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

\begin{equation*}\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}}, \end{equation*}

where δ 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 $u = U u_{\rm sph}$. 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,

\begin{equation*}\limsup_{|x| \rightarrow \infty} |x|^{\frac{n-4}{2}} u(x) \lt \infty.\end{equation*}

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

\begin{equation*} \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.\end{equation*}

Thus

\begin{equation*}\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 \}. \end{equation*}

In the spherical setting, the linearized operator has the form

(8)\begin{equation} 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, \end{equation}

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

\begin{equation*}\mathcal{H}_\delta (u) = (-\Delta_0)^2 u - \frac{n(n-4)(n^2-4)}{16} u^{\frac{n+4}{n-4}}=0, \end{equation*}

which in turn implies

\begin{align*}\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 } \right.\\ &\qquad\left. \limsup_{|x| \rightarrow \infty} |x|^{\frac{n-4}{2}} u(x) \lt \infty \right \}. \end{align*}

In the Euclidean setting the linearized operator has the form

(9)\begin{equation} L_g(v) = (-\Delta_0)^2 v - \frac{n(n+4)(n^2-4)}{16} u^{\frac{8}{n-4}} v. \end{equation}

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 $u = U u_{\rm sph}$. Using (4) we see that $u: \mathbf{R}^n \backslash \{0 \} \rightarrow (0,\infty)$ satisfies

(10)\begin{equation} \mathcal H_\delta(u)=0. \end{equation}

Frank and König [Reference Frank and König5] classified all the solutions of (10), and we describe them here. First we perform the Emden-Fowler change of coordinates, defining

(11)\begin{align} \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). \nonumber\\ \end{align}

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

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

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

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

we see

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

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

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

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

(12)\begin{equation} \mathcal H_{\rm cyl} (v)=P_{\rm cyl} (v) - \frac{n(n-4)(n^2-4)}{16} v^{\frac{n+4}{n-4}}=0, \end{equation}

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

(13)\begin{equation} P_{\rm cyl} = (-\Delta_{\rm cyl})^2 - \frac{n(n-4)}{2} \Delta_{\rm cyl} - 4 \partial_t^2 + \frac{n^2(n-4)^2}{16} \qquad\qquad\qquad\qquad\qquad\end{equation}

\begin{align*} &= \partial_t^4 + \Delta_{\mathbf{S}^{n-1}}^2+ 2\Delta_{\mathbf{S}^{n-1}} \partial_t^2 -\frac{n(n-4)+8}{2} \partial_t^2 - \frac{n(n-4)}{2} \Delta_{\mathbf{S}^{n-1}}\qquad \quad\ \\ & + \frac{n^2(n-4)^2}{16}\ \qquad \end{align*}

is the Paneitz operator of the cylindrical metric. Note that $\Delta_{\rm cyl} = \partial_t^2 + \Delta_{\mathbf{S}^{n-1}}$. C. S. Lin [Reference Lin9] used a moving planes argument to prove that solutions of (10) are rotationally invariant, reducing (12) to the ODE

(14)\begin{equation} \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. \end{equation}

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

(15)\begin{equation} \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}}. \end{equation}

We denote the nonzero constant solution of (14) by

\begin{equation*}\overline{\varepsilon} = \left ( \frac{n(n-4)}{n^2-4} \right )^{\frac{n-4}{8}}. \end{equation*}

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 ɛ.

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

(16)\begin{equation} 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}. \end{equation}

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_ɛ 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 β > 1 so that

(17)\begin{align} 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 )\right.\nonumber\\ &\quad +\mathcal{O}(|x|^\beta) \bigg). \end{align}

This expansion combines the local asymptotic expansions in [Reference Jin and Xiong7] and in [Reference Ratzkin12]. 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

(18)\begin{equation} 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}), \end{equation}

where $T= -\log R$.

