# Morse Index Stability for Critical Points to Conformally Invariant
Lagrangians

Francesca Da Lio, Matilde Gianocca and Tristan Rivière111Department of
Mathematics, ETH Zentrum, CH-8093 Zürich, Switzerland.

Abstract :We prove the upper-semi-continuity of the Morse index plus nullity
of critical points to general conformally invariant Lagrangians in dimension 2
under weak convergence. Precisely we establish that the sum of the Morse
indices and the nullity of an arbitrary sequence of weakly converging critical
points to a general conformally invariant Lagrangians of maps from an
arbitrary closed surface into an arbitrary closed smooth manifold passes to
the limit in the following sense : it is asymptotically bounded from above by
the sum of the Morse indices plus the nullity of the weak limit and the
bubbles, while it was well known that the sum of the Morse index of the weak
limit with the Morse indices of the bubbles is asymptotically bounded from
above by the Morse indices of the weakly converging sequence. The main result
is then extended to the case of sequences of maps from sequences of domains
degenerating to a punctured Riemann surface assuming that the lengths of the
images by the maps of the collars associated to this degeneration stay below
some critical length.

Keywords. Prescribed mean curvature surfaces, minimal surfaces, Morse index
theory, harmonic maps, conformally invariant variational problems.

MSC 2020. 58E05, 58E12, 53A10, 53C43, 58E20

##  I Introduction

###  I.1 General Framework and Main Result

Let $(\Sigma,h)$ be an arbitrary smooth closed and oriented 2-dimensional
Riemannian manifold and let $({\mathcal{N}}^{n},g)$ be an arbitrary closed
smooth Riemannian manifold of arbitrary dimension $n$ that we can assume to be
isometrically embedded in some Euclidian Space ${\mathbb{R}}^{m}$ thanks to
Nash theorem.

We consider the general form of conformally invariant Lagrangians of maps from
$(\Sigma,h)$ into $({\mathcal{N}}^{n},g)$ given by Grüter222A classical result
of Grüter [11] asserts that any strictly elliptic lagrangian of quadratic
growth from $\Sigma$ into ${\mathcal{N}}^{n}$ is given by (I.1) for some
metric $g$ in ${\mathcal{N}}^{n}$.

$\mathfrak{L}(u):=\frac{1}{2}\int_{\Sigma}|du|^{2}_{h}\
dvol_{h}+u^{\ast}\alpha$ (I.1)

where $\alpha$ is an arbitrary 2-form of ${\mathcal{N}}^{n}$. Observe that
${\mathfrak{L}}(u)$ is unchanged by replacing $h$ by any conformally
equivalent metric on $\Sigma$. Hence ${\mathfrak{L}}$ can be considered as a
lagrangian of maps from a given Riemann Surface into $({\mathcal{N}}^{n},g)$
and very often in the literature $h$ is chosen to have constant $0,+1,-1$
Gauss curvature thanks to the uniformization theorem. We are interested in
critical points to $\mathfrak{L}$ within the non linear Sobolev Space

$W^{1,2}(\Sigma,{\mathcal{N}}^{n}):=\left\\{u\in
W^{1,2}(\Sigma,{\mathbb{R}}^{m})\ ;\
u(x)\in{\mathcal{N}}^{n}\quad\mbox{a.e.}\right\\}.$

Let $H$ be the section of $TN\otimes\wedge^{2}T^{\ast}N$ such that

$\forall\,X,Y,Z\in T_{z}{\mathcal{N}}^{n}\quad\
d\alpha(X,Y,Z):=\left<X,H_{z}(Y,Z)\right>$ (I.2)

where $<\cdot,\cdot>$ denotes the scalar multiplication in ${\mathbb{R}}^{m}$.
The corresponding Euler-Lagrange equation is given by

$\Delta_{h}u+H_{u}(du\wedge du)_{h}=-\,{\mathbb{I}}_{u}(du,du)_{h}\quad\mbox{
in }{\mathcal{D}}^{\prime}(\Sigma)\ .$ (I.3)

where $\Delta_{h}$ is denoting the positive Laplace Beltrami operator and
${\mathbb{I}}_{z}(X,Y)$ is the second fundamental form of the embedding
${\mathcal{N}}^{n}\hookrightarrow{\mathbb{R}}^{m}$ at the point
$z\in{\mathcal{N}}^{n}$ and taken over a pair of vectors $X,Y\in
T_{z}{\mathcal{N}}^{n}$. The non linearity ${\mathbb{I}}_{u}(du,du)_{h}$ is
equal to $\mbox{Tr}_{h}({\mathbb{I}}_{u}(du\otimes du))$. Using local
coordinates in $\Sigma$ it is given by

${\mathbb{I}}_{u}(du,du)_{h}:=\sum_{i,j=1}^{2}h^{ij}\,{\mathbb{I}}_{u}(\partial_{x_{i}}u,\partial_{x_{j}}u)\
.$

and we define for any maps $f,g$ from $\Sigma$ into ${\mathbb{R}}^{m}$ using
local coordinates

$H(df\wedge
dg)=\left[H_{u}(\partial_{x_{1}}f,\partial_{x_{2}}g)-H_{u}(\partial_{x_{2}}f,\partial_{x_{1}}g)\right]\,dx_{1}\wedge
dx_{2}\ ,$

and333Observe that
$H_{u}(\partial_{x_{1}}f,\partial_{x_{2}}g)-H_{u}(\partial_{x_{2}}f,\partial_{x_{1}}g)=H_{u}(\partial_{x_{1}}f,\partial_{x_{2}}g)+H_{u}(\partial_{x_{1}}g,\partial_{x_{2}}f)$
which gives in particular $H_{u}(df\wedge dg)=H_{u}(dg\wedge df)$.

$H(df\wedge dg)_{h}\ dvol_{h}:=H(df\wedge dg)\ .$

If one denotes $P_{z}$ the $m\times m$ real symmetric matrix444We shall denote
$S_{m}({\mathbb{R}})$ the space of real $m\times m$ real symmetric matrices.
given by the orthogonal projection of ${\mathbb{R}}^{m}$ onto
$T_{z}{\mathcal{N}}^{n}$, for any map $u\in
W^{1,2}(\Sigma,{\mathcal{N}}^{n})$, the map $P_{u(x)}$ is clearly in
$W^{1,2}(\Sigma,S_{m}({\mathbb{R}}))$ and one proves (see [12]) that (I.3) is
equivalent to

$P_{u}\left[\Delta_{h}u+H_{u}(du\wedge du)_{h}\right]=0\quad\mbox{ in
}{\mathcal{D}}^{\prime}(\Sigma)\ .$ (I.4)

Solutions to (I.4) are known to be smooth [29]. Any solution to (I.4) which is
in addition conformal corresponds to a possibly branched immersion of $\Sigma$
with mean curvature vector $H(\vec{e}_{1},\vec{e}_{2})$ where
$(\vec{e}_{1},\vec{e}_{2})$ is an orthonormal basis of the tangent space in
$T{\mathcal{N}}^{n}$ to $u_{\ast}T\Sigma$. For that reason, equation (I.4) is
called Prescribed Mean Curvature Equation into ${\mathcal{N}}^{n}$. Solutions
to the Prescribed Mean Curvature Equation into manifolds that we will simply
call PMC solutions into ${\mathcal{N}}^{n}$ are relatively common mathematical
objects from geometric calculus of variations. They are playing a special role
in particular in the differential geometry of surfaces, Teichmüller space
theory and uniformization as well as in mathematical models from general
relativity or condensed matter physics.

Beside the value of it’s energy of course there are two other important
numbers attached to every harmonic map : the Morse index and the nullity. They
are respectively the dimension of the largest space on which the second
derivative of $\mathfrak{L}$ is negative and the dimension of it’s kernel.
Precisely, the second derivative of $\mathfrak{L}$ at a PMC map $u$ is defined
on the space of infinitesimal variations of $u$ which are nothing but the
sections of the pull-back by $u$ of the tangent bundle to ${\mathcal{N}}^{n}$
:

$V_{u}:=\Gamma(u^{-1}T{\mathcal{N}}^{n}):=\left\\{{w}\in
W^{1,2}(\Sigma,{\mathcal{N}}^{n})\ ;\ P_{u(x)}{w}=w\mbox{ for a. e.
}x\in\Sigma\right\\}\ .$ (I.5)

The second derivative of

$E(u):=\frac{1}{2}\int_{\Sigma}|du|^{2}_{h}\ dvol_{h}$

at $u$ is a quadratic form on $V_{u}$ given by

$D^{2}E_{u}(w)=\int_{\Sigma}|dw|_{h}^{2}\
dvol_{h}-\int_{\Sigma}\left<{\mathbb{I}}_{u}(du,du)_{h},{\mathbb{I}}_{u}(w,w)\right>\
dvol_{h}$ (I.6)

where $<\cdot,\cdot>$ denotes the scalar multiplication in ${\mathbb{R}}^{m}$.
In the sphere case for instance when
${\mathcal{N}}^{n}=S^{n}\subset{\mathbb{R}}^{n+1}$ the computations give

$D^{2}E_{u}(w)=\int_{\Sigma}|dw|^{2}_{h}-|du|^{2}_{h}\,|w|^{2}\,dvol_{h}\ .$
(I.7)

In the general case, for $\omega$ arbitrary,

$\begin{array}[]{l}\displaystyle
Q_{u}(w):=D^{2}\mathfrak{L}_{u}(w)=\int_{\Sigma}|dw|^{2}_{h}\
dvol_{h}-\int_{\Sigma}\left<{\mathbb{I}}_{u}(du,du)_{h},{\mathbb{I}}_{u}(w,w)\right>\
dvol_{h}\\\\[14.22636pt]
\displaystyle\quad+\int_{\Sigma}w\cdot\left[2\,H_{u}(dw\wedge
du)_{h}+\nabla_{w}H_{u}(du\wedge du)_{h}\right]\ dvol_{h}\ .\end{array}$ (I.8)

We define the Morse index of $u$ to be

$\mbox{Ind}_{\mathfrak{L}}(u):=\sup\left\\{\mbox{dim}(W)\ ;\ W\mbox{ is a sub
vector-space of }V\mbox{ s.t. }\left.Q_{u}\right|_{W}<0\right\\}$

and the Nullity

$\mbox{Null}_{\mathfrak{L}}(u):=\mbox{dim}\,Ker(B_{u})$

where $B_{u}$ is the bilinear form associated to $Q_{u}$.

The present paper is aiming at studying the behaviour of
$\mbox{Ind}_{L}(u_{k})+\mbox{Null}_{L}(u_{k})$ for sequences of harmonic maps
with uniformly bounded energy.

A classical result in concentration compactness theory (see for instance [17]
theorem 3) asserts that modulo extraction of a subsequence there exists a
limiting harmonic map $u_{\infty}$ from $\Sigma$ into ${\mathcal{N}}^{n}$,
there exist $p^{1}\ldots p^{Q}\in\Sigma$, there exist $x^{1}_{k}\ldots
x^{Q}_{k}\in\Sigma$ such that $x^{j}_{k}\rightarrow p^{j}$ for any $j=1\ldots
Q$, there exist $\delta^{1}_{k}\ldots\delta^{Q}_{k}$ positive numbers
converging to zero and there exist $v_{\infty}^{1}\ldots v_{\infty}^{Q}$
harmonic maps from ${\mathbb{C}}$ into ${\mathcal{N}}^{n}$ such that

$\lim_{k\rightarrow
0}\left\|\nabla\left(u_{k}-u_{\infty}-\sum_{j=1}^{Q}v^{j}_{k}\right)\right\|_{L^{2}(\Sigma)}=0\
,$ (I.9)

where, using normal coordinates for the metric $h$ around each $x^{j}_{k}$, we
denote $v_{k}^{j}(\delta^{j}_{k}y+x_{k}^{j}):=v^{j}_{\infty}(y)$. The harmonic
maps $v^{j}_{\infty}$ are called the bubbles and the convergence (I.9) which
is proper to conformally invariant variational problems is called bubble tree
convergence of $u_{k}$ towards $(u_{\infty},v_{\infty}^{1}\ldots
v_{\infty}^{Q})$.

Our first main result in the present work is the following

######  Theorem I.1.

Let $u_{k}$ be a sequence of solutions to the prescribed mean curvature
equation (I.4) from a closed oriented surface $(\Sigma,h)$ into a closed
arbitrary $C^{2}$ riemannian manifold ${\mathcal{N}}^{n}$ for $H\in C^{1}$.
Assume $u_{k}$ bubble tree converges towards $(u_{\infty},v_{\infty}^{1}\ldots
v_{\infty}^{Q})$. Then, for $k$ large enough one has

$\mbox{Ind}_{\mathfrak{L}}(u_{k})+\mbox{Null}_{\mathfrak{L}}(u_{k})\leq\mbox{Ind}_{\mathfrak{L}}(u_{\infty})+\mbox{Null}_{\mathfrak{L}}(u_{\infty})+\sum_{j=1}^{Q}\mbox{Ind}_{\mathfrak{L}}(v_{\infty}^{j})+\mbox{Null}_{\mathfrak{L}}(v_{\infty}^{j})\
.$ (I.10)

$\Box$

Observe that, in view of the explicit form (I.8) of $D^{2}{\mathfrak{L}}$, the
regularity assumptions we are making on ${\mathcal{N}}^{n}$ and $\alpha$ are
the most general ones for ensuring the continuity of $D^{2}\mathfrak{L}$.
Thanks to the method we are using, we don’t require more regularity on
$D^{2}\mathfrak{L}$.

Theorem I.1 has been first established in [33] and [34] in the case of
harmonic maps into $C^{3}$ manifolds based on pointwise estimates established
in [27] (see more comment on these estimates below).

Recall that classical results (see [25] or more recently [14] for
$\mathfrak{L}=E$) imply that for $k$ large enough one has the lower-semi-
continuity of the Morse index in the following sense

$\mbox{Ind}_{\mathfrak{L}}(u_{\infty})+\sum_{j=1}^{Q}\mbox{Ind}_{\mathfrak{L}}(v_{\infty}^{j})\leq\mbox{Ind}_{\mathfrak{L}}(u_{k})\
.$ (I.11)

The lower semi-continuity of the Morse index (I.11) is also true while
considering harmonic maps approximations and holds in general for limits of
sequences solving some relaxed min-max problem converging to possible non zero
Morse index harmonic maps. Inequality (I.11) is a rather general and “robust”
inequality. It holds in various frameworks like for instance in the viscosity
method for minimal surfaces, [30].

The upper-semi-continuity of the Morse Index plus nullity is a way more
delicate to obtain than the lower semi-continuity of the Morse index and does
not always hold in fact. It has been established in two remarkable works on
minimal surface theory respectively by O. Chodosh and C. Mantoulidis for
sequences of critical points to the Allen-Cahn Functional (see [3] Theorem
1.9) and by F.C. Marques and A. Neves for limits of Almgren-Pitts minmax
procedures (see [22] Theorem 1.3). In both cases a main assumption is that the
limiting minimal surface has multiplicity one. It has been disproved when the
limit has multiplicity higher than one (see [8] and [3] Example 5.2).

Inequality (I.10) is not expected to hold for instance for general Palais-
Smale harmonic map sequences or for sequences of harmonic maps from
degenerating Riemann Surfaces. Indeed, in both cases Energy quantization (I.9)
does not necessarily holds (see [26], [35]) and the neck energy could
contribute to the negative part of the spectrum of the Hessian $D^{2}E$. More
seriously, even when (I.9) holds for some Sacks-Uhlenbeck type
$\alpha-$harmonic maps, it is not excluded that the necks are given by
geodesics with non zero lengths (see [24] ). In such a case it could be that
these geodesics contribute to the negative part of the spectrum of the Hessian
$D^{2}E$. Our analysis throughout the present work is very much based on the
$C^{0}-$no neck property for weakly converging critical points of
$\mathfrak{L}$. This $C^{0}$-no neck property property is saying that there is
no distance between the bubbles $v^{j}_{\infty}$ and the main part
$u_{\infty}$, roughly speaking the bubbles $v_{\infty}^{j}$ are
“$L^{\infty}-$glued” directly to the weak limit $u_{\infty}$. It was first
proved in the special case of harmonic maps into manifolds in a remarkable
paper by J. Qing and G. Tian [27]. The main idea in this paper was to obtain
pointwise estimates of the gradient of the harmonic maps in the neck region by
bounding the difference between the harmonic map and geodesic maps into
${\mathcal{N}}^{n}$ (which are nothing but axially symmetric harmonic maps
from the neck regions into ${\mathcal{N}}^{n}$) thanks to a very refined three
circle theorem by L.Simon [31]. In a later work [21], F.H. Lin and C.Y. Wang
found a different strategy for proving the $C^{0}$-no neck property under the
same assumptions than the ones made in [27]. They used mostly the fact that
the angular energy of harmonic maps on surrounding circles in the neck regions
satisfy a remarkable ordinary differential inequality, as it had been
previously observed by T.Parker in [26]. Both $C^{0}-$no neck property proved
in [27] and [21] are valid for sequences of harmonic maps into manifolds which
are at least $C^{3}$ regular.

A more general situation where strong convergence of sequences of critical
points of $\mathfrak{L}$ fails, arises when instead of considering $u_{k}$
from a fixed Riemann Surface $(\Sigma,h)$ one considers sequences of critical
points $u_{k}$ of $\mathfrak{L}$ from a sequence of Riemann surfaces
$(\Sigma,h_{k})$ into $({\mathcal{N}}^{n},g)$ and when the sequence of
Constant Gauss Curvature metric $h_{k}$ is degenerating in the Moduli space of
$\Sigma$. The corresponding bubbling analysis in this situation has been
carefully studied by M. Zhu in [35] for ${\mathfrak{L}}=E$. According to the
general moduli space compactification theory (see for instance [13]) the
degeneration of the Riemann surface $(\Sigma,h_{k})$ is equivalent to the
convergence to zero of the length of the shortest geodesics. One can extract a
subsequence such that there exists a fixed number of pairwise disjoint closed
embedded geodesics $(L_{k}^{j})_{j=1\ldots P}$ of lengths
$(l_{k}^{j})_{j=1\ldots P}$ all tending to zero and conformal charts
$\psi^{j}_{k}$ from annuli ${\mathcal{C}}_{k}^{j}$ of degenerating conformal
classes containing $L_{k}^{j}$ and called the collars into $B_{1}(0)\setminus
B_{\delta_{k}^{j}}(0)$ where $\delta_{k}^{j}:=\exp(-1/{l_{k}^{j}})$. Away from
these collars the bubble tree analysis (I.9) holds locally with respect to the
nodal Riemann surface obtained by collapsing the geodesics $L_{k}^{j}$ to
points called punctures (see [35] in the harmonic map case with $C^{3}$
targets and [17] for the general case). In order to simplify the presentation
we assume that no bubble is formed in the collars that is to say, in the chart
$\psi_{k}^{j}$,

$\lim_{\eta\rightarrow
0}\,\lim_{k\rightarrow+\infty}\sup_{\rho\in[\eta^{-1}\,\delta_{k}^{j},\eta]}\int_{B_{2\rho}(0)\setminus
B_{\rho}(0)}|\nabla u_{k}|^{2}\ dx^{2}=0\ .$

The limiting average length  of the images by $u_{k}$ of the collar
${\mathcal{C}}_{k}^{j}$ is given by

$\Lambda^{j}:=\lim_{\eta\rightarrow
0}\,\limsup_{k\rightarrow+\infty}\int_{\eta^{-1}\,\delta_{k}^{j}}^{\eta}\frac{1}{2\pi}\int_{0}^{2\pi}\left|\frac{d{u}_{k}}{dr}\right|\
dr\ ,$

where the coordinates $(r,\theta)$ are referring to the charts $\psi_{k}^{j}$.
Our second main result in this work is the following theorem

######  Theorem I.2.

Let $u_{k}$ be a sequence of solutions to the prescribed mean curvature
equation (I.4) from a sequence of closed Riemann surfaces $(\Sigma,h_{k})$ of
genus larger than one into a closed arbitrary $C^{2}$ riemannian manifold
${\mathcal{N}}^{n}$ for $H\in C^{1}$. Assume $(\Sigma,h_{k})$ is degenerating
to a punctured surface with a fixed number of collars
$({\mathcal{C}}_{k}^{j})_{j=1\ldots P}$ at the limit. Assume $u_{k}$ bubble
tree converges away from the collars towards $(u_{\infty},v_{\infty}^{1}\ldots
v_{\infty}^{Q})$ and that no bubble is formed in the collars. There exists a
constant $\Lambda^{\ast}>0$ depending only on $m$, ${\mathcal{N}}^{n}$ and $H$
such that, if the limiting average length $\Lambda^{j}$ of the images by
$u_{k}$ of the collar ${\mathcal{C}}_{k}^{j}$ are less than $\Lambda^{\ast}$
then, for $k$ large enough one has

$\mbox{Ind}_{\mathfrak{L}}(u_{k})+\mbox{Null}_{\mathfrak{L}}(u_{k})\leq\mbox{Ind}_{\mathfrak{L}}(u_{\infty})+\mbox{Null}_{\mathfrak{L}}(u_{\infty})+\sum_{j=1}^{Q}\mbox{Ind}_{\mathfrak{L}}(v_{\infty}^{j})+\mbox{Null}_{\mathfrak{L}}(v_{\infty}^{j})\
.$ (I.12)

$\Box$

The result can be understood in the following sense. As explained in [35], in
the harmonic map case $E={\mathfrak{L}}$, the images of the collars converge
towards portions of geodesics of ${\mathcal{N}}^{n}$. It is a classical result
that below a critical length all geodesic arcs are stable and then the
corresponding collar does not contribute to the negative part of the spectrum.
This is shedding some light on the reason why the upper-semi-continuity of the
dimension of the negative part of the spectrum obtained for sequences of
critical points of ${\mathfrak{L}}$ from a fixed surface - i.e. (I.10) - does
extend to sequences of domains under the assumption that
$\Lambda^{j}<\Lambda^{\ast}$ for any $j$. We believe that this assumption is
optimal since geodesics beyond some length have negative Morse index and the
collars could then contribute to the negative part of the spectrum, which
could eventually break (I.12).

The role of interpolation spaces in neck/collar analysis : In the present work
the $C^{0}$-no neck property is derived from the main estimates established by
the 3rd author and P. Laurain in [17] (see the explanations in subsections
III.1 and IV.1 below)555See relevant estimates by T. Lamm and B. Sharp in
[16].. The $C^{0}$-no neck property in our approach is a consequence of the so
called $L^{2,1}-$energy quantization666The $L^{2,1}-$energy quantization (i.e.
lemma III.2 below) was first established in [17] for the “angular part of the
energy” in the necks. where $L^{2,1}$ refers to the Lorentz space pre-dual to
the Marcinkiewicz space $L^{2}-$weak also denoted $L^{2,\infty}$.

The idea to introduce the Lorentz space $L^{2,1}$ norm in neck analysis for
conformally invariant variational problems goes back to the two works of the
third authors with F.H.Lin ([19] and [20]) while the general derivation of
conservation laws for harmonic maps, which is also central in the present
work, has been derived by the third author in [29] .

There are strong reasons to expect that the Morse index plus nullity upper-
semi-continuity holds true for a large class of conformally invariant
variational problems for which conservation laws opening the way to $C^{0}$-no
neck property have been derived in the last two decades. We can quote for
instance half-harmonic map in one dimension and free boundary surfaces for
which almost conservation laws have been derived in [5], [6] by the first and
the third authors, in [4], by the first author and Pigati and in [7] by the
first author, Mazowiecka and Schikorra, Bi-harmonic maps in 4 dimensions where
the $C^{0}$-no neck property should follow from the analysis in [18] which it-
self is derived from the conservation laws established in [15], Yang-Mills
Fields in dimension 4 where the $L^{2,1}-$norm estimates of the curvature in
neck regions is established by the third author in [28], or Willmore surfaces
where the $C^{0}$-no neck property for the Gauss map has been proved recently
by the third author in collaboration with A. Michelat in [23].

One of the motivation of the present work is to ultimately establish the upper
semi-continuity of the Morse index plus nullity for solution to min-max
problems via the Ginzburg Landau type relaxation $\mathfrak{L}_{\varepsilon}$
of the energy $\mathfrak{L}$ of the form

${\mathfrak{L}}_{\varepsilon}(u):=\frac{1}{2}\int_{\Sigma}\left[|du|^{2}_{h}+\frac{1}{\varepsilon^{2}}F(u)\right]\
dvol_{h}+u^{\ast}\alpha\ ,$

for $u$ in $W^{1,2}(\Sigma,{\mathbb{R}}^{m})$ where $F$ is some smoothing of
the square of the distance to ${\mathcal{N}}^{n}$. This question is under
investigation by the authors. It would be interesting to see under which
assumption the $C^{0}$-no neck property holds. We expect the entropy estimates
of the form

$\frac{1}{\varepsilon^{2}}F(u)=o\left(\frac{1}{\log\varepsilon^{-1}}\right)\
,$

to play a role in this context. Such entropy estimate can be derived from the
refinement of Palais-Smale deformation theory due to M.Struwe also known as
“Struwe Monotonicity Trick”.

