# Moduli space theory for complete, constant $Q$-curvature metrics on finitely
punctured spheres

Rayssa Caju  ,  Jesse Ratzkin  and  Almir Silva Santos  Department of
Mathematical Engineering, University of Chile
Beauchef 851, Edificio Norte, Santiago, Chile<EMAIL_ADDRESS>Department
of Mathematics, Universität Würzburg
97070, Würzburg-BA, Germany<EMAIL_ADDRESS>Department of
Mathematics, Federal University of Sergipe
49107-230, Sao Cristovão-SE, Brazil<EMAIL_ADDRESS>
###### Abstract.

We study constant $Q$-curvature metrics conformal to the the round metric on
the sphere with finitely many point singularities. We show that the moduli
space of solutions with finitely many punctures in fixed positions, equipped
with the Gromov-Hausdorff topology, has the local structure of a real
algebraic variety with formal dimension equal to the number of the punctures.
If a nondegeneracy hypothesis holds, we show that a neighborhood in the moduli
spaces is actually a smooth, real-analytic manifold of the expected dimension.
We also construct a geometrically natural set of parameters, construct a
symplectic structure on this parameter space and show that in the smooth case
a small neighborhood of the moduli space embeds as a Lagrangian submanifold in
the parameter space.

###### Key words and phrases:

Paneitz–Branson operator, $Q$-curvature, moduli spaces

###### 2020 Mathematics Subject Classification:

53C18, 58D17, 58D27

RC was partially supported by FONDECYT grant number 11230872 and by Centro de
Modelamiento Matemático (CMM) BASAL fund FB210005 for center of excellence
from ANID-Chile. ASS was partially supported by CNPq grant number
403349/2021-4, 408834/2023-4 and 312027/2023-0

##  1\. Introduction

In this manuscript we study a fourth-order analog of the singular Yamabe
problem on finitely punctured spheres, as formulated by Schoen and Yau [12],
Mazzeo, Pollack, and Uhlenbeck [9] and others. Our main result characterizes
the local structure of the moduli space of solutions in the case of the round
metric on a finitely punctured sphere.

In general, if $(M,g)$ is a Riemannian manifold of dimension $n\geq 5$ one
defines the $Q$-curvature of $g$ as

$Q_{g}=-\frac{1}{2(n-1)}\Delta_{g}R_{g}-\frac{2}{(n-2)^{2}}|\operatorname{Ric}_{g}|^{2}+\frac{n^{3}-4n^{2}+16n-16}{8(n-1)^{2}(n-2)^{2}}R_{g}^{2},$
(1)

where $R_{g}$ and $\operatorname{Ric}_{g}$ are the scalar and Ricci curvatures
and $\Delta_{g}$ is the Laplace-Beltrami operator. A short computation
demonstrates that $\displaystyle Q_{\overset{\circ}{g}}=\frac{n(n^{2}-4)}{8}$,
where $\overset{\circ}{g}$ is the usual round metric on the sphere
$\mathbf{S}^{n}$. A longer computation shows that the task of finding a
conformal metric $\widetilde{g}=U^{\frac{4}{n-4}}g$ with $Q$-curvature equal
to $\displaystyle\frac{n(n^{2}-4)}{8}$ is equivalent to solving the nonlinear
partial differential equation

$\mathcal{H}_{g}(U):=P_{g}(U)-\frac{n(n-4)(n^{2}-4)}{16}U^{\frac{n+4}{n-4}}=0$
(2)

where

$P_{g}(U):=(-\Delta_{g})^{2}(U)+\operatorname{div}\left(\frac{4}{n-2}\operatorname{Ric}_{g}(\nabla
U,\cdot)-\frac{(n-2)^{2}+4}{2(n-1)(n-2)}R_{g}\nabla
U\right)+\frac{n-4}{2}Q_{g}U$ (3)

is the Paneitz-Branson operator. This operator is conformally covariant, in
that

$P_{U^{\frac{4}{n-4}}g}(\phi)=U^{-\frac{n+4}{n-4}}P_{g}(U\phi).$ (4)

Substituting $\phi=1$ into (4) we obtain the transformation law for
$Q$-curvature under a conformal change of metric, which is

$Q_{U^{\frac{4}{n-4}}g}=\frac{2}{n-4}U^{-\frac{n+4}{n-4}}P_{g}(U).$ (5)

In the case that the background metric is $\overset{\circ}{g}$ the Paneitz
operator factors as

$P_{\overset{\circ}{g}}=\left(-\Delta_{\overset{\circ}{g}}+\frac{(n-4)(n+2)}{4}\right)\left(-\Delta_{\overset{\circ}{g}}+\frac{n(n-2)}{4}\right).$
(6)

This factorization in some sense simplifies the analysis of (2), making the
study of the space of solutions in this setting seem more approachable.
However, the conformal invariance of (4) combined with the noncompactness of
the conformal group of the round metric leads to the blow-up of sequences of
solutions. For this reason we study the following singular problem: given a
closed subset $\Lambda\subset\mathbf{S}^{n}$, describe all the conformal
metrics $g=U^{\frac{4}{n-4}}\overset{\circ}{g}$ that are complete on
$\mathbf{S}^{n}\backslash\Lambda$ and have $\displaystyle
Q_{g}=\frac{n(n^{2}-4)}{8}$. We can reformulate this geometric problem as the
following infinite boundary value problem:

$U:\mathbf{S}^{n}\backslash\Lambda\rightarrow(0,\infty),\qquad\mathcal{H}_{\overset{\circ}{g}}(U)=0,\qquad\liminf_{p\rightarrow\Lambda}U(p)=\infty.$
(7)

We concentrate on the case that $\Lambda=\\{p_{1},\dots,p_{k}\\}$ is a finite
set of points and define the marked moduli space

$\mathcal{M}_{\Lambda}=\left\\{g\in[\overset{\circ}{g}]:Q_{g}=\frac{n(n^{2}-4)}{8}\mbox{
and }g\textrm{ is complete on }\Lambda\right\\}$

and the unmarked moduli space

$\mathcal{M}_{k}=\left\\{g\in[\overset{\circ}{g}]:Q_{g}=\frac{n(n^{2}-4)}{8},\,g\textrm{
is complete on }\Lambda,\,\\#\Lambda=k\right\\}.$

Here $[\overset{\circ}{g}]$ is the set of all conformal metric to
$\overset{\circ}{g}$. We equip both moduli spaces with the Gromov-Hausdorff
topology. The difference between the two is that in the marked moduli space we
fix the singular points, whereas in the unmarked moduli space we allow them to
vary, so long as they remain $k$ distinct points.

Let $\Lambda=\\{p_{1},\dots,p_{k}\\}\subset\mathbf{S}^{n}$ and let
$g\in\mathcal{M}_{\Lambda}$. Then $g$ admits a definite asymptotic structure
near each singular point $p_{i}$, and is asymptotic to one of the Delaunay
metrics described below in Section 2. These Delaunay metrics are, after an
appropriate change of variables, periodic and uniquely described by their
necksizes $\varepsilon\in(0,\overline{\varepsilon}]$, where the maximal
necksize $\overline{\varepsilon}$ depends only on the dimension $n$, which
allows one to assign asymptotic data to $g\in\mathcal{M}_{k}$, including the
asymptotic necksize $\varepsilon_{i}$ at the singular point $p_{i}$. We ask
the following question: how well does this asymptotic data determine a metric
$g\in\mathcal{M}_{\Lambda}$? Our results below form a first step in answering
this question. More precisely, we show that under some conditions one can use
the asymptotic data to parameterize a small neighborhood of moduli space near
$g\in\mathcal{M}_{\Lambda}$.

Our main theorem is the following result.

######  Theorem 1.

For each finite subset $\Lambda\subset\mathbf{S}^{n}$ with $\\#\Lambda=k\geq
3$ the moduli space $\mathcal{M}_{\Lambda}$ is locally a real analytic variety
of formal dimension $k$.

Our proof follows the road map developed by Kusner, Mazzeo, Pollack and
Uhlenbeck in [7] and [9], combining the implicit function theorem and the
Lyaponov-Schmidt process. The key technical part of our analysis is a fine
understanding the linearized operator $L_{g}$ of the operator
$\mathcal{H}_{g}$ defined in (2), which we describe in detail in Section 2.1.

As is usually the case, the analysis allows us to make a more precise
statement if $L_{g}$ is injective or surjective when acting on an appropriate
function space.

######  Definition 2.

A metric $g\in\mathcal{M}_{\Lambda}$ is nondegenerate if $w\in L^{2}$ and
$L_{g}(w)=0$ implies $w\equiv 0$.

######  Theorem 3.

If $g\in\mathcal{M}_{\Lambda}$ is nondegenerate then there exists an open
neighborhood $\mathcal{U}\subset\mathcal{M}_{\Lambda}$ of $g$ that is a real
analytic manifold of dimension $k$.

As a part of proving these two local regularity theorems we construct a
$2k$-dimensional parameter space $\mathcal{W}_{g}$ to parameterize all metrics
$\mathcal{M}_{\Lambda}$ nearby a given $g$. Once we construct
$\mathcal{W}_{g}$ and use it to understand the local regularity of the moduli
space, we construct a symplectic structure on the geometric parameter space
and show that, in the nondegenerate case, a small neighborhood of $g$ in
$\mathcal{M}_{\Lambda}$ embeds in this parameter space $\mathcal{W}_{g}$ as a
Lagrangian submanifold.