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

(19)\begin{equation} \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), \end{equation}

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

(20)\begin{equation} u(x) \simeq \mathfrak{F}^{-1} (v_{\varepsilon_i} (-\log |x-p_i| + T_i)) \qquad \textrm{near }p_i. \end{equation}

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

\begin{equation*}\| v\|^2_{L^2_\delta} = \int_0^\infty \int_{\mathbf{S}^{n-1}} e^{-2\delta t} |v(t,\theta)|^2 d\theta dt \lt \infty.\end{equation*}

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 $|v(t,\theta)| \leq C e^{\widetilde{\delta} t}$ for each $\widetilde{\delta} \lt \delta$ and t > 0, 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

\begin{eqnarray*} \int_{\mathbf{S}^{n-1}} \int_{t_1}^{t_2} e^{-2\delta t} |v(t,\theta)|^2 dt d\theta & = & -\int_{\mathbf{S}^{n-1}} \int_{e^{-t_1}}^{e^{-t_2}} r^{2\delta+n-5} |u(r\theta)|^2 dr d\theta \\ & = & \int_{r_2 \leq |x| \leq r_1} |x|^{2\delta - 4} |u(x)|^2 d\mu_0(x), \end{eqnarray*}

where $r_1 = e^{-t_1}$ and $r_2 = e^{-t_2}$. Here $d\mu_0$ is the Euclidean volume element. 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

\begin{equation*}\| u \|^2_{L^2_\delta} = \int_{\mathbf{B}_r(0) \backslash \{0 \}} |x|^{2\delta - 4} |u(x)|^2 d\mu_0 \lt \infty.\end{equation*}

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

\begin{equation*}\| u\|_{L^2_\delta} = \int_{\mathbf{R}^n \backslash \widetilde{\Lambda}} (\operatorname{dist}(x,\widetilde{\Lambda}))^{2\delta - 4} |u(x)|^2 d\mu_0 \lt \infty.\end{equation*}

Once again, we see that if $|u(x)| \leq C(\operatorname{dist} (x,\widetilde{\Lambda}))^{2-\widetilde{\delta}}$ near each singularity, for each $\widetilde{\delta} \lt \delta + \frac{n}{2}$, and $|u(x)|\leq C|x|^{\frac{4-n}{2}-\lambda}$ for $|x|$ sufficiently large and any $\lambda \gt \delta$, then $u \in L^2_\delta(\mathbf{R}^n \backslash \widetilde{\Lambda})$.

3.1. The linearization about a Delaunay solution

Here we study the linearized operator about a Delaunay solution, which we denote as L_ɛ, and some of its mapping properties.

Following Section 5.2 of [Reference Andrade, Caju, do O, Ratzkin and Silva Santos1] we write

\begin{equation*}L_\varepsilon = (-\Delta_0)^2 - \frac{n(n+4)(n^2-4)}{16} u_\varepsilon^{\frac{8}{n-4}} \end{equation*}

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

(21)\begin{equation} \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). \end{equation}

Some computation give us

(22)\begin{equation} \begin{aligned} \mathcal{L}_\varepsilon & = P_{\rm cyl} - \frac{n(n+4)(n^2-4)}{16} v_\varepsilon^{\frac{8}{n-4}}\\ & = \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}}\\ &\qquad- \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}}. \end{aligned}\end{equation}

Here P _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

(23)\begin{equation} w_0^+(\varepsilon) = \dot v_{\varepsilon} , \qquad w_0^-(\varepsilon) = \frac{d}{d\varepsilon}v_\varepsilon. \end{equation}

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 $\varepsilon \lt \overline{\varepsilon}$, but both Jacobi fields vanish in the cylindrical case. If $\varepsilon = \overline{\varepsilon}$ we define

(24)\begin{align} 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}.\nonumber\\ \end{align}

The analysis in Proposition 1 of [Reference Baraket and Rebhi4] 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 [Reference Ratzkin12].