###  I.2 Description of the various steps in the proof of theorem I.1.

We consider a weakly converging sequence of solution to (I.3). We assume for
simplicity of the presentation that there is exactly one bubble of
characteristic size $\delta_{k}\rightarrow 0$ and centered at
$x_{k}\rightarrow p\in\Sigma$. The so called “neck region” connecting the
bubble to the main part of the solution is given by the family of annuli with
degenerating conformal class of the form

$A(\eta,\delta_{k}):=B_{\eta}(x_{k})\setminus B_{\delta_{k}/\eta}(x_{k})\ .$

The absence of bubble in the neck region (by definition) says

$\lim_{\eta\rightarrow 0}\limsup_{k\rightarrow+\infty}\
\sup_{\delta_{k}/\eta<\rho<2\rho<\eta}\ \int_{B_{2\rho}(x_{k})\setminus
B_{\rho}(x_{k})}|du_{k}|^{2}_{h}\ dvol_{h}=0\ .$

The main argument in [17] is implying

$\lim_{\eta\rightarrow 0}\limsup_{k\rightarrow+\infty}\
\int_{A(\eta,\delta_{k})}|du_{k}|^{2}_{h}\ dvol_{h}=0\ .$ (I.13)

which is also called “$L^{2}-$energy quantization”.

One of the main contribution of [29] is the starting point of the present
work. It says that the Euler-Lagrange Equation (I.3) can be rewritten in the
form

$\Delta_{h}u_{k}=\Omega_{k}\cdot du_{k}\ ,\quad\mbox{ where
}\quad\Omega_{k}\in(u_{k})_{\ast}\,T_{x}\Sigma\otimes so(m)\quad\mbox{ and
}\quad|\Omega_{k}|\leq C\,|du_{k}|\ .$ (I.14)

Thanks to the small Dirichlet energy of $u_{k}$ in the neck, one of the main
result of [29] implies the existence of $A_{k}\in
W^{1,2}(A(\eta,\delta_{k}),Gl_{m}({\mathbb{R}}))$ and $B_{k}\in
W^{1,2}(A(\eta,\delta_{k}),M_{m}({\mathbb{R}}))$ such that the Euler Lagrange
Equation (I.14) can be rewritten in the form of a conservation law

$\mbox{div}\left(A_{k}\ \nabla u_{k}+B_{k}\ \nabla^{\perp}u_{k}\right)=0,$
(I.15)

where in (I.15) we are using local conformal/complex coordinates $z$ for
$(\Sigma,h)$. In [17], the conservative form of the equation is used to
“upgrade” the $L^{2}-$energy quantization (I.13) to the following
$L^{2,1}-$energy quantization

$\lim_{\eta\rightarrow 0}\limsup_{k\rightarrow+\infty}\
\int_{0}^{+\infty}\left|\left\\{x\in A(\eta,\delta_{k})\ ;\
|du_{k}|_{h}(x)>s\right\\}\right|^{1/2}\ ds=0\ .$ (I.16)

This $L^{2,1}-$energy quantization is a crucial step for proving the the neck
contributes in a positive way to the second derivative of the Lagrangian.
Indeed we first proceed to a Hodge type decomposition of $A_{k}\,\nabla u_{k}$
in the form

$A_{k}\,\nabla
u_{k}=\nabla\varphi_{k}+\nabla^{\perp}\psi_{k}+\nabla\mathfrak{h}_{k}\quad\mbox{
in }A(\eta,\delta_{k})\ ,$ (I.17)

where respectively

$\Delta\varphi_{k}=\nabla^{\perp}B_{k}\cdot\nabla
u_{k}\quad,\quad\Delta\psi_{k}=\nabla^{\perp}A_{k}\cdot\nabla u_{k}\quad\mbox{
and }\quad\Delta\mathfrak{h}_{k}=0\ .$ (I.18)

The harmonic part is decomposed as follows

$\mathfrak{h}_{k}=\mathfrak{h}_{k}^{+}+\mathfrak{h}_{k}^{-}+\mathfrak{h}_{k}^{0}\
\ \mbox{ where }\ \ \mathfrak{h}_{k}^{+}=\Re\sum_{n>0}h_{n,k}z^{n}\ \ ,\ \
\mathfrak{h}_{k}^{-}=\Re\sum_{n<0}h_{n,k}z^{n}\ \ \mbox{ and }\ \
\mathfrak{h}_{k}^{0}=\mathfrak{h}_{0,k}+C_{\eta,k}^{0}\,\log|z|\ .$

An important fact in the proof of the main result is to prove that the
“$L^{2,1}-$energy quantization” is implying the following control of the
constant $C_{\eta,k}^{0}$

$C_{\eta,k}^{0}=o_{\eta,k}\left(\frac{1}{\log\frac{\eta^{2}}{\delta_{k}}}\right)$
(I.19)

where we observe that $\log\frac{\eta^{2}}{\delta_{k}}$ is nothing but the
degenerating class of the annulus $A(\eta,\delta_{k})$. The estimate
(LABEL:).12) is optimal for our purposes in the following sense. The
contribution to the pointwise estimate of $|du_{k}|^{2}(x)$ of
$\mathfrak{h}_{k}^{0}$ is equal to $(C_{\eta,k}^{0})^{2}/|x|^{2}$ and the
following general result777We prove that the constant $\pi^{2}$ is optimal in
the first inequality (I.20).  holds

$\forall f\in
W^{1,2}_{0}(A(\eta,\delta_{k}))\quad\int_{A(\eta,\delta_{k})}|\nabla f|^{2}\
dx^{2}\geq\frac{\pi^{2}}{\log^{2}\frac{\eta^{2}}{\delta_{k}}}\,\int_{A(\eta,\delta_{k})}\frac{|f|^{2}}{|x|^{2}}\
dx^{2}>>\ \int_{A(\eta,\delta_{k})}|d\mathfrak{h}_{k}^{0}|^{2}\ |f|^{2}\
dx^{2}\ .$ (I.20)

For the other contributions to $\nabla u_{k}$ we use on one hand classical
pointwise estimates for harmonic functions with only positive or negative
Fourier frequencies and on the other hand we are using weighted Wente type
inequalities (see Lemma E.1) for controlling the contributions respectively
from $\varphi_{k}$ and $\psi_{k}$ whose Laplacians are both given by
Jacobians. Finally we conclude the pointwise estimates of the gradient of
$u_{k}$ by proving the existence of $\beta>0$ independent of $k$ such that

$|du_{k}|^{2}\leq\ o_{\eta,k}(1)\ \omega_{\eta,k}(x)\ ,$ (I.21)

where $\omega_{\eta,k}$ is the weight function given in $A(\eta,\delta_{k})$
explicitly by

$\omega_{\eta,k}(x)=\frac{1}{|x|^{2}}\,\left[\frac{|x|^{\beta}}{\eta^{\beta}}+\frac{\delta_{k}^{\beta}}{\eta^{\beta}\,|x|^{\beta}}+\frac{1}{\log^{2}\frac{\eta^{2}}{\delta_{k}}}\right]\
.$ (I.22)

We generalize (I.20) by proving that the first eigenvalue of the Laplacian for
Dirichlet boundary conditions on the degenerating annulus $A(\eta,\delta_{k})$
with respect to the $L^{2}-$norm weighted by $\omega_{\eta,k}$ is bounded from
above and below by positive constants independent of $k$ in particular we have

$\exists\ \lambda_{0}>0\ \quad\forall f\in
W^{1,2}_{0}(A(\eta,\delta_{k}))\quad\int_{A(\eta,\delta_{k})}|\nabla f|^{2}\
dx^{2}\geq\lambda_{0}\ \int_{A(\eta,\delta_{k})}\omega_{\eta,k}\ |f|^{2}\
dx^{2}$ (I.23)

where $\lambda_{0}$ is independent of $k$. As a consequence of (I.21) and
(I.23) we obtain that the neck are asymptotically not contributing to the
negativity of the second derivative of the Lagrangian in the following sense

$\forall\ w\in\Gamma(u_{k}^{-1}T{\mathcal{N}}^{n})\cap
W^{1,2}_{0}(A(\eta,\delta_{k}),{\mathbb{R}}^{m})\quad\quad
D^{2}{\mathfrak{L}}_{u_{k}}(w)\geq\lambda_{0}\
\int_{A(\eta,\delta_{k})}\omega_{\eta,k}\ |w|^{2}\ dx^{2}\ .$ (I.24)

The proof of the main theorem follows from (I.24).

The paper is organized as follows : Section II is devoted to the computation
of $D^{2}{\mathfrak{L}}$. The proof of the pointwise estimate on the gradient
of $u_{k}$ is the subject of section III. It is relying on several lemmae
contained in the appendix (D to H) among which the weighted Wente estimate
(lemma E.1). The section IV.1 is devoted to the introduction of the weight
function $\omega_{\eta,k}$ and the proof of (I.24) which is the contribution
of the necks to the positivity of the second derivative of ${\mathfrak{L}}$ is
based on the pointwise estimate of the gradient of $u_{k}$. The conclusion of
section IV.1 is given in lemma IV.2. The use of the Sylvester inertia
principle and the diagonalisation of $D^{2}{\mathfrak{L}}_{u_{k}}$ with
respect to the weights $\omega_{\eta,k}$ is given in subsection IV.2. Thee
proof of the main theorem I.1 is given in section IV.3. In section V it is
explained how the main arguments can be modified to cover the case of
degenerating underlying Riemann surfaces and to prove theorem I.2.

##  II Preliminaries and Notations

###  II.1 The Hessian to the Dirichlet Energy :
$\mathbf{D^{2}{\mathfrak{L}}}$.

The second derivative of the Dirichlet energy at harmonic maps has been
computed for several decades already (see for instance [32]). More generally,
the second derivative of ${\mathfrak{L}}$ at critical points can be found in
[9] but for normal perturbations only. These computations moreover are
involving the intrinsic Riemann tensors of ${\mathcal{N}}^{n}$. We present
below a general computation which involves the extrinsic tensor given by the
second fundamental form of ${\mathcal{N}}^{n}$ instead and which has the
advantage of being relatively easily computable for a given sub-manifold.

We first consider the case $\omega=0$ that is the case of the Dirichlet energy
and harmonic maps.

####  II.1.1 The harmonic map case.

Let $u$ be an harmonic map from $(\Sigma,h)$ into ${\mathcal{N}}^{n}$. We
consider a smooth perturbation of $u$ of the form $u_{t}=\pi_{N}(u+t\,\phi)$
where $\phi$ is a smooth map in $C^{\infty}(\Sigma,{\mathbb{R}}^{m})$ and
$\pi_{N}$ is the orthogonal projection onto ${\mathcal{N}}^{n}$ (which is
smooth in a tubular neighborhood of ${\mathcal{N}}^{n}$). We denote

$w:=\frac{du_{t}}{dt}(0)=d\pi_{u}\phi\ .$

Observe that $d\pi_{z}$ for $z\in{\mathcal{N}}^{n}$ coincide with the matrix
orthogonal projection $P_{z}$ of ${\mathbb{R}}^{m}$ onto
$T_{z}{\mathcal{N}}^{n}$. We have moreover in ${\mathbb{R}}^{m}$
$d\pi_{u+t\,\phi}=d\pi_{\pi(u+t\,\phi)}$. This gives finally

$\frac{du}{dt}(t)=P_{u_{t}}\phi\ .$ (II.1)

which gives

$\frac{d^{2}u}{dt^{2}}(0)=d_{w}P_{u}\phi\ .$ (II.2)

We first compute the

$\frac{dE(u_{t})}{dt}=\int_{\Sigma}du_{t}\cdot_{h}d\frac{du_{t}}{dt}\
dvol_{h}$ (II.3)

The harmonicity of $u$ is equivalent to

$\forall\ \phi\in C^{\infty}(\Sigma,{\mathbb{R}}^{m})\quad\quad
0=\frac{dE(u)}{dt}(0)=2\,\int_{\Sigma}du\cdot_{h}d(P_{u}\phi)\
dvol_{h}=2\,\int_{\Sigma}\Delta_{h}u\cdot_{h}P_{u}\phi\ dvol_{h}$ (II.4)

Using the symmetry of the matrix $P_{u}$ we deduce that this is equivalent to

$P_{u}\left[\Delta_{h}u\right]=0\quad\Longleftrightarrow\Delta_{h}u=-\,d[P_{u}]\cdot_{h}du$
(II.5)

where we have used $P_{u}du=du$ and where we recall that $\Delta_{h}$ denotes
the positive Laplace Beltrami operator. Combining (II.2) and (II.6) gives

$\begin{array}[]{l}\displaystyle\frac{d^{2}E(u_{t})}{dt^{2}}(0)=\int_{\Sigma}d\frac{du_{t}}{dt}\cdot_{h}d\frac{du_{t}}{dt}\
dvol_{h}+\int_{\Sigma}du_{t}\cdot_{h}d\frac{d^{2}u_{t}}{dt^{2}}\
dvol_{h}\\\\[14.22636pt] \displaystyle=\int_{\Sigma}dw\cdot_{h}dw\
dvol_{h}+\int_{\Sigma}\left<\Delta_{h}u,d_{w}P_{u}\phi\right>\
dvol_{h}\\\\[14.22636pt] \end{array}$ (II.6)

where $<\cdot,\cdot>$ denotes the scalar multiplication in ${\mathbb{R}}^{m}$.
Since $P_{u}[P_{u}\phi]=P_{u}\phi$ taking the differential along $w$ gives

$d_{w}P_{u}\phi=d_{w}P_{u}w+P_{u}d_{w}P_{u}\phi\ .$ (II.7)

Since $P_{u}[\Delta_{h}u]=0$ we deduce

$\left<\Delta_{h}u,d_{w}P_{u}\phi\right>=\left<\Delta_{h}u,d_{w}P_{u}w\right>=-\left<d[P_{u}]\cdot_{h}du,d_{w}P_{u}\,w\right>\
.$ (II.8)

Combining (II.6) and (II.8) gives finally

$Q_{u}(w)=\int_{\Sigma}|dw|_{h}^{2}\
dvol_{h}-\int_{\Sigma}\left<d[P_{u}]\cdot_{h}du,d_{w}P_{u}\,w\right>\
dvol_{h}$ (II.9)

Recall that for any extensions in an ${\mathbb{R}}^{n}$ neighborhood of a
point $z\in{\mathcal{N}}^{n}$ of a pair of vectors $X,Y$ in
$T_{z}{\mathcal{N}}^{n}$ one has

${\mathbb{I}}_{z}(X,Y)=d_{Y}X-P_{z}(d_{Y}X)$

Choosing an extension such that $P_{z}(X)=X$ for all $z\in{\mathcal{N}}^{n}$
this gives

${\mathbb{I}}_{z}(X,Y)=d_{Y}(P_{z}X)-P_{z}(d_{Y}X)=d_{Y}P_{z}X\ .$ (II.10)

We finally obtain

$Q_{u}(w)=\int_{\Sigma}|dw|_{h}^{2}\
dvol_{h}-\int_{\Sigma}\left<{\mathbb{I}}_{u}(du,du)_{h},{\mathbb{I}}_{u}(w,w)\right>\
dvol_{h}$ (II.11)

We shall denote by $S_{u}(du)_{h}$ the map from $\Sigma$ into the space of
symmetric matrices real $m\times m$ matrices such that

$\forall x\in\Sigma\quad\forall X,Y\in
T_{u(x)}{\mathcal{N}}^{n}\quad\left<S_{u}(du)_{h}\,X,Y\right>=\left<{\mathbb{I}}_{u}(du,du)_{h},{\mathbb{I}}_{u}(X,Y)\right>\
.$ (II.12)

Observe that with this definition one has

$|S_{u}(du)_{h}|\leq C\,|du|^{2}_{h}\ .$ (II.13)

where $C>0$ is independent of $u$ and $h$.

In the case of the sphere ${\mathcal{N}}^{n}=S^{n}$ for instance one has
$P_{u}=I_{n+1}-u\otimes u$ which implies

$dP_{u}=-\left[du\otimes u-u\otimes du\right]\quad\mbox{ and }\quad
d_{w}P_{u}=-w\otimes u-u\otimes w\ .$ (II.14)

This gives

$d[P_{u}]\cdot_{h}du=-\,u\,|du|^{2}_{h}\quad\mbox{ and }\quad
d_{w}P_{u}\,w=-u\,|w|^{2}\ .$ (II.15)

Combining (II.9) and (II.14) gives in this case

$Q_{u}(w)=\int_{\Sigma}|dw|_{h}^{2}\
dvol_{h}-\int_{\Sigma}|du|^{2}_{h}\,|w|^{2}\ dvol_{h}\ .$ (II.16)

####  II.1.2 The computation of $D^{2}{\mathfrak{L}}$ in the general case.

We extend $\alpha$ in a small neighborhood of ${\mathcal{N}}^{n}$ and we keep
denoting $\alpha$ for $\pi_{N}^{\ast}\alpha$ in this neighborhood.888In this
work we take the most current (but not exclusively adopted in the literature)
convention  $dz^{i}\wedge dz^{j}(X,Y)=X^{i}\,Y^{j}-Y^{i}\,X^{j}\ $ and
$dz^{k}\wedge dz^{i}\wedge
dz^{j}(X,Y,Z)=X^{k}\,(Y^{i}\,Z^{j}-Z^{i}\,Y^{j})+Y^{k}\,(Z^{i}\,X^{j}-Z^{j}\,X^{i})+Z^{k}\,(X^{i}\,Y^{j}-X^{j}\,Y^{j})\
.$ We write

$\alpha:=\frac{1}{2}\sum_{i,j=1}^{m}\alpha_{ij}(z)\ dz^{i}\wedge
dz^{j}=\sum_{1\leq i<j\leq m}\ \alpha_{ij}(z)\ dz^{i}\wedge dz^{j},$

in such a way that

$d\alpha=\frac{1}{2}\sum_{i,j,k=1}^{m}\partial_{z_{k}}\alpha_{ij}\
dz_{k}\wedge dz_{i}\wedge dz_{j}\ .$

which gives

$\begin{array}[]{l}\displaystyle
d\alpha(X,Y,Z)=\frac{1}{2}\sum_{i,j,k=1}^{m}\partial_{z_{k}}\alpha_{ij}\left(X^{k}\,(Y^{i}\,Z^{j}-Z^{i}\,Y^{j})+Y^{k}\,(Z^{i}\,X^{j}-Z^{j}\,X^{i})+Z^{k}\,(X^{i}\,Y^{j}-X^{j}\,Y^{j})\right)\\\\[14.22636pt]
\displaystyle\
=\frac{1}{2}\sum_{k=1}^{m}X^{k}\sum_{i,j=1}^{m}[\partial_{z_{k}}\alpha_{ij}+\partial_{z_{j}}\alpha_{ki}+\partial_{z_{i}}\alpha_{jk}]\
(Y^{i}\,Z^{j}-Z^{i}\,Y^{j})\end{array}$

We introduce then

$\begin{array}[]{l}\displaystyle
H(Y,Z):=\sum_{k=1}^{m}H^{k}(Y,Z)\,\partial_{z_{k}}:=\frac{1}{2}\,\sum_{i,j,k=1}^{m}[\partial_{z_{k}}\alpha_{ij}+\partial_{z_{j}}\alpha_{ki}+\partial_{z_{i}}\alpha_{jk}]\
(Y^{i}\,Z^{j}-Z^{i}\,Y^{j})\ \partial_{z_{k}}\\\\[14.22636pt] \displaystyle\
=\sum_{k=1}^{m}\sum_{i<j}^{m}[\partial_{z_{k}}\alpha_{ij}+\partial_{z_{j}}\alpha_{ki}+\partial_{z_{i}}\alpha_{jk}]\
(Y^{i}\,Z^{j}-Z^{i}\,Y^{j})\ \partial_{z_{k}}\end{array}$

With this notation one has

$d\alpha(X,Y,Z)=\left<X,H(Y,Z)\right>\ .$

We have

$\begin{array}[]{l}\displaystyle\frac{1}{2}\frac{d}{dt}\int_{\Sigma}u_{t}^{\ast}\alpha=\frac{1}{2}\frac{d}{dt}\int_{\Sigma}\sum_{i,j=1}^{m}\alpha_{ij}(u_{t})du^{i}_{t}\wedge
du^{j}_{t}\\\\[14.22636pt]
\displaystyle=\frac{1}{2}\int_{\Sigma}\sum_{i,j,k=1}^{m}\partial_{z_{k}}\alpha_{ij}(u_{t})\frac{du^{k}_{t}}{dt}\
du^{i}_{t}\wedge du^{j}_{t}+\sum_{i,j=1}^{m}\alpha_{ij}(u_{t})\
d\frac{du^{i}_{t}}{dt}\wedge du^{j}_{t}+\sum_{i,j=1}^{m}\alpha_{ij}(u_{t})\
du^{i}_{t}\wedge d\frac{du^{j}_{t}}{dt}\\\\[14.22636pt]
\displaystyle=\int_{\Sigma}\sum_{k=1}^{m}\frac{du^{k}_{t}}{dt}\sum_{i<j}[\partial_{z_{k}}\alpha_{ij}(u_{t})+\partial_{z_{j}}\alpha_{ki}(u_{t})+\partial_{z_{i}}\alpha_{jk}(u_{t})]\
du^{i}_{t}\wedge du^{j}_{t}\\\\[14.22636pt]
\displaystyle=\int_{\Sigma}\sum_{k=1}^{m}\frac{du^{k}_{t}}{dt}\sum_{i<j}H^{k}_{ij}(u_{t})\
du^{i}_{t}\wedge du^{j}_{t}\end{array}$ (II.17)

Combining (II.1), (II.3) and (II.17) we obtain first that $u$ is a critical
point of $\mathfrak{L}$ if and only if

$\forall\ l=1\ldots m\quad
0=\sum_{k=1}^{m}P^{lk}_{u}\left[\Delta_{h}u^{k}+\sum_{i<j}H^{k}_{ij}(u_{t})\
du^{i}\wedge du^{j}\right].$ (II.18)

Since $P_{u}(H_{u}(du\wedge du))=H_{u}(du\wedge du)$ we deduce

$\Delta_{h}u+H_{u}(du\wedge
du)=-\,d[P_{u}]\cdot_{h}du=-\,{\mathbb{I}}_{u}(du,du).$ (II.19)

Taking the now second derivative of ${\mathfrak{L}}-E$ gives

$\begin{array}[]{l}\displaystyle\frac{d^{2}({\mathfrak{L}}-E)}{dt^{2}}=\frac{1}{2}\frac{d^{2}}{dt^{2}}\int_{\Sigma}u_{t}^{\ast}\alpha=\int_{\Sigma}\sum_{k=1}^{m}\frac{d^{2}u^{k}_{t}}{dt^{2}}\sum_{i<j}H^{k}_{ij}(u_{t})\
du^{i}_{t}\wedge du^{j}_{t}\\\\[14.22636pt]
\displaystyle+\int_{\Sigma}\sum_{k,l=1}^{m}\frac{du^{k}_{t}}{dt}\,\frac{du^{l}_{t}}{dt}\
\sum_{i<j}\partial_{z_{l}}H^{k}_{ij}(u_{t})\ du^{i}_{t}\wedge
du^{j}_{t}\\\\[14.22636pt]
\displaystyle+\int_{\Sigma}\sum_{k=1}^{m}\frac{du^{k}_{t}}{dt}\sum_{i<j}H^{k}_{ij}(u_{t})\
\left[d\frac{du^{i}_{t}}{dt}\wedge du^{j}_{t}+du^{i}_{t}\wedge
d\frac{du^{j}_{t}}{dt}\right]\end{array}$ (II.20)