Our results form a natural progression of the current understanding of
constant $Q$-curvature metrics. Previously, C. S. Lin [8] showed that all
smooth metrics with constant $Q$-curvature in the conformal class of the round
metric must be the image of $\overset{\circ}{g}$ under a Möbius
transformation. In the language we established above,
$\mathcal{M}_{0}=SO(n+1,1)$, which is the Möbius group of global conformal
transformations of the sphere. In the same paper Lin proved there are no
solutions with a single puncture, i.e. $\mathcal{M}_{1}=\varnothing$ and that
(after a conformal motion) any solution with two punctures is rotationally
invariant. Afterwards Frank and König [5] characterized all the two ended
solutions, showing $\mathcal{M}_{p,q}\simeq(0,\overline{\varepsilon}]$ for
each $p\neq q$, where $\overline{\varepsilon}$ is a finite, positive number
depending only on the dimension $n$. We describe these solutions in some
detail below. In general, explicit gluing constructions demonstrate that the
moduli space $\mathcal{M}_{k}$ is nonempty, provided $k\geq 2$. Baraket and
Rebhi [4] constructed solutions with an even number of punctures by gluing
cylinders together, using small necks as a bridge. Andrade, Wei and Ye [3]
construct many examples in the conformal class of the sphere and the authors
of this paper together with Andrade and do Ò [1] construct many other examples
in the inhomogeneous setting, using a different gluing technique. Together
with Andrade and do Ò [2], the second author described a geometric
characterization of compact subsets of the moduli space.

The rest of the paper proceeds as follows. In Section 2 we discuss some
analytic preliminaries, such as the Delaunay solutions, the local asymptotics
of a singular Yamabe metric near a puncture and the appropriate functions
spaces. In Section 3 we analyze the mapping properties of the linearized
operator $L_{g}$ in various weighted function spaces and introduce the
deficiency space $\mathcal{W}_{g}$, a $2k$-dimensional vector space that will
serve as a parameter space to describe the asymptotic geometry of nearby
metrics in $\mathcal{M}_{\Lambda}$. We prove Theorem 3 in Section 4 and
complete the proof of Theorem 1 in Section 5. Finally, in Section 6 we discuss
a symplectic structure on the natural parameter space of
$\mathcal{M}_{\Lambda}$ and prove that, near smooth points, the moduli space
$\mathcal{M}_{\Lambda}$ is a Lagrangian submanifold of this parameter space.

##  2\. Preliminaries

###  2.1. The choice of a gauge

The choice of a gauge in formulating the moduli problem is equivalent to
choosing the background metric in a conformal class.

While we have thus far phrased this problem in the sphere, it will often be
useful to rewrite in Euclidean space after stereographic projection and to
transfer our analysis between the two settings. Let
$\operatorname{Pr}:\mathbf{R}^{n}\rightarrow\mathbf{S}^{n}\backslash\\{N\\}$
be (the inverse of) stereographic projection mapping Euclidean space to the
sphere minus a pole. It is now a standard exercise to verify that

$\overset{\circ}{g}=u_{\rm sph}^{\frac{4}{n-4}}\delta,\qquad u_{\rm
sph}(x)=\left(\frac{1+|x|^{2}}{2}\right)^{\frac{4-n}{2}},$

where $\delta$ is the Euclidean metric. Using this transformation we can
identify $g=U^{\frac{4}{n-4}}\overset{\circ}{g}$ with
$g=u^{\frac{4}{n-4}}\delta$ where $u=Uu_{\rm sph}$. We also denote the
preimage of the singular set by
$\widetilde{\Lambda}=\operatorname{Pr}^{-1}(\Lambda)$. Without loss of
generality we let the north pole $N$ be a smooth point of $g$, so that the
conformal factor $u$ decays at infinity. More precisely,

$\limsup_{|x|\rightarrow\infty}|x|^{\frac{n-4}{2}}u(x)<\infty.$

######  Remark 4.

Hereafter we adopt the convention that capital letters will denote conformal
factors relative the round metric and lower case letters will denote conformal
factors relative to the Euclidean metric. The two are always related as
described above, e.g. $u=Uu_{\rm sph}$.

Furthermore, we can also rephrase the condition that a metric lies in the
moduli space $\mathcal{M}_{\Lambda}$ in the language of PDEs. Recall that
$g\in[\overset{\circ}{g}]$ precisely when
$g=U^{\frac{4}{n-4}}\overset{\circ}{g}$, so that the condition
$Q_{g}=\frac{n(n^{2}-4)}{8}$ becomes

$\mathcal{H}_{\overset{\circ}{g}}(U)=P_{\overset{\circ}{g}}(U)-\frac{n(n-4)(n^{2}-4)}{16}U^{\frac{n+4}{n-4}}=0.$

Thus

$\mathcal{M}_{\Lambda}=\left\\{U:\mathbf{S}^{n}\backslash\Lambda\rightarrow(0,\infty):\mathcal{H}_{\overset{\circ}{g}}(U)=0\mbox{
and }\liminf_{p\rightarrow\Lambda}U(p)=\infty\right\\}.$

In the spherical setting, the linearized operator has the form

$L_{g}(v)=\left.\frac{d}{dt}\right|_{t=0}\mathcal{H}_{\overset{\circ}{g}}(U+tv)=P_{\overset{\circ}{g}}(v)-\frac{n(n+4)(n^{2}-4)}{16}U^{\frac{8}{n-4}}v,$
(8)

where the Paneitz operator $P_{\overset{\circ}{g}}$ can be factor as in (6).

The operators in question have an even simpler appearance in the Euclidean
setting. This time we use the fact that $g=u^{\frac{4}{n-4}}\delta$, so
$g\in\mathcal{M}_{g}$ is now equivalent to the PDE

$\mathcal{H}_{\delta}(u)=(-\Delta_{0})^{2}u-\frac{n(n-4)(n^{2}-4)}{16}u^{\frac{n+4}{n-4}}=0,$

which in turn implies

$\mathcal{M}_{\Lambda}=\left\\{u:\mathbf{R}^{n}\backslash\widetilde{\Lambda}\rightarrow(0,\infty):\mathcal{H}_{\delta}(u)=0,\,\liminf_{x\rightarrow\widetilde{\Lambda}}u(x)=\infty\mbox{
and }\limsup_{|x|\rightarrow\infty}|x|^{\frac{n-4}{2}}u(x)<\infty\right\\}.$

In the Euclidean setting the linearized operator has the form

$L_{g}(v)=(-\Delta_{0})^{2}v-\frac{n(n+4)(n^{2}-4)}{16}u^{\frac{8}{n-4}}v.$
(9)

In either setting, we refer to a function satisfying the PDE $L_{g}(v)=0$ as a
Jacobi field.

###  2.2. Delaunay metrics

The Delaunay metrics are all the constant $Q$-curvature metrics on a twice-
punctured sphere and, as we will see later, play an important role in
understanding the behavior of singular constant $Q$-curvature metrics with
isolated singularities.

Consider a metric $g=U^{\frac{4}{n-4}}\overset{\circ}{g}$ on
$\mathbf{S}^{n}\backslash\\{p,q\\}$ where $p$ and $q$ are distinct. After a
rotation and a dilation, we can assume $p=N$ is the north pole and $q=S$ is
the south pole. As in the previous section, we transfer now
$\mathbf{R}^{n}\backslash\\{0\\}$ using stereographic projection and let
$u=Uu_{\rm sph}$. Using (4) we see that
$u:\mathbf{R}^{n}\backslash\\{0\\}\rightarrow(0,\infty)$ satisfies

$\mathcal{H}_{\delta}(u)=0.$ (10)

Frank and König [5] classified all the solutions of (10), and we describe them
here. First we perform the Emden-Fowler change of coordinates, defining

$\mathfrak{F}:\mathcal{C}^{\infty}(\mathbf{B}_{r}(0)\backslash\\{0\\})\rightarrow\mathcal{C}^{\infty}((-\log
r,\infty)\times\mathbf{S}^{n-1}),\qquad\mathfrak{F}(u)(t,\theta)=e^{\frac{4-n}{2}t}u(e^{-t}\theta).$
(11)

We can of course invert $\mathfrak{F}$, obtaining

$\mathfrak{F}^{-1}(v)(x)=|x|^{\frac{4-n}{2}}v(-\log|x|,\theta).$

While the prefactor of $e^{\frac{4-n}{2}t}$ might look a little strange at
first, a short computation shows it is geometrically necessary. Letting

$\Upsilon:\mathbf{R}\times\mathbf{S}^{n-1}\rightarrow\mathbf{R}^{n}\backslash\\{0\\},\qquad\Upsilon(t,\theta)=e^{-t}\theta,$

we see

$\Upsilon^{*}(\delta)=e^{-2t}g_{\rm cyl}.$

where $g_{\rm cyl}=dt^{2}+d\theta^{2}$ is the cylindrical metric. If we now
consider a conformal metric $g=u^{\frac{4}{n-4}}\delta$, we see that

$\Upsilon^{*}(g)(t,\theta)=\mathfrak{F}(u)(t,\theta)^{\frac{4}{n-4}}g_{\rm
cyl}.$

After the Emden-Fowler change of coordinates, using (4), (10) becomes

$\mathcal{H}_{\rm cyl}(v)=P_{\rm
cyl}(v)-\frac{n(n-4)(n^{2}-4)}{16}v^{\frac{n+4}{n-4}}=0,$ (12)

where $v:\mathbf{R}\times\mathbf{S}^{n-1}\rightarrow(0,\infty)$ and

$\displaystyle P_{\rm cyl}$ $\displaystyle=(-\Delta_{\rm
cyl})^{2}-\frac{n(n-4)}{2}\Delta_{\rm
cyl}-4\partial_{t}^{2}+\frac{n^{2}(n-4)^{2}}{16}$ (13)
$\displaystyle=\partial_{t}^{4}+\Delta_{\theta}^{2}+2\Delta_{\theta}\partial_{t}^{2}-\frac{n(n-4)+8}{2}\partial_{t}^{2}-\frac{n(n-4)}{2}\Delta_{\theta}+\frac{n^{2}(n-4)^{2}}{16}$

is the Paneitz operator of the cylindrical metric. Note that $\Delta_{\rm
cyl}=\partial_{t}^{2}+\Delta_{\mathbf{S}^{n-1}}$. C. S. Lin [8] used a moving
planes argument to prove that solutions of (10) are rotationally invariant,
reducing (12) to the ODE

$\ddddot{v}-\frac{n(n-4)+8}{2}\ddot{v}+\frac{n^{2}(n-4)^{2}}{16}v-\frac{n(n-4)(n^{2}-4)}{16}v^{\frac{n+4}{n-4}}=0.$
(14)

