=\operatorname{d}A_{\varepsilon}$ is an exact form and,
in particular, it is orthogonal to all harmonic $2$-forms in $M$. Then, for
any $p\in(n,\,+\infty)$, the Gaffney inequality can be written in the form

$\left\|F_{\varepsilon}-F_{0}\right\|_{W^{1,p}(M)}\leq
C_{p}\left\|\operatorname{d}^{*}F_{\varepsilon}-\operatorname{d}^{*}F_{0}\right\|_{L^{p}(M)}$

for some constant $C_{p}$ that depend only on $p$ and $M$ (see e.g. [19,
Theorem 4.11]). Now, the form $F_{0}$ is smooth and does not depend on
$\varepsilon$, so

$\begin{split}\left\|F_{\varepsilon}\right\|_{W^{1,p}(M)}\leq
C_{p}\left(\left\|\operatorname{d}^{*}F_{\varepsilon}\right\|_{L^{p}(M)}+1\right)&\leq
C_{p}\,\varepsilon^{-1}\end{split}$

The lemma follows by Sobolev embedding. ∎

#### Compactness for gauge-invariant quantities.

In this section, we apply the bounds obtained above, as well as results from
[13], to prove compactness results for gauge-invariant quantitites, such as
the norm $\left|u_{\varepsilon}\right|$ and the pre-Jacobian
$j(u_{\varepsilon},\,A_{\varepsilon})$ of a sequence of critical points
$(u_{\varepsilon},\,A_{\varepsilon})$ that satisfies (8).

######  Proposition 1.7.

Let $\\{(u_{\varepsilon},\,A_{\varepsilon})\\}\subset(W^{1,2}\cap
L^{\infty})(M,\,E)\times W^{1,2}(M,\,\mathrm{T}^{*}M)$ be a sequence of
critical point of $G_{\varepsilon}$ satisfying the logarithmic energy bound
(8). Then, for any $p$ with $1\leq p<2$, there exists a constant $C_{p}$,
depending only on $\Lambda$, $p$, $M$, and $g$, such that

$\left\|\operatorname{d}(\left|u_{\varepsilon}\right|)\right\|_{L^{p}(M)}\leq
C_{p}.$ (1.16)

Moreover, there holds

$\int_{M}\left|\operatorname{d}(\left|u_{\varepsilon}\right|)\right|^{p}\,\operatorname{vol}_{g}\to
0\quad\mbox{as }\varepsilon\to 0.$ (1.17)

###### Proof.

We follow the argument of [7, Lemma X.13]. Set
$\rho_{\varepsilon}:=\left|u_{\varepsilon}\right|$ and rewrite (1.2) in the
form

$\frac{1}{2}\Delta\rho_{\varepsilon}^{2}=\frac{1}{\varepsilon^{2}}\rho^{2}_{\varepsilon}(\rho_{\varepsilon}^{2}-1)+\left|\operatorname{D}_{A_{\varepsilon}}u_{\varepsilon}\right|^{2}.$
(1.18)

Next, for any $0<\varepsilon<1$, we define:

$\widetilde{\rho}_{\varepsilon}:=\max\left\\{\rho_{\varepsilon},1-\frac{1}{\left|\log\varepsilon\right|^{2}}\right\\},\quad
M_{\varepsilon}:=\left\\{x\in
M:\widetilde{\rho_{\varepsilon}}(x)=\rho_{\varepsilon}(x)\right\\},\quad
N_{\varepsilon}:=M\setminus M_{\varepsilon}.$ (1.19)

Notice that on $M_{\varepsilon}$ there holds
$\rho_{\varepsilon}=\left|u_{\varepsilon}\right|\geq
1-\frac{1}{\left|\log\varepsilon\right|^{2}}$ for any $\varepsilon\in(0,1)$.
Moreover, as $\rho_{\varepsilon}$ belongs to $(L^{\infty}\cap W^{1,2})(M)$, it
follows that $\widetilde{\rho}_{\varepsilon}$ belongs to $(L^{\infty}\cap
W^{1,2})(M)$ as well, for any $\varepsilon\in(0,1)$. Since $M$ is compact,
also $1-\widetilde{\rho}_{\varepsilon}$ belongs to $(L^{\infty}\cap
W^{1,2})(M)$ and, by Remark 1.3, we can use $1-\widetilde{\rho}_{\varepsilon}$
as a test function in (1.18). Since $M$ is compact,
$\widetilde{\rho}_{\varepsilon}$ is bounded in $M$ and moreover constant in
$N_{\varepsilon}$, we obtain

$\int_{M_{\varepsilon}}\rho_{\varepsilon}\left|\operatorname{d}\rho_{\varepsilon}\right|^{2}\,\operatorname{vol}_{g}\leq
2\int_{M}(1-\widetilde{\rho}_{\varepsilon})\left|\operatorname{D}_{A_{\varepsilon}}u_{\varepsilon}\right|^{2}\,\operatorname{vol}_{g}.$
(1.20)

Since
$1-\widetilde{\rho}_{\varepsilon}\leq\frac{1}{\left|\log\varepsilon\right|^{2}}$,
by (1.20) and (8) we infer

$\int_{M_{\varepsilon}}\rho_{\varepsilon}\left|\operatorname{d}\rho_{\varepsilon}\right|^{2}\,\operatorname{vol}_{g}\lesssim\frac{1}{\left|\log\varepsilon\right|}\longrightarrow
0\quad\mbox{as }\varepsilon\to 0.$ (1.21)

Being $\rho_{\varepsilon}=\left|u_{\varepsilon}\right|\geq
1-\frac{1}{\left|\log\varepsilon\right|^{2}}$ on $M_{\varepsilon}$, (1.21)
gives

$\left\|\operatorname{d}(\left|u_{\varepsilon}\right|)\right\|_{L^{p}(M_{\varepsilon})}\lesssim
1$ (1.22)

(where the implicit constant depends only on $\Lambda$, $p$, $M$, and $g$) and

$\int_{M_{\varepsilon}}\left|\operatorname{d}\rho_{\varepsilon}\right|^{2}\,\operatorname{vol}_{g}\longrightarrow
0\quad\mbox{as }\varepsilon\to 0.$ (1.23)

Now, we show that the measure of the “bad” set $N_{\varepsilon}$ decreases at
a quantitative rate as $\varepsilon\to 0$. Indeed, the logarithmic energy
bound (8) and (1.3) imply

$\left|\log\varepsilon\right|\gtrsim\frac{1}{\varepsilon^{2}}\int_{N_{\varepsilon}}\left(1-\rho_{\varepsilon}^{2}\right)^{2}\,\operatorname{vol}_{g}\geq\frac{1}{\varepsilon^{2}}\int_{N_{\varepsilon}}(1-\rho_{\varepsilon})^{2}\,\operatorname{vol}_{g}\geq\frac{\left|N_{\varepsilon}\right|}{\left|\log\varepsilon\right|^{4}\varepsilon^{2}},$

whence

$\left|N_{\varepsilon}\right|\lesssim\left|\log\varepsilon\right|^{5}\varepsilon^{2}.$
(1.24)

By (1.1), it is straightforward to see that, for any $\varepsilon>0$,

$\left|\operatorname{d}\rho_{\varepsilon}\right|\leq\left|\operatorname{D}_{A_{\varepsilon}}u_{\varepsilon}\right|\quad\mbox{a.e.
on }M.$ (1.25)

Combining (1.24) with (1.6) (which holds a.e. on $M$), we get

$\left\|\operatorname{D}_{A_{\varepsilon}}u_{\varepsilon}\right\|_{L^{p}(N_{\varepsilon})}\lesssim\varepsilon^{\frac{2}{p}-1}\left|\log\varepsilon\right|^{\frac{5}{p}}\longrightarrow
0\quad\mbox{as }\varepsilon\to 0.$ (1.26)

Thus, by (1.26) and (1.25),

$\int_{N_{\varepsilon}}\left|\operatorname{d}\rho_{\varepsilon}\right|^{p}\,\operatorname{vol}_{g}\longrightarrow
0\quad\mbox{as }\varepsilon\to 0$ (1.27)

for any $p$ with $1\leq p<2$. Moreover, along with (1.21) and the definition
of $M_{\varepsilon}$, (1.25) and (1.26) imply (1.16). Finally, combining
(1.27) with (1.23) yields (1.17), completing the proof of the proposition. ∎

By combining the compactness result in [13, Theorem A] with the properties of
the London equation, and reasoning along the lines of [13, Corollary B], we
can extract a subsequence and find limits $J_{*}$, $F_{*}$ such that

$J(u_{\varepsilon},\,A_{\varepsilon})\to\pi J_{*}\quad\textrm{strongly in
}W^{-1,p}(M),\qquad F_{\varepsilon}\to F_{*}\quad\textrm{strongly in
}W^{1,p}(M)$ (1.28)

for any $p$ with $1\leq p<n/(n-1)$. In fact, $F_{*}$ and $J_{*}$ satisfy

$-\Delta F_{*}+F_{*}=2\pi J_{*}$ (1.29)

that is the London equation in (9). We claim that the curvature energy grows
less than logaritmically in $\varepsilon$. This fact will be useful to our
analysis of the properties of the energy concentration measure, in Section 5.

######  Lemma 1.8.

Let
$\\{(u_{\varepsilon},\,A_{\varepsilon})\\}_{\varepsilon>0}\subset(W^{1,2}\cap
L^{\infty})(M,\,E)\times W^{1,2}(M,\,\mathrm{T}^{*}M)$ be a sequence of
finite-energy solutions of (5)–(6) that satisfies (8). Then,

$\lim_{\varepsilon\to
0}\frac{1}{\left|\log\varepsilon\right|}\int_{M}\left|F_{\varepsilon}\right|^{2}\operatorname{vol}_{g}=0.$

The proof of Lemma 1.8 is very similar to arguments we used alredy in [13,
Section 4].

###### Proof of Lemma 1.8.

The curvature form $F_{\varepsilon}$ is, by definition, closed. Therefore, by
Hodge decomposition, we can write
$F_{\varepsilon}=\operatorname{d}\varphi_{\varepsilon}+\xi_{\varepsilon}$,
where $\xi_{\varepsilon}$ is a harmonic $2$-form and $\varphi_{\varepsilon}$
is a co-exact $1$-form such that

$\left\|\varphi_{\varepsilon}\right\|_{W^{1,2}(M)}+\left\|\xi_{\varepsilon}\right\|_{L^{2}(M)}\lesssim\left\|F_{\varepsilon}\right\|_{L^{2}(M)}\stackrel{{\scriptstyle\eqref{hp:logenergy}}}{{\lesssim}}\left|\log\varepsilon\right|^{1/2}$
(1.30)

(see all the discussion from [13, Proposition A.14]). As the space of harmonic
forms is finite-dimensional, the $W^{1,2}(M)$-norm of $\xi_{\varepsilon}$,
too, is of order $\left|\log\varepsilon\right|^{1/2}$ at most. From (6), we
obtain

$-\Delta\varphi_{\varepsilon}=\operatorname{d}^{*}\operatorname{d}\varphi_{\varepsilon}=\operatorname{d}^{*}F_{\varepsilon}=j(u_{\varepsilon},\,A_{\varepsilon})$

As $\varphi_{\varepsilon}$ is co-exact, hence orthogonal to all harmonic
$1$-forms, elliptic regularity theory implies that the $W^{2,2}(M)$-norm of
$\varphi_{\varepsilon}$ is bounded from above by the $L^{2}(M)$-norm of
$-\Delta\varphi_{\varepsilon}$. We deduce

$\left\|\varphi_{\varepsilon}\right\|_{W^{2,2}(M)}\lesssim\left\|j(u_{\varepsilon},\,A_{\varepsilon})\right\|_{L^{2}(M)}\lesssim\left\|\operatorname{D}_{A_{\varepsilon}}u_{\varepsilon}\right\|_{L^{2}(M)}\left\|u_{\varepsilon}\right\|_{L^{\infty}(M)}\lesssim\left|\log\varepsilon\right|^{1/2}$
(1.31)

(the last inequality follows from (8) and (1.3)). Therefore, we have

$\left\|F_{\varepsilon}\right\|_{W^{1,2}(M)}\leq\left\|\varphi_{\varepsilon}\right\|_{W^{2,2}(M)}+\left\|\xi_{\varepsilon}\right\|_{W^{1,2}(M)}\stackrel{{\scriptstyle\eqref{curvature1},\,\eqref{curvature3}}}{{\lesssim}}\left|\log\varepsilon\right|^{1/2}$

and hence, by Sobolev embedding,

$\left\|F_{\varepsilon}\right\|_{L^{q}(M)}\lesssim\left|\log\varepsilon\right|^{1/2}\qquad\textrm{with
}q:=2^{*}=\frac{2n}{n-2}$ (1.32)

From (1.28) and (1.32) we obtain, by interpolation,

$\left\|F_{\varepsilon}\right\|_{L^{2}(M)}\leq\left\|F_{\varepsilon}\right\|_{L^{q}(M)}^{\alpha}\left\|F_{\varepsilon}\right\|_{L^{1}(M)}^{1-\alpha}\lesssim\left|\log\varepsilon\right|^{\alpha/2}$
(1.33)

with $\alpha:=n/(n+2)\in(0,\,1)$. The lemma follows. ∎

Next, we notice that criticality and $L^{p}(M)$-compactness of
$\\{\operatorname{d}^{*}F_{A_{\varepsilon}}\\}$ for any $p$ with $1\leq
p<\frac{n}{n-1}$ (cf. [13, Proposition 3.11(i) and Remark 3.12]) imply
compactness of the prejacobians $j(u_{\varepsilon},\,A_{\varepsilon})$ in
$L^{p}(M,\,\mathrm{T}^{*}M)$, for any $1\leq p<\frac{n}{n-1}$.

######  Lemma 1.9.

Let $\\{(u_{\varepsilon},\,A_{\varepsilon})\\}\subset\mathcal{E}$ be a
sequence of critical points of $G_{\varepsilon}$ satisfying the logarithmic
energy bound (8). Then, there exists $j_{*}\in L^{p}(M,\,\mathrm{T}^{*}M)$ for
any $p$ with $1\leq p<\frac{n}{n-1}$, such that, up to a (not relabelled)
subsequence, there holds

$j(u_{\varepsilon},\,A_{\varepsilon})\longrightarrow j_{*}\quad\mbox{in
}L^{p}(M,\,\mathrm{T}^{*}M)\quad\mbox{as }\varepsilon\to 0,$ (1.34)

for any $p$ with $1\leq p<\frac{n}{n-1}$. In particular, there holds

$\left\|j(u_{\varepsilon},\,A_{\varepsilon})\right\|_{L^{p}(M)}\lesssim 1,$
(1.35)

for any $p$ with $1\leq p<\frac{n}{n-1}$, where the implicit constant in front
of the right-hand-side depends only on $\Lambda$, $p$, $M$ and $g$.

###### Proof.

Set, for short, $j_{\varepsilon}:=j(u_{\varepsilon},\,A_{\varepsilon})$ and
$F_{\varepsilon}:=F_{A_{\varepsilon}}$. To prove (1.34), it is enough to show
that $\\{j_{\varepsilon}\\}$ contains a Cauchy subsequence in (the Banach
space) $L^{p}(M,\,\mathrm{T}^{*}M)$, for any $p$ with $1\leq p<\frac{n}{n-1}$.
Due to [13, Proposition 3.11(i) and Remark 3.12], we can extract a subsequence
$\varepsilon_{k}\to 0$ such that $\\{F_{\varepsilon_{k}}\\}$ is a Cauchy
sequence in $W^{1,p}(M,\,\Lambda^{2}\mathrm{T}^{*}M)$, for any $p$ with $1\leq
p<\frac{n}{n-1}$. But then, by (6), for any $k$, $m\in\mathbb{N}$ we have

$\left\|j_{\varepsilon_{k}}-j_{\varepsilon_{m}}\right\|_{L^{p}(M)}=\left\|\operatorname{d}^{*}(F_{\varepsilon_{k}}-F_{\varepsilon_{m}})\right\|_{L^{p}(M)}.$

This proves (1.34). If (1.35) were false, there would be a (countable)
sequence of critical points
$\\{(u_{\varepsilon},\,A_{\varepsilon})\\}\subset\mathcal{E}$ that satisfy (8)
(for some $\varepsilon$-independent constant $\Lambda$) and are such that
$\left\|j(u_{\varepsilon},\,A_{\varepsilon})\right\|_{L^{p}(M)}\to+\infty$ as
$\varepsilon\to 0$. This contradicts (1.34). Therefore, (1.35) is proved. ∎

We apply Proposition 1.7 and Lemma 1.9 to prove a uniform $L^{p}(M)$-bound on
$\\{\operatorname{D}_{A_{\varepsilon}}u_{\varepsilon}\\}$.