Hence, for $u$ critical point of $\mathfrak{L}$, we have

$\begin{array}[]{l}\displaystyle\frac{d^{2}{\mathfrak{L}}(u)}{dt^{2}}(0)=\int_{\Sigma}|dw|^{2}_{h}\
dvol_{h}+\int_{\Sigma}\left<\left[\Delta_{h}u+H_{u}(du\wedge
du)_{h}\right],(I-P_{u})d_{w}P_{u}\phi\right>\ dvol_{h}\\\\[14.22636pt]
\displaystyle+\int_{\Sigma}\sum_{k,l=1}^{m}\,{w^{k}}\,{w^{l}}\
\sum_{i<j}\partial_{z_{l}}H^{k}_{ij}(u)\ du^{i}\wedge
du^{j}+\int_{\Sigma}\sum_{k=1}^{m}w^{k}\,\sum_{i<j}H^{k}_{ij}(u)\
\left[dw^{i}\wedge du^{j}+du^{i}\wedge dw^{j}\right]\\\\[14.22636pt]
\end{array}$ (II.21)

and using (II.7) and (II.10) we obtain

$\begin{array}[]{l}\displaystyle\frac{d^{2}{\mathfrak{L}}(u)}{dt^{2}}(0)=\int_{\Sigma}|dw|^{2}_{h}\
dvol_{h}+\int_{\Sigma}\left<\left[\Delta_{h}u+H_{u}(du\wedge
du)_{h}\right],(I-P_{u})d_{w}P_{u}\phi\right>\ dvol_{h}\\\\[14.22636pt]
\displaystyle+\int_{\Sigma}\sum_{k,l=1}^{m}\,{w^{k}}\,{w^{l}}\
\sum_{i<j}\partial_{z_{l}}H^{k}_{ij}(u)\ du^{i}\wedge
du^{j}+\int_{\Sigma}\sum_{k=1}^{m}w^{k}\,\sum_{i<j}H^{k}_{ij}(u)\
\left[dw^{i}\wedge du^{j}+du^{i}\wedge dw^{j}\right]\\\\[14.22636pt]
\end{array}$ (II.22)

Hence, using (II.19) this gives

$\begin{array}[]{l}\displaystyle\frac{d^{2}{\mathfrak{L}}(u)}{dt^{2}}(0)=\int_{\Sigma}|dw|^{2}_{h}\
dvol_{h}-\int_{\Sigma}\left<{\mathbb{I}}_{u}(du,du)_{h},{\mathbb{I}}_{u}(w,w)\right>\
dvol_{h}\\\\[14.22636pt]
\displaystyle\quad+\int_{\Sigma}w\cdot\left[2\,H_{u}(dw\wedge
du)_{h}+\nabla_{w}H_{u}(du,du)_{h}\right]\ dvol_{h}\end{array}$ (II.23)

where

$H_{u}(dw\wedge du)_{h}:=\frac{1}{2}\sum_{i,j=1}^{m}H^{k}_{ij}(u)\
\left[dw^{i}\wedge du^{j}+du^{i}\wedge dw^{j}\right]\ \partial_{z_{k}}\ .$

We introduce

$\begin{array}[]{rcl}\displaystyle{\mathcal{H}}_{u}(w)\
dvol_{h}:&=&\displaystyle\sum_{i<j}^{m}H_{ij}(u)\ [dw^{i}\wedge
du^{j}+du^{i}\wedge dw^{j}]+\sum_{l=1}^{m}{w^{l}}\
\sum_{i<j}^{m}\partial_{z_{l}}H_{ij}(u)\ du^{i}\wedge du^{j}\\\\[14.22636pt]
\displaystyle\hfil&=&\displaystyle 2\,H_{u}(dw\wedge
du)_{h}+\nabla_{w}H_{(}du\wedge du)_{h}\end{array}$ (II.24)

and we claim that ${\mathcal{H}}_{u}$ is self-adjoint on
$V_{u}=\Gamma(u^{-1}TN)$. We have

$\begin{array}[]{l}\displaystyle\int_{\Sigma}\sum_{k=1}^{m}w^{k}\,\sum_{i<j}H^{k}_{ij}(u)\
\left[dv^{i}\wedge du^{j}+du^{i}\wedge dv^{j}\right]\\\\[14.22636pt]
\displaystyle=-\frac{1}{2}\int_{\Sigma}\sum_{k=1}^{m}v^{i}\sum_{i,j}^{m}H^{k}_{ij}(u)\
dw^{k}\wedge
du^{j}+\frac{1}{2}\int_{\Sigma}\sum_{i,j}^{m}v^{j}\sum_{k=1}^{m}H^{k}_{ij}(u)\
dw^{k}\wedge du^{i}\\\\[14.22636pt]
\displaystyle-\frac{1}{2}\int_{\Sigma}\sum_{k,l=1}^{m}v^{i}\sum_{i,j}^{m}\partial_{z_{l}}H^{k}_{ij}(u)\
w^{k}\ du^{l}\wedge
du^{j}+\frac{1}{2}\int_{\Sigma}\sum_{i,j}^{m}v^{j}\sum_{k,l=1}^{m}\partial_{z_{l}}H^{k}_{ij}(u)\
w^{k}du^{l}\wedge du^{i}\\\\[14.22636pt]
\displaystyle=\int_{\Sigma}\sum_{k=1}^{m}v^{k}\sum_{i<j}^{m}H^{k}_{ij}(u)\
[dw^{i}\wedge du^{j}+du^{i}\wedge dw^{j}]\\\\[14.22636pt]
\displaystyle+\frac{1}{2}\int_{\Sigma}\sum_{k,l=1}^{m}v^{k}\sum_{i,j=1}^{m}\partial_{z_{l}}H^{k}_{ij}(u)\
w^{i}\ du^{l}\wedge
du^{j}-\frac{1}{2}\int_{\Sigma}\sum_{i,j=1}^{m}v^{k}\sum_{k,l=1}^{m}\partial_{z_{l}}H^{k}_{ij}(u)\
w^{j}du^{l}\wedge du^{i}\end{array}$ (II.25)

We have

$\begin{array}[]{l}\displaystyle\frac{1}{2}\int_{\Sigma}\sum_{k,l=1}^{m}\,{w^{k}}\,{v^{l}}\
\sum_{i,j=1}^{m}\partial_{z_{l}}H^{k}_{ij}(u)\ du^{i}\wedge
du^{j}+\frac{1}{2}\int_{\Sigma}\sum_{k,l=1}^{m}v^{k}\sum_{i,j=1}^{m}\partial_{z_{l}}H^{k}_{ij}(u)\
w^{i}\ du^{l}\wedge du^{j}\\\\[14.22636pt]
\displaystyle-\frac{1}{2}\int_{\Sigma}\sum_{i,j=1}^{m}v^{k}\sum_{k,l=1}^{m}\partial_{z_{l}}H^{k}_{ij}(u)\
w^{j}du^{l}\wedge du^{i}\\\\[14.22636pt]
\displaystyle=\frac{1}{2}\int_{\Sigma}\sum_{k,l=1}^{m}\,{w^{k}}\,{v^{l}}\
\sum_{i,j=1}^{m}\left[\partial_{z_{l}}H^{k}_{ij}(u)+\partial_{z_{i}}H^{l}_{kj}(u)-\partial_{z_{j}}H^{l}_{ki}-\partial_{z_{k}}H^{l}_{ij}\right]\
du^{i}\wedge du^{j}\\\\[14.22636pt]
\displaystyle+\frac{1}{2}\int_{\Sigma}\sum_{k,l=1}^{m}\,v^{k}\,{w^{l}}\
\sum_{i,j=1}^{m}\partial_{z_{l}}H^{k}_{ij}(u)\ du^{i}\wedge
du^{j}\\\\[14.22636pt]
\displaystyle=\frac{1}{2}\int_{\Sigma}\sum_{k,l=1}^{m}\,v^{k}\,{w^{l}}\
\sum_{i,j=1}^{m}\partial_{z_{l}}H^{k}_{ij}(u)\ du^{i}\wedge du^{j}\end{array}$
(II.26)

Combining (II.25) and (II.26) gives that the operator ${\mathcal{H}}_{u}$ is
self-adjoint on $V_{u}=\Gamma(u^{-1}T{\mathcal{N}}^{n})$.

We shall denote

${\mathcal{L}}_{u}w:=P_{u}\left[\Delta_{h}w-S_{u}(du)_{h}\,w+{\mathcal{H}}_{u}(w)\right]\
.$

and with this notation

$D^{2}{\mathfrak{L}}_{u}(w)=\int_{\Sigma}w\cdot{\mathcal{L}}_{u}w\ dvol_{h}\
.$

##  III Proof of the pointwise estimate of the gradient in the necks.

Let ${\mathcal{N}}^{n}$ be an arbitrary closed $C^{2}$ sub-manifold of
${\mathbb{R}}^{m}$ and $\alpha$ be an arbitrary $C^{2}$ 2-form on
${\mathcal{N}}^{n}$. For a sake of clarity we assume that we have exactly one
bubble. The general case is inducing more complicated notations but no change
at all in the arguments.

Hence from now on we shall consider a sequence of critical points $u_{k}$ of
${\mathfrak{L}}$ among maps from $(\Sigma,h)$ into ${\mathcal{N}}^{n}$ which
“bubble tree converges” to a pair of maps $(u_{\infty},v_{\infty})$
respectively from $\Sigma$ into ${\mathcal{N}}^{n}$ and from $\mathbb{C}$ into
${\mathcal{N}}^{n}$. More precisely there exists $p\in\Sigma$,
$x_{k}\rightarrow p$ and $\delta_{k}\rightarrow 0$ such that

  * i)

$u_{k}\longrightarrow u_{\infty}\quad\mbox{in
}C^{1}_{loc}(\Sigma\setminus\\{p\\})\ ,$ (III.1)

  * ii)

in local fixed conformal coordinates for $h$ around $p$

$u_{k}(\delta_{k}\,y+x_{k})\longrightarrow v_{\infty}(y)\quad\mbox{in
}C^{1}_{loc}({\mathbb{C}})\ ,$ (III.2)

  * iii)

$\lim_{\eta\rightarrow 0}\lim_{k\rightarrow+\infty}\
\sup_{\delta_{k}/\eta<\rho<2\rho<\eta}\int_{B_{2\rho}(x_{k})\setminus
B_{\rho}(x_{k})}|du_{k}|^{2}_{h}\ dvol_{h}=0$ (III.3)

The main arguments in [17] is implying

$\lim_{\eta\rightarrow 0}\limsup_{k\rightarrow+\infty}\
\int_{A(\eta,\delta_{k})}|du_{k}|^{2}_{h}\ dvol_{h}=0$ (III.4)

where we shall be using the following notation in the rest of the paper

$A(\eta,\delta_{k}):=B_{\eta}(x_{k})\setminus B_{\delta_{k}/\eta}(x_{k})\ .$

The annulus $A(\eta,\delta_{k})$ is called annular neck region and the
identity (III.4) is also known under $L^{2}-$energy quantization. The goal of
the present section is to convert the $L^{2}$-energy quantization given by
(III.4) into a pointwise quantization given by the following lemma.

######  Lemma III.1.

Let $u_{k}$ be a sequence of critical points of $\mathfrak{L}$ from
$(\Sigma,h)$ into ${\mathcal{N}}^{n}$, a $C^{2}$ sub-manifold of
${\mathbb{R}}^{m}$ and $\alpha$ is a $C^{2}$ 2-form of ${\mathcal{N}}^{n}$.
Assume $u_{k}$ “bubble tree converges” to a pair of maps
$(u_{\infty},v_{\infty})$ in the sense that (III.1), (III.2) and (III.3) are
satisfied. Then there exist $C>0$ and $0<\beta<1$ such that, for any $\eta$
small enough independent of $k$ in local conformal coordinates around $x_{k}$
such that $(x_{1}(x_{k}),x_{2}(x_{k}))=(0,0)$ there holds

$\begin{array}[]{l}\displaystyle\forall x\in
A(\eta,\delta_{k})\quad\quad\displaystyle|x|^{2}\ |\nabla u_{k}|^{2}(x)\leq C\
\left[\frac{|x|^{\beta}}{\eta^{\beta}}+\left(\frac{\delta_{k}}{\eta\,|x|}\right)^{\beta}\right]\
\int_{A(2\eta,\delta_{k})}|du_{k}|^{2}_{h}\ dvol_{h}+C_{\eta,k}\ ,\end{array}$
(III.5)

where

$\lim_{\eta\rightarrow 0}\limsup_{k\rightarrow+\infty}\
\log^{2}\left(\frac{\eta^{2}}{\delta_{k}}\right)\,C_{\eta,k}=0\ .$ (III.6)

and $C_{\eta,k}$ is uniformly bounded independent of $k$ and of
$\eta<\eta_{0}$ for some $\eta_{0}>0$. $\Box$

###  III.1 The $C^{0}$-no neck Property and $L^{2,1}-$Energy Quantization.

The $L^{2}$ energy quantization (III.4) is reinforced in [17] \- estimate (56)
- by the following

$\lim_{\eta\rightarrow 0}\lim_{k\rightarrow+\infty}\
\left\|\frac{1}{r}\frac{\partial
u_{k}}{\partial\theta}\right\|_{L^{2,1}(A(\eta,\delta_{k}))}=0\ .$ (III.7)

Where we recall that the $L^{2,1}$ norm of a function $f$ is equivalent to the
following quasi-norm

$|f|_{L^{2,1}(A(\eta,\delta_{k}))}=\int_{0}^{+\infty}\left|\left\\{x\in
A(\eta,\delta_{k})\ ;\ |f(x)|>s\right\\}\right|^{1/2}\ ds\ .$

Recall that, for any critical point of conformally invariant problems of the
form given by $\mathfrak{L}$, the Hopf differentials of $u_{k}$ given in local
conformal coordinates for $(\Sigma,h)$ by

$\mathfrak{H}(u_{k}):=\partial_{z}u_{k}\cdot\partial_{z}u_{k}\ dz\otimes dz$
(III.8)

defines an holomorphic quadratic differential of $(\Sigma,h)$. The space of
holomorphic quadratic differentials is finite dimensional and since
$\mathfrak{H}(u_{k})$ is uniformly bounded in $L^{1}$, it is pre-compact in
any norm. In particular it is uniformly bounded in $L^{2}$. We have

$\partial_{z}u_{k}=e^{-i\theta}\,(\partial_{\rho}u_{k}-i\,\rho^{-1}\,\partial_{\theta}u_{k})\
.$

Hence

$\Re\left(e^{2\,i\,\theta}\,\partial_{z}u_{k}\cdot\partial_{z}u_{k}\right):=|\partial_{\rho}u_{k}|^{2}-\rho^{-2}\,|\partial_{\theta}u_{k}|^{2}\
.$ (III.9)

Which gives

$|\partial_{\rho}u_{k}|\leq\,\rho^{-1}\,|\partial_{\theta}u_{k}|+\,\sqrt{\left|\Re\left(e^{2\,i\,\theta}\,\partial_{z}u_{k}\cdot\partial_{z}u_{k}\right)\right|}\
.$ (III.10)

Generalized Hölder inequality (see [10]) gives

$\begin{array}[]{rcl}\displaystyle\left\|\sqrt{\left|\Re\left(e^{2\,i\,\theta}\,\partial_{z}u_{k}\cdot\partial_{z}u_{k}\right)\right|}\right\|_{L^{2,1}(A(\eta,\delta_{k}))}&\leq&|A(\eta,\delta_{k}))|^{1/4}\
\left\|\sqrt{\left|\Re\left(e^{2\,i\,\theta}\,\partial_{z}u_{k}\cdot\partial_{z}u_{k}\right)\right|}\right\|_{L^{4}(A(\eta,\delta_{k}))}\\\\[14.22636pt]
\displaystyle\hfil&\leq&\sqrt{\eta}\,\sqrt{\|\mathfrak{H}(u_{k})\|_{L^{2}(\Sigma)}}\end{array}$
(III.11)

Combining (III.7) and (III.11) gives

$\lim_{\eta\rightarrow 0}\limsup_{k\rightarrow+\infty}\ \left\|\frac{\partial
u_{k}}{\partial r}\right\|_{L^{2,1}(A(\eta,\delta_{k}))}=0\ .$ (III.12)

Hence we have established the following lemma.

######  Lemma III.2.

Let $u_{k}$ be a sequence of critical points of $\mathfrak{L}$ satisfying
(III.1), (III.2) and (III.3). Then the following holds

$\lim_{\eta\rightarrow 0}\ \limsup_{k\rightarrow+\infty}\
\|du_{k}\|_{L^{2,1}(A(\eta,\delta_{k}))}=0\ .$ (III.13)

$\Box$

###  III.2 Proof of Lemma III.1.

As explained in [29] the Euler-Lagrange equation (I.3) can be rewritten in the
form

$\Delta_{h}u_{k}=\Omega_{k}\cdot du_{k}\quad\quad\mbox{ on }\Sigma\ ,$
(III.14)

where there exists $C>0$ such that

$\Omega_{k}\in\Gamma(T^{\ast}\Sigma\otimes so(m))\quad\mbox{ and
}\quad|\Omega_{k}|(x)\leq C\,|du_{k}|_{h}(x)\quad\forall x\in\Sigma\ .$
(III.15)

where $so(m)$ denotes the space of real $m\times m$ antisymmetric matrices.

By extending $\Omega_{k}$ outside $A(2\eta,\delta_{k})$ by 0 one produces
$\tilde{\Omega}_{k}\in L^{2}(\wedge^{1}B_{1}(0),so(m))$. For $\eta$ small
enough and $k$ large enough, using theorem I.4 of [29] we obtain the existence
of $A_{k}\in W^{1,2}(B_{1},Gl_{m}({\mathbb{R}}))$ and $B_{k}\in
W^{1,2}(B_{1},M_{m}({\mathbb{R}}))$ such that

$\nabla A_{k}-A_{k}\,\tilde{\Omega}_{k}=\nabla^{\perp}B_{k}\quad\quad\mbox{ in
}B_{1}(0)\ ,$ (III.16)

and

$\begin{array}[]{l}\displaystyle\int_{B_{1}}|\nabla A_{k}|^{2}+|\nabla
A_{k}^{-1}|^{2}\ dx^{2}+\int_{B_{1}}|\nabla B_{k}|^{2}\
dx^{2}+\left\|\mbox{dist}\left(\left\\{A_{k},A_{k}^{-1}\right\\},SO(m)\right)\right\|^{2}_{L^{\infty}(B_{1}(0))}\\\\[14.22636pt]
\displaystyle\leq C\ \int_{A(2\eta,\delta_{k})}|\Omega_{k}|^{2}\ dx^{2}\leq C\
\int_{A(2\eta,\delta_{k})}|\nabla u_{k}|^{2}\ dx^{2}\ .\end{array}$ (III.17)

and we have also obviously

$\int_{B_{1}}|\nabla B_{k}|^{2}\ dx^{2}\leq C\
\int_{A(2\eta,\delta_{k})}|\nabla u_{k}|^{2}\ dx^{2}\ .$ (III.18)

It is proved in [29] that the following conservation law (in conformal
coordinates for $h$) is satisfied

$\mbox{div}\left(A_{k}\,\nabla u_{k}\right)=\nabla^{\perp}B_{k}\cdot\nabla
u_{k}\quad\mbox{ in }A(\eta,\delta_{k})\ .$ (III.19)

We extend $u_{k}$, $A_{k}$ and $B_{k}$ outside $A(\eta,\delta_{k})$ in the
whole ${\mathbb{C}}$ using lemma C.1 and we denote respectively
$\tilde{A}_{k}$ and $\tilde{B}_{k}$ these extensions. Let $\varphi_{k}$ and
$\psi_{k}$ be the solutions in $\dot{W}^{1,2}({\mathbb{C}})$ respectively of

$\Delta\varphi_{k}=\nabla^{\perp}\tilde{B}_{k}\cdot\nabla\tilde{u}_{k}\quad\mbox{
in }{\mathbb{C}}\ ,$ (III.20)

and

$\Delta\psi_{k}=\nabla^{\perp}\tilde{A}_{k}\cdot\nabla\tilde{u}_{k}\quad\mbox{
in }{\mathbb{C}}\ .$ (III.21)

We have that $A_{k}du_{k}-d\varphi_{k}-\ast d\psi_{k}$ realizes an harmonic
1-form in $A(\eta,\delta_{k})$. Hence there exists $\mathfrak{h}_{k}$ harmonic
such that

$A_{k}\,\nabla
u_{k}-\nabla\varphi_{k}-\nabla^{\perp}\psi_{k}=\nabla\mathfrak{h}_{k}+C^{1}_{k}\,\nabla^{\perp}\log
r$ (III.22)

We have in particular for any $r\in[\delta_{k}/\eta,\eta]$ such that

$2\pi\,C_{k}^{1}=\int_{\partial B_{r}(0)}A_{k}\,\frac{1}{r}\frac{\partial
u_{k}}{\partial\theta}\ \,d\theta-\int_{\partial
B_{r}(0)}\partial_{\nu}\psi_{k}\
\,d\theta=\int_{B_{r}}\nabla\tilde{A}_{k}\cdot\nabla^{\perp}u_{k}-\Delta\psi_{k}\
dx^{2}=0$ (III.23)

Hence finally we have

$A_{k}\,\nabla
u_{k}=\nabla\varphi_{k}+\nabla^{\perp}\psi_{k}+\nabla\mathfrak{h}_{k}\ .$
(III.24)

The harmonic part $\mathfrak{h}_{k}$ is decomposed as follows

$\mathfrak{h}_{k}=\mathfrak{h}_{k}^{+}+\mathfrak{h}_{k}^{-}+\mathfrak{h}_{k}^{0}\
\ \mbox{ where }\ \ \mathfrak{h}_{k}^{+}=\Re\sum_{n>0}h_{n,k}z^{n}\ \ ,\ \
\mathfrak{h}_{k}^{-}=\Re\sum_{n<0}h_{n,k}z^{n}\ \ \mbox{ and }\ \
\mathfrak{h}_{k}^{0}=\mathfrak{h}_{0,k}+C_{\eta,k}^{0}\,\log|z|\ .$

Coifman-Lions-Meyer-Semmes estimate (see [12]) applies to both $\varphi_{k}$
and $\psi_{k}$ and says respectively

$\|\nabla\varphi_{k}\|^{2}_{L^{2,1}({\mathbb{C}})}\leq C\
\int_{\mathbb{C}}|\nabla\tilde{B}_{k}|^{2}\ dx^{2}\
\int_{\mathbb{C}}|\nabla\tilde{u}_{k}|^{2}\ dx^{2}\leq
C\,\left[\int_{A(\eta,\delta_{k})}|du_{k}|^{2}_{h}\ dvol_{h}\right]^{2}\ ,$
(III.25)

and

$\|\nabla\psi_{k}\|^{2}_{L^{2,1}({\mathbb{C}})}\leq C\
\int_{\mathbb{C}}|\nabla\tilde{A}_{k}|^{2}\ dx^{2}\
\int_{\mathbb{C}}|\nabla\tilde{u}_{k}|^{2}\ dx^{2}\leq
C\,\left[\int_{A(\eta,\delta_{k})}|du_{k}|^{2}_{h}\ dvol_{h}\right]^{2}\ ,$
(III.26)

where we have used (III.17) and (III.18). Hence we deduce from (III.13)

$\lim_{\eta\rightarrow 0}\ \limsup_{k\rightarrow+\infty}\
\|d\mathfrak{h}_{k}\|_{L^{2,1}(A(\eta,\delta_{k}))}=0\ .$ (III.27)

The following lemma due to A.Michelat and the third author [23] is going to be
used for controlling the $L^{2,1}-$norms of $d\mathfrak{h}_{k}^{\pm}$ in the
neck region.

######  Lemma III.3 ([23] Lemma 2.3).

Let $0<4\,r<R$ be fixed radii, $u\colon
B_{R}(0)\setminus\bar{B}_{r}(0)\to\mathbb{R}$ be a harmonic function such that
for some $\rho_{0}\in(r,R)$

$\int_{\partial B_{\rho_{0}}(0)}\partial_{\nu}u=0.$ (III.28)

Then for all $\left(\frac{r}{R}\right)^{1/2}<t<1$ we have

$\|\nabla u\|_{L^{2,1}(B_{tR}(0)\setminus B_{t^{-1}r}(0))}\leq
C\,\frac{t}{1-t}\|\nabla u\|_{L^{2}(B_{R}(0)\setminus B_{r}(0))}\ ,$ (III.29)

for some universal constant $C$ independent of $R$, $r$ and $\alpha$. $\Box$

We apply lemma III.3 to $\mathfrak{h}^{\pm}_{k}$ and this gives for any fixed
$t<1$

$\|d\mathfrak{h}^{\pm}_{k}\|_{L^{2,1}(A(t\,\eta,\delta_{k}))}\leq
C\,\|d\mathfrak{h}^{\pm}_{k}\|_{L^{2}(A(\eta,\delta_{k}))}\ .$ (III.30)

We have

$\begin{array}[]{l}\displaystyle\int_{A(\eta,\delta_{k})}|\nabla\mathfrak{h}_{k}^{0}|^{2}\
dx^{2}+\int_{A(\eta,\delta_{k})}|\nabla\mathfrak{h}_{k}^{+}|^{2}\
dx^{2}+\int_{A(\eta,\delta_{k})}|\nabla\mathfrak{h}_{k}^{-}|^{2}\
dx^{2}\\\\[14.22636pt] \displaystyle\leq 4\,\int_{A(\eta,\delta_{k})}|\nabla
u_{k}|^{2}\ dx^{2}+4\,\int_{A(\eta,\delta_{k})}|\nabla\varphi_{k}|^{2}\
dx^{2}+4\,\int_{A(\eta,\delta_{k})}|\nabla\psi_{k}|^{2}\ dx^{2}\ .\end{array}$
(III.31)

Combining (III.4), (III.25), (III.26), (III.27), (III.30) and (III.31) we
obtain

$\lim_{\eta\rightarrow 0}\ \lim_{k\rightarrow+\infty}\
\|d\mathfrak{h}^{0}_{k}\|_{L^{2,1}(A(t\,\eta,\delta_{k}))}=0\ .$ (III.32)

The following direct computations are illustrating the “screening” each of the
$L^{2,q}$ norms for $q=1,2,\infty$ is operating on the neck. It looks
elementary but this is nevertheless central in the present work.