Notice that one can find a first integral for this ODE defined as

$\mathcal{H}_{\varepsilon}=-\dot{v}_{\varepsilon}\dddot{v}_{\varepsilon}+\frac{1}{2}\ddot{v}_{\varepsilon}^{2}+\frac{n(n-4)+8}{4}\dot{v}_{\varepsilon}^{2}-\frac{n^{2}(n-4)^{2}}{32}v_{\varepsilon}^{2}+\frac{(n-4)^{2}(n^{2}-4)}{32}v_{\varepsilon}^{\frac{2n}{n-4}}.$
(15)

We denote the nonzero constant solution of (14) by

$\overline{\varepsilon}=\left(\frac{n(n-4)}{n^{2}-4}\right)^{\frac{n-4}{8}}.$

######  Theorem 5 (Frank and König [5]).

For each $\varepsilon\in(0,\overline{\varepsilon}]$ there exists a unique
$v_{\varepsilon}:\mathbf{R}\rightarrow(0,\infty)$ solving the ODE (14)
attaining its minimal value of $\varepsilon$ at $t=0$. All these solutions are
periodic. Furthermore, let
$v:\mathbf{R}\times\mathbf{S}^{n-1}\rightarrow(0,\infty)$ be a smooth solution
of the PDE (12). Then either $v(t,\theta)=(\cosh(t+T))^{\frac{4-n}{2}}$ for
some $T\in\mathbf{R}$ or there exist
$\varepsilon\in(0,\overline{\varepsilon}]$ and $T\in\mathbf{R}$ such that
$v(t,\theta)=v_{\varepsilon}(t+T)$.

Later in this paper we will use the fact that the set of Delaunay solutions is
ordered by the Hamiltonian energy $\mathcal{H}$. In other words, $\mathcal{H}$
is a strictly decreasing function of the necksize $\varepsilon$.

We can now write the Delaunay metric in Euclidean coordinates by reversing the
coordinate transformation (11), letting

$u_{\varepsilon}(x)=\mathfrak{F}^{-1}(v)(x)=|x|^{\frac{4-n}{2}}v_{\varepsilon}(-\log|x|),\qquad
g_{\varepsilon}=u_{\varepsilon}^{\frac{4}{n-4}}\delta=v_{\varepsilon}^{\frac{4}{n-4}}g_{\rm
cyl}.$ (16)

The geometric formulation of the Frank-König classification now reads: if
$g=U^{\frac{4}{n-4}}\overset{\circ}{g}$ is a constant $Q$-curvature metric on
$\mathbf{S}^{n}\backslash\\{p,q\\}$ then, after a global conformal
transformation, either $g$ extends to smoothly to the round metric or $g$ is
singular at both $p$ and $q$ and is the image of a Delaunay metric
$g_{\varepsilon}$ after said conformal transformation.

###  2.3. Local asymptotics

A metric $g=U^{\frac{4}{n-4}}\overset{\circ}{g}\in\mathcal{M}_{k}$ with
constant $Q$-curvature and finitely many singular points has a definite
asymptotic structure near each singular point. Let $p_{i}\in\Lambda$ be a
singular point of $g$ and choose stereographic coordinates $x$ centered at
$p_{i}$. With respect to these coordinates we have
$g=u^{\frac{4}{n-4}}\delta=(Uu_{\rm sph})^{\frac{4}{n-4}}\delta$ there exist
$\varepsilon\in(0,\overline{\varepsilon}]$, $R>0$, $a\in\mathbf{R}^{n}$ and
$\beta>1$ so that

$u(x)=R^{\frac{n-4}{2}}u_{\varepsilon}(Rx)+|x|^{\frac{4-n}{2}}\left(\langle
x,a\rangle\left(\frac{n-4}{2}v_{\varepsilon}(-\log(R|x|))-\dot{v}_{\varepsilon}(-\log(R|x|))\right)+\mathcal{O}(|x|^{\beta})\right).$
(17)

This expansion combines the local asymptotic expansions in [6] and in [11]. As
is usually the case, the asymptotic expansion (17) is more tractable in Emden-
Fowler coordinates. The transformed function $v=\mathfrak{F}(u)$ satisfies the
equation (12) on the half-infinite cylinder
$(T_{0},\infty)\times\mathbf{S}^{n-1}$ and the asymptotic expansion now reads

$v(t,\theta)=v_{\varepsilon}(t+T)+e^{-t}\langle
a,\theta\rangle\left(\frac{n-4}{2}v_{\varepsilon}(t+T)-\dot{v}_{\varepsilon}(t+T)\right)+\mathcal{O}(e^{-\beta
t}),$ (18)

where $T=-\log R$.

These asymptotic expansions (17) and (18) allow us to define an asymptotes map

$\mathcal{A}:\mathcal{M}_{\Lambda}\rightarrow(0,\overline{\varepsilon}]^{k}\times\mathbf{R}^{k},\qquad\mathcal{A}(g)=(\varepsilon_{1},\dots,\varepsilon_{k},T_{1},\dots,T_{k}),$
(19)

where $g=u^{\frac{4}{n-4}}\delta$ and

$u(x)\simeq\mathfrak{F}^{-1}(v_{\varepsilon_{i}}(-\log|x-p_{i}|+T_{i}))\qquad\textrm{
near }p_{i}.$ (20)

We will see later on, in the proofs of Theorems 1 and 3, that the asymptotes
maps provides us with local coordinates for the moduli space in the
nondegenerate setting.

###  2.4. Weighted function spaces

We perform most of our analysis below on weighted Sobolev spaces. We first
define these weighted spaces on a half-infinite cylinder, and then transfer
the definition to a punctured ball (and thereafter to a finitely punctured
sphere) using the Emden-Fowler change of coordinates.

######  Definition 6.

Let $\delta\in\mathbf{R}$ and let $v\in L^{2}_{\rm
loc}((0,\infty)\times\mathbf{S}^{n-1})$. We say $v\in
L^{2}_{\delta}((0,\infty)\times\mathbf{S}^{n-1})$ if

$\|v\|^{2}_{L^{2}_{\delta}}=\int_{0}^{\infty}\int_{\mathbf{S}^{n-1}}e^{-2\delta
t}|v(t,\theta)|^{2}d\theta dt<\infty.$

One can similarly define the Sobolev spaces
$W^{k,2}_{\delta}((0,\infty)\times\mathbf{S}^{n-1})$ for any natural number
$k$.

Observe that if $|v(t,\theta)|\leq Ce^{\widetilde{\delta}t}$ for each
$\widetilde{\delta}<\delta$ then $v\in
L^{2}_{\delta}((0,\infty)\times\mathbf{S}^{n-1})$. Next we undo the Emden-
Fowler change of coordinates, letting $u=\mathfrak{F}^{-1}(v)$ to see

$\displaystyle\int_{\mathbf{S}^{n-1}}\int_{t_{1}}^{t_{2}}e^{-2\delta
t}|v(t,\theta)|^{2}dtd\theta$ $\displaystyle=$
$\displaystyle-\int_{\mathbf{S}^{n-1}}\int_{e^{-t_{1}}}^{e^{-t_{2}}}r^{2\delta+n-5}|u(r\theta)|^{2}drd\theta$
$\displaystyle=$ $\displaystyle\int_{r_{2}\leq|x|\leq
r_{1}}|x|^{2\delta-4}|u(x)|^{2}d\mu(x),$

where $r_{1}=e^{-t_{1}}$ and $r_{2}=e^{-t_{2}}$. Thus we have the following
definition.

######  Definition 7.

Let $\delta\in\mathbf{R}$, let $r>0$ and let $u\in L^{2}_{\rm
loc}(\mathbf{B}_{r}(0)\backslash\\{0\\})$. We say $u\in
L^{2}_{\delta}(\mathbf{B}_{r}(0)\backslash\\{0\\})$ if

$\|u\|^{2}_{L^{2}_{\delta}}=\int_{\mathbf{B}_{r}(0)\backslash\\{0\\}}|x|^{2\delta-4}|u(x)|^{2}d\mu<\infty.$

More generally we let $\widetilde{\Lambda}\subset\mathbf{R}^{n}$ be a finite
set and $u\in L^{2}_{\rm loc}(\mathbf{R}^{n}\backslash\widetilde{\Lambda})$.
We say $u\in L^{2}_{\delta}(\mathbf{R}^{n}\backslash\widetilde{\Lambda})$ if

$\|u\|_{L^{2}_{\delta}}=\int_{\mathbf{R}^{n}\backslash\widetilde{\Lambda}}(\operatorname{dist}(x,\widetilde{\Lambda}))^{2\delta-4}|u(x)|^{2}d\mu<\infty.$

Once again, we see that if $|u(x)|\leq
C(\operatorname{dist}(x,\widetilde{\Lambda}))^{2-\widetilde{\delta}}$ for each
$\widetilde{\delta}<\delta$ then $u\in
L^{2}_{\delta}(\mathbf{R}^{n}\backslash\widetilde{\Lambda})$.

##  3\. Linear analysis

###  3.1. The linearization about a Delaunay solution

Here we study the linearized operator about a Delaunay solution, which we
denote as $L_{\varepsilon}$, and some of its mapping properties.

Following Section 5.2 of [1] we write

$L_{\varepsilon}=(-\Delta_{0})^{2}-\frac{n(n+4)(n^{2}-4)}{16}u_{\varepsilon}^{\frac{8}{n-4}}$

and promptly transform to Emden-Fowler coordinates using (11), obtaining the
operator $\mathcal{L}_{\varepsilon}$ defined by

$\mathcal{L}_{\varepsilon}(w)(t,\theta)=e^{\frac{4-n}{2}t}L_{\varepsilon}(\mathfrak{F}^{-1}(w))\circ\Upsilon(t,\theta)=\mathfrak{F}(L_{\varepsilon}(\mathfrak{F}^{-1}(w)))(t,\theta).$
(21)

Some computation gives us

$\displaystyle\mathcal{L}_{\varepsilon}$ $\displaystyle=$ $\displaystyle
P_{\rm cyl}-\frac{n(n+4)(n^{2}-4)}{16}v_{\varepsilon}^{\frac{8}{n-4}}$
$\displaystyle=$
$\displaystyle\partial_{t}^{4}+\Delta_{\mathbf{S}^{n-1}}^{2}+2\Delta_{\mathbf{S}^{n-1}}\partial_{t}^{2}-\frac{n(n-4)}{2}\Delta_{\mathbf{S}^{n-1}}$
$\displaystyle-\frac{n(n-4)+8}{2}\partial_{t}^{2}+\frac{n^{2}(n-4)^{2}}{16}-\frac{n(n+4)(n^{2}-4)}{16}v_{\varepsilon}^{\frac{8}{n-4}}.$

Here $P_{\rm cyl}$ is given by (13).