######  Lemma 1.10.

Let $\\{(u_{\varepsilon},\,A_{\varepsilon})\\}\subset\mathcal{E}$ be a
sequence of critical points of $G_{\varepsilon}$ satisfying the logarithmic
energy bound (8). Then, for any $p$ with $1\leq p<\frac{n}{n-1}$ there exists
a constant $C_{p}>0$ (depending on $\Lambda$, $p$, $M$ and $g$) such that

$\left\|\operatorname{D}_{A_{\varepsilon}}u_{\varepsilon}\right\|_{L^{p}(M)}\leq
C_{p}$ (1.36)

for any $\varepsilon>0$. Moreover, there exists a (non-relabelled) subsequence
such that

$\left\|\operatorname{D}_{A_{\varepsilon}}u_{\varepsilon}\right\|_{L^{p}(M)}\to\left\|j_{*}\right\|_{L^{p}(M)}\quad\mbox{for
any }1\leq p<\frac{n}{n-1},\quad\mbox{as }\to 0,$ (1.37)

where $j_{*}$ is given by Lemma 1.9.

###### Proof.

Define the sets $M_{\varepsilon}$ and $N_{\varepsilon}$ as in (1.19).

_Proof of ( 1.36)._ Let $p$ be fixed, such that $1\leq p<\frac{n}{n-1}$. If
(1.36) were false, we could extract a (non-relabelled) subsequence) in such a
way that

$\left\|\operatorname{D}_{A_{\varepsilon}}u_{\varepsilon}\right\|_{L^{p}(M)}\to+\infty\qquad\textrm{as
}\varepsilon\to 0$ (1.38)

Let $\varepsilon_{2}>0$ be such that
$\frac{1}{\left|\log\varepsilon_{2}\right|^{2}}=\frac{1}{2}$ and notice that
$\left|u_{\varepsilon}\right|>\frac{1}{2}$ on $M_{\varepsilon}$ for any
$\varepsilon$ with $0<\varepsilon<\varepsilon_{2}$. Then, by (1.1),

$\left|\operatorname{D}_{A_{\varepsilon}}u_{\varepsilon}\right|\leq
2\left(\left|\operatorname{d}(\left|u_{\varepsilon}\right|)\right|+\left|j(u_{\varepsilon},\,A_{\varepsilon})\right|)\right)\qquad\mbox{on
}M_{\varepsilon},$

whence

$\left|\operatorname{D}_{A_{\varepsilon}}u_{\varepsilon}\right|^{p}\leq
2^{p}\left(\left|\operatorname{d}(\left|u_{\varepsilon}\right|)\right|^{p}+\left|j(u_{\varepsilon},\,A_{\varepsilon})\right|^{p}\right)\qquad\mbox{on
}M_{\varepsilon},$ (1.39)

for any $\varepsilon$ with $0<\varepsilon<\varepsilon_{2}$. Combining (1.39)
with (1.16) and (1.35), we obtain that for any $p$ with $1\leq
p<\frac{n}{n-1}$ there exists a constant $C_{p}^{\prime}>0$ such that

$\left\|\operatorname{D}_{A_{\varepsilon}}u_{\varepsilon}\right\|_{L^{p}(M_{\varepsilon})}\leq
C_{p}^{\prime}$ (1.40)

for any $\varepsilon>0$. By (1.40) and (1.26), we deduce that
$\operatorname{D}_{A_{\varepsilon}}u_{\varepsilon}$ is bounded in $L^{p}(M)$.
This contradicts (1.38) and proves (1.36).

_Proof of ( 1.37)._ By Lemma 1.9, (1.36), Lemma 1.11, and the Dominated
Convergence Theorem,

$\begin{split}\int_{M}\left|j_{*}\right|^{p}\,\operatorname{vol}_{g}&\leq\liminf_{\varepsilon\to
0}\int_{M_{\varepsilon}}\frac{\left|j(u_{\varepsilon},\,A_{\varepsilon})\right|^{p}}{\left|u_{\varepsilon}\right|^{p}}\,\operatorname{vol}_{g}\\\
&\leq\limsup_{\varepsilon\to
0}\int_{M_{\varepsilon}}\frac{\left|j(u_{\varepsilon},\,A_{\varepsilon})\right|^{p}}{\left|u_{\varepsilon}\right|^{p}}\,\operatorname{vol}_{g}\\\
&\leq\limsup_{\varepsilon\to
0}\left(1-\frac{1}{\left|\log\varepsilon\right|^{2}}\right)\int_{M_{\varepsilon}}\left|j(u_{\varepsilon},\,A_{\varepsilon})\right|^{p}\,\operatorname{vol}_{g}\\\
&\leq\lim_{\varepsilon\to
0}\int_{M}\left|j(u_{\varepsilon},\,A_{\varepsilon})\right|^{p}\,\operatorname{vol}_{g}=\int_{M}\left|j_{*}\right|^{p}\,\operatorname{vol}_{g}.\end{split}$
(1.41)

Thus, by (1.1) and (1.41),

$\begin{split}\left\|j_{*}\right\|_{L^{p}(M)}&=\lim_{\varepsilon\to
0}\left(\int_{M_{\varepsilon}}\frac{\left|j(u_{\varepsilon},\,A_{\varepsilon})\right|^{p}}{\left|u_{\varepsilon}\right|^{p}}\,\operatorname{vol}_{g}\right)^{1/p}\\\
&\leq\lim_{\varepsilon\to
0}\left(\int_{M_{\varepsilon}}\left|\operatorname{D}_{A_{\varepsilon}}u_{\varepsilon}\right|^{p}\,\operatorname{vol}_{g}\right)^{1/p}\\\
&\leq\lim_{\varepsilon\to
0}\left\\{\left(\int_{M_{\varepsilon}}\left|\operatorname{d}\rho_{\varepsilon}\right|^{p}\,\operatorname{vol}_{g}\right)^{1/p}+\left(1-\frac{1}{\left|\log\varepsilon\right|^{2}}\right)\left(\int_{M_{\varepsilon}}\left|j(u_{\varepsilon},\,A_{\varepsilon})\right|^{p}\,\operatorname{vol}_{g}\right)^{1/p}\right\\}\\\
&\stackrel{{\scriptstyle\eqref{eq:conv-Lp-rho-eps}}}{{=}}\lim_{\varepsilon\to
0}\left(\int_{M_{\varepsilon}}\left|j(u_{\varepsilon},\,A_{\varepsilon})\right|^{p}\,\operatorname{vol}_{g}\right)^{1/p}\\\
&\stackrel{{\scriptstyle\eqref{eq:Lp-conv-
j_eps-j*}}}{{=}}\left\|j_{*}\right\|_{L^{p}(M)},\end{split}$ (1.42)

for any $p$ with $1\leq p<\frac{n}{n-1}$. But, by (1.26) and the Dominated
Convergence Theorem,

$\lim_{\varepsilon\to
0}\left(\int_{M_{\varepsilon}}\left|\operatorname{D}_{A_{\varepsilon}}u_{\varepsilon}\right|^{p}\,\operatorname{vol}_{g}\right)^{1/p}=\lim_{\varepsilon\to
0}\left(\int_{M}\left|\operatorname{D}_{A_{\varepsilon}}u_{\varepsilon}\right|^{p}\,\operatorname{vol}_{g}\right)^{1/p},$

for any $p$ with $1\leq p<\frac{n}{n-1}$, hence (1.37) follows by (1.41). ∎

Finally, the following lemma is classical in Ginzburg-Landau theory.

######  Lemma 1.11.

Let $\\{(u_{\varepsilon},\,A_{\varepsilon})\\}\subset\mathcal{E}$ be a
sequence satisfying the logarithmic bound (8). Then,

$\lim_{\varepsilon\to
0}\int_{M}\left(1-\left|u_{\varepsilon}\right|^{2}\right)^{2}\,\operatorname{vol}_{g}=0.$
(1.43)

In particular, $\left|u_{\varepsilon}\right|\to 1$ in $L^{p}(M)$ as
$\varepsilon\to 0$, for any finite $p\geq 1$.

###### Proof.

Equation (1.43) is an immediate consequence of (8). The $L^{p}(M)$-convergence
$\left|u_{\varepsilon}\right|\to 1$ follows by (8) and (1.3), by
interpolation. ∎

##  2 Compactness up to gauge equivalence

In this section, we prove Theorem 2. We assume that
$\\{(u_{\varepsilon},\,A_{\varepsilon})\\}\subset\mathcal{E}$ is a sequence of
critical points of $G_{\varepsilon}$ satisfying (8) and that each
$A_{\varepsilon}$ takes the form (10), i.e., that each $A_{\varepsilon}$
writes as

$A_{\varepsilon}=\operatorname{d}^{*}\psi_{\varepsilon}+\zeta_{\varepsilon}$

for an exact 2-form $\psi_{\varepsilon}$ and a harmonic form
$\zeta_{\varepsilon}$ satisfying
$\left\|\zeta_{\varepsilon}\right\|_{L^{\infty}(M)}\leq C_{M}$, where $C_{M}$
depends only on $M$. As already mentioned in the Introduction, this is always
possible. Indeed, a family of gauge transformations
$\\{\Phi_{\varepsilon}\\}\subset W^{2,2}(M,\,\mathbb{S}^{1})$ with this
property can be obtained by applying, for each $\varepsilon>0$, Lemma 2.10 in
[13]. As shown by (the proof of) Proposition 3.1 in [13], if $A_{\varepsilon}$
takes the form (10), we can find $A_{*}\in W^{2,p}(M,\,\mathrm{T}^{*}M)$ for
any $p$ with $1\leq p<\frac{n}{n-1}$ such that, possibly up to extraction of a
(not relabeled) subsequence,

$\Phi_{\varepsilon}\cdot A_{\varepsilon}\to A_{*}\quad\mbox{strongly in
}W^{2,p}(M)\quad\mbox{as }\varepsilon\to 0,\quad\mbox{for any }p\mbox{ with
}1\leq p<\frac{n}{n-1}.$ (2.1)

Thus, to prove Theorem 2, it is enough to prove that there exists $u_{*}$
belonging to $W^{1,p}(M,\,E)$ for any $p$ with $1\leq p<\frac{n}{n-1}$, such
that, up to a subsequence,

$\Phi_{\varepsilon}u_{\varepsilon}\to u_{*}\quad\mbox{strongly in
}W^{1,p}(M)\quad\mbox{as }\varepsilon\to 0,\quad\mbox{for any }p\mbox{ with
}1\leq p<\frac{n}{n-1}.$ (2.2)

In (2.1), (2.2), $\Phi_{\varepsilon}u_{\varepsilon}$, $\Phi_{\varepsilon}\cdot
A_{\varepsilon}$ are defined as in (3); i.e.,
$\Phi_{\varepsilon}u_{\varepsilon}$ is given by fibre-wise action of
$\mathbb{S}^{1}$ on $E$ and $\Phi_{\varepsilon}\cdot
A_{\varepsilon}=A_{\varepsilon}-i\Phi_{\varepsilon}^{-1}\operatorname{d}\Phi_{\varepsilon}$.
Since in the rest of this section we will always work in the gauge (10), we
will to omit the gauge transformations $\Phi_{\varepsilon}$ from the notation;
i.e., while writing $A_{\varepsilon}$ and $u_{\varepsilon}$, we will actually
mean $\Phi_{\varepsilon}\cdot A_{\varepsilon}$ and
$\Phi_{\varepsilon}u_{\varepsilon}$, respectively.

The following corollary is a straightforward consequence of Lemma 1.10.

######  Corollary 2.1.

Let $\\{(u_{\varepsilon},\,A_{\varepsilon})\\}\subset\mathcal{E}$ be a
sequence of critical points of $G_{\varepsilon}$ satisfying the logarithmic
energy bound (8). Assume, in addition, that (10) holds. Then, there exists a
section $u_{*}\in W^{1,p}(M,\,E)$ for any $p$ with $1\leq p<\frac{n}{n-1}$
satisfying $\left|u_{*}\right|=1$ almost everywhere on $M$ such that, up to
extraction of a (not relabelled) subsequence, for any $p$ with $1\leq
p<\frac{n}{n-1}$ there holds

$\displaystyle u_{\varepsilon}\rightharpoonup u_{*}\quad\mbox{weakly in
}W^{1,p}(M),$ (2.3) $\displaystyle u_{\varepsilon}\to u_{*}\quad\mbox{strongly
in }L^{p}(M),$ (2.4) $\displaystyle\left|u_{\varepsilon}\right|\to
1\quad\mbox{strongly in }W^{1,p}(M),$ (2.5)

as $\varepsilon\to 0$.

###### Proof.

By Lemma 1.10, (2.1), and the uniform bound
$\left\|u_{\varepsilon}\right\|_{L^{\infty}(M)}\leq 1$ (given by Proposition
1.2), it follows that the sequence $\\{\operatorname{D}_{0}u_{\varepsilon}\\}$
is uniformly bounded in $L^{p}(M,\,\mathrm{T}^{*}M\otimes E)$, for any $p$
with $1\leq p<\frac{n}{n-1}$. Consequently, the sequence
$\\{u_{\varepsilon}\\}$ is uniformly bounded in $W^{1,p}(M,\,E)$, for any $p$
with $1\leq p<\frac{n}{n-1}$. Hence, there exists a section $u_{*}\in
W^{1,p}(M,\,E)$ for any $p$ with $1\leq p<\frac{n}{n-1}$ such that up to a
(not relabelled) subsequence

$u_{\varepsilon}\rightharpoonup u_{*}\quad\mbox{weakly in
}W^{1,p}(M),\quad\mbox{for any }p\mbox{ with }1\leq
p<\frac{n}{n-1}\quad\mbox{as }\varepsilon\to 0,$

i.e., (2.3). Moreover, since $M$ is compact, by the Rellich-Kondrachov theorem
it follows that, up to another (not relabelled) subsequence,

$u_{\varepsilon}\to u_{*}\quad\mbox{strongly in }L^{p}(M),\quad\mbox{for any
}p\mbox{ with }1\leq p<\frac{n}{n-1}\quad\mbox{as }\varepsilon\to 0,$

i.e., (2.4), and so, up to still another (not relabelled) subsequence,
$\left|u_{\varepsilon}\right|\to\left|u_{*}\right|$ pointwise almost
everywhere on $M$ as $\varepsilon\to 0$. Then, Lemma 1.11 implies

$\left|u_{*}\right|=1\quad\mbox{a.e. on }M.$ (2.6)

By (2.6), $\operatorname{d}(\left|u_{*}\right|)=0$ almost everywhere on $M$.
Therefore,

$\left\|\operatorname{d}(\left|u_{*}\right|)-\operatorname{d}(\left|u_{\varepsilon}\right|)\right\|_{L^{p}(M)}=\left\|\operatorname{d}(\left|u_{\varepsilon}\right|)\right\|_{L^{p}(M)}\stackrel{{\scriptstyle\eqref{eq:conv-
Lp-rho-eps}}}{{\longrightarrow}}0\quad\mbox{as }\varepsilon\to 0,$

for any $p$ with $1\leq p<\frac{n}{n-1}$. Along with (2.4), this proves (2.5),
and the corollary follows. ∎

######  Remark 2.2.

By [13, Proposition 3.11 and Remark 3.12], the uniform bound (1.3), Corollary
2.1, and the dominated convergence theorem, it follows that

$A_{\varepsilon}u_{\varepsilon}\to A_{*}u_{*}\quad\mbox{strongly in
}L^{p}(M)\quad\mbox{as }\varepsilon\to 0,\quad\mbox{for any }p\mbox{ with
}1\leq p<\frac{n}{n-1}.$ (2.7)

Indeed, for any $p$ with $1\leq p<\frac{n}{n-1}$, we have

$\begin{split}\left\|A_{\varepsilon}u_{\varepsilon}-A_{*}u_{*}\right\|_{L^{p}(M)}&=\left\|(A_{\varepsilon}-A_{*})u_{\varepsilon}-A_{*}(u_{*}-u_{\varepsilon})\right\|_{L^{p}(M)}\\\
&\stackrel{{\scriptstyle\eqref{maxprinc}}}{{\leq}}\left\|A_{\varepsilon}-A_{*}\right\|_{L^{p}(M)}+\left\|\left|A_{*}\right|\left|u_{*}-u_{\varepsilon}\right|\right\|_{L^{p}(M)}.\end{split}$

The first term at right hand side vanishes as $\varepsilon\to 0$, because
$A_{\varepsilon}\to A_{*}$ in $W^{2,p}(M)$, for any $p$ with $1\leq
p<\frac{n}{n-1}$. On the other hand, since
$\left|A_{*}\right|\left|u_{*}-u_{\varepsilon}\right|\leq 2\left|A_{*}\right|$
(again by (1.3)) and $u_{\varepsilon}\to u_{*}$ pointwise a.e. in $M$ (by
(2.4)), the dominated convergence theorem yields that
$\left\|\left|A_{*}\right|\left|u_{*}-u_{\varepsilon}\right|\right\|_{L^{p}(M)}\to
0$ as $\varepsilon\to 0$, proving (2.7).