$\left\\{\begin{array}[]{l}\displaystyle\|\nabla\log|x|\|_{L^{2,\infty}(A(\eta,\delta_{k}))}\simeq\displaystyle\sup_{\lambda>0}\lambda\,\left|\left\\{x\in
A(\eta,\delta_{k}):|x|^{-1}>\lambda\right\\}\right|^{1/2}=\sqrt{\pi}\\\\[14.22636pt]
\displaystyle\|\nabla\log|x|\|_{L^{2}(A(\eta,\delta_{k}))}=\left[\int_{A(\eta,\delta_{k})}|\nabla\log|x||^{2}dx^{2}\right]^{1/2}=\sqrt{2\pi}\
\sqrt{\log\left(\frac{\eta^{2}}{\delta_{k}}\right)}\\\\[14.22636pt]
\displaystyle\|\nabla\log|x|\|_{L^{2,1}(A(\eta,\delta_{k}))}\simeq\displaystyle\int_{0}^{\infty}|x\in
A(\eta,\delta_{k}):~{}~{}|x|^{-1}\geq\lambda|^{1/2}d\lambda=\sqrt{2\pi}\,\log\left[\frac{\eta^{2}}{\delta_{k}}\right]\
.\end{array}\right.$ (III.33)

where the sign $\simeq$ is referring to the fact that we are giving equivalent
quasi-norms to the norms $L^{2,\infty}$ and $L^{2,1}$ respectively and the
constants behind are universal.

We deduce from (III.32) and (III.33) the following lemma999In the sphere case
one has that $A_{k}$ is the identity matrix, $\psi_{k}=0$ and moreover the
conservation law  $\mbox{div}(\nabla u_{k}+B_{k}\,\nabla^{\perp}u_{k})=0$ is
satisfied throughout the bubble in the whole domain $\Sigma$. As a
consequence, in this very special case, one can deduce by an integration by
part that  $C_{\eta,k}^{0}=0\ .$ (III.34) This might not be the case in the
general situation where (III.19) cannot a-priori be extended throughout
$B_{\delta_{k}/\eta}(x_{k})$..

######  Lemma III.4.

Under the above notations one has

$C_{\eta,k}^{0}=o_{{\eta},k}\left(\frac{1}{\log\left(\frac{\eta^{2}}{\delta_{k}}\right)}\right),~{}~{}\mbox{as
$k\to+\infty,~{}\eta\to 0$}.$ (III.35)

$\Box$

We introduce the following notation

$s_{1}:=[\log\eta^{-1}]\quad,\quad
s_{2}=[\log\left(\eta\,\delta_{k}^{-1}\right)]\quad\mbox{ and }\quad
A_{k}=B_{2^{-k}}(0)\setminus B_{2^{-k-1}}(0)\ .$ (III.36)

We adopt the notations of the previous subsection.

######  Proposition III.1.

There exists $0<\gamma<1$ such that for any $\gamma<\mu<1$ there exists $C>0$
such that for $\eta$ sufficiently small and $k$ large enough the following
holds. For all $j\in[s_{1},s_{2}]$ we have

$\begin{array}[]{l}\displaystyle\int_{A_{j}}|\nabla u_{k}|^{2}\ dx^{2}\leq
C\sum_{\ell=s_{1}}^{s_{2}}\mu^{|\ell-j|}{\int_{A_{\ell}}}|\nabla\mathfrak{h}_{k}|^{2}+C\,\sum_{\ell=s_{1}}^{s_{2}}\mu^{|\ell-j|}\gamma^{\ell}\left(\int_{B_{\eta}(0)}|\nabla\tilde{u}_{k}|^{2}\right)^{2}\\\\[14.22636pt]
\displaystyle+\,C\ \int_{B_{\eta}(0)}|\nabla\tilde{u}_{k}|^{2}\
dx^{2}\,\left(\mu^{j-s_{1}}\int_{B_{2\eta}(0)\setminus
B_{\eta}(0)}|\nabla\tilde{u}_{k}|^{2}\
dx^{2}+\mu^{s_{2}-j}\int_{B_{\frac{\delta_{k}}{\eta}}(0)\setminus
B_{\frac{\delta_{k}}{2\,\eta}}(0)}|\nabla\tilde{u}_{k}|^{2}\
dx^{2}\right)\end{array}$ (III.37)

Proof of Proposition III.1. We omit to explicitly write the subscript $k$
during the proof of the proposition. We can apply lemma F.2 to both $\varphi$
and $\psi$ in order to deduce for any $j\in\\{s_{1},\ldots,s_{2}\\}$

$\displaystyle{\int_{A_{j}}}|\nabla\varphi|^{2}\ dx^{2}$ $\displaystyle\leq$
$\displaystyle\gamma^{j}\,\int_{B_{\eta}(0)}|\nabla\varphi|^{2}\
dx^{2}+C\,\int_{B_{\eta}(0)}|\nabla\tilde{B}|^{2}\ dx^{2}\
\sum_{n=0}^{\infty}\gamma^{|n-j|}\int_{A_{n}}|\nabla\tilde{u}|^{2}\ dx^{2}\ ,$

and

$\displaystyle{\int_{A_{j}}}|\nabla\psi|^{2}\ dx^{2}$ $\displaystyle\leq$
$\displaystyle\gamma^{j}\,\int_{B_{\eta}(0)}|\nabla\psi|^{2}\
dx^{2}+C\,\int_{B_{\eta}(0)}|\nabla\tilde{A}|^{2}\ dx^{2}\
\sum_{n=0}^{\infty}\gamma^{|n-j|}\int_{A_{n}}|\nabla\tilde{u}|^{2}\ dx^{2}\ .$

Using the fact that

$\int_{B_{\eta}(0)}|\nabla\tilde{A}_{k}|^{2}+|\nabla\tilde{B}_{k}|^{2}\
dx^{2}\leq C\,\int_{A(\eta,\delta)}|\nabla u_{k}|^{2}\ dx^{2}$ (III.38)

we obtain respectively

$\displaystyle{\int_{A_{j}}}|\nabla\varphi|^{2}\ dx^{2}$ $\displaystyle\leq$
$\displaystyle\gamma^{j}\,\int_{B_{\eta}(0)}|\nabla\varphi|^{2}\
dx^{2}+C\,\int_{A(\eta,\delta)}|\nabla u|^{2}\ dx^{2}\
\sum_{n=0}^{\infty}\gamma^{|n-j|}\int_{A_{n}}|\nabla\tilde{u}|^{2}\ dx^{2}\ ,$

and

$\displaystyle{\int_{A_{j}}}|\nabla\psi|^{2}\ dx^{2}$ $\displaystyle\leq$
$\displaystyle\gamma^{j}\,\int_{B_{\eta}(0)}|\nabla\psi|^{2}\
dx^{2}+C\,\int_{A(\eta,\delta)}|\nabla u|^{2}\ dx^{2}\
\sum_{n=0}^{\infty}\gamma^{|n-j|}\int_{A_{n}}|\nabla\tilde{u}|^{2}\ dx^{2}\ .$

By using that $A\,\nabla
u=\nabla\varphi+\nabla^{\perp}\psi+\nabla\mathfrak{h}$ it also holds:

$\displaystyle{\int_{A_{j}}}|\nabla\tilde{u}|^{2}\ dx^{2}$ $\displaystyle\leq$
$\displaystyle 4{\int_{A_{j}}}|\nabla\mathfrak{h}|^{2}\
dx^{2}+4\,{\int_{A_{j}}}|\nabla\varphi|^{2}+|\nabla\psi|^{2}\ dx^{2}$
$\displaystyle\leq$ $\displaystyle 4\,{\int_{A_{j}}}|\nabla\mathfrak{h}|^{2}\
dx^{2}+4\,\gamma^{j}\left(\int_{B(0,\eta)}|\nabla\tilde{u}|^{2}\
dx^{2}\right)^{2}$ $\displaystyle+$ $\displaystyle
C\int_{B(0,\eta)}|\nabla\tilde{u}|^{2}\ dx^{2}\
\sum_{n=0}^{\infty}\gamma^{|n-j|}\int_{A_{n}}|\nabla\tilde{u}|^{2}\ dx^{2}\ .$

From Lemma C.1 implies

$\int_{B^{c}_{2\eta}(0)}|\nabla\tilde{u}|^{2}\ dx^{2}=0\quad\mbox{ and
}\quad\int_{B_{\frac{\delta}{2\eta}}(0)}|\nabla\tilde{u}|^{2}\ dx^{2}=0\ .$

If $\int_{B(0,\eta)}|\nabla\tilde{u}|^{2}<\varepsilon_{0}$ then by applying
lemma G.1 to the sequences

$a_{k}=\int_{A_{k}}|\nabla\tilde{u}|^{2}\ dx^{2}~{}~{}~{}\mbox{
and}~{}~{}~{}b_{k}=2{\int_{A_{k}}}|\nabla\mathfrak{h}|^{2}\
dx^{2}+2\,\gamma^{k}\left(\int_{B(0,\eta)}|\nabla\tilde{u}|^{2}\
dx^{2}\right)^{2}$

and using the fact that $a_{k}=0$ if $k\geq[\log(2\eta/\delta)]$ and
$k\leq[\log((2\eta)^{-1})]$ we obtain for $\eta$ small enough and $k$ large
enough, passing

$C\ \int_{B_{\eta}(0)}|\nabla\tilde{u}|^{2}\
dx^{2}\,\sum_{\ell=s_{1}}^{s_{2}}\mu^{|\ell-j|}\int_{A_{\ell}}|\nabla\tilde{u}|^{2}\
dx^{2}$

from the r.h.s to the l.h.s. of (G.2)

$\begin{array}[]{l}\displaystyle\sum_{\ell=s_{1}}^{s_{2}}\mu^{|\ell-j|}\int_{A_{\ell}}|\nabla\tilde{u}|^{2}\
dx^{2}\leq\,\sum_{\ell=s_{1}}^{s_{2}}\mu^{|\ell-j|}b_{\ell}+\,C\
\int_{B_{\eta}(0)}|\nabla\tilde{u}|^{2}\
dx^{2}\sum_{n=[\log\left(\frac{1}{2\eta}\right)]}^{s_{1}-1}\mu^{j-n}\,\int_{A_{n}}|\nabla\tilde{u}|^{2}\
dx^{2}\\\ \displaystyle+\,C\ \int_{B_{\eta}(0)}|\nabla\tilde{u}|^{2}\
\sum_{n=s_{2}+1}^{[\log(2\eta/\delta)]}\mu^{n-j}\,\int_{A_{n}}|\nabla\tilde{u}|^{2}\\\
\displaystyle\leq\,C\
\sum_{\ell=s_{1}}^{s_{2}}\mu^{|\ell-j|}{\int_{A_{\ell}}}|\nabla\mathfrak{h}|^{2}+C\sum_{\ell=s_{1}}^{s_{2}}\mu^{|\ell-j|}\gamma^{\ell}\left(\int_{B(0,\eta)}|\nabla\tilde{u}|^{2}\right)^{2}\\\
\
\displaystyle+\,C\,\left(\int_{B(0,\eta)}|\nabla\tilde{u}|^{2}\right)\left(\mu^{j-s_{1}}\int_{B(0,2\eta)\setminus
B(0,\eta)}|\nabla\tilde{u}|^{2}+\mu^{s_{2}-j}\int_{B(0,\delta/\eta)\setminus
B(0,\delta/2\eta)}|\nabla\tilde{u}|^{2}\right)\ .\end{array}$ (III.40)

In particular we get (III.37) and we conclude the proof of proposition III.1.
$\Box$

Next we estimate the Dirichlet energy of the harmonic part of $u_{k}$.

######  Lemma III.5.

There exists $4^{-1}<\mu<1$ and $C>0$ independent of $k$ and $\eta$ such that
for any $j\in\\{s_{1},\ldots,s_{2}\\}$:

$\displaystyle\sum_{\ell=s_{1}}^{s_{2}}\mu^{|\ell-j|}{\int_{A_{\ell}}}|\nabla\mathfrak{h}_{k}|^{2}\
dx^{2}$ $\displaystyle\leq$ $\displaystyle
C\left[\left(\frac{2^{-j}}{\eta}\right)^{\beta}+\left(\frac{\delta}{2^{-j}\,\eta}\right)^{\beta}\right]\int_{A(2\,\eta,\delta_{k})}|\nabla\tilde{u}_{k}|^{2}\
dx^{2}$ $\displaystyle+$ $\displaystyle
o_{\eta,k}\left(\frac{1}{\log^{2}\left(\frac{\eta^{2}}{\delta_{k}}\right)}\right),\
$

where $\lim_{\eta\rightarrow 0}\limsup_{k\rightarrow+\infty}o_{\eta,k}(1)=0$
and $\beta:=-\log_{2}\mu$. $\Box$

Proof of Lemma III.5. In this proof again we omit to write the subscript $k$.

1. Using (D.1) from lemma D.1 on $A(2\,\eta,\delta)$, $\mathfrak{h}^{+}$ satisfies for $\rho\in[2^{-s_{2}},2^{-s_{1}}]$

$\displaystyle|\nabla\mathfrak{h}^{+}(\rho,\theta)|^{2}$
$\displaystyle\leq\frac{C}{\eta^{2}}\int_{A(\eta,\delta)}|\nabla\mathfrak{h}^{+}|^{2}\
dx^{2}\leq\frac{C}{\eta^{2}}\ \int_{A(2\,\eta,\delta)}|\nabla\tilde{u}|^{2}\
dx^{2}\ .$ (III.42)

Therefore for every $\ell\in\\{s_{1},\ldots,s_{2}\\}$ we have

$\int_{A_{\ell}}|\nabla\mathfrak{h}^{+}(\rho,\theta)|^{2}\
\rho\,d\rho\,d\theta\leq
2^{-2\ell}\frac{C}{\eta^{2}}\int_{A(2\,\eta,\delta)}|\nabla\tilde{u}|^{2}\
dx^{2}$

and

$\displaystyle\sum_{\ell=s_{1}}^{s_{2}}\mu^{|\ell-j|}{\int_{A_{\ell}}}|\nabla\mathfrak{h}^{+}|^{2}$
$\displaystyle\leq$ $\displaystyle\frac{C}{\eta^{2}}\
\left[\sum_{\ell=s_{1}}^{s_{2}}\mu^{|\ell-j|}2^{-2\ell}\right]\
\int_{A(2\eta,\delta)}|\nabla\tilde{u}|^{2}\ dx^{2}$ (III.43) $\displaystyle=$
$\displaystyle\frac{C}{\eta^{2}}\
\left[\sum_{\ell=s_{1}}^{j}\mu^{j-\ell}2^{-2\ell}+\sum_{\ell=j+1}^{s_{2}}\mu^{\ell-j}2^{-2\ell}\right]\
\int_{A(2\eta,\delta)}|\nabla\tilde{u}|^{2}\ dx^{2}$ $\displaystyle\leq$
$\displaystyle\frac{C}{\eta^{2}}\ \left(\mu^{j-s_{1}}\
2^{-2s_{1}}+2^{-2j}\right)\ \int_{A(2\eta,\delta)}|\nabla\tilde{u}|^{2}\
dx^{2}.$

and we have

$\mu^{j-s_{1}}\
2^{-2s_{1}}\leq\eta^{2}\,\mu^{j-s_{1}}=\eta^{2}\,2^{\log_{2}\mu(j-s_{1})}=\eta^{2}\,\left(\frac{2^{-j}}{\eta}\right)^{\beta}\
.$ (III.44)

Combining (III.43) and (III.44) and using the fact that
$\left(\frac{2^{-j}}{\eta}\right)^{2}\leq\left(\frac{2^{-j}}{\eta}\right)^{\beta}$
we obtain

$\displaystyle\sum_{\ell=s_{1}}^{s_{2}}\mu^{|\ell-j|}{\int_{A_{\ell}}}|\nabla\mathfrak{h}^{+}|^{2}$
$\displaystyle\leq$ $\displaystyle
C\,\left(\frac{2^{-j}}{\eta}\right)^{\beta}\
\int_{A(2\,\eta,\delta)}|\nabla\tilde{u}|^{2}\ dx^{2}\ .$ (III.45)

2. Using (D.2) from lemma D.1 on $A(2\,\eta,\delta)$ we deduce that $w^{-}$ satisfies

$\displaystyle|\nabla\mathfrak{h}^{-}(\rho,\theta)|^{2}$ $\displaystyle\leq
C\,\frac{\delta^{2}}{\eta^{2}\rho^{4}}\
\int_{A(2\eta,\delta)}|\nabla\mathfrak{h}^{-}|^{2}\ dx^{2}\leq
C\,\frac{\delta^{2}}{\eta^{2}\rho^{4}}\
\int_{A(2\eta,\delta)}|\nabla\tilde{u}|^{2}\ dx^{2}.$ (III.46)

Therefore for every $\ell\in\\{s_{1},\ldots,s_{2}\\}$ we have

$\int_{A_{\ell}}|\nabla\mathfrak{h}^{-}(\rho,\theta)|^{2}\ dx^{2}\leq
C\,\frac{\delta^{2}}{\eta^{2}\,2^{-2\ell}}\int_{A(2\eta,\delta)}|\nabla\tilde{u}|^{2}\
dx^{2}\ .$

Thus

$\displaystyle\sum_{\ell=s_{1}}^{s_{2}}\mu^{|\ell-j|}{\int_{A_{\ell}}}|\nabla\mathfrak{h}^{-}|^{2}\
dx^{2}$ $\displaystyle\leq$ $\displaystyle
C\,\frac{\delta^{2}}{\eta^{2}}\left[\sum_{\ell=s_{1}}^{s_{2}}\mu^{|\ell-j|}2^{2\ell}\right]\
\int_{A(2\eta,\delta)}|\nabla\tilde{u}|^{2}\ dx^{2}$ (III.47) $\displaystyle=$
$\displaystyle C\,\frac{\delta^{2}}{\eta^{2}}\
\left[\sum_{\ell=s_{1}}^{j}\mu^{j-\ell}2^{2\ell}+\sum_{\ell=j+1}^{s_{2}}\mu^{\ell-j}2^{2\ell}\right]\int_{A(2\eta,\delta)}|\nabla\tilde{u}|^{2}\
dx^{2}$ $\displaystyle\leq$ $\displaystyle C\ \frac{\delta^{2}}{\eta^{2}}\
\left[2^{2j}+\mu^{s_{2}-j}2^{2s_{2}}\right]\
\int_{A(2\eta,\delta)}|\nabla\tilde{u}|^{2}\ dx^{2}$

We have

$\mu^{s_{2}-j}2^{2s_{2}}\leq\frac{\eta^{2}}{\delta^{2}}\
2^{(s_{2}-j)\,\log_{2}\mu}=\frac{\eta^{2}}{\delta^{2}}\
\left(\frac{\delta}{\eta\,2^{-j}}\right)^{\beta}\ .$ (III.48)

Using the fact that
$\left(\frac{\delta}{\eta\,2^{-j}}\right)^{2}\leq\left(\frac{\delta}{\eta\,2^{-j}}\right)^{\beta}$
we obtain

$\displaystyle\sum_{\ell=s_{1}}^{s_{2}}\mu^{|\ell-j|}{\int_{A_{\ell}}}|\nabla\mathfrak{h}^{-}|^{2}\
dx^{2}$ $\displaystyle\leq$ $\displaystyle C\
\left(\frac{\delta}{\eta\,2^{-j}}\right)^{\beta}\
\int_{A(2\eta,\delta)}|\nabla\tilde{u}|^{2}\ dx^{2}\ .$ (III.49)

3. Finally for $w^{0}=C^{0}\log(|x|)$ we use lemma III.4:

$\displaystyle\sum_{\ell=s_{1}}^{s_{2}}\mu^{|\ell-j|}{\int_{A_{\ell}}}|\nabla\mathfrak{h}^{0}|^{2}\
dx^{2}$ $\displaystyle\lesssim$
$\displaystyle(C^{0})^{2}\,\sum_{\ell=s_{1}}^{s_{2}}\mu^{|\ell-j|}\leq(C^{0})^{2}=o\left(\frac{1}{\log^{2}\left(\frac{\eta^{2}}{\delta}\right)}\right)\
.$ (III.50)

Combining (III.45), (III.49) and (III.50) we obtain (III.5) and this concludes
the proof of the lemma III.5. $\Box$

Proof of Lemma III.1. We again omit to write explicitly the subscript $k$.
Combining proposition III.1 and lemma III.5 we obtain for any
$j\in\\{s_{1},\ldots,s_{2}\\}$

$\begin{array}[]{l}\displaystyle\int_{A_{j}}|\nabla\tilde{u}|^{2}\ dx^{2}\leq
C\left[\left(\frac{2^{-j}}{\eta}\right)^{\beta}+\left(\frac{\delta}{2^{-j}\,\eta}\right)^{\beta}\right]\int_{A(\eta,\delta)}|\nabla\tilde{u}|^{2}\
dx^{2}\\\\[14.22636pt]
\displaystyle+o_{\eta,k}\left(\frac{1}{\log^{2}\left(\frac{\eta^{2}}{\delta_{k}}\right)}\right)\
+C\,\sum_{\ell=s_{1}}^{s_{2}}\mu^{|\ell-j|}\gamma^{\ell}\left(\int_{B_{\eta}(0)}|\nabla\tilde{u}|^{2}\right)^{2}\\\\[14.22636pt]
\displaystyle+\,C\ \int_{B_{\eta}(0)}|\nabla\tilde{u}|^{2}\
dx^{2}\,\left(\mu^{j-s_{1}}\int_{B_{2\eta}(0)\setminus
B_{\eta}(0)}|\nabla\tilde{u}|^{2}\
dx^{2}+\mu^{s_{2}-j}\int_{B_{\frac{\delta}{\eta}}(0)\setminus
B_{\frac{\delta}{2\,\eta}}(0)}|\nabla\tilde{u}|^{2}\
dx^{2}\right).\end{array}$ (III.51)

The $\epsilon$-regularity in [29] implies that for every $\rho>0$ such that
$\frac{\delta}{\eta}<\rho/4<4\rho<\eta$ we have

$\rho^{2}\,\|\nabla
u\|^{2}_{L^{\infty}(B(0,2\rho)\setminus\bar{B}(0,\rho/2))}\leq C\,\|\nabla
u\|^{2}_{L^{2}(B(0,4\rho)\setminus\bar{B}(0,\rho/4))}.$ (III.52)

Combining (III.51) and (III.52) gives for $|x|\simeq 2^{-j}$

$\begin{array}[]{l}\displaystyle|x|^{2}\ |\nabla u|^{2}(x)\leq
C\left[\left(\frac{2^{-j}}{\eta}\right)^{\beta}+\left(\frac{\delta}{2^{-j}\,\eta}\right)^{\beta}\right]\int_{A(\eta,\delta)}|\nabla\tilde{u}|^{2}\
dx^{2}\\\\[14.22636pt]
\displaystyle+o_{\eta,k}\left(\frac{1}{\log^{2}\left(\frac{\eta^{2}}{\delta_{k}}\right)}\right)\
+C\,\sum_{\ell=s_{1}}^{s_{2}}\mu^{|\ell-j|}\gamma^{\ell}\left(\int_{B_{\eta}(0)}|\nabla\tilde{u}|^{2}\
dx^{2}\right)^{2}\\\\[14.22636pt] \displaystyle+\,C\
\int_{B_{\eta}(0)}|\nabla\tilde{u}|^{2}\
dx^{2}\,\left(\mu^{j-s_{1}}\int_{B_{2\eta}(0)\setminus
B_{\eta}(0)}|\nabla\tilde{u}|^{2}\
dx^{2}+\mu^{s_{2}-j}\int_{B_{\frac{\delta}{\eta}}(0)\setminus
B_{\frac{\delta}{2\,\eta}}(0)}|\nabla\tilde{u}|^{2}\
dx^{2}\right).\end{array}$ (III.53)

We estimate

$\sum_{\ell=s_{1}}^{s_{2}}\mu^{|\ell-j|}\,\gamma^{\ell}=\sum_{\ell=s_{1}}^{j}\mu^{j-\ell}\,\gamma^{\ell}+\sum_{\ell=j}^{s_{2}}\mu^{\ell-j}\,\gamma^{\ell}\leq\mu^{j}\
\left[\frac{\gamma}{\mu}\right]^{s_{1}}+\gamma^{j}.$ (III.54)

Now we use the fact that

$\displaystyle\displaystyle\int_{B(0,2\eta)\setminus
B(0,\eta)}|\nabla\tilde{u}|^{2}+\int_{B(0,\delta/\eta)\setminus
B(0,\delta/2\eta)}|\nabla\tilde{u}|^{2}$ $\displaystyle\leq$
$\displaystyle\int_{A(\eta,\delta)}|\nabla\tilde{u}|^{2}$ (III.55)
$\displaystyle\displaystyle\int_{B(0,\eta)}|\nabla\tilde{u}|^{2}$
$\displaystyle\leq$ $\displaystyle\int_{A(\eta,\delta)}|\nabla\tilde{u}|^{2}.$
(III.56)

Combining (III.53), (III.54) and (III.55) we obtain (III.5) and this concludes
the proof of Lemma III.1. $\Box$

##  IV Proof of the main result theorem I.1.

Once the pointwise estimate of the gradient is established the rest of the
proof of the main result, theorem I.1, is formally exactly the same under the
general hypothesis of the theorem as for the special case $\alpha=0$ and
${\mathcal{N}}^{n}=S^{n}$ . We will hence be considering this later special
case exclusively starting from subsection IV.2 on until the end of section IV
in order to have simplified notations.