We isolate two specific Jacobi fields of a Delaunay solution: the Jacobi field
$w_{0}^{+}(\varepsilon)$ generating translations along the axis and the Jacobi
field $w_{0}^{-}(\varepsilon)$ generating changes to the necksize. In Emden-
Fowler coordinates these are given by

$w_{0}^{+}(\varepsilon)=\dot{v}_{\varepsilon},\qquad
w_{0}^{-}(\varepsilon)=\frac{d}{d\varepsilon}v_{\varepsilon}.$ (23)

Differentating the relation
$v_{\varepsilon}(t+T_{\varepsilon})=v_{\varepsilon}(t)$ it is straight-forward
to verify that $w_{0}^{+}$ is bounded and periodic while $w_{0}^{-}$ grows
linearly. The formulation of $w_{0}^{\pm}$ above is well-formed in the case
that $\varepsilon<\overline{\varepsilon}$, but both Jacobi fields vanish in
the cylindrical case. If $\varepsilon=\overline{\varepsilon}$ we define

$w_{0}^{+}=\sin(\sqrt{\mu}t),\qquad
w_{0}^{-}=\cos(\sqrt{\mu}t),\qquad\mu=\frac{\sqrt{n^{4}-64n+64}-(n^{2}-4n+8)}{4}.$
(24)

The analysis in Proposition 1 of [4] shows these two Jacobi fields play the
role of varying the necksize and translation parameter on the cylinder.

One can find the following results and their proofs in Section 3.6 of [11].

We first write a Jacobi field in Fourier series. Recall that the $j$th
eigenvalue of $-\Delta_{\mathbf{S}^{n-1}}$ is $\lambda_{j}=j(n-1+j)$ and it
has multiplicity

$m_{j}=\binom{n-1+j}{j}+\binom{n-3+j}{j-2},$

and so we can expand $w$ in Fourier series as

$w(t,\theta)=\sum_{j=0}^{\infty}\sum_{l=1}^{m_{j}}w_{j,l}(t)E_{j,l}(\theta),$

where $\\{E_{j,1},\dots,E_{j,m_{j}}\\}$ is an orthonormal basis of the
eigenspace of $-\Delta_{\mathbf{S}^{n-1}}$ with eigenvalue $\lambda_{j}$. Thus
the restriction of the operator $\mathcal{L}_{\varepsilon}$ to the eigenspace

$\operatorname{Span}\\{E_{j,1},\dots,E_{j,m_{j}}\\}$

is the ordinary differential operator

$\mathcal{L}_{\varepsilon,j}=\frac{d^{4}}{dt^{4}}-\frac{n(n-4)+8+4\lambda_{j}}{2}\frac{d^{2}}{dt^{2}}+\frac{n^{2}(n-4)^{2}}{16}+\frac{n(n-4)}{2}\lambda_{j}+\lambda_{j}^{2}-\frac{n(n+4)(n^{2}-4)}{16}v_{\varepsilon}^{\frac{8}{n-4}}.$

######  Lemma 8.

For each $j\geq 1$ we have
$0\not\in\operatorname{spec}(\mathcal{L}_{\varepsilon,j})$

The two functions $w_{0}^{\pm}(\varepsilon)$ described above both lie in the
kernel of $\mathcal{L}_{\varepsilon,0}$, and so
$0\in\operatorname{spec}(\mathcal{L}_{\varepsilon,0})$ for each
$\varepsilon\in(0,\overline{\varepsilon}]$.

For proof of the next proposition see [11, Proposition 28].

######  Proposition 9.

There exists a discrete set of real numbers

$\Gamma_{\varepsilon}=\\{\dots,-\gamma_{2}(\varepsilon)<-\gamma_{1}(\varepsilon)<0<\gamma_{1}(\varepsilon)<\gamma_{2}(\varepsilon),\dots\\}$
(25)

with $\gamma_{j}(\varepsilon)\rightarrow\infty$ as $j\rightarrow\infty$ such
that the operator

$\mathcal{L}_{\varepsilon}:W^{4,2}_{\delta}((0,\infty)\times\mathbf{S}^{n-1})\rightarrow
L^{2}_{\delta}((0,\infty)\times\mathbf{S}^{n-1})$

is Fredholm provided $\delta\not\in\Gamma_{\varepsilon}$. In particular, for
any $\delta\in(0,\gamma_{1}(\varepsilon))$

$\mathcal{L}_{\varepsilon}:W^{4,2}_{-\delta}((0,\infty)\times\mathbf{S}^{n-1})\rightarrow
L^{2}_{-\delta}((0,\infty)\times\mathbf{S}^{n-1})$

is injective and

$\mathcal{L}_{\varepsilon}:W^{4,2}_{\delta}((0,\infty)\times\mathbf{S}^{n-1})\rightarrow
L^{2}_{\delta}((0,\infty)\times\mathbf{S}^{n-1})$

is surjective.

One calls $\gamma_{j}(\varepsilon)$ the $j$th indicial root of the Jacobi
operator $\mathcal{L}_{\varepsilon}$ and $\Gamma_{\varepsilon}$ the set of
indicial roots associated to the Delaunay solution $v_{\varepsilon}$.

######  Proposition 10.

Let $\phi\in\mathcal{C}^{\infty}_{0}((0,\infty)\times\mathbf{S}^{n-1})$ and
let $\mathcal{L}_{\varepsilon}(v)=\phi$. Then $v$ satisfies the asymptotic
expansion $v(t,\theta)\simeq\sum_{j=0}^{\infty}v_{j}(t,\theta)$ where each
$v_{j}$ is a Jacobi field, i.e. $L_{\varepsilon}(v_{j})=0$, and $v_{j}$ decays
like a polynomial times $e^{-\gamma_{j}t}$, where $\gamma_{j}>0$ is the $j$th
indicial root.

######  Corollary 11 (Linear Decomposition Lemma I).

Let $\delta\in(0,\gamma_{1}(\varepsilon))$, let $v\in
W^{4,2}_{\delta}((0,\infty)\times\mathbf{S}^{n-1})$ and let
$\phi\in\mathcal{C}^{\infty}((0,\infty)\times\mathbf{S}^{n-1})\cap
L^{2}_{-\delta}((0,\infty)\times\mathbf{S}^{n-1})$ be such that
$\mathcal{L}_{\varepsilon}(v)=\phi$. Then there exist $z\in
W^{4,2}_{-\delta}((0,\infty)\times\mathbf{S}^{n-1})$ and
$w\in\operatorname{Span}(w_{0}^{+}(\varepsilon),w_{0}^{-}(\varepsilon))$ such
that $v=z+w$.

For reasons that will become apparent later in the paper, we call
$\mathcal{W}_{\varepsilon}=\operatorname{Span}(w_{0}^{+}(\varepsilon),w_{0}^{-}(\varepsilon))$
the deficiency space associated to the Delaunay metric with necksize
$\varepsilon$.

###  3.2. The linearization about a singular Yamabe metric

We transfer the mapping properties of the linearization about a Delaunay
solution to study the mapping properties of $L_{g}$, where
$g\in\mathcal{M}_{\Lambda}$ is a conformally flat, singular, constant
$Q$-curvature metric with $k$ prescribed singularities. We denote the
asymptotic necksize of the puncture $p_{j}$ by $\varepsilon_{j}$, and define
the indicial set

$\Gamma_{g}=\bigcup_{i=1}^{k}\Gamma_{\varepsilon_{i}}.$

It follows directly from Proposition 9 that

$L_{g}:W^{4,2}_{\delta}(\mathbf{R}^{n}\backslash\widetilde{\Lambda})\rightarrow
L^{2}_{\delta-4}(\mathbf{R}^{n}\backslash\widetilde{\Lambda})$

is Fredholm if and only if $\delta\not\in\Gamma_{g}$.

######  Definition 12.

Let $g\in\mathcal{M}_{\Lambda}$ and choose $r_{0}>0$ sufficiently small such
that $\mathbf{B}_{2r_{0}}(p_{i})\cap\mathbf{B}_{2r_{0}}(p_{j})=\varnothing$
for each distinct pair of punctures. We define the deficiency space
$\mathcal{W}_{g}$ by

$\mathcal{W}_{g}=\operatorname{Span}\\{\chi\mathfrak{F}^{-1}(w_{0}^{+}(\varepsilon_{i})),\chi\mathfrak{F}^{-1}(w_{0}^{-}(\varepsilon_{i})):i=1,\dots,k\\},$

where $\chi$ is a fixed cut-off function such that

$\chi(x)=\left\\{\begin{array}[]{rl}1&|x|<r_{0}\\\
0&|x|>3r_{0}/2\end{array}\right.,\qquad\|\nabla^{k}\chi\|_{\mathcal{C}^{0}}\leq
cr^{-k}.$

######  Proposition 13 (Linear decomposition lemma II).

Let $0<\delta<\min_{1\leq i\leq k}\gamma_{1}(\varepsilon_{i})$ and let $u\in
W^{4,2}_{\delta}(\mathbf{R}^{n}\backslash\widetilde{\Lambda})$ and $\phi\in
L^{2}_{-\delta-4}(\mathbf{R}^{n}\backslash\widetilde{\Lambda})$ satisfying
$L_{g}(u)=\phi$. Then there exist $w\in\mathcal{W}_{g}$ and $v\in
W^{4,2}_{-\delta}(\mathbf{R}^{n}\backslash\widetilde{\Lambda})$ such that
$u=w+v$.