We are now in the position to prove Theorem 2.

###### Proof of Theorem 2.

Since $A_{\varepsilon}$ satisfies (10), Equations (2.3) and (2.7) imply

$\operatorname{D}_{A_{\varepsilon}}u_{\varepsilon}\rightharpoonup\operatorname{D}_{A_{*}}u_{*}\quad\mbox{weakly
in }L^{p}(M)\quad\mbox{for any }p\mbox{ with }1\leq
p<\frac{n}{n-1}\quad\mbox{as }\varepsilon\to 0.$ (2.8)

We claim that

$\int_{M}\left|\operatorname{D}_{A_{\varepsilon}}u_{\varepsilon}\right|^{p}\,\operatorname{vol}_{g}\to\int_{M}\left|\operatorname{D}_{A_{*}}u_{*}\right|^{p}\,\operatorname{vol}_{g}\quad\mbox{for
any }p\mbox{ with }1\leq p<\frac{n}{n-1},\quad\mbox{as }\varepsilon\to 0,$
(2.9)

The condition (2.9) implies the strong convergence
$\operatorname{D}_{A_{\varepsilon}}u_{\varepsilon}\to\operatorname{D}_{A_{*}}u_{*}$
in $L^{p}(M)$ for any $1\leq p<\frac{n}{n-1}$, and, in view of (2.7), that
$\operatorname{D}_{0}u_{\varepsilon}\to\operatorname{D}_{0}u_{*}$ strongly in
$L^{p}(M)$, for any $1\leq p<\frac{n}{n-1}$.

Towards the proof of (2.9), we observe that, by Lemma 1.1 and since
$\left|u_{*}\right|=1$ a.e. on $M$ (by Corollary 2.1), there holds

$\operatorname{D}_{A_{*}}u_{*}=ij(u_{*},\,A_{*})u_{*}\quad\mbox{a.e. on }M,$

where
$j(u_{*},\,A_{*})=\left\langle\operatorname{D}_{A_{*}}u_{*},\,iu_{*}\right\rangle$.
On the other hand, Equations (2.8), (2.4), and the uniform bound
$\left\|\operatorname{D}_{A_{\varepsilon}}u_{\varepsilon}\right\|_{L^{p}(M)}\leq
C_{p}$ given, for any $p$ with $1\leq p<\frac{n}{n-1}$, by Lemma 1.10, yield

$j(u_{\varepsilon},\,A_{\varepsilon})=\left\langle\operatorname{D}_{A_{\varepsilon}}u_{\varepsilon},\,iu_{\varepsilon}\right\rangle\rightharpoonup\left\langle\operatorname{D}_{A_{*}}u_{*},\,iu_{*}\right\rangle=j(u_{*},\,A_{*})$
(2.10)

weakly in $L^{p}(M)$ for any $p$ with $1\leq p<\frac{n}{n-1}$ as
$\varepsilon\to 0$. By the uniqueness of the weak limit, we must have
$j(u_{*},\,A_{*})=j_{*}$, where $j_{*}$ is given by Lemma 1.9. Recalling
(1.37), we deduce (2.9).

By passing to the limit in the Euler-Lagrange equation (6), with the help of
(1.28) and (2.10), we see that

$j(u_{*},\,A_{*})=\operatorname{d}^{*}F_{*}$ (2.11)

in the sense of distributions in $M$ — that is, Equation (11) holds. In order
to complete the proof, it only remains to show that $u_{*}$ and $A_{*}$ are
smooth in $M\setminus\operatorname{spt}\mu_{*}$, where $\mu_{*}$ is the limit
of the rescaled energy densities, given by Theorem 1. By passing to the limit
as $\varepsilon\to 0$ in (10) (with the help of [33, Proposition 5.6] or, say,
[13, Proposition A.13]), we see that $A_{*}$ can be written in the form

$A_{*}=\operatorname{d}^{*}\psi_{*}+\zeta_{*},$ (2.12)

where $\psi_{*}$ is an exact $2$-form and $\zeta_{*}$ a harmonic $1$-form on
$M$. By taking the differential of both sides, we obtain

$-\Delta\psi_{*}=\operatorname{d}\operatorname{d}^{*}\psi_{*}=\operatorname{d}A_{*}=F_{*}-F_{0},$

where $F_{0}$ is the curvature of the reference connection
$\operatorname{D}_{0}$. Now, $F_{*}$ is a solution of the London equation
(1.29), whose right-hand side $2\pi J_{*}$ is a measure concentrated on
$\operatorname{spt}\mu_{*}$, due to [13, Theorem A]. By elliptic regularity,
it follows that $F_{*}$ and $\psi_{*}$ are smooth in
$M\setminus\operatorname{spt}\mu_{*}$ and hence, $A_{*}$ is. Now, let
$B\subset\\!\subset M\setminus\operatorname{spt}\mu_{*}$ be a ball. We claim
that there holds

$\int_{B}\left|\operatorname{D}_{A_{*}}u_{*}\right|^{2}\,\operatorname{vol}_{g}<+\infty$
(2.13)

Equation (2.13) is an almost immediate consequence of Proposition 4.1, which
we will prove later on in Section 4, combined with a covering argument. As
$A_{*}$ is smooth in a neighbourhood of $\bar{B}$, from (2.13) we deduce that
$u_{*}\in W^{1,2}(B,\,E)$. Therefore, up to identifying $u_{*|B}$ with a
complex-valued map, we can write $u_{*}=e^{i\theta_{*}}$ in $B$ for some
scalar function $\theta_{*}\in W^{1,2}(B,\,\mathbb{R})$. (This claim follows
by lifting results that were originally proven in [11]; see also [12, Theorem
1.1].) The reference connection can locally be written as
$\operatorname{D}_{0}=\operatorname{d}-i\gamma_{0}$ on $B$, for some (smooth)
$1$-form $\gamma_{0}$. Then, a direct computation shows that

$j(u_{*},\,A_{*})=\operatorname{d}\theta_{*}-\gamma_{0}-A_{*}$

However, $j(u_{*},\,A_{*})$ is smooth in $B$, due to (2.11). Therefore,
$\operatorname{d}\theta_{*}$ is smooth in $B$, and hence
$u_{*}=e^{i\theta_{*}}$ is. This completes the proof of the theorem. ∎

######  Remark 2.3.

Let $\varphi\in C^{\infty}_{\mathrm{c}}(M\setminus\operatorname{spt}\mu_{*})$
be a test function. From (2.11), we know that
$\operatorname{d}^{*}j(u_{*},\,A_{*})=0$ and hence, by integrating by parts,

$\begin{split}0=\int_{M}\left\langle
j(u_{*},\,A_{*}),\,\operatorname{d}\varphi\right\rangle\,\operatorname{vol}_{g}&=\int_{M}\left\langle\operatorname{D}_{A_{*}}u_{*},\,\operatorname{d}\varphi\otimes
iu_{*}\right\rangle\,\operatorname{vol}_{g}\\\
&=\int_{M}\left\langle\operatorname{D}_{A_{*}}u_{*},\,\operatorname{D}_{A_{*}}(i\varphi
u_{*})\right\rangle\,\operatorname{vol}_{g}=\int_{M}\left\langle\operatorname{D}_{A_{*}}^{*}\operatorname{D}_{A_{*}}u_{*},\,iu_{*}\right\rangle\varphi\,\operatorname{vol}_{g}\end{split}$

where $\operatorname{D}_{A_{*}}^{*}$ is the $L^{2}$-adjoint of
$\operatorname{D}_{A_{*}}$. As $\varphi$ is arbitrary, it follows that
$\operatorname{D}_{A_{*}}^{*}\operatorname{D}_{A_{*}}u_{*}=\lambda u_{*}$ in
$M\setminus\operatorname{spt}\mu_{*}$, for some (real-valued) function
$\lambda\colon M\setminus\operatorname{spt}\mu_{*}\to\mathbb{R}$. Since
$\left|u_{*}\right|=1$, we obtain that
$\lambda=\left\langle\operatorname{D}_{A_{*}}^{*}\operatorname{D}_{A_{*}}u_{*},\,u_{*}\right\rangle$
and hence, by Lemma 1.4,

$\operatorname{D}_{A_{*}}^{*}\operatorname{D}_{A_{*}}u_{*}=\left|\operatorname{D}_{A_{*}}u_{*}\right|^{2}u_{*}\qquad\textrm{in
}M\setminus\operatorname{spt}\mu_{*}.$

In other words, $u_{*}$ is an $A_{*}$-harmonic unit section away from the
singular set $\operatorname{spt}\mu_{*}$.

##  3 Monotonicity formula

This section is devoted to the proof of the monotonicity formula, Theorem 3.
For the reader’s convenience, we first recall some notation. Let
$(u_{\varepsilon},\,A_{\varepsilon})\in\mathcal{E}$ be a critical point of
$G_{\varepsilon}$, fixed once and for all throughout this section. For
$x_{0}\in M$ and $r\in(0,\,\operatorname{inj}(M))$, we define

$E_{\varepsilon}(x_{0},\,\rho):=\int_{\mathcal{B}_{\rho}(x_{0})}e_{\varepsilon}(u_{\varepsilon},\,A_{\varepsilon})\,\operatorname{vol}_{g}.$
(3.1)

Moreover, if $\nu$ is the exterior unit normal field to
$\partial\mathcal{B}_{\rho}(x_{0})$, we set

$X_{\varepsilon}(x_{0},\,\rho):=\int_{\partial\mathcal{B}_{\rho}(x_{0})}\left(\left|\operatorname{D}_{A_{\varepsilon},\nu}u_{\varepsilon}\right|^{2}+\left|{\rm
i}_{\nu}F_{\varepsilon}\right|^{2}\right)\operatorname{vol}_{\hat{g}}+\frac{1}{2\varepsilon^{2}\,\rho}\int_{\mathcal{B}_{\rho}(x_{0})}\left(1-\left|u_{\varepsilon}\right|^{2}\right)^{2}\operatorname{vol}_{g}$
(3.2)

where $F_{\varepsilon}:=F_{A_{\varepsilon}}$, $\mathrm{i}_{\nu}$ denotes the
contraction against $\nu$ (as defined in (17)) and $\hat{g}$ is the metric
induced by $g$ on $\partial\mathcal{B}_{\rho}(x_{0})$.

We will achieve the proof of Theorem 3 through a series of lemmas. The first
one is a well-known consequence of the inner variation equation for critical
points (see e.g. [27, Section 4]).

######  Lemma 3.1.

Let $(u_{\varepsilon},\,A_{\varepsilon})\in\mathcal{E}$ be any critical point
of $G_{\varepsilon}$. For any $x_{0}\in M$ and any
$\rho\in(0,\,\operatorname{inj}(M))$, there holds

$\rho\int_{\partial\mathcal{B}_{\rho}(x_{0})}e_{\varepsilon}(u_{\varepsilon},\,A_{\varepsilon})\,\operatorname{vol}_{\hat{g}}\geq\rho
X_{\varepsilon}(x_{0},\,\rho)+(n-2-C(M)\rho^{2})E_{\varepsilon}(x_{0},\,\rho)\\\
-\int_{\mathcal{B}_{\rho}(x_{0})}\left|F_{\varepsilon}\right|^{2}\,\operatorname{vol}_{g}$
(3.3)

where $C(M)>0$ is a constant depending only on $M$ and $g$.

Lemma 3.1 follows by the very same computations contained in, e.g., [27] (cf.
the computations starting at the beginning of Section 4 and ending right above
Equation (4.6) there). Therefore, we skip its proof.

The second lemma provides an estimate on the term
$\int_{\mathcal{B}_{\rho}(x_{0})}\left|F_{\varepsilon}\right|^{2}\,\operatorname{vol}_{g}$
in (3.3).

######  Lemma 3.2.

There exists $\bar{\rho}\in(0,\operatorname{inj}(M))$ and a constant
$\lambda>0$, both depending only on $M$ and $g$, such that the following
holds. If $(u_{\varepsilon},\,A_{\varepsilon})\in\mathcal{E}$ is any critical
point of $G_{\varepsilon}$, then for any $x_{0}\in M$ and any
$\rho\in(0,\,\bar{\rho})$ there holds

$\left\|F_{\varepsilon}\right\|_{L^{2}(\mathcal{B}_{\rho}(x_{0}))}\leq
K\rho\left\|\operatorname{D}_{A_{\varepsilon}}u_{\varepsilon}\right\|_{L^{2}(\mathcal{B}_{\rho}(x_{0}))}+\lambda\sqrt{\rho}\left\|{\rm
i}_{\nu}F_{\varepsilon}\right\|_{L^{2}(\partial\mathcal{B}_{\rho}(x_{0}))},$
(3.4)

where $K>0$ is a constant depending only on $M$ and $g$.

###### Proof.

The idea is to exploit both the Euler-Lagrange equations and gauge
transformations to rewrite
$\int_{\mathcal{B}_{\rho}(x_{0})}\left|F_{\varepsilon}\right|^{2}\,\operatorname{vol}_{g}$
in terms of quantities that can be estimated more easily. To this purpose, we
proceed step by step.

_Step 1._ Let $(u_{\varepsilon},\,A_{\varepsilon})\in(W^{1,2}\cap
L^{\infty})(M,\,E)\times W^{1,2}(M,\,\mathrm{T}^{*}M)$ be any critical point
of $G_{\varepsilon}$. Then, the curvature
$F_{\varepsilon}:=F_{A_{\varepsilon}}$ satisfies the Euler-Lagrange equation

$\operatorname{d}^{*}F_{\varepsilon}=j(u_{\varepsilon},\,A_{\varepsilon})$
(3.5)

in the sense of distributions on $M$, and hence in particular on
$\mathcal{B}_{\rho}(x_{0})$, for any fixed $x_{0}\in M$. Recall that, by
definition,
$\operatorname{D}_{A_{\varepsilon}}=\operatorname{D}_{0}-iA_{\varepsilon}$,
and that locally we may write
$\operatorname{D}_{0}=\operatorname{d}-i\gamma_{0}$, where $\gamma_{0}$ is a
closed 1-form. Thus, on the geodesic ball $\mathcal{B}_{\rho}(x_{0})$ we can
write
$\operatorname{D}_{A_{\varepsilon}}=\operatorname{d}-i\widetilde{A}_{\varepsilon}$,
where $\widetilde{A}_{\varepsilon}:=A_{\varepsilon}+\gamma_{0}$, and it holds
$\operatorname{d}\widetilde{A}_{\varepsilon}=F_{\varepsilon}$. From now on, we
shall work locally and, for convenience, we will drop the tilde, writing
$A_{\varepsilon}$ to mean, actually, $\widetilde{A}_{\varepsilon}$.