###  IV.1 Introduction of the weight functions and the proof of the positive
contribution of the necks to $D^{2}{\mathfrak{L}}$

The following elementary lemma will be central in the proof of the positive
contribution of the necks to $D^{2}{\mathfrak{L}}$ .

######  Lemma IV.1.

There exists $c_{0}>0$ independent of $k$ and $\eta$, such that, for all $f\in
W^{1,2}_{0}(A(\eta,\delta_{k}))$ we have respectively

$\int_{A(\eta,\delta_{k})}|\nabla
f|^{2}-\frac{c_{0}}{\displaystyle\log^{2}\left(\frac{\eta^{2}}{\delta_{k}}\right)}\frac{1}{|x|^{2}}|f|^{2}\geq
0$ (IV.1)

moreover

$\int_{A(\eta,\delta_{k})}|\nabla
f|^{2}-\frac{\delta_{k}^{\beta}}{\eta^{\beta}\,|x|^{\beta}}\,\frac{c_{0}}{|x|^{2}}|f|^{2}\
dx^{2}\geq 0$ (IV.2)

and finally

$\int_{A(\eta,\delta_{k})}|\nabla
f|^{2}-\frac{|x|^{\beta}}{\eta^{\beta}}\,\frac{c_{0}}{|x|^{2}}|f|^{2}\
dx^{2}\geq 0.$ (IV.3)

$\Box$

Proof of lemma IV.1

Proof of (IV.1) We consider the minimization problem

$\inf\left\\{\int_{A(\eta,\delta_{k})}|\nabla f|^{2}\ dx^{2}\ ;\ f\in
W^{1,2}_{0}(A(\eta,\delta_{k}))\quad\mbox{ and
}\quad\int_{A(\eta,\delta_{k})}\frac{|f|^{2}}{|x|^{2}}\ dx^{2}=1\right\\}$
(IV.4)

Since $|x|^{-2}$ is a $C^{\infty}$ function on $A(\eta,\delta_{k})$, the
infimum is achieved. Let $f$ be such a minimum. Then for some
$\lambda\in\mathbb{R}$ it solves the following Euler Lagrange equation

$\left\\{\begin{array}[]{l}\displaystyle-\Delta f=\frac{\lambda}{|x|^{2}}\
f\quad\mbox{ in }A(\eta,\delta_{k})\\\\[14.22636pt] \displaystyle
f=0\quad\mbox{ on }\partial A(\eta,\delta_{k})\end{array}\right.$ (IV.5)

and introduce

$f^{\ast}(r):=\sqrt{\frac{1}{2\pi}\int_{0}^{2\pi}|f(r,\theta)|^{2}\ d\theta}\
.$

We have clearly

$\int_{A_{\eta,\delta}}\frac{|f|^{2}}{|x|^{2}}\
dx^{2}=\int_{A_{\eta,\delta}}\frac{|f^{\ast}|^{2}}{|x|^{2}}\ dx^{2}$

Moreover

$\begin{array}[]{l}\displaystyle|\nabla
f^{\ast}|^{2}=\left|\frac{df^{\ast}}{dr}\right|^{2}=\left|\displaystyle\frac{\frac{1}{2\pi}\displaystyle\int_{0}^{2\pi}2\,\frac{\partial|f|}{\partial
r}\,|f|(r,\theta)\ d\theta}{\displaystyle
2\,\sqrt{\frac{1}{2\pi}\int_{0}^{2\pi}|f(r,\theta)|^{2}\
d\theta}}\right|^{2}\leq\left|\sqrt{\frac{1}{2\pi}\int_{0}^{2\pi}|\partial_{r}|f||^{2}\
d\theta}\right|^{2}\\\\[28.45274pt]
\displaystyle\quad=\frac{1}{2\pi}\int_{0}^{2\pi}|\partial_{r}|f||^{2}\
d\theta\ \leq\frac{1}{2\pi}\int_{0}^{2\pi}|\nabla f|^{2}(r,\theta)\
d\theta\end{array}$ (IV.6)

This implies

$\int_{A(\eta,\delta_{k})}|\nabla f^{\ast}|^{2}\
dx^{2}\leq\int_{A(\eta,\delta_{k})}|\nabla f|^{2}\ dx^{2}$ (IV.7)

Hence (IV.4) admits an axially symmetric minimizer which satisfies the ODE

$-{f}^{\prime\prime}-\frac{{f}^{\prime}}{r}-\frac{\lambda}{r^{2}}f=0\quad\mbox{
and }\quad f(\delta_{k}/\eta)=f(\eta)=0\ .$ (IV.8)

Let $Y(s):=f(e^{s})$. One has

$\dot{Y}(s)=e^{s}\,f^{\prime}(e^{s})=r\,f^{\prime}(r)\mbox{ and
}\ddot{Y}(s)=e^{s}\,f^{\prime}(e^{s})+e^{2s}\,f^{\prime\prime}(e^{s})=r\,f^{\prime}(r)+r^{2}\,f^{\prime\prime}(r)$

Multiplying by $r^{2}$ (IV.8) and proceeding to the change of variable
$r=e^{s}$, the ODE becomes

$-\ddot{Y}(s)-\lambda\,Y(s)=0\quad
Y(\log(\delta_{k}/\eta))=0\quad\mbox{and}\quad Y(\log(\eta))=0\ .$ (IV.9)

Hence $Y$ has the form

$Y(s)=A\,\cos(\sqrt{\lambda}s)+B\,\sin(\sqrt{\lambda}s)$ (IV.10)

The smallest $\lambda>0$ for which there exists a solution which vanishes both
at $\log(\delta/\eta)$ and $\log(\eta)$ is the one for which

$\sqrt{\lambda}\,[\log\eta-\log(\delta/\eta)]=\pi$

Therefore the minimizer we are considering satisfies

$\left\\{\begin{array}[]{l}\displaystyle-\Delta
f=\frac{\pi^{2}}{|x|^{2}\log^{2}\frac{\eta^{2}}{\delta_{k}^{2}}}\ f\quad\mbox{
in }A(\eta,\delta_{k})\\\\[14.22636pt] \displaystyle f=0\quad\mbox{ on
}\partial A(\eta,\delta_{k})\end{array}\right.$ (IV.11)

Multiplying by $f$ and integrating by parts gives then

$\int_{A(\eta,\delta_{k})}|\nabla f|^{2}\
dx^{2}=\frac{\pi^{2}}{\log^{2}\frac{\eta^{2}}{\delta_{k}^{2}}}\int_{A(\eta,\delta_{k})}\frac{|f|^{2}}{|x|^{2}}\
dx^{2}\ .$ (IV.12)

Since $f$ is a minimizer we obtain (IV.1) for $c_{0}=\pi^{2}$ in fact.

Proof of (IV.2) Let

$y:=\frac{\delta}{\eta}\frac{x}{|x|^{2}}$

and $\tilde{f}(y)=f(x)$. We have in particular

$|y|=\frac{\delta}{\eta\,|x|}\quad\mbox{ and }\quad
c_{1}\,\left(\frac{\delta}{\eta}\right)^{2}\ \frac{dx^{2}}{|x|^{4}}=dy^{2}$

Hence

$\int_{A_{\eta,\delta_{k}}}|\nabla
f|^{2}-\frac{\delta_{k}^{\beta}}{\eta^{\beta}\,|x|^{\beta}}\,\frac{1}{|x|^{2}}|f|^{2}\
dx^{2}=\int_{A_{1,\delta_{k}/\eta^{2}}}|\nabla\tilde{f}|^{2}-|y|^{\beta}\frac{c}{|y|^{2}}|\tilde{f}|^{2}\
dy^{2}$ (IV.13)

We have that $W^{1,2}_{0}(B_{1}(0))$ embeds continuously into $L^{2q}$ for any
$1<q<+\infty$ (Poincaré Sobolev inequality). For such a $q$, extending
$\tilde{f}$ by $0$ inside $B_{\delta/\eta^{2}}(0)$, we have in one hand

$\int_{B_{1}(0)}|\nabla\tilde{f}|^{2}(y)\ dy^{2}\geq
C_{q}\left[\int_{B_{1}(0)}|\tilde{f}|^{2q}(y)\ dy^{2}\right]^{1/q}$ (IV.14)

and, choosing $q$ large enough in such a way that $(2-\beta)q^{\prime}<2$
where $(q^{\prime})^{-1}=1-q^{-1}$

$\int_{B_{1}(0)}|y|^{\beta}\frac{c}{|y|^{2}}|\tilde{f}|^{2}\
dy^{2}\leq\left[\int_{0}^{1}\frac{s}{s^{2q^{\prime}-\beta q^{\prime}}}\
ds\right]\left[\int_{B_{1}(0)}|\tilde{f}|^{2q}(y)\
dy^{2}\right]^{1/q}\leq\tilde{C}_{q}\ \int_{B_{1}(0)}|\nabla\tilde{f}|^{2}(y)\
dy^{2}$ (IV.15)

Hence combining (IV.13) and (IV.15) we obtain (IV.2) for
$c_{0}:=c/\tilde{C}_{q}$.

Proof of (IV.3) We introduce $\tilde{f}(y):=f(y\,\eta)$ hence we have

$\int_{A(1,\delta_{k}/\eta^{2})}|\nabla\tilde{f}|^{2}-|y|^{\beta}\frac{c_{0}}{|y|^{2}}|\tilde{f}|^{2}(y)\
dy^{2}=\int_{A(\eta,\delta_{k})}|\nabla
f|^{2}-\frac{|x|^{\beta}}{\eta^{\beta}}\,\frac{c_{0}}{|x|^{2}}|f|^{2}\ dx^{2}$
(IV.16)

and (IV.3) follows from (IV.15).

This concludes the proof of lemma IV.1. $\Box$

We introduce the following weight function101010We could have simply extended
$\omega_{\eta,k}$ in $B_{\delta_{k}}{\eta}(0)$ by
$\frac{\eta^{2}}{\delta_{k}^{2}}\,\left[1+\frac{\delta_{k}^{\beta}}{\eta^{2\,\beta}}+\frac{1}{\log^{2}\frac{\eta^{2}}{\delta_{k}}}\right]\
.$ The reason why we extend $1$ by
$\frac{1}{\eta^{4}}\frac{(1+\eta^{2})^{2}}{(1+\delta_{k}^{-2}\,|x|^{2})^{2}}$
starting at $|x|=\delta_{k}/\eta$ inside $B_{\delta_{k}}{\eta}(0)$ is because,
after the composition with the stereographic projection$\pi$ it will become
simply a constant in $\pi^{-1}(B_{\delta_{k}}{\eta}(0))$ which offers some
advantages in later computations. Both choices would lead in any case to the
same conclusion.

$\omega_{\eta,k}(x)\left\\{\begin{array}[]{l}\displaystyle=\frac{1}{|x|^{2}}\,\left[\frac{|x|^{\beta}}{\eta^{\beta}}+\frac{\delta_{k}^{\beta}}{\eta^{\beta}\,|x|^{\beta}}+\frac{1}{\log^{2}\frac{\eta^{2}}{\delta_{k}}}\right]\quad\mbox{
in }A(\eta,\delta_{k})\\\\[22.76219pt]
\displaystyle=\frac{1}{\eta^{2}}\,\left[1+\frac{\delta_{k}^{\beta}}{\eta^{2\,\beta}}+\frac{1}{\log^{2}\frac{\eta^{2}}{\delta_{k}}}\right]\quad\mbox{
in }\Sigma\setminus B_{\eta}(0)\\\\[22.76219pt]
\displaystyle=\frac{\eta^{2}}{\delta_{k}^{2}}\left[\frac{1}{\eta^{4}}\frac{(1+\eta^{2})^{2}}{(1+\delta_{k}^{-2}\,|x|^{2})^{2}}+\frac{\delta_{k}^{\beta}}{\eta^{2\beta}}+\frac{1}{\log^{2}\frac{\eta^{2}}{\delta_{k}}}\right]\quad\mbox{
in }B_{\delta_{k}/\eta}(0)\end{array}\right.$ (IV.17)

Combining Lemma III.1 and lemma IV.1 we obtain the following lemma

######  Lemma IV.2.

There exists $\lambda_{0}>0$ independent of $k$ and there exists $\eta_{0}$
such that for any $0<\eta<\eta_{0}$ and for $k$ large enough the following
holds

$\forall\ {w}\,\in V_{u_{k}}\quad\mbox{ s. t. }\quad{w}=0\quad\mbox{ in
}\quad\Sigma\setminus A(\eta,\delta_{k})\quad\quad
Q_{u_{k}}(w)\geq\,\lambda_{0}\,\int_{\Sigma}\omega_{\eta,k}\ w^{2}\ dvol_{h}\
.$ (IV.18)

$\Box$

###  IV.2  The diagonalization of $D^{2}{\mathfrak{L}}$ with respect to the
weights $\omega_{\eta,k}$.

We denote by

$<w,v>_{\omega_{\eta,k}}=\int_{\Sigma}\ w\,v\ \omega_{\eta,k}\,dvol_{h}\ .$

We consider the diagonalization of the self-adjoint operator
${\mathcal{L}}_{\eta,k}$ with respect to $<\ ,\ >_{\omega_{\eta,k}}$ given by

$Q_{u_{k}}(w)=<{\mathcal{L}}_{\eta,k}w,w>_{\omega_{\eta,k}}=\int_{\Sigma}\left[\omega_{\eta,k}^{-1}\,\Delta_{h}w-\omega_{\eta,k}^{-1}\,|du_{k}|_{h}^{2}\,w\right]w\
\omega_{\eta,k}\ dvol_{h}.$

We consider the diagonalization of ${\mathcal{L}}_{\eta,k}$ and we denote by
${\mathcal{E}}_{k,\eta}(\lambda)$ the eigenspace for the eigenvalue $\lambda$.
We have the following lemma

######  Lemma IV.3.

$\mbox{Ind}_{E}(u_{k})=\mbox{dim}\left[\bigoplus_{\lambda<0}{\mathcal{E}}_{\eta,k}(\lambda)\right]\
.$ (IV.19)

$\Box$

Proof of lemma IV.3. On the one hand, the restriction of $Q_{u_{k}}$ to
$W:=\oplus_{\lambda<0}{\mathcal{E}}_{\eta,k}(\lambda)$ is strictly negative
hence
$\mbox{Ind}_{E}(u_{k})\geq\mbox{dim}\left[\oplus_{\lambda<0}{\mathcal{E}}_{\eta,k}(\lambda)\right]$.
Vice versa, let $W$ be a sub-vectorspace of $L^{2}$ on which $Q$ is strictly
negative, we obviously have

$W\cap\bigoplus_{\lambda\geq 0}{\mathcal{E}}_{\eta,k}(\lambda)=\\{0\\}\ .$
(IV.20)

This implies that
dim$W\leq\mbox{dim}\left[\oplus_{\lambda<0}{\mathcal{E}}_{\eta,k}(\lambda)\right]$.
Hence we have proved lemma IV.3. $\Box$

We shall be using the following Lemma

######  Lemma IV.4.

There exist two constants $\eta_{0}>0$ and $\mu_{0}>0$ in dependent of $k$ and
a family of constants $\mu_{\eta,k}>0$ satisfying

$\lim_{\eta\rightarrow 0}\limsup_{k\rightarrow+\infty}\mu_{\eta,k}=0\
\quad\mbox{ and }\quad\forall
k\in{\mathbb{N}},~{}~{}\forall\eta\in(0,\eta_{0})\quad\mbox{ one has }\quad
0<\mu_{\eta,k}<\mu_{0}\ ,$ (IV.21)

and such that for any element $\lambda\in{\mathbb{R}}$ with

$\mbox{dim}\left({\mathcal{E}}_{\eta,k}(\lambda)\right)>0\quad\Longrightarrow\quad\lambda\geq-\,\mu_{\eta,k}\geq-\mu_{0}\
.$ (IV.22)

$\Box$

Proof of lemma IV.4.

Observe that, thanks to lemma III.1 and using respectively (III.1) and
(III.2), there exists a positive constant $C_{0}$ independent of $\eta$ small
enough ($\eta<\eta_{0}$) and $k$ such that

$\forall\ x\in A(\eta,\delta_{k})\quad|du_{k}|^{2}_{h}(x)\leq C_{0}\
\omega_{\eta,k}(x)\ ,$ (IV.23)

Moreover

$\lim_{\eta\rightarrow
0}\limsup_{k\rightarrow+\infty}\left\|\frac{|du_{k}|^{2}_{h}}{\omega_{\eta,k}}\right\|_{L^{\infty}(A(\eta,\delta_{k}))}=0\
.$ (IV.24)

Claim:

$\sup_{0<\eta<\eta_{0}}\sup_{k\in\mathbb{N}}\left\|\frac{|du_{k}|^{2}_{h}}{\omega_{\eta,k}}\right\|_{L^{\infty}(\Sigma)}<+\infty.$
(IV.25)

Proof of Claim. We prove (IV.25) by contradiction. Suppose that there is
sequence $\eta_{n}\to 0$ as $n\to+\infty$ such that

$\sup_{k\in\mathbb{N}}\left\|\frac{|du_{k}|^{2}_{h}}{\omega_{\eta_{n},k}}\right\|_{L^{\infty}(\Sigma)}=+\infty.$
(IV.26)

Because of (IV.23), if (IV.25) would not be true we would contradict the
strong $C^{1}$ convergence respectively of $u_{k}$ towards $u_{\infty}$ on
$\Sigma\setminus B_{\eta_{n}}(x_{k})$ and of
$v_{k}(y):=u_{k}(\delta_{k}\,y+x_{k})$ towards $v_{\infty}$. Hence (IV.25)
holds true. We conclude the proof of the Claim.

We show (IV.21).

We introduce respectively on $\Sigma$

$\omega_{\eta,\infty}(x)=\left\\{\begin{array}[]{l}\displaystyle\frac{1}{\eta^{2}}\quad\mbox{
in }\Sigma\setminus B_{\eta}(p)\\\\[14.22636pt]
\displaystyle\frac{|x|^{\beta}}{\eta^{\beta}}\,\frac{1}{|x|^{2}}\quad\mbox{ in
}B_{\eta}(p)\ ,\end{array}\right.$ (IV.27)

and on $\mathbb{C}$

$\hat{\omega}_{\eta,\infty}(x)=\left\\{\begin{array}[]{l}\displaystyle\frac{1}{\eta^{2}}\,\frac{(1+\eta^{2})^{2}}{(1+|y|^{2})^{2}}\quad\mbox{
for }|y|\leq\eta^{-1}\\\\[14.22636pt]
\displaystyle\frac{1}{\eta^{\beta}}\frac{1}{|y|^{2+\beta}}\quad\mbox{ for
}|y|\geq\eta^{-1}\ .\end{array}\right.$ (IV.28)

We have on the one hand

$\limsup_{k\rightarrow+\infty}\left\|\frac{|du_{k}|^{2}_{h}}{\omega_{\eta,k}}\right\|_{L^{\infty}(\Sigma\setminus
B_{\eta}(x_{k}))}=\left\|\frac{|du_{\infty}|^{2}_{h}}{\omega_{\eta,\infty}}\right\|_{L^{\infty}(\Sigma\setminus
B_{\eta}(x_{k}))}=O(\eta^{2})\ ,$ (IV.29)

and on the other hand

$\begin{array}[]{l}\displaystyle\limsup_{k\rightarrow+\infty}\left\|\frac{|du_{k}|^{2}_{h}}{\omega_{\eta,k}}\right\|_{L^{\infty}(B_{\delta_{k}\,\eta^{-1}}(x_{k}))}=\limsup_{k\rightarrow+\infty}\left\|\frac{\delta_{k}^{2}\,|du_{k}|^{2}_{h}}{\delta_{k}^{2}\,\omega_{\eta,k}}\right\|_{L^{\infty}(B_{\delta_{k}\,\eta^{-1}}(x_{k}))}\\\\[22.76219pt]
\displaystyle=\limsup_{k\rightarrow+\infty}\left\|\frac{|\nabla
v_{k}|^{2}}{\delta_{k}^{2}\,\omega_{\eta,k}(\delta_{k}\,y)}\right\|_{L^{\infty}(B_{\eta^{-1}}(0))}=\left\|\frac{|\nabla
v_{\infty}|^{2}}{\hat{\omega}_{\eta,\infty}}\right\|_{L^{\infty}(B_{\eta^{-1}}(0))}=O(\eta^{2})\
.\end{array}$ (IV.30)

where we use the fact that, thanks to the point removability on $S^{2}$ we
have $|\nabla v_{\infty}|^{2}\leq C\,(1+|y|^{2})^{-2}$.

Then if we set

$\mu_{\eta,k}:=\left\|\frac{|du_{k}|^{2}_{h}}{\omega_{\eta,k}}\right\|_{L^{\infty}(\Sigma)}\
.$

the sequence $\mu_{\eta,k}$ satisfies (IV.21).

We next show that (IV.22) holds as well.