We now define the bounded null space. Once again we fix a number $\delta$ such
that $0<\delta<\min_{1\leq i\leq k}\gamma_{1}(\varepsilon_{i})$. Each element
of the bounded null space is, strictly speaking, an equivalence class of
functions, that is

$\mathcal{B}_{g}=\frac{\ker(L_{g}:W^{4,2}_{\delta}\rightarrow
W^{0,2}_{\delta-4})}{\ker(L_{g}:W^{4,2}_{-\delta}\rightarrow
W^{0,2}_{-\delta-4})}.$

Using the Hilbert space structure of $W^{k,2}_{\delta}$ we can identify

$\mathcal{B}_{g}\simeq\\{\ker(L_{g}:W^{4,2}_{\delta}\rightarrow
W^{0,2}_{\delta-4})\\}\cap\\{\ker(L_{g}:W^{4,2}_{-\delta}\rightarrow
W^{0,2}_{-\delta-4})\\}^{\perp}.$

Combining this characterization with the linear decomposition lemma we see
that one can identify any $v=w+\phi$ for any $v\in\mathcal{B}$, where
$w\in\mathcal{W}_{g}$ and $\phi\in
W^{4,2}_{-\delta}(\mathbf{S}^{n}\backslash\Lambda)$ decays at each puncture.

Applying Melrose’s relative-index calculus we show the following dimension
count.

######  Theorem 14.

$\dim(\mathcal{B}_{g})=k$

The proof below is more or less the same as the proof of Theorem 4.24 in [9].

###### Proof.

We compute the relative index of $L_{g}$ acting on the appropriate weighted
function spaces. Recall that the index of

$L_{g}:W^{4,2}_{\delta}(\mathbf{R}^{n}\backslash\widetilde{\Lambda})\rightarrow
W^{0,2}_{\delta-4}(\mathbf{R}^{n}\backslash\widetilde{\Lambda})$

is

$\operatorname{ind}(\delta)=\dim(\ker(L_{g}))-\dim(\operatorname{coker}(L_{g})).$

Integration by parts shows that the $L^{2}$-adjoint of $L_{g}$ acting on
$W^{4,2}_{\delta}$ is $L_{g}$ acting on $W^{4,2}_{-\delta}$, and so it follows

$\operatorname{ind}(-\delta)=-\operatorname{ind}(\delta)$

provided $\delta\not\in\Gamma_{g}$. (In this case, reversing the sign of the
weight $\delta$ exchanges the kernel and the cokernel.) Next recall that,
provided $\delta_{1},\delta_{2}\not\in\Gamma_{g}$, the relative index is
defined as

$\operatorname{rel-
ind}(\delta_{1},\delta_{2})=\operatorname{ind}(\delta_{1})-\operatorname{ind}(\delta_{2}).$

We use duality once more (i.e. the operator $L_{g}$ is formally self-adjoint
in $L^{2}$) to see so long as $0<\delta<\min_{1\leq i\leq
k}\gamma_{1}(\varepsilon_{i})$ we have

$\displaystyle\operatorname{rel-ind}(\delta,-\delta)$ $\displaystyle=$
$\displaystyle\operatorname{ind}(\delta)-\operatorname{ind}(-\delta)=2\operatorname{ind}(\delta)$
$\displaystyle=$ $\displaystyle
2\left(\dim(\ker(\left.L_{g}\right|_{W^{4,2}_{\delta}}))-\dim(\ker(\left.L_{g}\right|_{W^{4,2}_{-\delta}}))\right)=2\dim\mathcal{B}_{g}.$

Thus it suffices to show that $\operatorname{rel-ind}(\delta,-\delta)=2k$ for
an appropriate choice of $\delta$. We choose $0<\delta<\min_{1\leq i\leq
k}\gamma_{1}(\varepsilon_{i})$.

We compute this relative index theorem using the Melrose’s relative index
theorem (see Theorem 6.5 of [10]). We first decompose

$\mathbf{S}^{n}\backslash\\{p_{1},\dots,p_{k}\\}=\Omega^{c}\cup\left(\bigcup_{i=1}^{k}\mathbf{B}_{r}(p_{i})\backslash\\{p_{i}\\}\right).$

We can now write $L_{g}$ as the sum of restrictions

$L_{g}=\left.L_{g}\right|_{\Omega^{c}}+\sum_{i=1}^{k}\left.L_{g}\right|_{\mathbf{B}_{r}(p_{i})\backslash\\{p_{i}\\}}$

and compute the relative index of each restriction separately. It might appear
that we first have to take the boundary data of the restrictions into account,
but since

$\partial\Omega^{c}=\bigcup_{i=1}^{k}\partial\mathbf{B}_{r}(p_{i}),$

except for the opposite orientations, the boundary contributions in the
relative indices will cancel out. Thus it suffices to use the Dirichlet
boundary data on the $k$ spheres $\\{\partial\mathbf{B}_{r}(p_{i})\\}$.

The operator $L_{g}$ is elliptic and has index zero on $\Omega^{c}$, so now
we’re left with computing the relative index of the restriction
$\left.L_{g}\right|_{\mathbf{B}_{r}(p_{i})\backslash\\{p_{i}\\}}$ for some
$i\in\\{1,\dots,k\\}$. Next we observe that, because the relative index is a
topological invariant, we can deform the metric of $g$ to be exactly Delaunay
in a small neighborhood of each puncture $p_{j}$. After transforming to
cylindrical coordinates using the Emden-Fowler change of coordinates
$\mathfrak{F}$, we finally arrive at the problem of computing the relative
index of

$\mathcal{L}_{\varepsilon_{i}}:W^{4,2}_{\delta}((0,\infty)\times\mathbf{S}^{n-1})\rightarrow
W^{0,2}_{\delta}((0,\infty)\times\mathbf{S}^{n-1}).$

This is where we use Melrose’s machinery, as developed in Chapters 4,5 and 6
of [10]. To do so we introduce the Fourier-Laplace transform

$\widehat{v}(t,\zeta,\theta)=\mathcal{F}_{\varepsilon_{i}}(v)(t,\zeta,\theta)=\sum_{m=-\infty}^{\infty}e^{-im\zeta}v(t+mT_{\varepsilon_{i}},\theta)$
(27)

and the twisted operator

$\widetilde{\mathcal{L}_{\varepsilon_{i}}}:W^{4,2}(\mathbf{S}^{1}_{T_{\varepsilon_{i}}}\times\mathbf{S}^{n-1})\rightarrow
W^{0,2}(\mathbf{S}^{1}_{T_{\varepsilon_{i}}}\times\mathbf{S}^{n-1})$

defined by

$\widetilde{\mathcal{L}_{\varepsilon_{i}}}(\zeta)(\widehat{v})=e^{i\zeta}\mathcal{F}_{\varepsilon_{i}}\circ\mathcal{L}_{\varepsilon_{i}}\circ\mathcal{F}_{\varepsilon_{i}}^{-1}(e^{-i\zeta}\widehat{v}).$
(28)

We make several observations before continuing. First observe that
$\zeta\in\mathbf{C}$ is a parameter in the Fourier-Laplace transform, and the
sum in (27) converges precisely when $\zeta$ is in the half-space
$\\{\zeta:\mathrm{Im}(\zeta)<-\delta T_{\varepsilon_{i}}\\}$. Next observe
that $\widetilde{\mathcal{L}_{\varepsilon_{i}}}$ is now a family of operators
defined between the fixed function spaces
$W^{4,2}(\mathbf{S}^{1}_{T_{\varepsilon_{i}}}\times\mathbf{S}^{n-1})$ and
$W^{0,2}(\mathbf{S}^{1}_{T_{\varepsilon_{i}}}\times\mathbf{S}^{n-1})$ that
depends holomorphically on the complex parameter $\zeta$. This allows us to
use the analytic Fredholm theorem to conclude that
$\widetilde{\mathcal{L}_{\varepsilon_{i}}}$ is Fredholm so long as $\zeta$
avoids a discrete set in the complex plan, which in turn allows us to define a
right-inverse $\widetilde{\mathcal{G}_{\varepsilon_{i}}}(\zeta)$ for
$\widetilde{\mathcal{L}_{\varepsilon_{i}}}$. This right inverse
$\widetilde{\mathcal{G}_{\varepsilon_{i}}}$ has a meromorphic extenstion to
$\mathbf{C}$ with poles at $\widetilde{\Gamma}_{\varepsilon_{i}}$. In fact,
the indicial roots $\Gamma_{\varepsilon_{i}}$ are precisely the imaginary
parts of the points in $\widetilde{\Gamma}_{\varepsilon_{i}}$.

Melrose’s relative index theorem states in this context that the relative
index is given by a contour integral of the resolvent
$(\widetilde{\mathcal{L}_{\varepsilon_{i}}}-\zeta)^{-1}$ about a contour
surrounding the pole corresponding to the weight $0$, as described in the
proof of Proposition 26 of [11] and the proof of Proposition 4.15 of [9]. This
contour integral counts the number of tempered, non-decaying Jacobi fields
with subexponential growth on a Delaunay end. However, we already know there
are only two such Jacobi fields, namely $w_{0}^{+}(\varepsilon_{i})$ and
$w_{0}^{-}(\varepsilon_{i})$. We conclude that

$\displaystyle 2\dim(\mathcal{B}_{g})$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{k}\operatorname{ind}(\mathcal{L}_{\varepsilon_{i}}:W^{2,2}_{\delta}((0,\infty)\times\mathbf{S}^{n-1})\rightarrow
W^{-2,2}_{\delta}((0,\infty)\times\mathbf{S}^{n-1})$
$\displaystyle-\sum_{i=1}^{k}\operatorname{ind}(\mathcal{L}_{\varepsilon_{i}}:W^{2,2}_{-\delta}((0,\infty)\times\mathbf{S}^{n-1})\rightarrow
W^{-2,2}_{-\delta}((0,\infty)\times\mathbf{S}^{n-1})$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{k}2=2k,$

as we claimed. ∎

##  4\. Local structure in the nondegenerate case

In this section we prove local regularity of the moduli space near
nondegenerate points, as stated in Theorem 3. We first recall the statement of
the theorem, namely that if $g\in\mathcal{M}_{\Lambda}$ is nondegenerate then
there exists an open neighborhood $\mathcal{U}\subset\mathcal{M}_{\Lambda}$ of
$g$ that is a smooth $k$-dimensional manifold.