We fix Coulomb gauge in $\mathcal{B}_{\rho}(x_{0})$ (without changing
notation), so that $A_{\varepsilon}$ satisfies

$\left\\{\begin{aligned} &\operatorname{d}^{*}A_{\varepsilon}=0&&\mbox{in
}\mathcal{B}_{\rho}(x_{0}),\\\ &{\rm i}_{\nu}A_{\varepsilon}=0&&\mbox{on
}\partial\mathcal{B}_{\rho}(x_{0}),\\\
&\operatorname{d}A_{\varepsilon}=F_{\varepsilon}.\end{aligned}\right.$ (3.6)

(Such a gauge exists: see, e.g. [26, Proposition 2.1].) Moreover,
$A_{\varepsilon}$ is smooth up to the boundary of $\mathcal{B}_{\rho}(x_{0})$,
and hence so is $F_{\varepsilon}$ (recall that $F_{\varepsilon}$ is gauge-
invariant, so it is actually globally smooth).

_Step 2._ Using (3.6), we write

$\left|F_{\varepsilon}\right|^{2}\,\operatorname{vol}_{g}=\left\langle
F_{\varepsilon},\,F_{\varepsilon}\right\rangle=\left\langle
F_{\varepsilon},\,\operatorname{d}A_{\varepsilon}\right\rangle.$

Then, by the well-known integration-by-parts formula for differential forms
(see, for instance, [19, (2.22) and (2.34)]), we obtain

$\int_{\mathcal{B}_{\rho}(x_{0})}\left|F_{\varepsilon}\right|^{2}\,\operatorname{vol}_{g}=\int_{\mathcal{B}_{\rho}(x_{0})}\left\langle
j(u_{\varepsilon},\,A_{\varepsilon}),\,A_{\varepsilon}\right\rangle+\int_{\partial\mathcal{B}_{\rho}(x_{0})}A_{\varepsilon}\wedge(\star
F_{\varepsilon})_{\mathrm{T}}$ (3.7)

where $(\star F_{\varepsilon})_{\mathrm{T}}$ denotes the part of $\star
F_{\varepsilon}$ tangent to $\partial\mathcal{B}_{\rho}(x_{0})$. We are going
to estimate the terms at the right hand side of (3.7). Before doing this, we
notice that the Poincaré and Gaffney inequalities yield the estimate

$\left\|A_{\varepsilon}\right\|_{L^{2}(\mathcal{B}_{\rho}(x_{0}))}\lesssim\rho\left\|F_{\varepsilon}\right\|_{L^{2}(\mathcal{B}_{\rho}(x_{0}))},$
(3.8)

up to constant depending only on $M$ and $g$ (see Proposition A.1 in the
appendix for more details).

_Step 3._ Let
$\varphi=(x^{1},\dots,x^{n}):\mathcal{B}_{\rho}(x_{0})\to\mathbb{R}^{n}$ be
coordinates so that $\operatorname{d}{x^{1}},\dots,\operatorname{d}{x^{n-1}}$
are 1-forms tangent to $\partial\mathcal{B}_{\rho}(x_{0})$ and
$\operatorname{d}{x^{n}}$ is normal to $\mathcal{B}_{\rho}(x_{0})$, in the
sense of [19]. We may further assume that
$\\{\operatorname{d}{x^{1}},\dots,\operatorname{d}{x^{n}}\\}$ is an
orthonormal frame for $\mathrm{T}^{*}_{p}M$ at some _prescribed_
$p\in\partial\mathcal{B}_{\rho}(x_{0})$. (The existence of such coordinate
systems is well-known; see, for instance, [19, p. 49].) Then, we can split

$\star F_{\varepsilon}=\star\left(\sum_{1\leq i<j\leq
n-1}F_{ij}\operatorname{d}x^{i}\wedge\operatorname{d}x^{j}\right)+\star\left(\sum_{j=1}^{n-1}F_{jn}\operatorname{d}x^{j}\wedge\operatorname{d}x^{n}\right)$
(3.9)

In particular, at $p$,

$\begin{split}\star F_{\varepsilon}(p)=&\sum_{1\leq i<j\leq
n-1}F_{ij}(p)\,\savestack{\tmpbox}{\stretchto{\scaleto{\scalerel*[\widthof{\operatorname{d}x^{i}_{p}\wedge
dx^{j}_{p}}]{\kern-0.6pt\bigwedge\kern-0.6pt}{\rule[-505.89pt]{4.30554pt}{505.89pt}}}{}}{0.5ex}}\stackon[1pt]{\operatorname{d}x^{i}_{p}\wedge
dx^{j}_{p}}{\tmpbox}+\sum_{j=1}^{n-1}F_{jn}(p)\,\savestack{\tmpbox}{\stretchto{\scaleto{\scalerel*[\widthof{\operatorname{d}x^{j}_{p}\wedge
dx^{n}_{p}}]{\kern-0.6pt\bigwedge\kern-0.6pt}{\rule[-505.89pt]{4.30554pt}{505.89pt}}}{}}{0.5ex}}\stackon[1pt]{\operatorname{d}x^{j}_{p}\wedge
dx^{n}_{p}}{\tmpbox},\end{split}$ (3.10)

where the caret denotes omission (i.e.,
$\savestack{\tmpbox}{\stretchto{\scaleto{\scalerel*[\widthof{\operatorname{d}x^{i}_{p}\wedge
dx^{j}_{p}}]{\kern-0.6pt\bigwedge\kern-0.6pt}{\rule[-505.89pt]{4.30554pt}{505.89pt}}}{}}{0.5ex}}\stackon[1pt]{\operatorname{d}x^{i}_{p}\wedge
dx^{j}_{p}}{\tmpbox}$ is the wedge product of $\operatorname{d}x^{k}_{p}$ over
all indices $k$, in increasing order, except $k=i$ and $k=j$). Thus, the first
terms at right hand side of (3.10) is the normal part of $\star
F_{\varepsilon}(p)$ to $\partial\mathcal{B}_{\rho}(x_{0})$ while the second
one is its tangent part, and we have

$\sum_{j=1}^{n-1}F_{jn}(p)\,\savestack{\tmpbox}{\stretchto{\scaleto{\scalerel*[\widthof{\operatorname{d}x^{j}_{p}\wedge\operatorname{d}x^{n}_{p}}]{\kern-0.6pt\bigwedge\kern-0.6pt}{\rule[-505.89pt]{4.30554pt}{505.89pt}}}{}}{0.5ex}}\stackon[1pt]{\operatorname{d}x^{j}_{p}\wedge\operatorname{d}x^{n}_{p}}{\tmpbox}=\star\left({\rm
i}_{\nu}F_{\varepsilon}\wedge\nu^{\flat}\right)_{p},$ (3.11)

Since $p$ can be prescribed arbitrarily on $\partial\mathcal{B}_{\rho}(x_{0})$
and $\star F_{\varepsilon}$ is invariant under changes of coordinates, we
deduce that

$(\star F_{\varepsilon})_{\rm T}=\star\left({\rm
i}_{\nu}F_{\varepsilon}\wedge\nu^{\flat}\right)$ (3.12)

on $\partial\mathcal{B}_{\rho}(x_{0})$.

_Step 4._ By (3.12), we obtain

$\begin{split}\int_{\partial\mathcal{B}_{\rho}(x_{0})}A_{\varepsilon}\wedge(\star
F_{\varepsilon})_{\mathrm{T}}&=\int_{\partial\mathcal{B}_{\rho}(x_{0})}\left\langle
A_{\varepsilon},\,({\rm
i}_{\nu}F_{\varepsilon})\wedge\nu^{\flat}\right\rangle\\\
&\leq\left\|A_{\varepsilon}\right\|_{L^{2}(\partial\mathcal{B}_{\rho}(x_{0}))}\left\|{\rm
i}_{\nu}F_{\varepsilon}\right\|_{L^{2}(\partial\mathcal{B}_{\rho}(x_{0}))}.\end{split}$
(3.13)

On the other hand, the trace inequality and Gaffney’s inequality imply that
there exists a constant $\lambda>0$, depending only on $(M,\,g)$, such that

$\left\|A_{\varepsilon}\right\|_{L^{2}(\partial\mathcal{B}_{\rho}(x_{0}))}\leq\lambda\sqrt{\rho}\left\|F_{\varepsilon}\right\|_{L^{2}(\mathcal{B}_{\rho}(x_{0}))}.$
(3.14)

(see (A.3) in the appendix for details). Recalling the easy estimate

$\left\|j(u_{\varepsilon},\,A_{\varepsilon})\right\|_{L^{2}(\mathcal{B}_{\rho}(x_{0}))}\leq\left\|\operatorname{D}_{A_{\varepsilon}}u_{\varepsilon}\right\|_{L^{2}(\mathcal{B}_{\rho}(x_{0}))},$
(3.15)

and combining (3.7), (3.13), (3.14), and (3.15) with (3.7), we obtain

$\left\|F_{\varepsilon}\right\|^{2}_{L^{2}(\mathcal{B}_{\rho}(x_{0}))}\leq\left(K\rho\left\|\operatorname{D}_{A_{\varepsilon}}u_{\varepsilon}\right\|_{L^{2}(\mathcal{B}_{\rho}(x_{0}))}+\lambda\sqrt{\rho}\left\|{\rm
i}_{\nu}F_{\varepsilon}\right\|_{L^{2}(\partial
B_{\rho}(x_{0}))}\right)\left\|F_{\varepsilon}\right\|_{L^{2}(\mathcal{B}_{\rho}(x_{0}))}.$
(3.16)

As (3.4) is obvious if
$\left\|F_{\varepsilon}\right\|_{L^{2}(\mathcal{B}_{\rho}(x_{0}))}=0$, we may
assume $\left\|F_{\varepsilon}\right\|_{L^{2}(\mathcal{B}_{\rho}(x_{0}))}>0$.
Then, dividing both sides of (3.16) by
$\left\|F_{\varepsilon}\right\|_{L^{2}(\mathcal{B}_{\rho}(x_{0}))}$ yields the
conclusion. ∎

######  Proposition 3.3.

There exist a number $\alpha\in(0,\,2]$, a number
$\bar{\rho}\in(0,\,\operatorname{inj}(M))$ and a constant $C>0$, depending
only on $M$ and $g$, such that the following holds. If
$(u_{\varepsilon},\,A_{\varepsilon})\in\mathcal{E}$ is any critical point of
$G_{\varepsilon}$, then for any $x_{0}$ in $M$ and any
$\rho\in(0,\,\bar{\rho})$, there holds

$\frac{\operatorname{d}}{\operatorname{d}\rho}\left(\frac{e^{C\rho^{2}}\,E_{\varepsilon}(x_{0},\,\rho)}{\rho^{n-4+\alpha}}\right)>0$
(3.17)

where $E_{\varepsilon}(x_{0},\,\rho)$ is defined by (3.1).

###### Proof.

Assume, for the moment being, $\alpha$ is any (fixed) nonnegative number.
Using (3.3), we compute

$\begin{split}&\frac{\operatorname{d}}{\operatorname{d}\rho}\left(\frac{e^{C\rho^{2}}\,E_{\varepsilon}(x_{0},\,\rho)}{\rho^{n-4+\alpha}}\right)\\\
&=\frac{e^{C\rho^{2}}}{\rho^{n-4+\alpha}}\left[\left(2C\rho-\frac{n-4+\alpha}{\rho}\right)E_{\varepsilon}(x_{0},\,\rho)+\int_{\partial\mathcal{B}_{\rho}(x_{0})}e_{\varepsilon}(u_{\varepsilon},\,A_{\varepsilon})\right]\\\
&\stackrel{{\scriptstyle\eqref{eq:PS-
ineq}}}{{\geq}}\frac{e^{C\rho^{2}}}{\rho^{n-4+\alpha}}\left\\{\vphantom{\int_{\mathcal{B}_{\rho}(x_{0})}\frac{\left(1-\left|u_{\varepsilon}\right|^{2}\right)^{2}}{4\varepsilon^{2}}\,\operatorname{vol}_{g}}\left(2C-C(M)\right)\rho
E_{\varepsilon}(x_{0},\,\rho)+X_{\varepsilon}(x_{0},\,\rho)\right.\\\
&\left.+\frac{2-\alpha}{\rho}\int_{\mathcal{B}_{\rho}(x_{0})}\left(\frac{1}{2}\left|\operatorname{D}_{A_{\varepsilon}}u_{\varepsilon}\right|^{2}+\frac{\left(1-\left|u_{\varepsilon}\right|^{2}\right)^{2}}{4\varepsilon^{2}}\right)\,\operatorname{vol}_{g}-\frac{\alpha}{2\rho}\int_{\mathcal{B}_{\rho}(x_{0})}\left|F_{\varepsilon}\right|^{2}\,\operatorname{vol}_{g}\right\\},\end{split}$

for any $\rho\in(0,\,\operatorname{inj}(M))$. Then, by (3.4) and Young’s
inequality we obtain

$\begin{split}&\frac{\operatorname{d}}{\operatorname{d}\rho}\left(\frac{e^{C\rho^{2}}}{\rho^{n-4+\alpha}}\int_{B_{\rho}(x_{0})}e_{\varepsilon}(u_{\varepsilon},\,A_{\varepsilon})\,\operatorname{vol}_{g}\right)\\\
&\geq\frac{e^{C\rho^{2}}}{\rho^{n-4+\alpha}}\left\\{\vphantom{\int_{\mathcal{\partial
B}_{\rho}(x_{0})}\left|{\rm
i}_{\nu}F_{\varepsilon}\right|^{2}\,\operatorname{vol}_{\hat{g}}}\left(2C-C(M)\right)\rho
E_{\varepsilon}(x_{0},\,\rho)+X_{\varepsilon}(\rho,\,x_{0})\right.\\\
&\left.+\frac{2-\alpha}{\rho}\int_{\mathcal{B}_{\rho}(x_{0})}\frac{1}{2}\left|\operatorname{D}_{A_{\varepsilon}}u_{\varepsilon}\right|^{2}+\frac{\left(1-\left|u_{\varepsilon}\right|^{2}\right)^{2}}{4\varepsilon^{2}}\,\operatorname{vol}_{g}\right.\\\
&\left.-2\alpha
K^{2}\rho\int_{\mathcal{B}_{\rho}(x_{0})}\left|\operatorname{D}_{A_{\varepsilon}}u_{\varepsilon}\right|^{2}\,\operatorname{vol}_{g}-2\alpha\lambda^{2}\int_{\mathcal{\partial
B}_{\rho}(x_{0})}\left|{\rm
i}_{\nu}F_{\varepsilon}\right|^{2}\,\operatorname{vol}_{\hat{g}}\right\\},\end{split}$

for any $\rho\in(0,\,\bar{\rho})$, with $\bar{\rho}$ as in Lemma 3.2, and
where $K$ and $\lambda$ are the same constants as in Lemma 3.2 (both of which
depend only on $M$ and $g$). Thus, recalling the definition (3.2) of
$X_{\varepsilon}(x_{0},\,\rho)$ and taking
$\alpha=\min\left\\{2,\,\frac{1}{2\lambda^{2}},\,\frac{C(M)}{2K^{2}}\right\\}$
and (for instance) $C=2C(M)$, the proposition follows. ∎

With Proposition 3.3 at hand, we can prove a decay estimate for the
$L^{2}$-norm of the curvatures $F_{\varepsilon}$ in sufficiently small
geodesic balls. In turn, such decay property is the key point to obtain the
$(n-2)$-monotonicity formula (15).

######  Lemma 3.4.

There exist a number $\alpha\in(0,\,2)$ and a number
$R_{0}\in(0,\,\operatorname{inj}(M))$, both depending only on $M$ and $g$,
such that the following holds. If
$(u_{\varepsilon},\,A_{\varepsilon})\in\mathcal{E}$ is any critical point of
$G_{\varepsilon}$, then for any $x_{0}\in M$ and any $0<r<R<R_{0}$, there
holds

$\int_{\mathcal{B}_{r}(x_{0})}\left|F_{\varepsilon}\right|^{2}\,\operatorname{vol}_{g}\lesssim\frac{r^{n-2+\alpha}}{R^{n-2+\alpha}}\int_{\mathcal{B}_{R}(x_{0})}e_{\varepsilon}(u_{\varepsilon},\,A_{\varepsilon})\,\operatorname{vol}_{g},$
(3.18)

where the implicit constant in of the right-hand-side depends only on $M$ and
$g$.