Let $w$ be a non zero eigenfunction for a negative eigenvalue $\lambda$ for
${\mathcal{L}}_{\eta,k}$. One has

$\int_{\Sigma}|dw|^{2}_{h}-|du_{k}|^{2}_{h}\,w^{2}\
dvolg_{h}=\lambda\,\int_{\Sigma}\omega_{\eta,k}\,w^{2}\ dvol_{h}$ (IV.31)

We have from (IV.31)

$\lambda\,\int_{\Sigma}\omega_{\eta,k}\,w^{2}\
dvol_{h}\geq-\int_{\Sigma}\,|du_{k}|^{2}_{h}\,w^{2}\
dvolg_{h}\geq-\,\mu_{\eta,k}\,\int_{\Sigma}\omega_{\eta,k}\,w^{2}\ dvol_{h}$
(IV.32)

This implies (IV.22) and lemma IV.4 is proved. $\Box$

We define

$<w,v>_{\omega_{\eta,\infty}}=\int_{\Sigma}\ w\,v\
\omega_{\eta,\infty}\,dvol_{h}\ .$

Let $w\in V_{u_{\infty}}$. We consider the diagonalization of the self-adjoint
operator ${\mathcal{L}}_{\eta,\infty}$ with respect to $<\ ,\
>_{\omega_{\eta,\infty}}$ given by

$Q_{u_{\infty}}(w)=<{\mathcal{L}}_{\eta,\infty}w,w>_{\hat{\omega}_{\eta,\infty}}=\int_{\Sigma}\left[\omega_{\eta,\infty}^{-1}\,\Delta_{h}w-\omega_{\eta,\infty}^{-1}\,|du_{\infty}|_{h}^{2}\,w\right]w\
\omega_{\eta,\infty}\ dvol_{h}.$

Let $w\in V_{v_{\infty}}$. We consider the self-adjoint operator
$\hat{\mathcal{L}}_{\infty}$ with respect to $<\ ,\
>_{\hat{\omega}_{\eta,\infty}}$ given by

$\begin{array}[]{l}\displaystyle
Q_{v_{\infty}}(w)=<\hat{\mathcal{L}}_{\eta,\infty}w,w>_{\hat{\omega}_{\eta,\infty}}=\int_{{\mathbb{C}}}\left[\hat{\omega}_{\eta,\infty}^{-1}\,\Delta
w-\hat{\omega}_{\eta,\infty}^{-1}\,|dv_{\infty}|^{2}_{\mathbb{C}}\,w\right]w\
\hat{\omega}_{\eta,\infty}\ dy^{2}\\\\[14.22636pt]
\displaystyle=\tilde{Q}_{\tilde{v}_{\infty}}(\tilde{w})=<\tilde{\mathcal{L}}_{\infty}\tilde{w},\tilde{w}>_{\omega_{\infty}}=\int_{S^{2}}\left[\tilde{\omega}_{\infty}^{-1}\,\Delta_{S^{2}}\tilde{w}-\tilde{\omega}_{\infty}^{-1}\,|d\tilde{v}_{\infty}|_{S^{2}}^{2}\,\tilde{w}\right]\tilde{w}\
\tilde{\omega}_{\infty}\ dvol_{S^{2}}\ ,\end{array}$ (IV.33)

where $\tilde{v}_{\infty}:=v_{\infty}\circ\pi$ and $\tilde{w}:=w\circ\pi$
where $\pi$ is the stereographic projection from $S^{2}$ into $\mathbb{C}$
sending the south pole to infinity and

$\tilde{\omega}_{\eta,\infty}:=[\hat{\omega}_{\eta,\infty}(y)\
(1+|y|^{2})^{2}]\circ\pi$

Denote by $z=(z_{1},z_{2},z_{3})$ the coordinates on $S^{2}$, because of
(IV.28), since

$\pi^{-1}(y_{1},y_{2})=\left(\frac{y_{1}}{1+|y|^{2}},\frac{y_{2}}{1+|y|^{2}},\frac{1-|y|^{2}}{1+|y|^{2}}\right)$

we have in a neighborhood of the south pole denoting
$r^{2}=z_{1}^{2}+z_{2}^{2}\simeq\mbox{dist}^{2}(z,\mbox{South})$

$\mbox{dist}^{2}(z,\mbox{South})\simeq\frac{|y|^{2}}{(1+|y|^{2})^{2}}\simeq\frac{1}{|y|^{2}}\
.$ (IV.34)

This gives thanks to (IV.28)

$0<C_{\eta}^{-1}\leq\tilde{\omega}_{\eta,\infty}(z)\leq C_{\eta}\
\mbox{dist}^{-2+\beta}(z,\mbox{South}).$ (IV.35)

Hence we can conclude with the following result:

######  Lemma IV.5.

The conclusion of lemma B.1 holds respectively for
$(\Sigma,\omega_{\eta,\infty},{\mathcal{L}}_{\eta,\infty})$ and for
$(S^{2},\tilde{\omega}_{\eta,\infty},\tilde{\mathcal{L}}_{\eta,\infty})$.
$\Box$

We next introduce

${\mathcal{E}}^{0}_{\eta,\infty}:=\left\\{w\in V_{u_{\infty}}\ ;\
w\in\bigoplus_{\lambda\leq 0}{\mathcal{E}}_{\eta,\infty}(\lambda)\right\\}\ ,$

where

${\mathcal{E}}_{\eta,\infty}(\lambda):=\left\\{w\in V_{u_{\infty}}\ ;\
\Delta_{h}w-|du_{\infty}|^{2}_{h}\,w=\lambda\ \omega_{\eta,\infty}\,w\
\right\\}$

Because of lemma A.1, ${\mathcal{E}}^{0}_{\eta,\infty}$ is a finite (hence
closed) sub-vector space of $L^{2}_{\omega_{\eta,\infty}}(\Sigma)$ and it is a
sub-space on which $Q_{u_{\infty}}\leq 0$. Hence

$\mbox{dim}\left({\mathcal{E}}^{0}_{\eta,\infty}\right)\leq\mbox{Ind}_{E}(u_{\infty})+\mbox{Null}_{E}(u_{\infty})\
,$ (IV.36)

where $\mbox{Null}_{E}(u_{\infty})$ corresponds to the kernel of the bilinear
form associated to $Q_{u_{\infty}}$.

We denote

$V_{v_{\infty}}=\left\\{w\in W^{1,2}({\mathbb{C}},S^{n-1})\ ;\ w\cdot
v_{\infty}\equiv 0\quad\mbox{ on }{\mathbb{C}}\right\\}$

and

$\hat{\mathcal{E}}^{0}_{\eta,\infty}:=\left\\{w\in V_{v_{\infty}}\ ;\
w\in\bigoplus_{\lambda\leq
0}\hat{\mathcal{E}}_{\eta,\infty}(\lambda)\right\\}\ ,$

where

$\hat{\mathcal{E}}_{\eta,\infty}(\lambda):=\left\\{\phi\in V_{v_{\infty}}\ ;\
\Delta_{h}\phi-|dv_{\infty}|^{2}_{h}\,\phi=\lambda\
\hat{\omega}_{\eta,\infty}\,w\ \right\\}.$

We have the following result:

######  Proposition IV.1.

$\mbox{dim}\left(\hat{\mathcal{E}}^{0}_{\eta,\infty}\right)\leq\mbox{Ind}_{E}(\tilde{v}_{\infty})+\mbox{Null}_{E}(\tilde{v}_{\infty})\
.$ (IV.37)

where $\mbox{Null}_{E}(\tilde{v}_{\infty})$ corresponds to the kernel of the
bilinear form associated to $Q_{\tilde{v}_{\infty}}$ with
$\tilde{v}_{\infty}:=v_{\infty}\circ\pi$. $\Box$

Proof of proposition IV.1. Recall that the composition
$\tilde{v}_{\infty}:=v_{\infty}\circ\pi$, where $\pi$ is the stereographic
projection sending the south pole to infinity, extends as a smooth harmonic
map over the whole $S^{2}$. Moreover, for any $w\in V_{v_{\infty}}$, thanks to
conformal invariance the map $\tilde{w}:=w\circ\pi$ has finite $W^{1,2}$
energy in $S^{2}\setminus B_{\varepsilon}(\\{South\\})$ and

$\limsup_{\varepsilon\rightarrow 0}\int_{S^{2}\setminus
B_{\varepsilon}(\mbox{South})}|d\tilde{w}|^{2}_{S^{2}}\,dvol_{S^{2}}=\int_{{\mathbb{C}}}|\nabla
w|^{2}(y)\ dy^{2}<+\infty$

On $S^{2}$ we define

$\tilde{w}_{\varepsilon}(x):=\left[1-\chi\left(\frac{|x|}{\varepsilon}\right)\right]\tilde{w}-\chi\left(\frac{|x|}{\varepsilon}\right)\frac{1}{|B_{2\varepsilon}(0)\setminus
B_{\varepsilon}(0)|}\ \int_{B_{2\varepsilon}(0)\setminus
B_{\varepsilon}(0)}\tilde{w}\ dvol_{S^{2}}$

where $|x|$ is the geodesic distance to the south pole (we are taking normal
coordinates around the south pole). We have

$\begin{array}[]{l}\displaystyle\int_{S^{2}}|d\tilde{w}_{\varepsilon}|^{2}_{S^{2}}\
dvol_{S^{2}}\\\\[14.22636pt] \displaystyle\leq 2\ \int_{S^{2}\setminus
B_{\varepsilon}(0)}|d\tilde{w}|^{2}_{S^{2}}\,dvol_{S^{2}}+2\
\|\chi^{\prime}\|_{\infty}\ \frac{1}{\varepsilon^{2}}\
\int_{B_{2\varepsilon}(0)\setminus
B_{\varepsilon}(0)}|\tilde{w}-\overline{w}_{\varepsilon}|^{2}\
dvol_{S^{2}}\end{array}$ (IV.38)

where

$\overline{w}_{\varepsilon}:=\frac{1}{|B_{2\varepsilon}(0)\setminus
B_{\varepsilon}(0)|}\ \int_{B_{2\varepsilon}(0)\setminus
B_{\varepsilon}(0)}\tilde{w}\ dvol_{S^{2}}$

Using Poincaré inequality we finally obtain

$\begin{array}[]{l}\displaystyle\int_{S^{2}}|d\tilde{w}_{\varepsilon}|^{2}_{S^{2}}\
dvol_{S^{2}}\leq C\ \int_{{\mathbb{C}}}|\nabla w|^{2}(y)\ dy^{2}\end{array}$
(IV.39)

Hence $\tilde{w}_{\varepsilon}$ is uniformly bounded in
$W^{1,2}(S^{2},{\mathbb{R}}^{n})$ and it converges almost everywhere to
$\tilde{w}$ over $S^{2}$ hence it weakly converges $\tilde{w}$ and this
implies the $\tilde{w}\in W^{1,2}(S^{2},{\mathbb{R}}^{n})$. We have moreover
that $\tilde{w}\cdot\tilde{v}_{\infty}=0$ almost everywhere on $S^{2}$ hence
$\tilde{w}\in V_{\tilde{v}_{\infty}}$. We have proved that the composition
with the stereographic projection is realizing a continuous isomorphism $\Pi$
between $(V_{v_{\infty}},\dot{W}^{1,2}({\mathbb{C}},{\mathbb{R}}^{n}))$ into
$(V_{\tilde{v}_{\infty}},\dot{W}^{1,2}(S^{2},{\mathbb{R}}^{n}))$. Observe that

$Q_{v_{\infty}}(w)=Q_{\tilde{v}_{\infty}}(\tilde{w})\ ,$ (IV.40)

hence

$W:=\Pi\left[\hat{\mathcal{E}}^{0}_{\eta,\infty}\right]$ (IV.41)

realizes a sub-vectorspace of $V_{\tilde{v}_{\infty}}$ on which
$Q_{\tilde{v}_{\infty}}\leq 0$ and one has

$\mbox{dim}\left(\hat{\mathcal{E}}^{0}_{\eta,\infty}\right)\leq\mbox{Ind}_{E}(\tilde{v}_{\infty})+\mbox{Null}_{E}(\tilde{v}_{\infty})\
.$ (IV.42)

where $\mbox{Null}_{E}(\tilde{v}_{\infty})$ corresponds to the kernel of the
bilinear form associated to $Q_{\tilde{v}_{\infty}}$. $\Box$

###  IV.3  The proof of the main result theorem I.1.

We prove now the following lemma which implies theorem I.1 in the case
${\mathcal{N}}^{n}=S^{n}$ and $\alpha=0$.

######  Lemma IV.6.

Under the previous notations for $\eta$ sufficiently small and $k$ large
enough we have

$\mbox{Ind}_{E}(u_{k})+\mbox{Null}_{E}(u_{k})=\mbox{
dim}\left(\bigoplus_{\lambda\leq
0}{\mathcal{E}}_{\eta,k}(\lambda)\right)\leq\mbox{Ind}_{E}({u}_{\infty})+\mbox{Null}_{E}(u_{\infty})+\mbox{Ind}_{E}(\tilde{v}_{\infty})+\mbox{Null}_{E}(\tilde{v}_{\infty})\
.$ (IV.43)

$\Box$

Proof of lemma IV.6. We consider the finite dimensional sphere given by

${\mathcal{S}}_{\eta,k}:=\left\\{w\in\bigoplus_{\lambda\leq
0}{\mathcal{E}}_{\eta,k}(\lambda)\ ;\ <w,w>_{\omega_{\eta,k}}=1\right\\}$

Let us take $w_{k}\in{\mathcal{S}}_{\eta,k}$. We have thanks to lemma IV.4

$\left\\{\begin{array}[]{l}\displaystyle\int_{\Sigma}\omega_{\eta,k}\
w_{k}^{2}\ dvol_{h}=1\ \quad\mbox{ and }\\\\[14.22636pt] \displaystyle
0\geq\int_{\Sigma}|dw_{k}|^{2}_{h}-|du_{k}|^{2}_{h}\,w_{k}^{2}\
dvol_{h}\geq-\mu_{0}\,\int_{\Sigma}\omega_{\eta,k}\ w_{k}^{2}\
dvol_{\Sigma}=-\mu_{0}\ .\end{array}\right.$ (IV.44)

Because of (IV.25) we have

$\int_{\Sigma}|du_{k}|^{2}_{h}\,w_{k}^{2}\ dvol_{h}\leq\mu_{0}\ ,$ (IV.45)

hence combining (IV.44) and (IV.45) gives

$\int_{\Sigma}|dw_{k}|^{2}_{h}\ dvol_{h}\leq\mu_{0}\ .$ (IV.46)

From (IV.46) it follows (up to sequence) that

$w_{k}\rightharpoonup w_{\infty}\quad\mbox{ in }W^{1,2}(\Sigma,h)\quad\mbox{
and }\quad w_{k}(\delta_{k}y+x_{k})\rightharpoonup\sigma_{\infty}\quad\mbox{
in }W^{1,2}_{loc}({\mathbb{C}})\ .$ (IV.47)

Claim 1 : $w_{\infty}\neq 0$ or $\sigma_{\infty}\neq 0$.

Proof of the claim 1 : Let $(\phi_{k}^{j})_{j=1\ldots N_{k}}$ be an
orthonormal basis111111It is very important to develop the argument without
assuming that $N_{k}$ is uniformly bounded because it is not known a-priori
and it will come as a direct consequence of the main result. with respect to
$L^{2}_{\omega_{\eta,k}}$ of $\bigoplus_{\lambda\leq
0}{\mathcal{E}}_{\eta,k}(\lambda)$ made of eigenfunctions of
${\mathcal{L}}_{\eta,k}$ :

$P_{u_{k}}\left[\Delta_{h}\phi_{k}^{j}\right]-|du_{k}|^{2}_{h}\
\phi_{k}^{j}=\lambda_{k}^{j}\ \omega_{\eta,k}\
\phi_{k}^{j}\quad\quad\forall\,j=1\ldots N_{k}\ .$ (IV.48)

where $P_{u_{k}}$ is the pointwise orthogonal projection onto
$T_{u_{k}}S^{n}$. Since $\phi_{k}^{j}$ is section of $u_{k}^{-1}TS^{n-1}$ we
have $P_{u_{k}}(\phi_{k}^{j})=\phi_{k}^{j}$ and since $\Delta_{h}$ is a
positive laplacian

$\begin{array}[]{l}\displaystyle
P_{u_{k}}\left[\Delta_{h}\phi_{k}^{j}\right]=\Delta_{h}\phi_{k}^{j}+2\,dP_{u_{k}}\cdot_{h}d\phi_{k}^{j}-\Delta_{h}P_{u_{k}}\,\phi_{k}^{j}\end{array}$
(IV.49)

Since $w_{k}\in{\mathcal{S}}_{\eta,k}$ we have the existence of $c_{k}^{\,j}$
such that

$w_{k}=\sum_{j=1}^{N_{k}}c_{k}^{\,j}\ \phi_{k}^{\,j}\quad\mbox{ where
}\quad\sum_{j=1}^{N_{k}}(c_{k}^{\,j})^{2}=1\ .$ (IV.50)

We have

$\Delta_{h}w_{k}+2\,dP_{u_{k}}\cdot_{h}dw_{k}-\Delta_{h}P_{u_{k}}\,w_{k}-|du_{k}|^{2}_{h}\
w_{k}=\sum_{j=1}^{N_{k}}\lambda_{k}^{\,j}\ \omega_{\eta,k}\ c_{k}^{\,j}\
\phi_{k}^{\,j}$ (IV.51)

Since $\lambda_{k}^{\,j}\in[-C_{0}/\eta^{2},0]$, this gives

$\begin{array}[]{l}\displaystyle\int_{\Sigma}\frac{1}{\omega_{\eta,k}}\left|\Delta_{h}w_{k}+2\,dP_{u_{k}}\cdot_{h}dw_{k}-\Delta_{h}P_{u_{k}}\,w_{k}-|du_{k}|^{2}_{h}\
w_{k}\right|^{2}\ dvol_{h}\\\\[14.22636pt]
\displaystyle\quad=\int_{\Sigma}\sum_{i,j=1}^{N_{k}}\lambda_{k}^{j}\
\lambda_{k}^{i}\ \omega_{\eta,k}\,c_{k}^{i}\ \phi_{k}^{i}\ c_{k}^{j}\
\phi_{k}^{j}\ dvol_{h}=\sum_{j=1}^{n}(\lambda_{k}^{j})^{2}\
(c_{k}^{j})^{2}\leq\mu_{0}^{2}\end{array}$ (IV.52)

Hence there exists $f_{k}$ uniformly bounded in $L^{2}$ such that

$\Delta_{h}w_{k}+2\,dP_{u_{k}}\cdot_{h}dw_{k}-\Delta_{h}P_{u_{k}}\,w_{k}-|du_{k}|^{2}_{h}\
w_{k}=\sqrt{\omega_{\eta,k}}\ f_{k}\ $ (IV.53)

Because of (III.1), using classical elliptic estimates we obtain

$\|\nabla^{2}w_{k}\|_{L^{2}(\Sigma\setminus B_{\eta}(0))}\leq C_{\eta}$
(IV.54)

Let $v_{k}(y):=u_{k}(\delta_{k}\,y+x_{k})$ and
$\sigma_{k}(y):=w_{k}(\delta_{k}\,y+x_{k})$. We have

$\Delta_{\mathbb{C}}\sigma_{k}+2\,dP_{v_{k}}\cdot
d\sigma_{k}-\Delta_{\mathbb{C}}P_{v_{k}}\,\sigma_{k}-|dv_{k}|^{2}_{h}\
\sigma_{k}=\delta_{k}^{2}\ \sqrt{\omega_{\eta,k}}\ f_{k}(\delta_{k}\,y+x_{k})\
.$ (IV.55)

We have

$\int_{B_{1/\eta}(0)}\delta_{k}^{4}\ {\omega_{\eta,k}}\
f^{2}_{k}(\delta_{k}\,y+x_{k})\ dy^{2}\leq C_{\eta}\
\int_{B_{\delta_{k}/\eta}(x_{k})}|f_{k}|^{2}\ dvol_{h}\ .$ (IV.56)

Hence, thanks to (III.2), we have

$\|\nabla^{2}_{y}\sigma_{k}\|_{L^{2}(B_{1/\eta}(0))}=\|\nabla^{2}_{y}(w_{k}(\delta_{k}\,y+x_{k}))\|_{L^{2}(B_{1/\eta}(0))}\leq
C_{\eta}\ .$ (IV.57)

Hence

$w_{k}\rightharpoonup w_{\infty}\quad\mbox{ in
}W^{2,2}_{loc}(\Sigma\setminus\\{0\\})\quad\mbox{ and
}\quad\sigma_{k}(y)=w_{k}(\delta_{k}y+x_{k})\rightharpoonup\sigma_{\infty}\quad\mbox{
in }W^{2,2}_{loc}({\mathbb{C}})\ .$ (IV.58)

Assume $w_{\infty}=0$ and $\sigma_{\infty}=0$. Let

$\check{w}_{k}:=w_{k}\
\chi\left(2\,\frac{|x-x_{k}|}{\eta}\right)\,\left(1-\chi\left(\eta\,\frac{|x-x_{k}|}{\delta_{k}}\right)\right)\in
W^{1,2}_{0}(A_{\eta,k},{\mathbb{R}}^{n})\ .$ (IV.59)

Because of (IV.58) and assuming $w_{\infty}=0$ and $\sigma_{\infty}=0$ we have

$\lim_{k\rightarrow+\infty}\|\nabla(w_{k}-\check{w}_{k})\|_{L^{2}(\Sigma)}=0\
.$ (IV.60)

Observe that on the one hand

$\begin{array}[]{l}\displaystyle\left|Q_{u_{k}}(w_{k})-Q_{u_{k}}(\check{w}_{k})\right|\leq\left|\int_{\Sigma\setminus
B_{\eta/2}(x_{k})}|\nabla
w_{k}|^{2}-|\nabla\check{w}_{k}|^{2}+|du_{k}|^{2}_{h}\,[w^{2}_{k}-\check{w}^{2}_{k}]\
dvol_{h}\right|\\\\[14.22636pt]
\displaystyle\quad+\left|\int_{B_{2\delta_{k}/\eta}(x_{k})}|\nabla
w_{k}|^{2}-|\nabla\check{w}_{k}|^{2}+|du_{k}|^{2}_{h}\,[w^{2}_{k}-\check{w}^{2}_{k}]\
dvol_{h}\right|\ ,\end{array}$ (IV.61)

and on the other hand

$\begin{array}[]{l}\displaystyle\left|\int_{\Sigma}\omega_{k,\eta}\,w_{k}^{2}\
\ dvol_{h}-\int_{\Sigma}\omega_{k,\eta}\,\check{w}_{k}^{2}\ \
dvol_{h}\right|\leq\left|\int_{\Sigma\setminus
B_{\eta/2}(x_{k})}\omega_{k,\eta}\,[w_{k}^{2}-\check{w}_{k}^{2}]\ \
dvol_{h}\right|\\\\[14.22636pt]
\displaystyle\quad+\left|\int_{B_{2\delta_{k}/\eta}(x_{k})}\omega_{k,\eta}\,[w_{k}^{2}-\check{w}_{k}^{2}]\
\ dvol_{h}\right|\end{array}$ (IV.62)

Since we have supposed that $w_{\infty}=\sigma_{\infty}=0$ then the last two
integrals in (IV.62) go to zero as $k\to+\infty$. Therefore combining (IV.60),
(IV.61) and (IV.62) gives

$\lim_{k\rightarrow+\infty}\left|Q_{u_{k}}(w_{k})-Q_{u_{k}}(\check{w}_{k})\right|=0\quad\mbox{
and
}\quad\lim_{k\rightarrow+\infty}\int_{\Sigma}\omega_{k,\eta}\,\check{w}_{k}^{2}\
\ dvol_{h}=1\ .$ (IV.63)

Since $\check{w}_{k}\in W^{1,2}_{0}(A_{\eta,k},{\mathbb{R}}^{m})\cap
V_{u_{k}}$, we have thanks to lemma IV.2, for any $0<c<c_{0}$,

$Q_{u_{k}}(\check{w}_{k})\geq c\
\int_{\Sigma}\omega_{k,\eta}\,\check{w}_{k}^{2}\ \ dvol_{h}\quad.$ (IV.64)

Hence, combining (IV.63) and (IV.64) gives

$\liminf_{k\rightarrow+\infty}Q_{u_{k}}({w}_{k})\geq c>0\ ,$ (IV.65)

which contradicts the fact that $Q_{u_{k}}({w}_{k})\leq 0$. Hence the claim 1
is proved and we have either $w_{\infty}\neq 0$ or $\sigma_{\infty}\neq 0$.