###### Proof.

We begin by prescribing the singular set
$\Lambda=\\{p_{1},p_{2},\dots,p_{k}\\}$ and choosing a nondegenerate metric
$g\in\mathcal{M}_{\Lambda}$. Using the Euclidean gauge, we write $g$ as
$g=u^{\frac{4}{n-4}}\delta$, where

$u:\mathbf{R}^{n}\backslash\Lambda\rightarrow(0,\infty),\quad(-\Delta_{0})^{2}u=\frac{n(n-4)(n^{2}-4)}{16}u^{\frac{n+4}{n-4}},\quad\liminf_{x\rightarrow\Lambda}u(x)=\infty.$

Nondegeneracy of $g$ states that the linearized operator

$L_{g}=(-\Delta_{0})^{2}-\frac{n(n+4)(n^{2}-4)}{16}u^{\frac{8}{n-4}}$

acting on $W^{4,2}(\mathbf{R}^{n}\backslash\\{p_{1},\dots,p_{k}\\})$ has no
kernel. By the linear decomposition lemma, this is equivalent to the condition
that

$\ker(L_{g}:W^{4,2}_{-\delta}\oplus\mathcal{W}_{g}\rightarrow
W^{0,2}_{-\delta})=\mathcal{B}_{g},$ (29)

whenever $\delta>0$ is sufficiently small. The bounded null space
$\mathcal{B}_{g}$ always lies in the kernel of (29), but in the degenerate
case the kernel will also contain a finite-dimensional space of decaying
Jacobi fields.

Intuitively, we would like to describe the metrics in $\mathcal{M}_{\Lambda}$
near $g$ as

$\mathcal{U}=\left\\{g_{v}=(u+v)^{\frac{4}{n-4}}\delta:\mathcal{H}_{\delta}(u+v)=(-\Delta_{0})^{2}(u+v)-\frac{n(n-4)(n^{2}-4)}{16}(u+v)^{\frac{n+4}{n-4}}=0\right\\},$

where $v$ is small with respect to an appropriate norm. If we only allow $v$
to decay, the linearized operator does not have any kernel by our hypothesis,
and so it would be an exercise in futility to construct a solution set this
way. Furthermore, we should allow the nearby metrics to have slightly
different asymptotic data, which we cannot encode with a decaying perturbing
function $v$. On the other hand, if we allow perturbing functions $v$ with any
order of growth (or even non-decay), it is difficult to analyze the zero-set
of the operator $\mathcal{H}$, and in particular it is impossible to relate
the kernel of the linearization to this zero-set. We remedy this problem by
deforming the asymptotic data according to an element of the deficiency space
$\mathcal{W}_{g}$, as described below.

We denote the asymptotic necksize of $g$ at the puncture $p_{i}$ by
$\varepsilon_{i}$. Choose $\delta$ such that

$0<\delta<\min_{1\leq i\leq k}\gamma_{1}(\varepsilon_{i}).$

We can identify conformally-related, constant $Q$-curvature metrics in a
neighborhood of $g$ with

$\mathcal{Z}=\\{(v,w)\in\mathcal{V}_{1}\oplus\mathcal{V}_{2}\subset
W^{4,2}_{-\delta}\oplus\mathcal{W}_{g}:\mathcal{H}(v,w)=0\\},$ (30)

where $\mathcal{V}_{1}$ and $\mathcal{V}_{2}$ are small neighborhoods of the
origin. To make sense of this, we should describe the mapping

$\mathcal{H}:W^{4,2}_{-\delta}\oplus\mathcal{W}_{g}\rightarrow
W^{0,2}_{-\delta}$ (31)

in some detail. By the expansion (17) (or, equivalently (18)) there exist
parameters $\varepsilon_{i}\in(0,\overline{\varepsilon})$,
$T_{i}\in\mathbf{R}$ and a decaying function $z\in
W^{4,2}_{-\delta}(\mathbf{B}_{r}(0))$ such that

$u(x-p_{i})=\mathfrak{F}^{-1}\left(v_{\varepsilon_{i}}(-\log|\cdot-
p_{i}|+T_{i})\right)+z(x-p_{i}).$

Now let $v\in W^{4,2}_{-\delta}$ and let $w\in\mathcal{W}_{g}$. By definition,

$w=\sum_{i=1}^{k}(a_{i}^{+}w_{0}^{+}(\varepsilon_{i})+a_{i}^{-}w_{0}^{-}(\varepsilon_{i})),$

where $a_{i}^{\pm}\in\mathbf{R}$. We define the metric
$\widetilde{g}=\widetilde{u}^{\frac{4}{n-4}}\delta$, where

$\widetilde{u}(x)=\left\\{\begin{array}[]{cc}u(x)+v(x)&|x-p_{i}|>2r_{0}\\\
v(x)+(1-\chi(x))u(x)+&\\\
\chi(x)(\mathfrak{F}^{-1}(v_{\varepsilon_{i}+a_{i}^{-}}(-\log|\cdot-
p_{i}|+T_{i}+a_{i}^{+}))+z(x-p_{i}))&r_{0}<|x-p_{i}|<2r_{0}\\\
\mathfrak{F}^{-1}(v_{\varepsilon_{i}+a_{i}^{-}}(-\log|\cdot-
p_{i}|+T_{i}+a_{i}^{+})+z(x-p_{i})+v(x-p_{i})&0<|x-p_{i}|<r_{0},\end{array}\right.$
(32)

where $r_{0}$ and $\chi$ are as in Definition 12. Observe that the
coefficients $\\{a_{i}^{+},a_{i}^{-}\\}$ uniquely determine function
$w\in\mathcal{W}_{g}$, so the dependence of $\widetilde{u}$ on $w$ is given in
how we deform the geometric asymptotic data of $g=u^{\frac{4}{n-4}}\delta$.
The construction of $\widetilde{g}$ is well-defined so long as
$\varepsilon_{i}<\overline{\varepsilon}$, but we must adjust it slightly if
$\varepsilon_{i}=\overline{\varepsilon}$. In this case we replace

$v_{\varepsilon_{i}+a_{i}^{-}}(-\log|\cdot-p_{i}|+T_{i}+a_{i}^{+})$

with

$\widetilde{v}(t,\theta)=v_{\overline{\varepsilon}}+a_{i}^{-}\cos(\sqrt{\mu}(t+T_{i}+a_{i}^{+})+\mathcal{O}(e^{-t}),$

as constructed in Proposition 1 of [4].

Finally, we identify

$\displaystyle\mathcal{H}(v,w)$ $\displaystyle=$
$\displaystyle\mathcal{H}_{g}(\widetilde{u})=\widetilde{u}^{\frac{n+4}{n-4}}\left(P_{\widetilde{g}}(1)-\frac{n(n-4)(n^{2}-4)}{16}\right)$
$\displaystyle=$
$\displaystyle\frac{(n-4)}{2}\widetilde{u}^{\frac{n+4}{n-4}}\left(Q_{\widetilde{g}}-\frac{n(n^{2}-4)}{8}\right).$

We also observe that, by construction, $\mathcal{H}(0,0)=0$.

With this definition, we see that the zero-set $\mathcal{Z}$ is exactly the
set of constant $Q$-curvature metrics whose asymptotic data are close to that
of $g$. Observe that we should not expect $\mathcal{H}(0,w)=0$ for any nonzero
element of the deficiency space $\mathcal{W}_{g}$. This is because we
construct elements of $\mathcal{W}_{g}$ using a cut-off function $\chi$ to
transfer deformations of the Delaunay asymptotes to the background metric $g$,
and so the $Q$-curvature is non-constant in the transition region, where
$\nabla\chi\neq 0$. However, the quantity $\displaystyle
Q_{\widetilde{g}}-\frac{n(n^{2}-4)}{8}$ is small (assuming $w$ is small) and
compactly supported. Thus we expect to be able correct the $Q$-curvature with
a decaying function $v\in W^{4,2}_{-\delta}$, exactly as described above.
Additionally, the linearization of the operator $\mathcal{H}_{g}$ as applied
to $W^{4,2}_{-\delta}\oplus\mathcal{W}_{g}$ is

$L_{g}:W^{4,2}_{-\delta}\oplus\mathcal{W}_{g}\rightarrow W^{0,2}_{-\delta-4}$

and

$\ker\left(L_{g}:W^{4,2}_{-\delta}\oplus\mathcal{W}_{g}\rightarrow
W^{0,2}_{-\delta-4}\right)=\mathcal{B}_{g}\oplus\ker\left(L_{g}:W^{4,2}_{-\delta}\rightarrow
W^{0,2}_{-\delta-4}\right).$ (34)

However, since $g$ is nondegenerate the second summand on the right hand side
of (34) is just to $0$ function. Thus the kernel of $L_{g}$ is precisely
$\mathcal{B}_{g}$, which has dimension $k$, which is also its minimal possible
dimension. Thus $\dim(\ker(L_{\widetilde{g}}))=k$ on an open neighborhood of
$g$ in $\mathcal{M}_{\Lambda}$, and so by the implicit function theorem an
open neighborhood of $\mathcal{Z}$ containing $g$ is a smooth, $k$-dimensional
manifold. ∎

##  5\. Local structure in the degenerate case

Our purpose is to discuss the local structure of the moduli space
$\mathcal{M}_{\Lambda}$ without the hypothesis that the linearization has a
trivial $L^{2}$-nullspace. In this context, we apply the Lyapunov-Schmidt
argument as presented in [7]. The key idea goes back to Simon’s proof of an
infinite-dimensional version of the Łojasiewicz inequality, see Theorem 3 of
[13].

######  Theorem 15.

The space $\mathcal{M}_{\Lambda}$ is locally a finite dimensional real
analytic variety.