###### Proof.

We start by fixing $x_{0}\in M$ and considering two geodesic balls
$\mathcal{B}_{r}:=\mathcal{B}_{r}(x_{0})$ and
$\mathcal{B}_{s}:=\mathcal{B}_{s}(x_{0})$, with $0<r<s<R$ and
$R\in(0,\,\operatorname{inj}(M))$ small enough (to be further specified
later). Up to subtracting a (smooth, $\varepsilon$-independent) 1-form to
$A_{\varepsilon}$, we can write

$\operatorname{d}A_{\varepsilon}=F_{\varepsilon}\qquad\mbox{in
}\mathcal{B}_{R}.$

Moreover, we can fix Coulomb gauge (without changing notation), i.e., we can
assume that

$\operatorname{d}^{*}A_{\varepsilon}=0\quad\mbox{in }\mathcal{B}_{R},\quad{\rm
i}_{\nu}A_{\varepsilon}=0\quad\mbox{on }\partial\mathcal{B}_{R},$

where $\nu$ is the unit normal field to $\partial\mathcal{B}_{R}$. Then, the
Euler-Lagrange equation (6) becomes

$-\Delta A_{\varepsilon}=j(u_{\varepsilon},\,A_{\varepsilon})\qquad\mbox{in
}\mathcal{B}_{R}.$

Standard decay estimates for elliptic equations with a right-hand side (see
e.g. Proposition B.4 for more details) imply that there exists $R_{*}>0$
(depending only on $M$ and $g$) so that, for $0<r<s<R<R_{*}$, there holds

$\int_{\mathcal{B}_{r}}\left(\left|\operatorname{d}A_{\varepsilon}\right|^{2}+\left|\operatorname{d}^{*}A_{\varepsilon}\right|^{2}\right)\operatorname{vol}_{g}\lesssim\frac{r^{n}}{s^{n}}\int_{\mathcal{B}_{s}}\left(\left|\operatorname{d}A_{\varepsilon}\right|^{2}+\left|\operatorname{d}^{*}A_{\varepsilon}\right|^{2}\right)\operatorname{vol}_{g}+s^{2}\int_{\mathcal{B}_{s}}\left|j(u_{\varepsilon},\,\operatorname{D}_{A_{\varepsilon}})\right|^{2}\,\operatorname{vol}_{g},$

up to a constant depending only on $M$ and $g$. Keeping in mind that
$\operatorname{d}^{*}A_{\varepsilon}=0$ and
$\operatorname{d}A_{\varepsilon}=F_{\varepsilon}$ in $\mathcal{B}_{R}$, and
recalling the pointwise estimate
$\left|j(u_{\varepsilon},\,A_{\varepsilon})\right|\leq\left|\operatorname{D}_{A_{\varepsilon}}u_{\varepsilon}\right|$,
it follows

$\int_{\mathcal{B}_{r}}\left|F_{\varepsilon}\right|^{2}\,\operatorname{vol}_{g}\lesssim\frac{r^{n}}{s^{n}}\int_{\mathcal{B}_{s}}\left|F_{\varepsilon}\right|^{2}\,\operatorname{vol}_{g}+s^{2}\int_{\mathcal{B}_{s}}\left|\operatorname{D}_{A_{\varepsilon}}u_{\varepsilon}\right|^{2}\,\operatorname{vol}_{g},$
(3.19)

up to a constant depending only on $M$ and $g$.

Taking now $R_{0}=\min\\{R_{*},\,\bar{\rho}\\}$, where $\bar{\rho}$ is the
distinguished radius (depending only on $M$ and $g$) given by Proposition 3.3
and shrinking $R$ so that $0<r<s<R<R_{0}$, by (3.19) and (3.17) we obtain

$\begin{split}\int_{\mathcal{B}_{r}}\left|F_{\varepsilon}\right|^{2}\,\operatorname{vol}_{g}&\lesssim\frac{r^{n}}{s^{n}}\int_{\mathcal{B}_{s}}\left|F_{\varepsilon}\right|^{2}\,\operatorname{vol}_{g}+s^{2}\int_{\mathcal{B}_{s}}e_{\varepsilon}(u_{\varepsilon},\,A_{\varepsilon})\,\operatorname{vol}_{g}\\\
&\lesssim\frac{r^{n}}{s^{n}}\int_{\mathcal{B}_{s}}\left|F_{\varepsilon}\right|^{2}\,\operatorname{vol}_{g}+\frac{s^{n-2+\alpha}}{R^{n-4+\alpha}}\int_{\mathcal{B}_{R}}e_{\varepsilon}(u_{\varepsilon},\,A_{\varepsilon})\,\operatorname{vol}_{g},\end{split}$
(3.20)

for $\alpha\in(0,2]$ as given by Proposition 3.3.

Assume $\alpha\in(0,\,2)$. Then, by a classical iteration lemma (e.g., [18,
Lemma III.2.1] or [6, Lemma B.3]), it follows

$\begin{split}\int_{\mathcal{B}_{r}}\left|F_{\varepsilon}\right|^{2}\,\operatorname{vol}_{g}&\lesssim\frac{r^{n-2+\alpha}}{s^{n-2+\alpha}}\int_{\mathcal{B}_{s}}\left|F_{\varepsilon}\right|^{2}\operatorname{vol}_{g}+\frac{r^{n-2+\alpha}}{R^{n-4+\alpha}}\int_{\mathcal{B}_{R}}e_{\varepsilon}(u_{\varepsilon},\,A_{\varepsilon})\,\operatorname{vol}_{g}\\\
&\lesssim\frac{r^{n-2+\alpha}}{s^{n-2+\alpha}}\int_{\mathcal{B}_{s}}\left|F_{\varepsilon}\right|^{2}\,\operatorname{vol}_{g}+\frac{r^{n-2+\alpha}}{R^{n-2+\alpha}}\int_{\mathcal{B}_{R}}e_{\varepsilon}(u_{\varepsilon},\,A_{\varepsilon})\,\operatorname{vol}_{g}\end{split}$
(3.21)

for any $0<r<s<R<R_{0}$. (The second inequality holds because $R$ is certainly
bounded from above.) In particular, by taking the limit $s\to R$, we get the
desired decay estimate (3.18). ∎

######  Remark 3.5.

The restriction $\alpha<2$ is necessary in the argument from (3.20) to (3.21),
as otherwise we could not use [18, Lemma III.2.1] or [6, Lemma B.3]. This
inconvenience will produce the “error term”, i.e., the term depending on
$\alpha$, in (15). Nevertheless, the monotonicity formula (15) is enough for
our purposes.

We are now ready to prove Theorem 3.

###### Proof of Theorem 3.

Thanks to Lemma 3.4, the inequality (3.3) yields

$\rho E^{\prime}_{\varepsilon}(x_{0},\,\rho)\geq\rho
X_{\varepsilon}(x_{0},\,\rho)+(n-2)E_{\varepsilon}(x_{0},\,\rho)+C\rho^{2}E_{\varepsilon}(x_{0},\,\rho)-\frac{C\rho^{n-2+\alpha}}{R^{n-2+\alpha}}E_{\varepsilon}(x_{0},\,R)$
(3.22)

for any $0<\rho<R<R_{0}$, where $R_{0}$ is the distinguished radius in
$(0,\,\operatorname{inj}(M))$ given by Lemma 3.4, any $x_{0}\in M$, and some
sufficiently large constant $C$ (depending on $M$, $g$ only).

Dividing both sides of (3.22) by $\rho^{n-1}$, we obtain

$\begin{split}\rho^{2-n}E^{\prime}_{\varepsilon}(x_{0},\,\rho)&-(n-2)\rho^{1-n}E_{\varepsilon}(x_{0},\,\rho)+C\rho^{3-n}E_{\varepsilon}(x_{0},\,\rho)\\\
&\geq\rho^{2-n}X_{\varepsilon}(x_{0},\,\rho)-\frac{C\rho^{\alpha-1}}{R^{n-2+\alpha}}E_{\varepsilon}(x_{0},\,R)\end{split}$
(3.23)

Multiplying (3.23) by $\exp(C\rho^{2}/2)$, where $C$ is exactly the same
constant as in (3.23), we get

$\frac{\operatorname{d}}{\operatorname{d}\rho}\left\\{e^{C\rho^{2}/2}\rho^{2-n}E_{\varepsilon}(x_{0},\,\rho)\right\\}\geq
e^{C\rho^{2}/2}\rho^{2-n}X_{\varepsilon}(x_{0},\,\rho)-\frac{Ce^{C\rho^{2}/2}\rho^{\alpha-1}}{R^{n-2+\alpha}}E_{\varepsilon}(x_{0},\,R),$
(3.24)

and the conclusion follows by integrating both sides of (3.24) over
$\rho\in(r,\,R)$ and recalling that $\alpha$ depends only on $(M,\,g)$. ∎

######  Remark 3.6.

Multiplying (3.24) by $\rho$ and recalling the definition (3.2) of
$X_{\varepsilon}$, we obtain the following variant of (3.24): for any
$0<\rho<R<R_{0}$,

$\rho\,\frac{\operatorname{d}}{\operatorname{d}\rho}\left(\frac{e^{C\rho^{2}}E(x_{0},\,\rho)}{\rho^{n-2}}\right)\geq\frac{1}{2\varepsilon^{2}\rho^{n-2}}\int_{\mathcal{B}_{\rho}}\left(1-\left|u_{\varepsilon}\right|^{2}\right)^{2}\operatorname{vol}_{g}-\frac{C^{\prime}\,\rho^{\alpha}\,E(x_{0},\,R)}{R^{n-2+\alpha}}$
(3.25)

where $\alpha\in(0,\,2)$, $C$, $C^{\prime}$ are absolute constants that depend
only on $M$ and $g$. We will use (3.25) in the proof of the clearing-out, in
Section 4.

As a byproduct of our arguments, we obtain the following bound on the
curvature in terms of the rescaled energy:

######  Proposition 3.7.

For any $q\in(0,\,n)$ there exist constant $C_{q}>0$, depending only on
$(M,g)$ and $q$, such that the follwing statement holds. If
$(u_{\varepsilon},\,A_{\varepsilon})\in\mathcal{E}$ is any critical point of
$G_{\varepsilon}$, then

$\int_{\mathcal{B}_{r}(x_{0})}\left|F_{\varepsilon}\right|^{2}\,\operatorname{vol}_{g}\leq\frac{C_{q}\,r^{q}}{R^{n-2}}E_{\varepsilon}(x_{0},\,R)$
(3.26)

holds for any $x_{0}\in M$ and $0<r<R$.

###### Proof.

By a covering argument, there is no loss of generality in assuming that
$R<R_{0}$, where $R_{0}$ is a uniform constant. For instance, we can take
$R_{0}$ exactly equal to the constant given by Lemma 3.4. Let
$\alpha\in(0,\,2]$ be given by Proposition 3.3, and let $0<r<s<R<R_{*}$. The
estimate (3.20), obtained in the proof of Lemma 3.4, and the (almost)
monotonicity formula (15) imply

$\begin{split}\int_{\mathcal{B}_{r}}\left|F_{\varepsilon}\right|^{2}\,\operatorname{vol}_{g}&\lesssim\frac{r^{n}}{s^{n}}\int_{\mathcal{B}_{s}}\left|F_{\varepsilon}\right|^{2}\,\operatorname{vol}_{g}+s^{2}\int_{\mathcal{B}_{s}}e_{\varepsilon}(u_{\varepsilon},\,A_{\varepsilon})\,\operatorname{vol}_{g}\\\
&\lesssim\frac{r^{n}}{s^{n}}\int_{\mathcal{B}_{s}}\left|F_{\varepsilon}\right|^{2}\,\operatorname{vol}_{g}+\frac{s^{n}}{R^{n-2}}\int_{\mathcal{B}_{R}}e_{\varepsilon}(u_{\varepsilon},\,A_{\varepsilon})\,\operatorname{vol}_{g},\end{split}$

The proposition follows by an iteration argument, based on [18, Lemma
III.2.1]. ∎

Finally, we conclude this section by stating and proving a corollary of the
monotonicity formula, which will be useful in Section 4.

######  Corollary 3.8.

Let $\varepsilon>0$ and let
$(u_{\varepsilon},\,A_{\varepsilon})\subset\mathcal{E}$ be any critical point
of $G_{\varepsilon}$. Moreover, let $\alpha\in(0,\,2)$,
$R_{0}\in(0,\,\operatorname{inj}(M))$ and $C>0$ be the numbers (depending only
on $(M,\,g)$) given by Theorem 3. Then, for any $x_{0}\in M$ and any
$R\in(0,\,R_{0})$, there holds

$\frac{1}{\varepsilon^{2}}\int_{B_{R}(x_{0})}\frac{\left(1-\left|u_{\varepsilon}(y)\right|^{2}\right)^{2}}{\operatorname{dist}^{n-2}(y,\,x_{0})}\,\operatorname{vol}_{g}(y)\lesssim
R^{2-n}E_{\varepsilon}(x_{0},\,R),$ (3.27)

where the implicit constant in front of the right hand side depends only on
$(M,\,g)$.

###### Proof.

By Proposition 1.2, Lemma 1.5 and Lemma 1.6, the energy density is a bounded
function (and its $L^{\infty}(M)$-norm is of order
$\mathrm{O}(\varepsilon^{-2})$). Consequently, there holds

$\lim_{r\to 0}r^{2-n}E_{\varepsilon}(x_{0},\,r)=0.$ (3.28)

Let us set

$W_{\varepsilon}(x_{0},\,\rho):=\int_{\mathcal{B}_{\rho}(x_{0})}\frac{\left(1-\left|u_{\varepsilon}(y)\right|^{2}\right)^{2}}{2\varepsilon^{2}}\operatorname{vol}_{g}(y).$

By (3.24) and (3.2), we clearly obtain

$\frac{\operatorname{d}}{\operatorname{d}\rho}\left\\{e^{C\rho^{2}/2}\rho^{2-n}E_{\varepsilon}(x_{0},\,\rho)\right\\}\geq\rho^{1-n}\int_{\mathcal{B}_{\rho}(x_{0})}\frac{\left(1-\left|u_{\varepsilon}(y)\right|^{2}\right)^{2}}{2\varepsilon^{2}}\operatorname{vol}_{g}(y)-\frac{Ce^{C\rho^{2}/2}\rho^{\alpha-1}}{R^{n-2+\alpha}}E_{\varepsilon}(x_{0},\,R).$

From now on, we work in polar coordinates in $\mathcal{B}_{\rho}$ (centered at
$x_{0}$). Since the function $(0,\,R)\ni\rho\mapsto
W_{\varepsilon}(x_{0},\,\rho)$ is absolutely continuous, for a.e.
$\rho\in(0,R)$ we have

$W^{\prime}_{\varepsilon}(x_{0},\,\rho)=\int_{\partial\mathcal{B}_{\rho}(x_{0})}\frac{\left(1-\left|u_{\varepsilon}(y)\right|^{2}\right)^{2}}{2\varepsilon^{2}}\operatorname{vol}_{\hat{g}}(y)$

as well as

$\rho^{1-n}W_{\varepsilon}(x_{0},\,\rho)=\frac{1}{n-2}\left\\{-\frac{\operatorname{d}}{\operatorname{d}\rho}\left(\rho^{2-n}W_{\varepsilon}(x_{0},\,\rho)\right)+\rho^{2-n}W^{\prime}_{\varepsilon}(x_{0},\,\rho)\right\\}.$

Hence,

$\begin{split}\frac{\operatorname{d}}{\operatorname{d}\rho}\Big{\\{}e^{C\rho^{2}/2}\rho^{2-n}E_{\varepsilon}(x_{0},\,\rho)&+\frac{1}{n-2}\rho^{2-n}W_{\varepsilon}(x_{0},\,\rho)\Big{\\}}\geq\\\
&\frac{\rho^{2-n}}{n-2}W^{\prime}_{\varepsilon}(x_{0},\,\rho)-\frac{Ce^{C\rho^{2}/2}\rho^{\alpha-1}}{R^{n-2+\alpha}}E_{\varepsilon}(x_{0},\,R)\end{split}$
(3.29)

for a.e. $\rho\in(0,\,R)$. Integrating both sides of (3.29) over $(0,\,R)$,
and taking (3.28) into account, yields

$\begin{split}\frac{1}{n-2}&\int_{\mathcal{B}_{R}(x_{0})}\frac{1}{2\varepsilon^{2}}\frac{\left(1-\left|u_{\varepsilon}(y)\right|^{2}\right)^{2}}{\operatorname{dist}(x_{0},y)^{n-2}}\,\operatorname{vol}_{g}(y)\\\
&\leq\left(1+\frac{C}{\alpha}\right)e^{CR^{2}/2}R^{2-n}E_{\varepsilon}(x_{0},\,R)+\frac{R^{2-n}}{n-2}\int_{\mathcal{B}_{R}(x_{0})}\frac{\left(1-\left|u_{\varepsilon}(y)\right|^{2}\right)^{2}}{2\varepsilon^{2}}\,\operatorname{vol}_{g}(y)\\\
&\leq\left(\frac{n}{n-2}+\frac{C}{\alpha}\right)R^{2-n}E_{\varepsilon}(x_{0},\,R).\end{split}$

Recalling that $\alpha$ depends only on $(M,\,g)$, the conclusion follows. ∎

##  4 Clearing-out and its consequences

The aim of this section is to prove Proposition 4.1 below, which will be
essential in characterising the support of the energy-concentration measure
$\mu_{*}$, defined in (5.1) (see Section 5 and Lemma 5.6 in particular). As in
the previous section, given a finite-energy pair
$(u_{\varepsilon},\,A_{\varepsilon})$, a point $x_{0}\in M$ and a radius
$\rho>0$, we write
$E_{\varepsilon}(x_{0},\,\rho):=G_{\varepsilon}(u_{\varepsilon},\,A_{\varepsilon};\,\mathcal{B}_{\rho}(x_{0}))$.