End of the proof of lemma IV.6. Let $N>0$ and $(\phi_{k}^{j})_{j=1\ldots N}$
be a free orthonormal family for $L^{2}_{\omega_{\eta,k}}(\Sigma)$ of
eigenfunctions of ${\mathcal{L}}_{\eta,k}$ where we choose a subsequence that
we still index by $k$ . Because of the claim 1 we have, modulo extraction of a
subsequence, for each $j=1\ldots N$

$\phi_{k}^{j}\rightharpoonup\phi_{\infty}^{j}\quad\mbox{ weakly in
}W^{2,2}_{loc}(\Sigma\setminus\\{p\\})\quad\mbox{ and
}\quad\phi_{k}^{j}(\delta_{k}^{j}\,y+x_{k}^{j})\rightharpoonup\sigma_{\infty}^{j}\quad\mbox{
weakly in }W^{2,2}_{loc}({\mathbb{C}})$ (IV.66)

Passing to the limit as $k\rightarrow+\infty$ we obtain that the maps
$\phi_{\infty}^{j}$ satisfy for some $\lambda_{\infty}^{j}\leq 0$

$P_{u_{\infty}}\left[\Delta_{h}\phi^{j}_{\infty}\right]-|du_{\infty}|^{2}_{h}\,\phi^{j}_{\infty}=\lambda_{\infty}^{j}\
\omega_{\eta,\infty}\ \phi^{j}_{\infty}\ ,$ (IV.67)

and $\sigma_{\infty}^{j}$ satisfies

$P_{v_{\infty}}\left[\Delta_{\mathbb{C}}\sigma^{j}_{\infty}\right]-|\nabla
v_{\infty}|^{2}\ \sigma^{j}_{\infty}=\lambda_{\infty}^{j}\
\hat{\omega}_{\eta,\infty}\,\sigma^{j}_{\infty}\ .$ (IV.68)

Because of claim 1 for all $j=1\ldots N$ we have
$(\phi^{j}_{\infty},\sigma^{j}_{\infty})\neq(0,0)$. Assume

$N>\mbox{dim}\,{\mathcal{E}}^{0}_{\eta,\infty}+\mbox{dim}\,\hat{\mathcal{E}}^{0}_{\eta,\infty}$
(IV.69)

Then thee family $(\phi^{j}_{\infty},\sigma^{j}_{\infty})_{j=1\ldots N}$ is
linearly dependent and there exists
$(c^{1}_{\infty},\ldots,c^{N}_{\infty})\neq(0,\ldots,0)$ such that

$\sum_{j=1}^{N}c^{j}_{\infty}\,\phi^{j}_{\infty}=0\quad\mbox{ and
}\quad\sum_{j=1}^{N}c^{j}_{\infty}\,\sigma^{j}_{\infty}=0\ .$ (IV.70)

Considering $w_{k}:=\sum_{j=1}^{N}c_{\infty}^{j}\,\phi_{k}^{j}$ we obtain a
contradiction. Hence we have proved
$N\leq\mbox{dim}\,{\mathcal{E}}^{0}_{\eta,\infty}+\mbox{dim}\,\hat{\mathcal{E}}^{0}_{\eta,\infty}$
which implies, using (IV.36) and (IV.42), lemma IV.6. $\Box$

##  V Proof of theorem I.2

We first prove the following result

######  Lemma V.1.

Let $u_{k}$ be a sequence of critical points of ${\mathfrak{L}}$ from
$A(\eta,\delta_{k})=B_{\eta}(0)\setminus B_{\delta_{k}/\eta}(0)$ into
${\mathcal{N}}^{n}$ such that

$\lim_{\eta\rightarrow 0}\lim_{k\rightarrow+\infty}\
\sup_{\delta_{k}/\eta<\rho<2\rho<\eta}\ \int_{B_{2\rho}(x_{k})\setminus
B_{\rho}(x_{k})}|du_{k}|^{2}_{h}\ dvol_{h}=0\ .$ (V.1)

Suppose

$0\leq\Lambda:=\lim_{\eta\rightarrow
0}\,\limsup_{k\rightarrow+\infty}\int_{\eta^{-1}\,\delta_{k}}^{\eta}\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{0}^{2\pi}\left|\frac{d{u}_{k}}{dr}\right|\
d\theta\,dr\ .$ (V.2)

Then there exists $\eta_{0}>0$ there exits $0<\beta<2$ and $C>0$ independent
of $k$ such that for any $\eta<\eta_{0}$ and any $x\in A(\eta,\delta_{k})$

$\begin{array}[]{l}\displaystyle|x|^{2}\ |\nabla u_{k}|^{2}(x)\leq C\
\left[\frac{|x|^{\beta}}{\eta^{\beta}}+\left(\frac{\delta_{k}}{\eta\,|x|}\right)^{\beta}\right]\
\int_{A(2\eta,\delta_{k})}|du_{k}|^{2}_{h}\
dvol_{h}+C\,\frac{\Lambda^{2}}{\log^{2}\left(\frac{\eta^{2}}{\delta_{k}}\right)}.\end{array}$
(V.3)

$\Box$

Proof of lemma V.1.

Claim 1: Quantization of $L^{2}$ norm of $u_{k}$ in $A(\eta,\delta_{k})$:  The
assumptions (V.1) and (V.2) imply that

$\lim_{\eta\rightarrow
0}\limsup_{k\rightarrow+\infty}\int_{A(\eta,\delta_{k})}|\nabla
u_{k}|^{2}dx=0.$ (V.4)

Proof of the Claim 1.

Case 1. We first assume that

$\|\nabla u_{k}\|_{L^{2}(A(\eta,\delta_{k}))}\leq\varepsilon_{0}$ (V.5)

where $\varepsilon_{0}$ is the constant appearing in Theorem I.4 in [29]. As
in subsection III.2, in the collar $A(\eta,\delta_{k})$ we decompose

$A_{k}\,\nabla\tilde{u}_{k}=\nabla\varphi_{k}+\nabla^{\perp}\psi_{k}+\nabla{\mathfrak{h}}_{k}$

where $\tilde{u}_{k}$ is the Whitney extension of $u_{k}$ as in Lemma C.1 and

$\displaystyle\Delta\varphi_{k}$ $\displaystyle=$
$\displaystyle\nabla^{\perp}B_{k}\cdot\nabla\tilde{u}_{k}~{}~{}\mbox{in
$\mathbb{C}$}$ (V.6) $\displaystyle\Delta\psi_{k}$ $\displaystyle=$
$\displaystyle\nabla^{\perp}A_{k}\cdot\nabla\tilde{u}_{k}~{}~{}\mbox{in
$\mathbb{C}$}$ (V.7) $\displaystyle\Delta\mathfrak{h}_{k}$ $\displaystyle=$
$\displaystyle 0~{}~{}\mbox{in $A(\eta,\delta_{k})$}$ (V.8)

The matrix $B_{k}$ satisfies the condition in (I.15). The generalized Wente
inequality (see [1]) implies that

$\displaystyle\|\nabla\varphi_{k}\|_{L^{2}(\mathbb{C})}$ $\displaystyle\leq$
$\displaystyle C\|\nabla
B_{k}\|_{L^{2}(\mathbb{C})}\|\nabla\tilde{u}_{k}\|_{L^{2,\infty}(\mathbb{C})}$
(V.9) $\displaystyle\|\nabla\psi_{k}\|_{L^{2}(\mathbb{C})}$
$\displaystyle\leq$ $\displaystyle C\|\nabla
A_{k}\|_{L^{2}(\mathbb{C})}\|\nabla\tilde{u}_{k}\|_{L^{2,\infty}(\mathbb{C})}$
(V.10)

Since

$\|\nabla\tilde{u}_{k}\|_{L^{2,\infty}(\mathbb{C})}\leq
C(\|\nabla{u}_{k}\|_{L^{2,\infty}(A(\eta/2,\delta_{k})}+\|\nabla\tilde{u}_{k}\|_{L^{2}(B_{\eta}\setminus
B_{\eta/2})}+\|\nabla\tilde{u}_{k}\|_{L^{2}(B_{2\delta_{k}/\eta}\setminus
B_{\delta_{k}/\eta})}),$ (V.11)

then the $\varepsilon$-regularity on the annular domain $A(\eta,\delta_{k})$
(see Lemma 13 in [18]) and the assumption (V.1) imply that

$\lim_{\eta\rightarrow
0}\limsup_{k\rightarrow+\infty}\|\nabla\varphi_{k}\|_{L^{2}(A(\eta,\delta_{k}))}+\|\nabla\psi_{k}\|_{L^{2}(A(\eta,\delta_{k}))}=0$
(V.12)

Lemma 2.2 in [23] (or Lemma 10 in [18]) imply that for all $t\in(0,1)$ there
exists $C_{t}$ such that

$\lim_{\eta\rightarrow
0}\limsup_{k\rightarrow+\infty}\|\nabla\mathfrak{h}^{\pm}_{k}\|_{L^{2}(t\eta,\delta_{k})}\leq
C_{t}\lim_{\eta\rightarrow
0}\limsup_{k\rightarrow+\infty}\|\nabla\mathfrak{h}^{\pm}_{k}\|_{L^{2,\infty}(\eta,\delta_{k})}=0.$
(V.13)

The following estimate holds:

$\displaystyle\lim_{\eta\rightarrow
0}\limsup_{k\rightarrow+\infty}\left|\int_{A(\eta,\delta_{k})}\nabla
u_{k}\cdot\nabla\log(|x|)dx\right|$ $\displaystyle\leq$
$\displaystyle\lim_{\eta\rightarrow
0}\limsup_{k\rightarrow+\infty}\int_{\delta_{k}/\eta}^{\eta}\int_{0}^{2\pi}\left|\frac{\partial
u}{\partial r}\right|=:\Lambda<+\infty$ (V.14)

Because of (V.12) and (V.13) we also have that

$\displaystyle\lim_{\eta\rightarrow
0}\limsup_{k\rightarrow+\infty}\left|\int_{A(\eta,\delta_{k})}\nabla\varphi_{k}\cdot\nabla\log(|x|)dx\right|<+\infty$
(V.15) $\displaystyle\lim_{\eta\rightarrow
0}\limsup_{k\rightarrow+\infty}\left|\int_{A(\eta,\delta_{k})}\nabla\psi_{k}\cdot\nabla\log(|x|)dx\right|<+\infty$
(V.16) $\displaystyle\lim_{\eta\rightarrow
0}\limsup_{k\rightarrow+\infty}\left|\int_{A(\eta,\delta_{k})}\nabla\mathfrak{h}^{\pm}_{k}\cdot\nabla\log(|x|)dx\right|<+\infty.$
(V.17)

By combining (V.14)-(V.17) we finally get

$\lim_{\eta\rightarrow
0}\limsup_{k\rightarrow+\infty}\int_{A(\eta,\delta_{k})}\nabla\mathfrak{h}^{0}_{k}\cdot\nabla\log(|x|)dx=\lim_{\eta\rightarrow
0}\limsup_{k\rightarrow+\infty}C^{0}_{\eta,k}\log(\frac{\eta^{2}}{\delta_{k}})<+\infty.$
(V.18)

In particular it follows that

$\displaystyle\lim_{\eta\rightarrow
0}\limsup_{k\rightarrow+\infty}\|\nabla\mathfrak{h}^{0}_{k}\|_{L^{2}(A(\eta,\delta_{k})}=\lim_{\eta\rightarrow
0}\limsup_{k\rightarrow+\infty}C^{0}_{\eta,k}\log^{1/2}(\frac{\eta^{2}}{\delta_{k}})=0$
(V.19)

From (III.16) and (III.17) it follows that
$\|A_{k}^{-1}\|_{L^{\infty}(A(\eta,\delta_{k})}\leq C$ for some $C$
independent on $k$ and $\eta.$ Hence by combining (V.12), (V.13) and (V.19) we
get (V.4) and we can conclude the proof of Claim 1.

Case 2. In the general case where

$\|du_{k}\|_{L^{2}(A(1,\delta_{k}))}>\varepsilon_{0}$ (V.20)

one proceeds as in the proof of Theorem 2 in [17]. The rest of the proof of
lemma V.1 follows word by word from the second part of section III. $\Box$

For $C\,\Lambda^{2}<c_{0}$ where $c_{0}$ is the constant in (IV.1) we obtain
lemma IV.2. The rest of the proof of theorem I.2 is deduced from lemma IV.2
similarly as in the non degenerating case. $\Box$

## Appendices

##  Appendix A The finiteness of the Morse index

######  Lemma A.1.

Let $u$ be a critical point of ${\mathfrak{L}}$ from a closed Riemann surface
$(\Sigma,h)$ into a closed sub-manifold ${\mathcal{N}}^{n}$ of an euclidian
space ${\mathbb{R}}^{m}$. Denote by

$Ind_{\mathfrak{L}}(u):=\sup\left\\{\mbox{dim}(W)\ ;\ W\mbox{ is a sub vector-
space of }V\mbox{ s.t. }\left.Q_{u}\right|_{W}<0\right\\}$

where $Q_{u}$ is the second derivative of $\mathfrak{L}$ at $u$ and is given
by

$Q_{u}(w):=D^{2}{\mathfrak{L}}_{u}(w)=\int_{\Sigma}|dw|^{2}-<w,S_{u}(du)_{h}w>+<w,{\mathcal{H}}_{u}w>\,dvol_{h}\
.$

where $S_{u}(du)_{h}$ and ${\mathcal{H}}$ are given respectively by (II.12)
and (II.24). This defines a quadratic form defined on
$\Gamma(u^{-1}T{\mathcal{N}}^{n})$, the sections of the pull-back bundle by
$u$ of the tangent bundle to ${\mathcal{N}}^{n}$. We have

$Ind_{\mathfrak{L}}(u)<+\infty\ .$ (A.1)

Similarly, denote $\mbox{Null}_{\mathfrak{L}}(u)$ the dimension of the Kernel
of the bilinear form associated to $Q_{u}$, there holds

$\mbox{Null}_{\mathfrak{L}}(u)<+\infty\ .$ (A.2)

$\Box$

Proof of lemma A.1 Denote by ${\mathcal{E}}(\lambda)$ the eigenspaces of the
Laplace Beltrami operator defined on $W^{1,2}(\Sigma,{\mathbb{R}})$ :

${\mathcal{E}}(\lambda):=\left\\{\phi\in W^{1,2}(\Sigma,{\mathbb{R}})\ ;\
\Delta_{h}\phi=\lambda\phi\right\\}\ .$

For any $\Lambda\in{\mathbb{R}}$ we shall denote

${\mathcal{E}}^{\Lambda}:=\bigoplus_{\lambda\leq\Lambda}{\mathcal{E}}(\lambda)$

It is well known that, for any $\Lambda\in{\mathbb{R}}$,
dim$({\mathcal{E}}^{\Lambda})<+\infty$.121212 We observe that
$\Delta^{-1}_{h}$ is a compact operator from $L^{2}(\Sigma)$ to
$L^{2}(\Sigma)$, being $\Sigma$ a compact manifold. Hence for every
$\lambda\in\mathbb{R}$, dim$({\mathcal{E}}(\lambda))<+\infty$.  We choose
$\Lambda>C\,\||du|_{h}\|_{L^{\infty}(\Sigma)}^{2}$ where $C$ is chosen such
that

$\left|\int_{\Sigma}-<w,S_{u}(du)_{h}w>+<w,{\mathcal{H}}_{u}w>\,dvol_{h}\right|\leq\frac{1}{2}\int_{\Sigma}|dw|^{2}\
+C\ |du|^{2}_{h}\ |w|^{2}\,dvol_{h}$

and we introduce

$V^{\Lambda}_{u}:=\left\\{w\in V_{u}\ ;\ \int_{\Sigma}w_{i}\,\phi\ dvol_{h}=0\
,\ \forall\phi\in{\mathcal{E}}^{\Lambda}\ ,\ i=1\ldots n\right\\}\ .$ (A.3)

We have in one hand that the space

$\left\\{V_{u}\ni w\rightarrow\int_{\Sigma}w_{i}\,\phi\ dvol_{h}\ ;\
\phi\in{\mathcal{E}}^{\Lambda}\ ,\ i=1\ldots n\right\\}$

is a finite dimensional subspace of $(V_{u})^{\ast}$ and hence
$V^{\Lambda}_{u}$ is finite co-dimensional. Moreover

$\begin{array}[]{l}\displaystyle\forall w\in
V^{\Lambda}_{u}\setminus\\{0\\}\quad
Q_{u}(w)\geq\frac{1}{2}\sum_{i=1}^{n}\int_{\Sigma}\left[{\Lambda}-C\,|du|^{2}_{h}\right]\,|w_{i}|^{2}\
dvol_{h}\geq\frac{1}{2}\,\left[{\Lambda}-C\,\||du|^{2}_{h}\|_{\infty}\right]\
\|w\|_{L^{2}}^{2}>0\end{array}$

For that reason, for any $W\mbox{ sub vector-space of }V_{u}\mbox{ s.t.
}\left.Q_{u}\right|_{W}<0$ one has and we deduce

$W\cap V^{\Lambda}_{u}=\\{0\\}\ .$ (A.4)

Since $V^{\Lambda}_{u}$ is finite co-dimensional we deduce the projection
parallel to $V^{\Lambda}_{u}$ of $W$ onto any finite dimensional supplement of
$V^{\Lambda}_{u}$ in $V_{u}$ must be injective and we deduce that

$\mbox{dim}(W)\leq\mbox{codim}(V^{\Lambda}_{u})\ .$ (A.5)

A similar argument can be applied to prove (A.2). This ends the proof of lemma
A.1. $\Box$

##  Appendix B The discreteness of the spectrum of $D^{2}{\mathfrak{L}}$

######  Lemma B.1.

Let $u$ be a smooth map from a closed oriented Riemannian 2-dimensional
manifold $(\Sigma,h)$ into a closed $C^{2}$ sub-manifold ${\mathcal{N}}^{n}$
of ${\mathbb{R}}^{m}$. Let $\alpha$ is an arbitrary $C^{2}$ 2-form of
${\mathcal{N}}^{n}$. On $\Sigma$ we consider a positive function ${\omega}$,
smooth away from one point $p$ and such that there exists $0<\beta<1$ such
that

$0<C^{-1}_{0}\,\leq\omega(x)\leq\frac{C_{0}}{\mbox{dist}(x,p)^{2-\beta}}\
\quad\mbox{ and }\quad|\nabla^{l}\omega^{-1}|(x)|\leq
C_{l}\,\mbox{dist}(x,p)^{2-l-\beta}\quad\forall\,l=0,1,2\ .$ (B.1)

Denote

$L^{2}_{\omega}(\Sigma):=\left\\{\phi\in L^{2}(\Sigma)\ ;\
\int_{\Sigma}\omega\,\phi^{2}\ dvol_{h}<+\infty\right\\}\ .$

Denote for any $z\in{\mathcal{N}}^{n}$ by $P_{z}$ the $m\times m$ matrix
corresponding to the orthogonal projection onto $T_{z}{\mathcal{N}}^{n}$. The
composition $P\circ u$ will simply be denoted $P_{u}$. Let

$W_{u}:=\left\\{w\in L^{2}_{\omega}(\Sigma,{R}^{m})\ ;\
P_{u}(w(x))=w(x)\quad\mbox{ for a. e. }x\in\Sigma\right\\}\ .$

On $W_{u}$ we consider the operator

${\mathcal{L}}w:=\omega^{-1}\,P_{u}\left[\Delta_{h}w+{\mathcal{H}}_{u}w-S_{u}(du)_{h}w\right]\
.$

where $S_{u}(du)_{h}$ and ${\mathcal{H}}_{u}$ are given respectively by
(II.12) and (II.24). Then there exists an Hilbert basis of $W_{u}$ for the
$L^{2}_{\omega}$ scalar product made of the eigenfunctions of $w$ and the
eigenvalues statisfy

$\lambda_{1}<\lambda_{2}<\lambda_{3}.....\rightarrow+\infty\,.$

$\Box$

Proof of lemma B.1. Observe first that ${\mathcal{L}}w$ has a distributional
sense for any $w\in W_{u}$. Indeed

${\mathcal{L}}w=\omega^{-1}\,\Delta_{h}w-2\,\omega^{-1}\,dP_{u}\cdot_{h}dw-\,\omega^{-1}\,\Delta_{h}P_{u}\,w+\omega^{-1}\,P_{u}\left[{\mathcal{H}}_{u}w\right]-\omega^{-1}\,S_{u}(du)_{h}w$
(B.2)

We recall that $P_{u}\left[H_{u}(dw\wedge du)\right]=H_{u}(dw\wedge du).$
Therefore we can write

$\displaystyle P_{u}\left[{\mathcal{H}}w\right]$ $\displaystyle=$
$\displaystyle P_{u}\left[2\,H_{u}(dw\wedge du)+\nabla_{w}H(du,du)\right]$
(B.3) $\displaystyle=$ $\displaystyle 2\,H_{u}(dw\wedge
du)+P_{u}\left[\nabla_{w}H(du,du)\right]$ (B.4) $\displaystyle=$
$\displaystyle 2d(H_{u}(w\wedge
du))-2\partial_{\xi_{\ell}}H_{i,j}^{k}(u)du^{\ell}\wedge w^{i}\wedge
du^{j}+P_{u}\left[\nabla_{w}H(du,du)\right]$ (B.5) $\displaystyle=$
$\displaystyle 2\,d\left[H_{u}(w\wedge du)\right]+K_{u}(du)_{h}w\ ,$ (B.6)

where

$K_{u}(du)_{h}w=-2\partial_{\xi_{\ell}}H_{i,j}^{k}(u)du^{\ell}\wedge
w^{i}\wedge du^{j}+P_{u}\left[\nabla_{w}H(du,du)\right]$

with

$|K_{u}(du)_{h}|\leq C\,|du|^{2}_{h}\ .$ (B.7)

Because of the hypothesis (B.1) we have that

$\left\\{\begin{array}[]{l}\displaystyle|\Delta_{h}\omega^{-1}|(x)\leq
C_{2}\,\mbox{dist}(x,p)^{-\beta}\in L^{2}(\Sigma)\ ,\\\\[8.53581pt]
d^{\ast}[\omega^{-1}\,dP_{u}]\in L^{2}(\Sigma)\\\\[14.22636pt]
\omega^{-1}\,\Delta_{h}P_{u}\,+\omega^{-1}\,[K_{u}(du)-S_{u}(du)_{h}]\in
L^{2}(\Sigma).\end{array}\right.$ (B.8)

Hence ${\mathcal{L}}w\in{\mathcal{D}}^{\prime}(\Sigma)$.

Denote ${\mathcal{L}}_{\lambda}={\mathcal{L}}+\lambda\,Id$.

Claim 1 : There exists $\lambda_{0}>0$ such that for any
$\lambda\geq\lambda_{0}$ the following holds

$\left\\{\begin{array}[]{l}\displaystyle\forall f\in W_{u}\quad\exists\,!\
w\in W_{u}\cap W^{1,2}(\Sigma,{\mathbb{R}}^{n})\quad\mbox{ s. t.
}\quad{\mathcal{L}}_{\lambda}w=f\\\\[14.22636pt]
\displaystyle\int_{\Sigma}[|dw|^{2}_{h}+\omega\,|w|^{2}]\ dvol_{h}\leq
2\|f\|^{2}_{L^{2}_{\omega}(\Sigma)}\end{array}\right.$ (B.9)

Proof of Claim 1 We start first by establishing a-priori estimates.
Multiplying the equation by $w\in W_{u}\cap W^{1,2}(\Sigma,{\mathbb{R}}^{m})$
and integrating by parts gives

$\int_{\Sigma}|dw|^{2}_{h}\
dvol_{h}+[\lambda\,\omega\,w^{2}-<w,S_{u}(du)_{h}w>+w{\mathcal{H}}w]\
dvol_{h}=\int_{\Sigma}\omega\,f\,w\
dvol_{h}\leq\|w\|_{L^{2}_{\omega}(\Sigma)}\ \|f\|_{L^{2}_{\omega}(\Sigma)}\ $
(B.10)

Because of (B.3) and (B.7), for any $\varepsilon>0$ there exists
$C_{\varepsilon}>0$ such that

$\left|\int_{\Sigma}w{\mathcal{H}}w\
dvol_{h}\right|\leq\varepsilon\int_{\Sigma}|dw|^{2}_{h}\
dvol_{h}+C_{\varepsilon}\,\int_{\Sigma}|du|^{2}_{h}\,|w|^{2}\ dvol_{h}\ .$
(B.11)

Hence for $\lambda\geq C\,\||du|^{2}_{h}\|_{\infty}$ for $C>0$ large enough
independent of $w$, we deduce

$\int_{\Sigma}[|dw|^{2}_{h}+\omega\,|w|^{2}]\ dvol_{h}\leq
2\,\|w\|_{L^{2}_{\omega}(\Sigma)}\ \|f\|_{L^{2}_{\omega}(\Sigma)}\ $ (B.12)

which implies

$\int_{\Sigma}[|dw|^{2}_{h}+\omega\,|w|^{2}]\ dvol_{h}\leq
4\,\|f\|^{2}_{L^{2}_{\omega}(\Sigma)}$ (B.13)

The previous a-priori estimate is giving the uniqueness of $w$ in $W_{u}\cap
W^{1,2}(\Sigma,{\mathbb{R}}^{m})$.

Regarding the proof of the existence, we consider on $W_{u}\cap
W^{1,2}(\Sigma,{\mathbb{R}}^{m})$ the minimization of

$\begin{array}[]{l}\displaystyle
L_{\lambda}(w):=\int_{\Sigma}[|dw|^{2}_{h}+\lambda\,\omega\,w^{2}-<w,S_{u}(du)_{h}w>+w{\mathcal{H}}w-2\,\omega\,f\cdot
w]\ dvol_{h}\\\\[14.22636pt]
\displaystyle\quad\geq\frac{1}{2}\int_{\Sigma}[|dw|^{2}_{h}+2\,\omega\,w^{2}]\
dvol_{h}-2\,\|f\|_{L^{2}_{\omega}(\Sigma)}\,\|w\|_{L^{2}_{\omega}(\Sigma)}\\\\[14.22636pt]
\displaystyle\quad\geq\frac{1}{2}\|dw\|_{L^{2}(\Sigma)}^{2}+\left[\|w\|_{L^{2}_{\omega}(\Sigma)}-\|f\|_{L^{2}_{\omega}(\Sigma)}\right]^{2}-\|f\|^{2}_{L^{2}_{\omega}(\Sigma)}\end{array}$
(B.14)

for $\lambda\geq C\,\||du|^{2}_{h}\|_{\infty}$. For a minimizing sequence
$w_{k}$ the energy $\|w_{k}\|_{W^{1,2}(\Sigma)}^{2}$ remains uniformly bounded
we can extract a subsequence such that $w_{k}\rightharpoonup w_{\infty}$ in
$W^{1,2}$ and, thanks to Rellich Kondrachov theorem, the weak $W^{1,2}$
convergence implies that the pointwise condition $P_{u}(w_{k})=0$ passes to
the limit and we obtain that $w_{\infty}\in W_{u}\cap
W^{1,2}(\Sigma,{\mathbb{R}}^{m})$.