###### Proof.

Once again, our problem can be reduced to the understanding of the zero set of

$\mathcal{H}:W^{4,2}_{-\delta}\oplus\mathcal{W}_{g}\rightarrow
W^{0,2}_{-\delta},$

where $\mathcal{H}$ is defined in (32) and (4). This time, however, the kernel
of the linearization that we shall denote as
$K\equiv\ker\left(L_{g}:W^{4,2}_{-\delta}\rightarrow
W^{0,2}_{-\delta-4}\right)$ is nontrivial in $\subset W^{4,2}_{-\delta}$, and
it can identify by duality with the cokernel of
$L_{g}:W^{4,2}_{\delta}\rightarrow W^{0,2}_{\delta-4}.$

Following [7] we define

$\widetilde{\mathcal{H}}:W^{4,2}_{-\delta}\oplus\mathcal{W}_{g}\oplus
K\rightarrow
W^{0,2}_{-\delta},\qquad\widetilde{\mathcal{H}}(v,w,\phi)=\mathcal{H}(v,w)+\phi,$

so that

$\displaystyle\mathcal{Z}$
$\displaystyle=\\{(v,w)\in\mathcal{V}_{1}\oplus\mathcal{V}_{2}:\mathcal{H}(v,w)=0\\}$
$\displaystyle=\\{(v,w,\phi)\in\mathcal{V}_{1}\oplus\mathcal{V}_{2}\oplus\mathcal{V}_{3}:\widetilde{\mathcal{H}}(v,w,\phi)=\phi\\}.$

where $\mathcal{V}_{1}\subset W^{4,2}_{-\delta}$,
$\mathcal{V}_{2}\subset\mathcal{W}_{g}$ and $\mathcal{V}_{3}\subset K$ are
small neighborhoods of the origin in each respective Banach space.

We now see that

$\displaystyle\mathcal{Z}$ $\displaystyle\subset$
$\displaystyle\widetilde{\mathcal{Z}}=\\{(v,w,\phi)\in\mathcal{V}_{1}\oplus\mathcal{V}_{2}\oplus\mathcal{V}_{3}:\widetilde{\mathcal{H}}(v,w,\phi)\in
K_{(1)}\\}$ $\displaystyle=$
$\displaystyle\\{(v,w,\phi)\in\mathcal{V}_{1}\oplus\mathcal{V}_{2}\oplus\mathcal{V}_{3}:\Pi^{\perp}(\widetilde{\mathcal{H}}(v,w,\phi))=0\\}=\ker(\Pi^{\perp}\circ\widetilde{\mathcal{H}}),$

where $\Pi^{\perp}$ is the orthogonal projection of $W^{0,2}_{-\delta}$ onto
$K^{\perp}$. The linearization of this operator is given by

$\Pi^{\perp}\circ L_{g}:K^{\perp}\oplus\mathcal{W}_{g}\oplus K\rightarrow
W^{0,2}_{-\delta},$

which is now a surjective operator. Furthermore, we can characterize the
kernel of the linearization as

$\ker(\Pi^{\perp}\circ L_{g})=\\{(v,w,\phi)\in
W^{4,2}_{-\delta}\oplus\mathcal{W}_{g}\oplus K:L_{g}(v+w)\in K\\}\simeq
K\oplus\mathcal{B}_{g}.$

Thus by the implicit function theorem, there is a real-analytic function

$\Psi:K\oplus\mathcal{B}_{g}\oplus
K\rightarrow(W^{4,2}_{-\delta}\oplus\mathcal{W}_{g})/(K\oplus\mathcal{B}_{g}),\qquad\Psi(v,w,\phi)=(\psi_{1}(v,w,\phi),\psi_{2}(v,w,\phi))$

such that

$\widetilde{\mathcal{Z}}=\\{(\psi_{1}(v,w,\phi),\psi_{2}(v,w,\phi),\phi):(v,w,\phi)\in
K\oplus\mathcal{B}_{g}\oplus K\\}.$

Unraveling these definitions we see

$\mathcal{Z}\simeq\\{(v,w,\phi)\in K\oplus\mathcal{B}_{g}\oplus
K:\mathcal{H}(v+\psi_{1}(v,w,\phi),w+\psi_{2}(v,w,\phi))=0\\},$

which is indeed the zero set of an analytic function acting on a finite-
dimensional vector space. This proves a small neighborhood
$\mathcal{U}\subset\mathcal{M}_{\Lambda}$ containing $g$ is indeed a real-
analytic variety. ∎

##  6\. Symplectic structure

Here we discuss the asymptotes mapping from the marked moduli space
$\mathcal{M}_{\Lambda}$ into a fixed configuration space
$\mathbb{M}_{\Lambda}=(0,\overline{\varepsilon}]^{k}\times\mathbf{R}^{k}$,
where each pair $(\varepsilon_{i},T_{i})$ characterizes the Delaunay asymptote
at the puncture $p_{i}$. We further show that if $g\in\mathcal{M}_{\Lambda}$
is nondegenerate then this local mapping is a Lagrangian embedding with
respect to the standard symplectic structure. One can construct a similar
asymptotes map for the unmarked moduli space, and much of the properties we
prove below carry through, but in this latter case, the configuration spaces
are larger and constructing the mapping is more involved.

First we construct a symplectic form on $\mathcal{M}_{\Lambda}$. Let
$g=U^{\frac{4}{n-4}}\overset{\circ}{g}\in\mathcal{M}_{\Lambda}$ and transfer
$g$ to $\mathbf{R}^{n}\backslash\widetilde{\Lambda}$ using stereographic
projection, rewriting $g=u^{\frac{4}{n-4}}\delta$ with $u=Uu_{\rm sph}$. For
any sufficiently small $r>0$ we define

$\Omega_{r}=\mathbf{R}^{n}\backslash\left(\bigcup_{i=1}^{k}\mathbf{B}_{r}(p_{i})\right)$

and

$\omega(v,w)=\lim_{r\searrow
0}\int_{\Omega_{r}}(vL_{g}(w)-wL_{g}(v))d\mu_{0}.$ (36)

Here $d\mu_{0}$ is the Euclidean volume element, $v,w\in\mathcal{W}_{g}$ lie
in the deficiency space of $g$ (See Definition 12) and $L_{g}$ is the Jacobi
operator of $g$, which is defined in (9).

######  Theorem 16.

The form $\omega$ defined in (36) is a symplectic form on the $2k$-dimensional
vector space $\mathcal{W}_{g}$.

###### Proof.

Our first order of business is to show that $\omega$ is well-defined, i.e.
that the limit in (36) exists. By (9) observe that

$vL_{g}(w)-wL_{g}(v)=v\Delta_{0}^{2}w-w\Delta_{0}^{2}v.$

Next, we recall that the outer unit normal of $\Omega_{r}$ is $-\partial_{r}$
on each boundary sphere $\partial\mathbf{B}_{r}(p_{i})$ and integrate by parts
to see

$\int_{\Omega_{r}}v\Delta_{0}^{2}wd\mu_{0}=\int_{\Omega_{r}}(\Delta_{0}v)(\Delta_{0}w)d\mu_{0}-\sum_{i=1}^{k}\int_{\partial\mathbf{B}_{r}(p_{i})}(v\partial_{r}\Delta_{0}w-\partial_{r}v\Delta_{0}w)d\sigma_{0},$

and so (36) becomes

$\omega(v,w)=\lim_{r\searrow
0}\sum_{i=1}^{k}\int_{\partial\mathbf{B}_{r}(p_{i})}(w\partial_{r}\Delta_{0}v-v\partial_{r}\Delta_{0}w+\partial_{r}v\Delta_{0}w-\partial_{r}w\Delta_{0}v)d\sigma_{0}.$
(37)

Next, we change variables using (11), letting $\widetilde{v}=\mathfrak{F}(v)$
and $\widetilde{w}=\mathfrak{F}(w)$. Under this change of variables

$\partial_{r}v=-e^{\frac{n-2}{2}t}\left(\partial_{t}\widetilde{v}+\frac{n-4}{2}\widetilde{v}\right),\qquad\Delta_{0}v=e^{\frac{n}{2}t}\left(\partial_{t}^{2}\widetilde{v}-2\partial_{t}\widetilde{v}-\frac{n(n-4)}{4}\widetilde{v}+\Delta_{\theta}\widetilde{v}\right)$
(38)

and

$\partial_{r}\Delta_{0}v=-e^{\frac{n+2}{2}t}\left(\partial_{t}^{3}\widetilde{v}+\frac{n-4}{2}\partial_{t}^{2}\widetilde{v}-\frac{n^{2}}{4}\partial_{t}\widetilde{v}-\frac{n^{2}(n-4)}{8}\widetilde{v}+\Delta_{\theta}\partial_{t}\widetilde{v}+\frac{n}{2}\Delta_{\theta}\widetilde{v}\right).$
(39)