######  Proposition 4.1.

There exists constants $\eta_{0}>0$, $R_{*}\in(0,\operatorname{inj}(M))$ and
$\varepsilon_{*}>0$, depending on $(M,g)$ only, such that the following
statement holds. Let $x_{0}\in M$, $R\in(0,R_{*})$ and let
$\\{(u_{\varepsilon},\,A_{\varepsilon})\\}\subset(W^{1,2}\cap
L^{\infty})(M,\,E)\times W^{1,2}(M,\,\mathrm{T}^{*}M)$ be a sequence of
critical points of $G_{\varepsilon}$ that satisfy (8) and

$E_{\varepsilon}(x_{0},R)\leq\eta_{0}\,R^{n-2}\log\frac{R}{\varepsilon}.$
(4.1)

Then, for any $\varepsilon<\varepsilon_{*}$, there holds

$\textrm{for any
}x\in\mathcal{B}_{3R/4}(x_{0}),\quad\left|u_{\varepsilon}(x)\right|\geq\frac{1}{2}.$
(4.2)

Moreover, there holds

$\sup_{\varepsilon>0}E_{\varepsilon}(x_{0},R/2)<+\infty.$ (4.3)

Proposition 4.1 is the counterpart of [8, Proposition VII.1]. For its proof,
we shall follow the same strategy as in [8], involving essentially three
ingredients: energy decay estimates, a clearing-out property, and elliptic
estimates.

The pivotal point of the argument is the “clearing-out” property, Proposition
4.6. Indeed, this property implies that Equation (6) is uniformly elliptic in
any geodesic ball in which (4.1) holds. In turn, uniform ellipticity implies
several strong elliptic estimates that allow to achieve the conclusion of
Proposition 4.1.

According to [8], the main concern towards the proof of the clearing-out
property lies in obtaining a suitable, quantitative decay of the rescaled
energy in small balls with respect to the radius. This is done in Section 4.1
below.

Throughout Section 4, we will consider a sequence of critical points
$\\{(u_{\varepsilon},\,A_{\varepsilon})\\}\subset(W^{1,2}\cap
L^{\infty})(M,\,E)\times W^{1,2}(M,\,\mathrm{T}^{*}M)$, a point $x_{0}\in M$
and a radius $R>0$ (smaller than the injectivity radius of $M$). We will write
indifferently $\mathcal{B}_{r}$ or $\mathcal{B}_{r}(x_{0})$ for a geodesic
ball of center $x_{0}$ and radius $r>0$.

###  4.1 An energy decay estimate

Towards the proof of Proposition 4.1, the first step is an energy decay
estimate, analogous to [8, Theorem 3].

######  Proposition 4.2.

There exist numbers $R_{*}>0$ and $\alpha\in(0,\,2)$, depending on $(M,\,g)$
only, such that the following statement holds. Let $x_{0}\in M$,
$0<\varepsilon<R<R_{*}$ and $0<\delta<1/10$. Let
$(u_{\varepsilon},\,A_{\varepsilon})$ be a critical point of
$G_{\varepsilon}$. Define

$p_{\varepsilon}:=\frac{1}{\varepsilon^{2}\,R^{n-2}}\int_{\mathcal{B}_{R}(x_{0})}\left(1-\left|u_{\varepsilon}\right|^{2}\right)^{2}\operatorname{vol}_{g}$
(4.4)

Then, there holds

$\begin{split}\frac{E_{\varepsilon}(x_{0},\,\delta
R)}{R^{n-2}}\lesssim\left(p_{\varepsilon}^{1/3}+\delta^{n-2+\alpha}\right)\frac{E_{\varepsilon}(x_{0},\,R)}{R^{n-2}}+p_{\varepsilon}^{2/3}\end{split}$

where

The rest of the section is devoted to the proof of Proposition 4.2. Let
$x_{0}\in M$, $0<R<\operatorname{inj}(M)$, $\varepsilon>0$ and $0<\delta<1/10$
be fixed, and let $(u_{\varepsilon},\,A_{\varepsilon})$ be a critical point of
$G_{\varepsilon}$. First of all, we choose a radius $r\in(R/10,\,R/5)$ such
that

$\displaystyle\int_{\partial\mathcal{B}_{r}}\left|\operatorname{D}_{A_{\varepsilon}}u_{\varepsilon}\right|^{2}\operatorname{d}\mathcal{H}^{n-1}$
$\displaystyle\leq\frac{40}{R}\int_{\mathcal{B}_{R}}\left|\operatorname{D}_{A_{\varepsilon}}u_{\varepsilon}\right|^{2}\operatorname{vol}_{g}$
(4.5)
$\displaystyle\int_{\partial\mathcal{B}_{r}}\left(\left|u_{\varepsilon}\right|^{2}-1\right)^{2}\operatorname{d}\mathcal{H}^{n-1}$
$\displaystyle\leq\frac{40}{R}\int_{\mathcal{B}_{R}}\left(\left|u_{\varepsilon}\right|^{2}-1\right)^{2}\operatorname{vol}_{g}$
(4.6)

Such a radius exist, because the set of radii $r\in(R/10,\,R/5)$ that do not
satisfy either of the inequalities (4.5) has length $R/40$ at most. For such a
choice of $r$, we have $r>\delta R$. Let $\eta_{r}$ be the indicator function
of the ball $\mathcal{B}_{r}$ (i.e, $\eta_{r}(x):=1$ if $x\in\mathcal{B}_{r}$
and $\eta_{r}(x):=0$ otherwise). By Hodge decomposition, there exist
$\varphi_{\varepsilon}\in W^{1,2}(M)$, $\psi_{\varepsilon}\in
W^{1,2}(M,\,\Lambda^{2}\mathrm{T}^{*}M)$ and
$\xi_{\varepsilon}\in\operatorname{Harm}^{1}(M)$ such that

$\eta_{r}\,j(u_{\varepsilon},\,A_{\varepsilon})=\operatorname{d}\varphi_{\varepsilon}+\operatorname{d}^{*}\psi_{\varepsilon}+\xi_{\varepsilon}$
(4.7)

where the three terms in the right-hand side are orthogonal in $L^{2}(M)$.
Moreover, there is no loss of generality in assuming that $\psi_{\varepsilon}$
is exact (if not, we replace $\psi_{\varepsilon}$ by its projection onto exact
forms). We prove decay estimates on $\varphi_{\varepsilon}$,
$\psi_{\varepsilon}$ and $\xi_{\varepsilon}$ separately.

######  Lemma 4.3.

We have

$\int_{\mathcal{B}_{\delta
R}}\left(\left|\operatorname{d}\varphi_{\varepsilon}\right|^{2}+\left|\xi_{\varepsilon}\right|^{2}\right)\operatorname{vol}_{g}\lesssim\delta^{n}E_{\varepsilon}(x_{0},\,R)$

###### Proof.

We consider $\varphi_{\varepsilon}$ first. In the interior of
$\mathcal{B}_{r}$, we have

$-\Delta\varphi_{\varepsilon}=\operatorname{d}^{*}\operatorname{d}\varphi_{\varepsilon}=\operatorname{d}^{*}j(u_{\varepsilon},\,A_{\varepsilon})=\operatorname{d}^{*}\operatorname{d}^{*}F_{\varepsilon}=0$

By differentiating both sides of $-\Delta\varphi_{\varepsilon}=0$, we deduce
that $-\Delta(\operatorname{d}\varphi_{\varepsilon})=0$ in the interior of
$\mathcal{B}_{R}$. Then, Theorem B.3 implies

$\int_{\mathcal{B}_{\delta
R}}\left|\operatorname{d}\varphi_{\varepsilon}\right|^{2}\,\operatorname{vol}_{g}\lesssim\delta^{n}\int_{\mathcal{B}_{R}}\left|\operatorname{d}\varphi_{\varepsilon}\right|^{2}\,\operatorname{vol}_{g}\lesssim\delta^{n}\int_{M}\left|\operatorname{d}\varphi_{\varepsilon}\right|^{2}\,\operatorname{vol}_{g}$

As the decomposition (4.7) is orthogonal in $L^{2}(M)$, we deduce

$\int_{\mathcal{B}_{\delta
R}}\left|\operatorname{d}\varphi_{\varepsilon}\right|^{2}\,\operatorname{vol}_{g}\lesssim\delta^{n}\int_{\mathcal{B}_{R}}\left|j(u_{\varepsilon},\,A_{\varepsilon})\right|^{2}\,\operatorname{vol}_{g}\lesssim\delta^{n}E_{\varepsilon}(x_{0},\,R)$

where we have used that
$\left|j(u_{\varepsilon},\,A_{\varepsilon})\right|\leq\left|\operatorname{D}_{A_{\varepsilon}}u_{\varepsilon}\right|$
for the last inequality. An analogous argument applies to the harmonic form
$\xi_{\varepsilon}$. ∎

The proof of the decay estimate for $\psi_{\varepsilon}$ is more technical. It
is convenient to further decompose $\psi_{\varepsilon}$ as a sum of several
contributions, as in [8, Proof of Theorem 3, Step 2]. Let $\beta\in(0,\,1/4)$
be a parameter, to be chosen later. Let $f\colon[0,\,+\infty)\to[0,\,+\infty)$
be a smooth function, such that

$\displaystyle f(t)=1\quad\textrm{if }0\leq t\leq 1-2\beta,\qquad$
$\displaystyle f(t)=\frac{1}{t}\quad\textrm{if }t\geq 1-\beta,$ (4.8)
$\displaystyle 1\leq f(t)\leq\frac{1}{t}\quad\textrm{if
}1-2\beta<t<1-\beta,\qquad$ $\displaystyle\left|f^{\prime}(t)\right|\leq
4\qquad\textrm{for any }t\geq 0$ (4.9)

We define

$\rho_{\varepsilon}:=f(\left|u_{\varepsilon}\right|),\qquad
v_{\varepsilon}:=\rho_{\varepsilon}u_{\varepsilon}$ (4.10)

By construction, the function $\rho_{\varepsilon}$ satisfies

$0\leq\rho_{\varepsilon}^{2}-1\leq 4\beta\qquad\textrm{in }M.$ (4.11)

Moreover, writing the reference connection as
$\operatorname{D}_{0}=\operatorname{d}-i\gamma_{0}$ for some some (locally
defined) $1$-form $\gamma_{0}$, we have

$j(v_{\varepsilon},\,A_{\varepsilon})=\left\langle\operatorname{d}\rho_{\varepsilon}\otimes
u_{\varepsilon}+\rho_{\varepsilon}\,\operatorname{d}u_{\varepsilon}-i(\gamma_{0}+A_{\varepsilon})u_{\varepsilon},\,i\rho_{\varepsilon}u_{\varepsilon}\right\rangle=\rho_{\varepsilon}^{2}\,j(u_{\varepsilon},\,A_{\varepsilon})$
(4.12)

Now, we consider the Hodge decomposition (4.7). As we have seen above, we can
assume with no loss of generality that $\psi_{\varepsilon}$ is exact. Then, by
differentiating both sides of (4.7), we obtain

$\begin{split}-\Delta\psi_{\varepsilon}=\operatorname{d}\operatorname{d}^{*}\psi_{\varepsilon}&=\operatorname{d}\left(\eta_{r}\,j(u_{\varepsilon},\,A_{\varepsilon})\right)\\\
&\stackrel{{\scriptstyle\eqref{v_BBO}}}{{=}}\operatorname{d}\left(\eta_{r}\,j(v_{\varepsilon},\,A_{\varepsilon})\right)+\operatorname{d}\left(\eta_{r}\,(1-\rho_{\varepsilon}^{2})\,j(u_{\varepsilon},\,A_{\varepsilon})\right)\\\
&=\eta_{r}\,\operatorname{d}j(v_{\varepsilon},\,A_{\varepsilon})-\sigma_{r}\,\nu^{\flat}\wedge
j(v_{\varepsilon},\,A_{\varepsilon})+\operatorname{d}\left(\eta_{r}\,(1-\rho_{\varepsilon}^{2})\,j(u_{\varepsilon},\,A_{\varepsilon})\right)\end{split}$

where $\sigma_{r}$ denotes the $(n-1)$-dimensional Hausdorff measure on
$\partial\mathcal{B}_{r}$ and $\nu$ is the outward unit vector to
$\partial\mathcal{B}_{r}$. Recalling the definition of Jacobian, Equation (4),
we deduce

$\begin{split}-\Delta\psi_{\varepsilon}&=2\eta_{r}\,J(v_{\varepsilon},\,A_{\varepsilon})-\eta_{r}\,F_{\varepsilon}-\sigma_{r}\,\nu^{\flat}\wedge
j(v_{\varepsilon},\,A_{\varepsilon})+\operatorname{d}\left(\eta_{r}\,(1-\rho_{\varepsilon}^{2})\,j(u_{\varepsilon},\,A_{\varepsilon})\right)\end{split}$

Therefore, we can write

$-\Delta\psi_{\varepsilon}=\omega_{1}+\omega_{2}+\omega_{3}+\omega_{4},$
(4.13)

where

$\displaystyle\omega_{1}:=2\eta_{r}\,J(v_{\varepsilon},\,A_{\varepsilon}),\qquad$
$\displaystyle\omega_{2}:=-\eta_{r}\,F_{\varepsilon}$ (4.14)
$\displaystyle\omega_{3}:=-\sigma_{r}\,\nu^{\flat}\wedge
j(v_{\varepsilon},\,A_{\varepsilon}),\qquad$
$\displaystyle\omega_{4}:=\operatorname{d}\left(\eta_{r}\,(1-\rho_{\varepsilon}^{2})\,j(u_{\varepsilon},\,A_{\varepsilon})\right)$
(4.15)

We have $\omega_{k}\in W^{-1,2}(M,\,\Lambda^{2}\mathrm{T}^{*}M)$ for each $k$
— in fact, $\omega_{1}$, $\omega_{2}$ belong to
$L^{2}(M,\,\Lambda^{2}\mathrm{T}^{*}M)$, $\omega_{4}$ is the differential of
an $L^{2}$-form, and $\omega_{3}\in W^{-1,2}(M,\,\Lambda^{2}\mathrm{T}^{*}M)$
by continuity of the trace operator $W^{1,2}(M)\to
L^{2}(\partial\mathcal{B}_{r})$. Let $H\colon
W^{-1,2}(M,\,\Lambda^{2}\mathrm{T}^{*}M)\to\operatorname{Harm}^{2}(M)$ be the
orthogonal projection onto the space of harmonic $2$-forms, defined by

$H(\omega):=\sum_{j=1}^{\ell}\left\langle\omega,\,\zeta_{j}\right\rangle_{\mathcal{D}^{\prime}(M),\,\mathcal{D}(M)}\zeta_{j}$
(4.16)

where $(\zeta_{1},\,\ldots,\,\zeta_{\ell})$ is an orthonormal basis of the
finite-dimensional space $\operatorname{Harm}^{2}(M)$. By Lax-Milgram lemma,
for each $k$ there exists a unique $\psi_{k}\in
W^{1,2}(M,\,\Lambda^{2}\mathrm{T}^{*}M)$ such that

$\begin{cases}-\Delta\tau_{k}=\omega_{k}-H(\omega_{k})\qquad&\textrm{in the
sense of distributions in }M\\\\[5.0pt]
\displaystyle\int_{M}\left\langle\tau_{k},\,\zeta\right\rangle\,\operatorname{vol}_{g}=0&\textrm{for
all harmonic $2$-forms }\zeta.\end{cases}$ (4.17)

in the sense of distributions on $M$. We have
$\psi_{\varepsilon}=\psi_{1}+\psi_{2}+\psi_{3}+\psi_{4}$ because, by
construction, $\psi_{\varepsilon}$ is exact and hence, orthogonal to all
harmonic $2$-forms. We shall prove a decay estimate for each
$\operatorname{d}^{*}\psi_{k}$ separately.