Moreover, the Euler-Lagrange variational theory implies that the following
holds

$\forall\,\varphi\in W_{u}\cap
W^{1,2}(\Sigma,{\mathbb{R}}^{n})\quad\int_{\Sigma}d\varphi\cdot_{h}dw_{\infty}+[\lambda\,\omega\,w_{\infty}-S(du)_{h}w_{\infty}+{\mathcal{H}}w_{\infty}]\cdot\varphi-\omega\,f\cdot\varphi\
dvol_{h}=0\ .$ (B.15)

and we deduce that ${\mathcal{L}}_{\lambda}w_{\infty}=f$. This concludes the
proof of the Claim 1.

Claim 2.

${\mathcal{K}}:=\left\\{w\in W_{u}\cap W^{1,2}(\Sigma,{\mathbb{R}}^{m})\ ;\
\int_{\Sigma}[|dw|^{2}_{h}+\omega\,|w|^{2}]\ dvol_{h}\leq 1\right\\}$ (B.16)

is a compact subset of $W_{u}$.

Proof of Claim 2. We consider a sequence $w_{k}\in{\mathcal{K}}$. It is weakly
pre-compact in $W^{1,2}(\Sigma,{\mathbb{R}}^{n})$ and we can then extract a
subsequence such that $w_{k}\rightharpoonup w_{\infty}$ weakly in $W^{1,2}$.
Clearly, by lower semi-continuity of the norm $L^{2}_{\omega}(\Sigma)$ for
such a convergence and since the pointwise constraint $P_{u}(w_{k})$ passes to
the limit we have $w_{\infty}$. We have for any $1<p<2/(2-\beta)$

$\int_{\Sigma}\omega\,|w_{k}-w_{\infty}|^{2}\
dvol_{h}\leq\|\omega\|_{L^{p}(\Sigma)}\
\|w_{k}-w_{\infty}\|^{2}_{L^{2p^{\prime}}(\Sigma)}\ .$ (B.17)

Rellich Kondrachov’s theorem gives the compactness of the continuous embedding
$W^{1,2}(\Sigma)\hookrightarrow L^{q}(\Sigma)$ for any $q<+\infty$ since
$\Sigma$ is compact. Because of (B.1) $\|\omega\|_{L^{p}}<+\infty$ for any
$p<2/(2-\beta)$. Thus we deduce that

$\lim_{k\rightarrow+\infty}\|w_{k}-w_{\infty}\|_{L^{2}_{\omega}(\Sigma)}=0\ .$
(B.18)

Any sequence of ${\mathcal{K}}$ posses a subsequence which strongly converges
for the metric topology induced by $L^{2}_{\omega}(\Sigma)$ this implies that
${\mathcal{K}}$ is a compact subset of $W_{u}$ and Claim 2 is proved.

We have proved so far that the self adjoint operator
${\mathcal{L}}^{-1}_{\lambda}$ for the $L^{2}_{\omega}$ norm is compact.
$L^{2}_{\omega}$ defines a separable Hilbert space. Theorem 6.11 of [2] is
implying the conclusion of lemma B.1 and this is ending the proof. $\Box$

##  Appendix C A Whitney type extension lemma on annuli.

######  Lemma C.1.

Let $0<2r<R$ and $a\in W^{1,2}(B_{R}(0)\setminus B_{r}(0))$ then there exists
$\tilde{a}\in W^{1,2}({\mathbb{C}})$ with

$\mbox{Supp}(\nabla\tilde{a})\subset B_{2R}(0)\setminus B_{r/2}(0)\ ,$

such that

$\int_{\mathbb{C}}|\nabla\tilde{a}|^{2}\ dx^{2}\leq C\ \int_{B_{R}(0)\setminus
B_{r}(0)}|\nabla a|^{2}\ dx^{2}\ ,$ (C.1)

where $C$ is independent of $r$ and $R$. We can also impose moreover

$\int_{B_{2R}(0)\setminus B_{R}(0)}|\nabla\tilde{a}|^{2}\ dx^{2}\leq C\
\int_{B_{R}(0)\setminus B_{R/2}(0)}|\nabla a|^{2}\ dx^{2}\ ,$ (C.2)

and

$\int_{B_{r}(0)\setminus B_{r/2}(0)}|\nabla\tilde{a}|^{2}\ dx^{2}\leq C\
\int_{B_{2r}(0)\setminus B_{r}(0)}|\nabla a|^{2}\ dx^{2}.$ (C.3)

$\Box$

Proof of lemma C.1. Let $\chi$ be a function in $C^{\infty}({\mathbb{R}}_{+})$
such that $\chi\equiv 1$ on $[0,1]$ and $\chi\equiv 0$ on $[2,+\infty)$. We
denote for any $t>0$

$\forall\,x\in{\mathbb{C}}\quad\quad\chi_{t}(x):=\chi\left(\frac{|x|}{t}\right)\
.$

For $|x|\geq r$ we define

$\hat{a}(x):=\chi_{r}(x)\,a(x)+(1-\chi_{r}(x))\,\overline{a}_{r}\quad\mbox{
where }\quad\overline{a}_{r}:=\frac{1}{|B_{2r}\setminus
B_{r}|}\int_{B_{2r}\setminus B_{r}}a(x)\ dx$

Because of Poincaré inequality there holds

$\int_{{\mathbb{C}}\setminus B_{r}}|\nabla\hat{a}|^{2}\ dx^{2}\leq C\
\int_{B_{2r}(0)\setminus B_{r}(0)}|\nabla a|^{2}\ dx^{2}\ .$ (C.4)

Now, we extend $a$ inside $B_{r}$ by taking

$\tilde{a}(x):=\hat{a}\left(r^{2}\,\frac{x}{|x|^{2}}\right)$

Since the inversion is conformally invariant we have

$\int_{B_{r}}|\nabla\tilde{a}|^{2}\ dx^{2}=\int_{{\mathbb{C}}\setminus
B_{r}}|\nabla\hat{a}|^{2}\ dx^{2}\ .$ (C.5)

We do something similar in order to extend $a$ outside $B_{R}$. For $|x|<R$ we
define

$\breve{a}(x):=(1-\chi_{R}(x))\,a(x)+\chi_{R}(x)\,\overline{a}_{R}\quad\mbox{
where }\quad\overline{a}_{R}:=\frac{1}{|B_{R}\setminus
B_{R/2}|}\int_{B_{R}\setminus B_{R/2}}a(x)\ dx$

Because of Poincaré inequality there holds

$\int_{B_{R}}|\nabla\breve{a}|^{2}\ dx^{2}\leq C\ \int_{B_{R}(0)\setminus
B_{R/2}(0)}|\nabla a|^{2}\ dx^{2}\ .$ (C.6)

Now, we extend $a$ outside $B_{R}$ by taking

$\tilde{a}(x):=\hat{a}\left(R^{2}\,\frac{x}{|x|^{2}}\right)$

Since the inversion is conformally invariant we have

$\int_{{\mathbb{C}}\setminus B_{R}}|\nabla\tilde{a}|^{2}\
dx^{2}=\int_{B_{R}}|\nabla\breve{a}|^{2}\ dx^{2}\ .$ (C.7)

The proof of lemma C.1 is complete. $\Box$

##  Appendix D Pointwise gradient estimates of harmonic functions on annuli

######  Lemma D.1.

Let $\eta^{2}>4\,\delta$. Let $\mathfrak{h}$ be a real harmonic functions
defined on $A(\eta,\delta):=B_{\eta}(0)\setminus B_{\delta/\eta}(0)$ given in
Fourier by

$\mathfrak{h}=\mathfrak{h}^{+}+\mathfrak{h}^{-}\quad\mbox{ where }\
\mathfrak{h}^{+}=\Re\left[\sum_{n>0}\mathfrak{h}_{n}z^{n}\right]\quad\mbox{and}\quad\mathfrak{h}^{-}=\Re\left[\sum_{n<0}\mathfrak{h}_{n}z^{n}\right]$

then for any $\rho\in[2\delta/\eta,\eta/2]$ and $\theta\in[0,2\pi]$ there
holds

$|\nabla\mathfrak{h}^{+}(\rho,\theta)|^{2}\leq\frac{C}{\eta^{2}}\
\int_{A(\eta,\delta)}|\nabla\mathfrak{h}^{+}|^{2}dx^{2}\ ,$ (D.1)

and

$|\nabla\mathfrak{h}^{-}(\rho,\theta)|^{2}\leq\frac{C\,\delta^{2}}{\eta^{2}}\
\frac{1}{\rho^{4}}\,\int_{A(\eta,\delta)}|\nabla\mathfrak{h}^{-}|^{2}dx^{2}\
.$ (D.2)

$\Box$

Proof of lemma D.1 Let $\mathfrak{f}^{+}(z):=\sum_{n>0}\mathfrak{h}_{n}x^{n}$
and $\mathfrak{f}^{-}(z):=\sum_{n<0}\mathfrak{h}_{n}x^{n}$. We have

$2\,|\nabla\mathfrak{h}^{\pm}|^{2}=2\,|\nabla\Re(\mathfrak{f}^{\pm})|^{2}=2\,|\nabla\Im(\mathfrak{f}^{\pm})|^{2}=|\nabla\mathfrak{f}^{\pm}|^{2}\
.$ (D.3)

We have

$\displaystyle\int_{A(\eta,\delta)}|\nabla\mathfrak{f}^{+}|^{2}$
$\displaystyle=4\pi\sum_{n>0}|\mathfrak{h}_{n}|^{2}n^{2}\int_{\delta/\eta}^{\eta}\rho^{2n-1}d\rho$
$\displaystyle=2\pi\sum_{n>0}|\mathfrak{h}_{n}|^{2}n\bigg{(}(\eta)^{2n}-\left(\frac{\delta}{\eta}\right)^{2n}\bigg{)}$
$\displaystyle\geq\pi\sum_{n>0}|\mathfrak{h}_{n}|^{2}\,n\ \eta^{2n}$ (D.4)

where we have used in the last inequality that
$\delta\leq\frac{1}{2}\eta^{2}$. For $\rho<\eta/2$ we have

$\displaystyle|\nabla\mathfrak{f}^{+}(\rho,\theta)|^{2}$
$\displaystyle\leq\left(2\sum_{n>0}|\mathfrak{h}_{n}|n\rho^{n-1}\right)^{2}$
$\displaystyle\leq\sum_{n>0}|\mathfrak{h}_{n}|^{2}\,n\,\eta^{2n}\sum_{n>0}\frac{\rho^{2n-2}}{\eta^{2n}}n$
$\displaystyle=\left(\sum_{n>0}|\mathfrak{h}_{n}|^{2}\,n\,\eta^{2n}\right)\frac{1}{\eta^{2}}\sum_{n>0}n\left(\frac{\rho^{2}}{\eta^{2}}\right)^{n-1}$
$\displaystyle\leq\frac{C}{\eta^{2}}\int_{A(\eta,\delta)}|\nabla\mathfrak{f}^{+}|^{2}$
(D.5)

Combining (D.3) and (D) gives (D.1).

$\displaystyle\int_{A(\eta,\delta)}|\nabla\mathfrak{f}^{-}|^{2}$
$\displaystyle=4\pi\sum_{n<0}|\mathfrak{h}_{n}|^{2}n^{2}\int_{\delta/\eta}^{\eta}\rho^{2n-1}d\rho$
(D.6) $\displaystyle=2\pi\sum_{n<0}|\mathfrak{h}_{n}|^{2}\,n\
\left(\eta^{2n}-\left(\frac{\delta}{\eta}\right)^{2n}\right)$ (D.7)
$\displaystyle\geq 2\,\pi\sum_{n<0}|\mathfrak{h}_{n}|^{2}\ |n|\
\left(\left(\frac{\eta}{\delta}\right)^{2|n|}-\eta^{2n}\right)\geq\pi\sum_{n<0}|\mathfrak{h}_{n}|^{2}\
|n|\ \left(\frac{\eta}{\delta}\right)^{2|n|}$ (D.8)

where we assume in the last inequality that $\delta\leq\frac{1}{2}\eta^{2}$.
For $\rho>2\,\delta/\eta$ we have

$\displaystyle|\nabla\mathfrak{f}^{-}(\rho,\theta)|^{2}$
$\displaystyle\leq\left(2\sum_{n<0}|\mathfrak{h}_{n}|\,n\,\rho^{n-1}\right)^{2}$
$\displaystyle\leq\sum_{n<0}|\mathfrak{h}_{n}|^{2}|n|\,\left(\frac{\delta}{\eta}\right)^{2n}\sum_{n>0}\frac{\rho^{-2n-2}}{\left(\frac{\delta}{\eta}\right)^{-2n}}n$
$\displaystyle=\left(\sum_{n<0}|\mathfrak{h}_{n}|^{2}\ |n|\
\left(\frac{\delta}{\eta}\right)^{2n}\right)\frac{\delta^{2}}{\rho^{4}\eta^{2}}\sum_{n>0}n\left(\frac{\delta^{2}}{\eta^{2}\rho^{2}}\right)^{n-1}$
$\displaystyle\leq
C\,\frac{\delta^{2}}{\rho^{4}\eta^{2}}\int_{A(\eta,\delta)}|\nabla\mathfrak{f}^{-}|^{2}$
(D.9)

Combining (D.3) and (D) gives (D.2). This concludes the proof of lemma D.1.
$\Box$

##  Appendix E Weighted Wente Estimates

The following lemma is giving a Wente type estimate with weight. The weight
$f$ in the r.h.s. of (E.2)might not be optimal but is sufficient for later
purposes.

######  Lemma E.1.

Let $\varphi$ be a solution of

$\left\\{\begin{array}[]{l}-\Delta\varphi=\partial_{x_{1}}a\
\partial_{x_{2}}b-\partial_{x_{2}}a\ \partial_{x_{1}}b\quad\quad~{}~{}\mbox{in
$B_{1}(0)$}\\\\[14.22636pt] \varphi=0\quad\quad~{}~{}\mbox{on $\quad\partial
B_{1}(0)$}\end{array}\right.$ (E.1)

Then

$\int_{B(0,1)}|x|^{2}|\nabla\varphi|^{2}\leq C\ \int_{B_{1}(0)}f(|x|)|\nabla
b|^{2}\ dx^{2}\ \int_{B_{1}(0))}|\nabla a|^{2}\ dx^{2}\ ,$ (E.2)

where

$f(r)=r^{2}\log^{2}\left(1+\frac{1}{r}\right)\log\left(1+\log\frac{1}{r}\right)\
,$ (E.3)

and $C>0$ is a universal constant. $\Box$

In order to prove lemma E.1 we need to decompose the Jacobian
$\nabla^{\perp}b\cdot\nabla a$ into a countable sum of Jacobians supported
each on a dyadic annulus of the form $A_{k}:=B_{2^{-k}}(0)\setminus
B_{2^{-k-1}}(0)$. We will thus first prove the following lemma

######  Lemma E.2.

Let $b$ be in $W^{1,2}(B_{1}(0),{\mathbb{R}})$. There exists $b_{k}$ such that

  * i)

$\mbox{supp}(\nabla b_{k})\ \subset A_{k}\cup A_{k+1}\ $ (E.4)

  * ii)

$\nabla b=\sum_{k=0}^{\infty}\nabla b_{k}$ (E.5)

where the convergence of the partial sum to the series has to be understood in
$L^{2}$

  * iii)

$\int_{B_{1}(0)}|\nabla b_{k}|^{2}\ dx^{2}\leq C\,\int_{A_{k}\cup
A_{k+1}}|\nabla b|^{2}\ dx^{2}$ (E.6)

where $C$ is a universal constant.

$\Box$

Proof of lemma E.2. Let $\chi$ be a smooth compactly supported function in
${\mathbb{R}}_{+}$ such that

$\begin{array}[]{l}\displaystyle\chi\equiv 1\quad\mbox{ in }[0,1/2]\quad\mbox{
and }\quad\chi\equiv 0\quad\mbox{ in }[1,+\infty)\end{array}$

For any $\rho<1$ we write $\chi_{\rho}(x):=\chi(|x|/\rho)$. With this notation
we have in particular

$\mbox{Supp}(\nabla\chi_{\rho})\subset B_{\rho}\setminus B_{\rho/2}\ .$

For any $\rho<1/2$ and any $c\in W^{1,2}(B_{1})$ such that supp$(\nabla
c)\subset B_{2\rho}$ we proceed to the following decomposition

$\nabla c:=\underbrace{\nabla\bigg{(}\chi_{\rho}\
({c}-\overline{c}_{\rho})\bigg{)}}_{\textbf{(I) }\text{supported in
}B_{\rho}}+\underbrace{\nabla\bigg{(}(1-\chi_{\rho})(c-\overline{c}_{\rho})\bigg{)}}_{\textbf{(II)
}\text{supported in }B_{2\rho}\setminus B_{\rho/2}}.$ (E.7)

where we take

$\overline{c}_{\rho}:=\frac{1}{|B_{\rho}\setminus
B_{\rho/2}|}\int_{B_{\rho}\setminus B_{\rho/2}}c(x)\ dx$

We denote

$T_{\rho}\ :\ c\in W^{1,2}(B_{1})\ \longrightarrow T_{\rho}(c):=\chi_{\rho}\
({c}-\overline{c}_{\rho})$ (E.8)

Because of Poincaré inequality we have

$\int_{B_{\rho}\setminus B_{\rho/2}}|\nabla T_{\rho}(c)|^{2}\ dx^{2}\leq C\
\int_{B_{\rho}\setminus B_{\rho/2}}|\nabla c|^{2}\ dx^{2}$ (E.9)

We define inductively

$b^{0}:=b\quad\mbox{ , }\quad b^{k}:=T_{2^{-k}}(b^{k-1})\quad\mbox{ for }k\geq
1\quad\mbox{ and }\quad b_{k}:=b^{k}-b^{k+1}\quad\mbox{ for }k\geq 0\ .$
(E.10)

By definition we have

$\mbox{Supp}(\nabla b^{k})\subset B_{2^{-k}}\quad\mbox{ and }\quad\nabla
b^{k}=\nabla b^{k-1}=\nabla b\quad\mbox{ in }\quad B_{2^{-k-1}}\quad.$ (E.11)

This gives for $k\geq 0$

$\mbox{Supp}(\nabla b_{k})\subset B_{2^{-k}}\setminus B_{2^{-k-2}}$ (E.12)

Combining (E.9) and (E.11) gives moreover

$\begin{array}[]{l}\displaystyle\int_{B_{2^{-k}}\setminus B_{2^{-k-1}}}|\nabla
b_{k}|^{2}\ dx^{2}\leq 2\,\int_{B_{2^{-k}}\setminus B_{2^{-k-1}}}|\nabla
b^{k}|^{2}+|\nabla b^{k+1}|^{2}\ dx^{2}\\\\[14.22636pt]
\displaystyle=2\,\int_{B_{2^{-k}}\setminus B_{2^{-k-1}}}|\nabla
T_{2^{-k}}(b^{k-1})|^{2}\ dx^{2}\leq C\ \int_{B_{2^{-k}}\setminus
B_{2^{-k-1}}}|\nabla b^{k-1}|^{2}\ dx^{2}\\\\[14.22636pt] \displaystyle=C\
\int_{B_{2^{-k}}\setminus B_{2^{-k-1}}}|\nabla b|^{2}\ dx^{2}\end{array}$
(E.13)

and

$\begin{array}[]{l}\displaystyle\int_{B_{2^{-k-1}}\setminus
B_{2^{-k-2}}}|\nabla b_{k}|^{2}\ dx^{2}\leq 2\,\int_{B_{2^{-k-1}}\setminus
B_{2^{-k-2}}}|\nabla b^{k}|^{2}+|\nabla b^{k+1}|^{2}\ dx^{2}\\\\[14.22636pt]
\displaystyle=2\,\int_{B_{2^{-k-1}}\setminus B_{2^{-k-2}}}|\nabla
b|^{2}+|\nabla T_{2^{-k}}(b^{k})|^{2}\ dx^{2}\leq
C\,\int_{B_{2^{-k-1}}\setminus B_{2^{-k-2}}}|\nabla b|^{2}+|\nabla b^{k}|^{2}\
dx^{2}\\\\[14.22636pt] \displaystyle=2\,C\ \int_{B_{2^{-k-1}}\setminus
B_{2^{-k-2}}}|\nabla b|^{2}\ dx^{2}\end{array}$ (E.14)

Combining (E.12), (E.13) and (E.14) give (E.4), (E.5) and (E.6) and lemma E.2
is proved. $\Box$

Before going to the proof of the weighted Wente estimate a further
intermediate lemma is needed.

######  Lemma E.3.

Let $j\in{\mathbb{N}}$ and $b\in W^{1,2}(B_{1}(0))$ such that Supp$(b)\subset
B_{2^{-j}}(0)$. Let $a\in W^{1,2}(B_{1}(0))$ and $\varphi$ be the solution of

$\left\\{\begin{array}[]{l}-\Delta\varphi=\partial_{x_{1}}a\
\partial_{x_{2}}b-\partial_{x_{2}}a\ \partial_{x_{1}}b\quad\quad~{}~{}\mbox{in
$B_{1}(0)$}\\\\[14.22636pt] \varphi=0\quad\quad~{}~{}\mbox{on $\quad\partial
B_{1}(0)$}\end{array}\right.$ (E.15)

Then for any $0\leq k<j$ there holds

$\int_{B_{2^{-k}}(0)\setminus B_{2^{-k-1}}(0)}|\nabla\varphi|^{2}\ dx^{2}\leq
C\ 2^{2k-2j}\ \int_{B_{1}(0)}|\nabla a|^{2}\ dx^{2}\ \int_{B_{1}(0)}|\nabla
b|^{2}\ dx^{2}$ (E.16)

where $C>0$ is a universal constant.

Assume now that Supp$(b)\subset B_{2^{-j}}(0)\setminus B_{2^{-j-1}}(0)$ then
for any $k>j$ there holds

$\int_{B_{2^{-k}}(0)}|\nabla\varphi|^{2}\ dx^{2}\leq C\ 2^{2j-2k}\
\int_{B_{1}(0)}|\nabla a|^{2}\ dx^{2}\ \int_{B_{1}(0)}|\nabla b|^{2}\ dx^{2}$
(E.17)

$\Box$

Proof of lemma E.3. We first prove (E.16). Under the hypothesis $\varphi$ is
harmonic in the annulus $B_{1}(0)\setminus B_{2^{-j}}(0)$. In this annulus
$\varphi$ takes the form

$\varphi=\varphi_{0}+C^{0}\,\log|x|+\sum_{n\in{\mathbb{Z}}^{\ast}}\varphi_{n}x^{n}+\overline{\varphi_{n}}\
\overline{x}^{n}\ .$

Since $\varphi=0$ on $\partial B_{1}(0)$ this gives $\varphi_{0}=0$ and
$\varphi_{n}=-\,\overline{\varphi_{-n}}$. Integrating (E.15) over $B_{r}(0)$
for $r>2^{-j}$ gives also

$\int_{\partial B_{r}(0)}\partial_{\nu}\varphi\ dl=0\ ,$ (E.18)

which implies that $C^{0}=0$. Hence we obtain

$\varphi=\varphi^{+}+\varphi^{-}\quad\mbox{ where
}\quad\varphi^{+}=\sum_{n>0}\varphi_{n}x^{n}+\overline{\varphi_{n}}\
\overline{x}^{n}\quad\mbox{ and
}\quad\varphi^{-}=-\sum_{n>0}\overline{\varphi_{n}}\ x^{-n}+{\varphi_{n}}\
\overline{x}^{-n}$ (E.19)

Using lemma D.1 we deduce that

$\begin{array}[]{l}\displaystyle\int_{B_{2^{-k}}(0)\setminus
B_{2^{-k-1}}(0)}|\nabla\varphi^{-}|^{2}\ dx^{2}\leq C\ 2^{2k-2j}\
\int_{B_{1}}|\nabla\varphi^{-}|^{2}\ dx^{2}\\\\[14.22636pt]
\displaystyle\quad\leq C\ 2^{2k-2j}\ \int_{B_{1}(0)}|\nabla a|^{2}\ dx^{2}\
\int_{B_{1}(0)}|\nabla b|^{2}\ dx^{2}\end{array}$ (E.20)

Observe that thanks to (E.19) one has for $2^{-j}<r<1$ since
$|\varphi_{n}|^{2}=|\varphi_{n}|^{2}$

$\int_{\partial B_{r}(0)}|\nabla\varphi^{+}|^{2}\ dl=4\pi\
\sum_{n>0}n^{2}\,|\varphi_{n}|^{2}r^{2n}\leq 4\pi\
\sum_{n>0}n^{2}\,|\varphi_{-n}|^{2}r^{-2n}=\int_{\partial
B_{r}(0)}|\nabla\varphi^{-}|^{2}\ dl$ (E.21)