Plugging (38) and (39) into (37) we obtain

$\displaystyle\int_{\partial\mathbf{B}_{r}(p_{i})}$
$\displaystyle\left(w\partial_{r}\Delta_{0}v-v\partial_{r}\Delta_{0}w+\partial_{r}v\Delta_{0}w-\partial_{r}w\Delta_{0}v\right)d\sigma_{0}$
$\displaystyle=$
$\displaystyle\int_{\mathbf{S}^{n-1}}(w\partial_{r}\Delta_{0}v-v\Delta_{0}w+\partial_{r}v\Delta_{0}w-\partial_{r}w\Delta_{0}v)(r\theta)r^{n-1}d\theta$
$\displaystyle=$
$\displaystyle\int_{\mathbf{S}^{n-1}}\left(-\widetilde{w}\partial_{t}^{3}\widetilde{v}-\frac{n-4}{2}\widetilde{w}\partial_{t}^{2}\widetilde{v}+\frac{n^{2}}{4}\widetilde{w}\partial_{t}\widetilde{v}+\frac{n^{2}(n-4)}{8}\widetilde{w}\widetilde{v}-\widetilde{w}\Delta_{\theta}\partial_{t}\widetilde{v}-\frac{n}{2}\widetilde{w}\Delta_{\theta}\widetilde{v}\right.$
$\displaystyle+\widetilde{v}\partial_{t}^{3}\widetilde{w}+\frac{n-4}{2}\widetilde{v}\partial_{t}^{2}\widetilde{w}-\frac{n^{2}}{4}\widetilde{v}\partial_{t}\widetilde{w}-\frac{n^{2}(n-4)}{8}\widetilde{v}\widetilde{w}+\widetilde{v}\Delta_{\theta}\partial_{t}\widetilde{w}+\frac{n}{2}\widetilde{v}\Delta_{\theta}\widetilde{w}$
$\displaystyle-\partial_{t}\widetilde{v}\partial_{t}^{2}\widetilde{w}+2\partial_{t}\widetilde{v}\partial_{t}\widetilde{w}+\frac{n(n-4)}{4}\widetilde{w}\partial_{t}\widetilde{v}-\partial_{t}\widetilde{v}\Delta_{\theta}\widetilde{w}-\frac{n-4}{2}\widetilde{v}\partial_{t}^{2}\widetilde{w}+(n-4)\widetilde{v}\partial_{t}\widetilde{w}$
$\displaystyle+\frac{n(n-4)^{2}}{8}\widetilde{v}\widetilde{w}-\frac{n-4}{2}\widetilde{v}\Delta_{\theta}\widetilde{w}$
$\displaystyle+\partial_{t}\widetilde{w}\partial_{t}^{2}\widetilde{v}-2\partial_{t}\widetilde{w}\partial_{t}\widetilde{v}-\frac{n(n-4)}{4}\widetilde{v}\partial_{t}\widetilde{w}+\partial_{t}\widetilde{w}\Delta_{\theta}\widetilde{v}+\frac{n-4}{2}\widetilde{w}\partial_{t}^{2}\widetilde{v}-(n-4)\widetilde{w}\partial_{t}\widetilde{v}$
$\displaystyle\left.-\frac{n(n-4)^{2}}{8}\widetilde{w}\widetilde{v}+\frac{n-4}{2}\widetilde{w}\Delta_{\theta}\widetilde{v}\right)d\theta$
$\displaystyle=$
$\displaystyle\int_{\mathbf{S}^{n-1}}\left(\widetilde{v}\partial_{t}^{3}\widetilde{w}-\widetilde{w}\partial_{t}^{3}\widetilde{v}+\partial_{t}\widetilde{w}\partial_{t}^{2}\widetilde{v}-\partial_{t}{v}\partial_{t}^{2}\widetilde{w}+\frac{n(n-4)+8}{2}(\widetilde{w}\partial_{t}\widetilde{v}-\widetilde{v}\partial_{t}\widetilde{w})\right)d\theta$

Observe that each element of the deficiency space is asymptotically radial
about each puncture point, so that the expansions

$\mathfrak{F}(v(\cdot-p_{i}))(t,\theta)=\overline{v}(t)+\mathcal{O}(e^{-\delta
t}),\qquad\mathfrak{F}(w(\cdot-
p_{i}))(t,\theta)=\overline{w}(t)+\mathcal{O}(e^{-\delta t})$

for each puncture $p_{i}$. Thus each term in the expansion above involving
derivatives with respect to $\theta$ will vanish in the limit.

Next we use the fact that both $v$ and $w$ lie in the definciency space
$\mathcal{W}_{g}$ (see Definition 12). This means that near each puncture
$p_{i}$ the functions $v$ and $w$ have asymptotic expansions of the form

$\widetilde{v}=\alpha_{i}^{+}w_{0}^{+}(\varepsilon_{i})+\alpha_{i}^{-}w_{0}^{-}(\varepsilon_{i}),\qquad\widetilde{w}=\beta_{i}^{+}w_{0}^{+}(\varepsilon_{i})+\beta_{i}^{-}w_{0}^{-}(\varepsilon_{i})$
(40)

for $t$ sufficiently large.

At each end, using bilinearity and skew-symmetry, we see that $\omega(v,w)$ is
the limit as $t\rightarrow\infty$ of

$\displaystyle(\alpha_{i}^{+}\beta_{i}^{-}-\alpha_{i}^{-}\beta_{i}^{+})\int_{\mathbf{S}^{n-1}}\left(w_{0}^{+}\dddot{w}_{0}^{-}-w_{0}^{-}\dddot{w}_{0}^{+}+\dot{w}_{0}^{-}\ddot{w}_{0}^{+}-\dot{w}_{0}^{+}\ddot{w}_{0}^{-}\right.$
(41)
$\displaystyle\qquad\qquad\qquad\qquad\qquad\left.+\frac{n(n-4)+8}{2}(w_{0}^{-}\dot{w}_{0}^{+}-w_{0}^{+}\dot{w}_{0}^{-})\right)d\theta$

If $A_{\varepsilon}$ is the integrand in (41), then

$\displaystyle\frac{d}{dt}A_{\varepsilon}$
$\displaystyle=w_{0}^{+}\left(\ddddot{w}_{0}^{-}-\frac{n(n-4)+8}{2}\ddot{w}_{0}^{-}\right)-w_{0}^{-}\left(\ddddot{w}_{0}^{+}-\frac{n(n-4)+8}{2}\ddot{w}_{0}^{+}\right)$
$\displaystyle=\left(\frac{n^{2}(n-4)^{2}}{16}-\frac{n(n+4)(n^{2}-4)}{16}v_{\varepsilon}^{\frac{8}{n-4}}\right)\left(w_{0}^{+}w_{0}^{-}-w_{0}^{-}w_{0}^{+}\right)=0.$

Thus $A_{\varepsilon}$ does not depend on $t$. Here we have used the ODE for
$w_{0}^{\pm}$, namely

$\ddddot{w}_{0}^{\pm}-\frac{n(n-4)+8}{2}\ddot{w}_{0}^{\pm}+\left(\frac{n^{2}(n-4)^{2}}{16}-\frac{n(n+4)(n^{2}-4)}{16}v_{\varepsilon}^{\frac{8}{n-4}}\right)w_{0}^{\pm}=0.$

Let us find the value of the integrand in (41) at $t=0$ using the definitions
of $w_{0}^{\pm}$ in (23). First observe that
$w_{0}^{+}(\varepsilon)=\dot{v}_{\varepsilon}$, so
$w_{0}^{+}(0)=\dot{v}_{\varepsilon}(0)=0$,
$\dot{w}_{0}^{+}(0)=\ddot{v}_{\varepsilon}(0)>0$,
$\ddot{w}_{0}^{+}(0)=\dddot{v}_{\varepsilon}(0)=0$ and by (14) we get

$\dddot{w}_{0}^{+}(0)=\ddddot{v}_{\varepsilon}(0)=\frac{n(n-4)+8}{2}\ddot{v}_{\varepsilon}(0)-\frac{n(n-4)}{16}\left(n(n-4)\varepsilon-(n^{2}-4)\varepsilon^{\frac{n+4}{n-4}}\right).$

Furthermore, since $\displaystyle
w_{0}^{-}=\frac{d}{d\varepsilon}v_{\varepsilon}$ and $v_{\varepsilon}$ assumes
its minimal value at $t=0$ we see $w_{0}^{-}(0)=1$. Therefore, at $t=0$ it
holds

$\displaystyle A_{\varepsilon}(0)$
$\displaystyle=-\dddot{w}_{0}^{+}(0)-\ddot{v}_{\varepsilon}(0)\ddot{w}_{0}^{-}+\frac{n(n-4)+8}{2}\ddot{v}_{0}^{+}(0)$
$\displaystyle=-\ddot{v}_{\varepsilon}(0)\ddot{w}_{0}^{-}+\frac{n(n-4)}{16}\left(n(n-4)\varepsilon-(n^{2}-4)\varepsilon^{\frac{n+4}{n-4}}\right)$

By (15) we have

$\mathcal{H}_{\varepsilon}=\frac{1}{2}\ddot{v}_{\varepsilon}(0)^{2}-\frac{n^{2}(n-4)^{2}}{32}\varepsilon^{2}+\frac{(n-4)^{2}(n^{2}-4)}{32}\varepsilon^{\frac{2n}{n-4}}.$
(42)

As we remarked earlier, the energy is a decreasing function of $\varepsilon$,
minimized by the cylinder, which has the largest possible necksize, so
differentiating (42) we see

$\displaystyle 0>\frac{d}{d\varepsilon}\mathcal{H}_{\varepsilon}$
$\displaystyle=\ddot{v}_{\varepsilon}(0)\frac{d}{d\varepsilon}\ddot{v}_{\varepsilon}(0)-\frac{n^{2}(n-4)^{2}}{16}\varepsilon+\frac{n(n-4)(n^{2}-4)}{16}\varepsilon^{\frac{n+4}{n-4}}$
(43)
$\displaystyle=\ddot{v}_{\varepsilon}(0)\ddot{w}_{0}^{-}(0)-\frac{n(n-4)}{16}(n(n-4)\varepsilon-(n^{2}-4)\varepsilon^{\frac{n+4}{n-4}}).$

This implies that

$A_{\varepsilon}(0)=-\frac{d}{d\varepsilon}\mathcal{H}_{\varepsilon}>0,$

and so $\omega$ is nondegenerate. ∎

######  Corollary 17.

Let $g\in\mathcal{M}_{\Lambda}$ be nondegenerate. Then there exists an open
neighborhood $\mathcal{U}$ of $g$ in $\mathcal{M}_{\Lambda}$ that embeds into
$\mathcal{W}_{g}$ as a Lagrangian submanifold, with respect to the symplectic
form given by (36).

###### Proof.

As in the proof of Theorem 3, we can identify the bounded null space
$\mathcal{B}_{g}$ as the tangent space $T_{g}\mathcal{M}_{\Lambda}$. In
particular, this identification shows $L_{g}(v)=0$ for each
$v\in\mathcal{B}_{g}$. On the other hand, the linear decomposition lemma
allows us to identify $\mathcal{B}_{g}$ as a $k$-dimensional subspace of
$\mathcal{W}_{g}$. The corollary now follows. ∎

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