#### Estimate for $\tau_{4}$.

We claim that

$\int_{M}\left|\operatorname{d}^{*}\tau_{4}\right|^{2}\,\operatorname{vol}_{g}\lesssim\beta^{2}\,E_{\varepsilon}(x_{0},\,R)$
(4.18)

Indeed, as $\omega_{4}$ is exact, it is orthogonal to all harmonic forms, i.e.
$H(\omega_{4})=0$. Then, by testing the equation for $\tau_{4}$ against
$\tau_{4}$, we obtain

$\int_{M}\left(\left|\operatorname{d}\tau_{4}\right|^{2}+\left|\operatorname{d}^{*}\tau_{4}\right|^{2}\right)\operatorname{vol}_{g}=\int_{\mathcal{B}_{r}}(1-\rho_{\varepsilon}^{2})\,\left\langle
j(u_{\varepsilon},\,A_{\varepsilon}),\,\operatorname{d}^{*}\tau_{4}\right\rangle\operatorname{vol}_{g}$

By applying the Young inequality at the right hand side, we deduce

$\begin{split}\int_{M}\left(\left|\operatorname{d}\tau_{4}\right|^{2}+\left|\operatorname{d}^{*}\tau_{4}\right|^{2}\right)\operatorname{vol}_{g}\lesssim\int_{\mathcal{B}_{r}}(1-\rho_{\varepsilon}^{2})^{2}\,\left|j(u_{\varepsilon},\,A_{\varepsilon})\right|^{2}\operatorname{vol}_{g}\stackrel{{\scriptstyle\eqref{rho_BBO_beta}}}{{\lesssim}}\beta^{2}\int_{\mathcal{B}_{r}}\left|j(u_{\varepsilon},\,A_{\varepsilon})\right|^{2}\operatorname{vol}_{g}\end{split}$

and (4.18) follows.

#### Estimate for $\tau_{3}$.

We shall prove that

$\int_{\mathcal{B}_{\delta
R}}\left|\operatorname{d}^{*}\tau_{3}\right|^{2}\,\operatorname{vol}_{g}\lesssim\delta^{n}E_{\varepsilon}(x_{0},\,R)$
(4.19)

To this end, we will need the following lemma.

######  Lemma 4.4.

For any $\tau\in W^{1,2}(M,\,\Lambda^{k}\mathrm{T}^{*}M)$, any $x_{0}\in M$
and any $r>0$ small enough, there holds

$\frac{1}{r^{2}}\int_{\partial\mathcal{B}_{r}(x_{0})}\left|\tau\right|^{2}\,\operatorname{d}\mathcal{H}^{n-1}\lesssim\int_{M}\left(\left|\operatorname{d}\tau\right|^{2}+\left|\operatorname{d}^{*}\tau\right|^{2}+\left|\tau\right|^{2}\right)\operatorname{vol}_{g}$

where the implicit constant at the right-hand side does not depend on $r$,
$x_{0}$.

###### Proof.

For simplicity of notation, we write $\mathcal{B}_{r}$ instead of
$\mathcal{B}_{r}(x_{0})$. We claim that, for any scalar function $f\in
W^{1,2}(\mathcal{B}_{r})$, there holds

$\frac{1}{r^{2}}\int_{\partial\mathcal{B}_{r}}f^{2}\,\operatorname{d}\mathcal{H}^{n-1}\lesssim\int_{\mathcal{B}_{r}}\left|\operatorname{d}f\right|^{2}\,\operatorname{vol}_{g}+\frac{1}{r^{2}}\int_{\mathcal{B}_{r}}f^{2}\,\operatorname{vol}_{g}$
(4.20)

The inequality (4.20) holds true if $\mathcal{B}_{r}$ is a ball in
$\mathbb{R}^{n}$, equipped with the Euclidean metric. In case
$\mathcal{B}_{r}$ is a geodesic ball in $M$ (with
$r<\operatorname{inj}(M)/2$), the estimate (4.20) follows by composition with
(normal geodesic) coordinate charts. The implicit constant in front of the
right-hand side of (4.20) depends only on the metric and is bounded uniformly
with respect to $x_{0}$, $r$, because $M$ is compact and smooth.

Now, take a form $\tau\in W^{1,2}(M,\,\Lambda^{k}\mathrm{T}^{*}M)$. Let
$(x^{1},\,\ldots,x^{n})$ be normal geodesic charts in
$\mathcal{B}_{r}(x_{0})$. Let us write
$\tau=\sum_{\alpha}\tau_{\alpha}\operatorname{d}x^{\alpha}$, where the sum is
taken over all multi-indices $\alpha$ of order $k$. By applying (4.20) to each
component $\tau_{\alpha}$, we deduce

$\frac{1}{r^{2}}\int_{\partial\mathcal{B}_{r}}\left|\tau\right|^{2}\,\operatorname{d}\mathcal{H}^{n-1}\lesssim\sum_{\alpha}\int_{\mathcal{B}_{r}}\left|\operatorname{d}\tau_{\alpha}\right|^{2}\,\operatorname{vol}_{g}+\frac{1}{r^{2}}\int_{\mathcal{B}_{r}}\left|\tau\right|^{2}\,\operatorname{vol}_{g}\lesssim\left\|\tau\right\|^{2}_{W^{1,2}(M)}+\frac{1}{r^{2}}\left\|\tau\right\|^{2}_{L^{2}(\mathcal{B}_{r})}$

The last term in the right-hand side can be estimated by applying the Hölder
inequality and Sobolev embeddings. Indeed, if $p:=2^{*}=2n/(n-2)$ then
$1/2=1/p+1/n$ and hence,

$\begin{split}\frac{1}{r^{2}}\left\|\tau\right\|^{2}_{L^{2}(\mathcal{B}_{r})}\lesssim\frac{1}{r^{2}}\left\|\tau\right\|^{2}_{L^{p}(M)}\left|\operatorname{vol}(\mathcal{B}_{r})\right|^{2/n}\lesssim\left\|\tau\right\|^{2}_{W^{1,2}(M)}\end{split}$

(where the implicit constants are uniform with respect to $r$, $x_{0}$). The
lemma now follows by applying Gaffney’s inequality (see e.g. [19, Theorem
4.8]). ∎

######  Remark 4.5.

If $\tau\in W^{1,2}(M,\,\Lambda^{k}\mathrm{T}^{*}M)$ is orthogonal to all
harmonic $k$-forms, then the $L^{2}(M)$-norm of $\tau$ is bounded by the
$L^{2}(M)$-norms of $\operatorname{d}\tau$ and $\operatorname{d}^{*}\tau$ (see
e.g. [19, Theorem 4.11]), and we obtain

$\frac{1}{r^{2}}\int_{\partial\mathcal{B}_{r}(x_{0})}\left|\tau\right|^{2}\,\operatorname{d}\mathcal{H}^{n-1}\lesssim\int_{M}\left(\left|\operatorname{d}\tau\right|^{2}+\left|\operatorname{d}^{*}\tau\right|^{2}\right)\operatorname{vol}_{g}$

Now, we proceed to the proof of (4.19). In the interior of $\mathcal{B}_{r}$,
the form $\tau_{3}$ satisfies $-\Delta\tau_{3}=0$. By taking the
codifferential of both sides of this equation, we deduce
$-\Delta(\operatorname{d}^{*}\tau_{3})=0$ in the interior of
$\mathcal{B}_{r}$. Then, standard decay estimates for elliptic equations (see
e.g. Theorem B.3 in the appendix) give

$\int_{\mathcal{B}_{\delta
R}}\left|\operatorname{d}^{*}\tau_{3}\right|^{2}\,\operatorname{vol}_{g}\lesssim\delta^{n}\int_{\mathcal{B}_{R}}\left|\operatorname{d}^{*}\tau_{3}\right|^{2}\,\operatorname{vol}_{g}$
(4.21)

It remains to estimate the right-hand side of (4.21). By testing the equation
for $\tau_{3}$ against $\tau_{3}$, we obtain

$\begin{split}\int_{M}\left(\left|\operatorname{d}\tau_{3}\right|^{2}+\left|\operatorname{d}^{*}\tau_{3}\right|^{2}\right)\operatorname{vol}_{g}&=-\int_{\partial\mathcal{B}_{r}}\left\langle\nu^{\flat}\wedge
j(v_{\varepsilon},\,A_{\varepsilon}),\,\tau_{3}\right\rangle\operatorname{d}\mathcal{H}^{n-1}+\int_{M}\left\langle
H(\omega_{3}),\,\tau_{3}\right\rangle\operatorname{vol}_{g}\\\
&\leq\frac{r}{2\lambda}\int_{\partial\mathcal{B}_{r}}\rho_{\varepsilon}^{4}\left|j(u_{\varepsilon},\,A_{\varepsilon})\right|^{2}\,\operatorname{d}\mathcal{H}^{n-1}+\frac{\lambda}{2r}\int_{\partial\mathcal{B}_{r}}\left|\tau_{3}\right|^{2}\operatorname{d}\mathcal{H}^{n-1}\\\
&\hskip
51.21504pt+\frac{1}{2\lambda}\int_{M}\left|H(\omega_{3})\right|^{2}\operatorname{vol}_{g}+\frac{\lambda}{2}\int_{M}\left|\tau_{3}\right|^{2}\operatorname{vol}_{g}\end{split}$

for any $\lambda>0$. If $\lambda$ is small enough, the last term in the right-
hand side can be absorbed into the left-hand side, because $\tau_{3}$ is
orthogonal to all harmonic $2$-forms (see e.g. [19, Theorem 4.11]). Moreover,
due to Lemma 4.4 and Remark 4.5, we can choose a constant $\lambda$ small
enough (uniformly with respect to $r$), so as to obtain

$\begin{split}\int_{M}\left(\left|\operatorname{d}\tau_{3}\right|^{2}+\left|\operatorname{d}^{*}\tau_{3}\right|^{2}\right)\operatorname{vol}_{g}&\lesssim
R\int_{\partial\mathcal{B}_{r}}\left|j(u_{\varepsilon},\,A_{\varepsilon})\right|^{2}\,\operatorname{d}\mathcal{H}^{n-1}+\int_{M}\left|H(\omega_{3})\right|^{2}\operatorname{vol}_{g}\end{split}$
(4.22)

Finally, we estimate the $L^{2}(M)$-norm of $H(\omega_{3})$. We observe that
$\omega_{3}$ is a bounded measure, whose total variation is given by

$\left|\omega_{3}\right|\\!(M)=\int_{\partial\mathcal{B}_{r}}\left|\nu^{\flat}\wedge
j(u_{\varepsilon},\,A_{\varepsilon})\right|\,\operatorname{d}\mathcal{H}^{n-1}\leq\left(\mathcal{H}^{n-1}(\partial\mathcal{B}_{r})\right)^{1/2}\left\|j(u_{\varepsilon},\,A_{\varepsilon})\right\|_{L^{2}(\partial\mathcal{B}_{r})}$

From the definition of the projection operator $H$, Equation (4.16), we
immediately obtain

$\left\|H(\omega_{3})\right\|_{L^{1}(M)}\lesssim\left|\omega_{3}\right|\\!(M)\lesssim
r^{n/2-1/2}\left\|j(u_{\varepsilon},\,A_{\varepsilon})\right\|_{L^{2}(\partial\mathcal{B}_{r})}$
(4.23)

Since all norms on the finite-dimensional space $\operatorname{Harm}^{2}(M)$
are equivalent, Equations (4.22) and (4.23) together imply

$\begin{split}\int_{M}\left(\left|\operatorname{d}\tau_{3}\right|^{2}+\left|\operatorname{d}^{*}\tau_{3}\right|^{2}\right)\operatorname{vol}_{g}&\lesssim\left(R+R^{n-1}\right)\int_{\partial\mathcal{B}_{r}}\left|j(u_{\varepsilon},\,A_{\varepsilon})\right|^{2}\,\operatorname{d}\mathcal{H}^{n-1}\stackrel{{\scriptstyle\eqref{average-r-1}}}{{\lesssim}}E_{\varepsilon}(x_{0},\,R)\end{split}$
(4.24)

By combining (4.21) with (4.24), the desired estimate (4.19) follows.

#### Estimate for $\tau_{2}$.

We claim that, for any $\mu\in(0,\,2)$, there exists a constant $C_{\mu}>0$
(independent on $\varepsilon$, $R$, $\delta$ and the point $x_{0}\in M$) such
that

$\int_{\mathcal{B}_{\delta
R}}\left|\operatorname{d}^{*}\tau_{2}\right|^{2}\,\operatorname{vol}_{g}\leq
C_{\mu}\,\delta^{n-\mu}\,E_{\varepsilon}(x_{0},\,R)$ (4.25)

We test the equation for $\tau_{2}$ against $\tau_{2}$. By applying Young’s
inequality at the right-hand side, we obtain

$\begin{split}\int_{M}\left(\left|\operatorname{d}\tau_{2}\right|^{2}+\left|\operatorname{d}^{*}\tau_{2}\right|^{2}\right)\operatorname{vol}_{g}\lesssim\frac{1}{2\lambda}\int_{\mathcal{B}_{r}}\left(\left|F_{\varepsilon}\right|^{2}+\left|H(\omega_{2})\right|^{2}\right)\operatorname{vol}_{g}+\lambda\int_{M}\left|\tau_{2}\right|^{2}\operatorname{vol}_{g}\end{split}$

Since $H$ is the $L^{2}(M)$-orthogonal projection onto the space of harmonic
forms, the $L^{2}(M)$-norm of $H(\omega_{2})$ is not greater than the
$L^{2}(M)$-norm of $\omega_{2}=-\eta_{r}F_{\varepsilon}$. Moreover, as
$\tau_{2}$ is orthogonal to all harmonic $2$-forms, the $L^{2}(M)$-norm of
$\tau_{2}$ can be estimated by the $L^{2}(M)$-norms of
$\operatorname{d}\tau_{2}$ and $\operatorname{d}^{*}\tau_{2}$ (see e.g. [19,
Theorem 4.11]). Therefore, choosing $\lambda$ small enough, we obtain

$\int_{M}\left(\left|\operatorname{d}\tau_{2}\right|^{2}+\left|\operatorname{d}^{*}\tau_{2}\right|^{2}\right)\operatorname{vol}_{g}\lesssim\int_{\mathcal{B}_{r}}\left|F_{\varepsilon}\right|^{2}\operatorname{vol}_{g}\lesssim
E_{\varepsilon}(x_{0},\,R)$ (4.26)

Now, let $s$, $t$ be numbers such that $0<s<t<r$. The decay estimate in
Proposition B.4 implies

$\begin{split}\int_{\mathcal{B}_{s}}\left(\left|\operatorname{d}\tau_{2}\right|^{2}+\left|\operatorname{d}^{*}\tau_{2}\right|^{2}\right)\operatorname{vol}_{g}\lesssim\frac{s^{n}}{t^{n}}\int_{\mathcal{B}_{t}}\left(\left|\operatorname{d}\tau_{2}\right|^{2}+\left|\operatorname{d}^{*}\tau_{2}\right|^{2}\right)\operatorname{vol}_{g}+t^{2}\int_{\mathcal{B}_{t}}\left|F_{\varepsilon}\right|^{2}\operatorname{vol}_{g}\end{split}$

(so long as we choose $R<R_{*}$, where $R_{*}$ is a small number that depends
on $M$ only, as given by Lemma A.2). The integral of
$\left|F_{\varepsilon}\right|^{2}$ at the right-hand side can be further
bounded by applying the decay estimate (3.21); we obtain

$\begin{split}\int_{\mathcal{B}_{s}}\left(\left|\operatorname{d}\tau_{2}\right|^{2}+\left|\operatorname{d}^{*}\tau_{2}\right|^{2}\right)\operatorname{vol}_{g}\leq\frac{C_{1}\,s^{n}}{t^{n}}\int_{\mathcal{B}_{t}}\left(\left|\operatorname{d}\tau_{2}\right|^{2}+\left|\operatorname{d}^{*}\tau_{2}\right|^{2}\right)\operatorname{vol}_{g}+\frac{C_{2}\,t^{n}\,E_{\varepsilon}(x_{0},\,R)}{R^{n-2}}\end{split}$
(4.27)

for some positive constants $C_{1}$, $C_{2}$ that depend only on $M$ (in
particular, not on $\varepsilon$, $R$, or $x_{0}$). Let $\mu\in(0,\,2)$ be
given. We choose a number $\theta\in(0,\,1)$ small enough that
$C_{1}\theta^{n}\leq\theta^{n-\mu/2}$. Then, the inequality (4.27) implies

$\begin{split}\int_{\mathcal{B}_{\theta
t}}\left(\left|\operatorname{d}\tau_{2}\right|^{2}+\left|\operatorname{d}^{*}\tau_{2}\right|^{2}\right)\operatorname{vol}_{g}\leq\theta^{n-\mu/2}\int_{\mathcal{B}_{t}}\left(\left|\operatorname{d}\tau_{2}\right|^{2}+\left|\operatorname{d}^{*}\tau_{2}\right|^{2}\right)\operatorname{vol}_{g}+\frac{C_{3}\,t^{n-\mu}\,E_{\varepsilon}(x_{0},\,R)}{R^{n-2}}\end{split}$

for any $t\in(0,\,r)$ and some constant $C_{3}$ that depends only on $C_{2}$,
$\mu$ and $R_{*}$. By an iteration argument (see e.g. [18, Lemma III.2.1] or
[6, Lemma B.3]), we deduce

$\begin{split}\int_{\mathcal{B}_{t}}\left(\left|\operatorname{d}\tau_{2}\right|^{2}+\left|\operatorname{d}^{*}\tau_{2}\right|^{2}\right)\operatorname{vol}_{g}&\leq\frac{C_{\mu}\,t^{n-\mu}}{r^{n-\mu}}\int_{\mathcal{B}_{r}}\left(\left|\operatorname{d}\tau_{2}\right|^{2}+\left|\operatorname{d}^{*}\tau_{2}\right|^{2}\right)\operatorname{vol}_{g}+\frac{C_{\mu}\,t^{n-\mu}\,E_{\varepsilon}(x_{0},\,R)}{R^{n-2}}\\\
&\stackrel{{\scriptstyle\eqref{tau2,1}}}{{\lesssim}}\frac{C_{\mu}\,t^{n-\mu}\,E_{\varepsilon}(x_{0},\,R)}{R^{n-2}}\end{split}$

for all $0<t<r$, where $C_{\mu}$ is a constant that depends only on $\mu$,
$n$, $\theta$ and $C_{3}$. The desired estimate (4.25) now follows by taking
$t=\delta R$.

#### Estimate for $\tau_{1}$.

We will prove that

$\int_{\mathcal{B}_{\delta
R}}\left|\operatorname{d}^{*}\tau_{1}\right|^{2}\lesssim\beta^{-4}E_{\varepsilon}(x_{0},\,R)\left(\frac{1}{\varepsilon^{2}\,R^{n-2}}\int_{\mathcal{B}_{R}}\left(1-\left|u_{\varepsilon}\right|^{2}\right)^{2}\operatorname{vol}_{g}\right)$
(4.28)

To this purpose, we need an estimate for the “Green function” of the Hodge
Laplacian, which is provided by Proposition B.1 below. Moreover, we will need
the basic bound on curvatures in Lemma 1.6.

We proceed exactly as in [8, pp. 453–454]. First of all, we observe that

$\left|\omega_{1}\right|\lesssim\frac{1}{\beta^{2}\,\varepsilon^{2}}\left(1-\left|u_{\varepsilon}\right|^{2}\right)^{2}\qquad\textrm{pointwise
in }\mathcal{B}_{r}.$ (4.29)

Indeed, in the open set
$\\{x\in\mathcal{B}_{r}\colon\left|u_{\varepsilon}(x)\right|>1-\beta\\}$, we
have $\left|v_{\varepsilon}\right|=1$ (by construction) and hence,
$\omega_{1}=2J(v_{\varepsilon},\,A_{\varepsilon})=0$. In the complement,
$\\{x\in\mathcal{B}_{r}\colon\left|u_{\varepsilon}(x)\right|\leq 1-\beta\\}$,
we have

$\left|\operatorname{D}_{A_{\varepsilon}}v_{\varepsilon}\right|\lesssim\varepsilon^{-1}$

because of (4.8)–(4.9) and the $L^{\infty}$ estimate (1.6) for
$\operatorname{D}_{A_{\varepsilon}}u_{\varepsilon}$. Taking Lemma 1.6 into
account, we deduce

$\left|\omega_{1}\right|\lesssim\left|J(u_{\varepsilon},\,A_{\varepsilon})\right|\lesssim\left|\operatorname{D}_{A_{\varepsilon}}u_{\varepsilon}\right|^{2}+\left|F_{\varepsilon}\right|\lesssim\frac{1}{\varepsilon^{2}}=\frac{1}{\beta^{2}\,\varepsilon^{2}}\cdot\beta^{2}\lesssim\frac{1}{\beta^{2}\,\varepsilon^{2}}\left(1-\left|u_{\varepsilon}\right|^{2}\right)^{2}$

at each point of
$\\{x\in\mathcal{B}_{r}\colon\left|u_{\varepsilon}(x)\right|\leq 1-\beta\\}$.
Therefore, (4.29) is proved.

Next, we show that

$\left\|\tau_{1}\right\|_{L^{\infty}(M)}\lesssim\frac{E_{\varepsilon}(x_{0},\,R)}{\beta^{2}\,R^{n-2}}$
(4.30)

Indeed, for any $x\in M$, Proposition B.1 implies

$\left|\tau_{1}(x)\right|\lesssim\int_{\mathcal{B}_{r}}\frac{\left|\omega_{1}(y)\right|}{\operatorname{dist}(x,\,y)^{n-2}}\operatorname{vol}_{g}(y)$
(4.31)

(we have used the fact that $\omega_{1}=0$ out of $\mathcal{B}_{r}$). If $x\in
M\setminus\mathcal{B}_{2r}$, then

$\left|\tau_{1}(x)\right|\lesssim\frac{1}{r^{n-2}}\int_{\mathcal{B}_{r}}\left|\omega_{1}(y)\right|\operatorname{vol}_{g}(y)\stackrel{{\scriptstyle\eqref{tau1,1}}}{{\lesssim}}\frac{1}{\beta^{2}\,\varepsilon^{2}\,R^{n-2}}\int_{\mathcal{B}_{r}}\left(1-\left|u_{\varepsilon}\right|^{2}\right)^{2}\operatorname{vol}_{g}\lesssim\frac{E_{\varepsilon}(x_{0},\,R)}{\beta^{2}\,R^{n-2}}$

On the other hand, if $x\in\mathcal{B}_{2r}$, then we have
$\mathcal{B}_{r}\subseteq\mathcal{B}_{3r}(x)$ and hence,

$\begin{split}\left|\tau_{1}(x)\right|\lesssim\int_{\mathcal{B}_{3r}}\frac{\left|\omega_{1}(y)\right|}{\operatorname{dist}(x,\,y)^{n-2}}\operatorname{vol}_{g}(y)\stackrel{{\scriptstyle\eqref{tau1,1}}}{{\lesssim}}\frac{1}{\beta^{2}\,\varepsilon^{2}}\int_{\mathcal{B}_{3r}(x)}\frac{(1-\left|u_{\varepsilon}(y)\right|^{2})^{2}}{\operatorname{dist}(x,\,y)^{n-2}}\,\operatorname{vol}_{g}(y)\end{split}$

By applying Corollary 3.8, we obtain

$\begin{split}\left|\tau_{1}(x)\right|\lesssim\frac{E_{\varepsilon}(u_{\varepsilon},\,A_{\varepsilon};\,\mathcal{B}_{3r}(x))}{\beta^{2}\,r^{n-2}}\lesssim\frac{E_{\varepsilon}(x_{0},\,R)}{\beta^{2}\,R^{n-2}}\end{split}$

(for the last inequality, we have used that
$\mathcal{B}_{3r}(x)\subseteq\mathcal{B}_{R}$, because $x\in\mathcal{B}_{2r}$
and $r<R/5$). Therefore, (4.30) is proved.

Finally, testing the equation for $\tau_{1}$ against $\tau_{1}$, we obtain

$\int_{M}\left(\left|\operatorname{d}\tau_{1}\right|^{2}+\left|\operatorname{d}^{*}\tau_{1}\right|^{2}\right)\operatorname{vol}_{g}\leq\left\|\tau_{1}\right\|_{L^{\infty}(M)}\left\|\omega_{1}-H(\omega_{1})\right\|_{L^{1}(M)}$

From the definition of the harmonic projector $H$, Equation (4.16), we
immediately deduce that
$\left\|H(\omega_{1})\right\|_{L^{1}(M)}\lesssim\left\|\omega_{1}\right\|_{L^{1}(M)}$.
Therefore, taking (4.29) and (4.30) into account, we obtain

$\int_{M}\left(\left|\operatorname{d}\tau_{1}\right|^{2}+\left|\operatorname{d}^{*}\tau_{1}\right|^{2}\right)\operatorname{vol}_{g}\lesssim\beta^{-4}E_{\varepsilon}(x_{0},\,R)\left(\frac{1}{\varepsilon^{2}\,R^{n-2}}\int_{\mathcal{B}_{R}}\left(1-\left|u_{\varepsilon}\right|^{2}\right)^{2}\operatorname{vol}_{g}\right)$

and (4.28) follows.

Combining Lemma 4.3 with (4.28), (4.25), (4.19) and (4.18), we obtain a decay
estimate for the prejacobian: for any $\delta\in(0,\,1/10)$, $\mu\in(0,\,2)$
and $\beta\in(0,\,1/4)$, there holds

$\int_{\delta
R}\left|j(u_{\varepsilon},\,A_{\varepsilon})\right|^{2}\operatorname{vol}_{g}\leq
C_{\mu}\left(\delta^{n-\mu}+\beta^{2}+\beta^{-4}\left(\frac{1}{\varepsilon^{2}\,R^{n-2}}\int_{\mathcal{B}_{R}}\left(1-\left|u_{\varepsilon}\right|^{2}\right)^{2}\operatorname{vol}_{g}\right)\right)E_{\varepsilon}(x_{0},\,R)$
(4.32)

where $C_{\mu}$ is a constant that depends only on $\mu$ and the ambient
manifold $M$ (not on $\varepsilon$, $\delta$, $\beta$, $R$).

#### Completing the proof of Proposition 4.2.

With (4.32) at our disposal, we can complete the proof of Proposition 4.2. The
arguments are largely similar to those in [8], so we omit some details. Let

$p_{\varepsilon}:=\frac{1}{\varepsilon^{2}R^{n-2}}\int_{\mathcal{B}_{R}}\left(1-\left|u_{\varepsilon}\right|^{2}\right)^{2}\operatorname{vol}_{g}$

By reasoning exactly as in [8, Proof of Theorem 3, Step 3, p. 455], we can
show that

$\begin{split}\int_{\mathcal{B}_{r}}\left|\operatorname{d}(\left|u_{\varepsilon}\right|^{2})\right|^{2}\operatorname{vol}_{g}&\lesssim\beta^{2}\int_{\mathcal{B}_{R}}\left|\operatorname{D}_{A_{\varepsilon}}u\right|^{2}\,\operatorname{vol}_{g}+\frac{1}{\beta^{2}\,\varepsilon^{2}}\int_{\mathcal{B}_{R}}\left(1-\left|u_{\varepsilon}\right|^{2}\right)\operatorname{vol}_{p}\\\
&\lesssim\beta^{2}E_{\varepsilon}(x_{0},\,R)+\beta^{-2}p_{\varepsilon}\,R^{n-2}\end{split}$
(4.33)

(the second inequality follows because, by assumption, $R$ is uniformly
bounded from above). Moreover, we have

$\int_{\mathcal{B}_{r}}\left(1-\left|u_{\varepsilon}\right|^{2}\right)\left|\operatorname{D}_{A_{\varepsilon}}u\right|^{2}\operatorname{vol}_{g}\lesssim\beta^{2}E_{\varepsilon}(x_{0},\,R)+\beta^{-2}p_{\varepsilon}\,R^{2-n}$
(4.34)

(as in Eq. (III.32) of [8]). From Lemma 1.1, we have

$\begin{split}\left|u_{\varepsilon}\right|^{2}\,\left|\operatorname{D}_{A_{\varepsilon}}u_{\varepsilon}\right|^{2}\lesssim\left|u_{\varepsilon}\right|^{2}\,\left|\operatorname{d}(\left|u_{\varepsilon}\right|)\right|^{2}+\left|j(u_{\varepsilon},\,A_{\varepsilon})\right|^{2}\lesssim\left|\operatorname{d}(\left|u_{\varepsilon}\right|^{2})\right|^{2}+\left|j(u_{\varepsilon},\,A_{\varepsilon})\right|^{2}\end{split}$

and hence,

$\begin{split}\left|\operatorname{D}_{A_{\varepsilon}}u_{\varepsilon}\right|^{2}\lesssim\left|\operatorname{d}(\left|u_{\varepsilon}\right|^{2})\right|^{2}+\left|j(u_{\varepsilon},\,A_{\varepsilon})\right|^{2}+\left(1-\left|u_{\varepsilon}\right|^{2}\right)^{2}\left|\operatorname{D}_{A_{\varepsilon}}u_{\varepsilon}\right|^{2}\end{split}$
(4.35)

Combining (4.35) with (4.32), (4.33) and (4.34), we deduce

$\int_{\mathcal{B}_{\delta
R}}\left|\operatorname{D}_{A_{\varepsilon}}u_{\varepsilon}\right|^{2}\operatorname{vol}_{g}\leq
C_{\mu}\left(\delta^{n-\mu}+\beta^{2}+\beta^{-4}\,p_{\varepsilon}\right)E_{\varepsilon}(x_{0},\,R)+\beta^{-2}\,p_{\varepsilon}\,R^{n-2}$
(4.36)

for all $\mu\in(0,\,2)$. On the other hand, by (3.18), there holds

$\int_{\mathcal{B}_{\delta
R}}\left|F_{\varepsilon}\right|^{2}\operatorname{vol}_{g}\lesssim\delta^{n-2+\alpha}\,E_{\varepsilon}(x_{0},\,R)$
(4.37)

for some $\alpha>0$ that depends only on $M$ (but is independent of $R$,
$\delta$, $\varepsilon$). Choosing $\mu=2-\alpha$ in (4.36), and taking (4.37)
into account, we immediately deduce

$E(\delta
R)\lesssim\left(\delta^{n-2+\alpha}+\beta^{2}+\beta^{-4}\,p_{\varepsilon}\right)E_{\varepsilon}(x_{0},\,R)+\beta^{-2}\,p_{\varepsilon}\,R^{n-2}$

Finally, choosing $\beta=\min\\{(p_{\varepsilon})^{1/6},\,1/8\\}$, the
proposition follows. ∎

###  4.2 Proof of Proposition 4.1

The energy decay estimate given by Proposition 4.2, and the monotonicity
formula (Theorem 3), together, imply a “clearing-out” or “$\eta$-ellipticity”
result (cf. [8, Theorem 2]).

######  Proposition 4.6.