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

\begin{equation*}m_j = \binom{n-1+j}{j} + \binom{n-3+j}{j-2},\end{equation*}

and so we can expand w in Fourier series as

\begin{equation*}w(t,\theta) = \sum_{j=0}^\infty \sum_{l=1}^{m_j} w_{j,l} (t) E_{j,l}(\theta),\end{equation*}

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

\begin{equation*}\operatorname{Span} \{E_{j,1}, \dots, E_{j, m_j}\}\end{equation*}

is the ordinary differential operator

\begin{align*}\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\\ &\quad - \frac{n(n+4)(n^2-4)}{16} v_\varepsilon^{\frac{8}{n-4}}.\end{align*}

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 [Reference Ratzkin12, Proposition 28].

Proposition 9. There exists a discrete set of real numbers

(25)\begin{equation} \Gamma_\varepsilon = \{\dots, -\gamma_2(\varepsilon) \lt -\gamma_1(\varepsilon) \lt 0 \lt \gamma_1(\varepsilon) \lt \gamma_2(\varepsilon), \dots \}, \end{equation}

with $\gamma_j(\varepsilon) \rightarrow \infty$ as $j \rightarrow \infty$ such that the operator

\begin{equation*}\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}) \end{equation*}

is Fredholm provided $\delta \not \in \Gamma_\varepsilon$. In particular, for any $\delta \in (0,\gamma_1(\varepsilon))$

\begin{equation*}\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}) \end{equation*}

is injective and

\begin{equation*}\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})\end{equation*}

is surjective.

One calls $\gamma_j(\varepsilon)$ the jth indicial root of the Jacobi operator $\mathcal{L}_\varepsilon$ and $\Gamma_\varepsilon$ the set of indicial roots associated to the Delaunay solution v_ɛ.

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 ɛ.

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 ɛ_j, and define the indicial set

\begin{equation*}\Gamma_g = \bigcup_{i=1}^k \Gamma_{\varepsilon_i}.\end{equation*}

It follows directly from Proposition 9 that

\begin{equation*}L_g : W^{4,2}_\delta (\mathbf{R}^n \backslash \widetilde{\Lambda}) \rightarrow L^2_{\delta-4} (\mathbf{R}^n \backslash \widetilde{\Lambda})\end{equation*}

is Fredholm if and only if $\delta \not \in \Gamma_g$.

Definition 12. Let $g\in \mathcal{M}_{\Lambda}$ and choose $r_0 \gt 0$ 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

\begin{equation*}\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\},\end{equation*}

where χ is a fixed cut-off function such that

\begin{equation*}\chi(x) = \left \{\begin{array}{rl} 1 & |x| \lt r_0 \\ 0 & |x| \gt 3r_0/2 \end{array} \right . , \qquad \|\nabla^k \chi \|_{\mathcal{C}^0} \leq c r^{-k}. \end{equation*}

We now define the bounded null space. Once again we fix a number δ such that $0 \lt \delta \lt \min_{1\leq i \leq k} \gamma_1(\varepsilon_i)$. Each element of the bounded null space is, strictly speaking, an equivalence class of functions, that is

\begin{equation*}\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})}.\end{equation*}

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

\begin{equation*}\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.\end{equation*}

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.

The proof below is more or less the same as the proof of Theorem 4.24 in [Reference Mazzeo, Pollack and Uhlenbeck10].

Proof. We compute the relative index of L_g acting on the appropriate weighted function spaces. Recall that the index of

\begin{equation*}L_g : W^{4,2}_\delta (\mathbf{R}^n \backslash \widetilde{\Lambda}) \rightarrow W^{0,2}_{\delta-4} (\mathbf{R}^n \backslash \widetilde{\Lambda})\end{equation*}

is

\begin{equation*}\operatorname{ind} (\delta) = \dim (\ker(L_g)) - \dim (\operatorname{coker}(L_g)). \end{equation*}

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

\begin{equation*}\operatorname{ind} (-\delta) = - \operatorname{ind}(\delta) \end{equation*}

provided $\delta \not \in \Gamma_g$. (In this case, reversing the sign of the weight δ exchanges the kernel and the cokernel.) Next recall that, provided $\delta_1, \delta_2 \not \in \Gamma_g$, the relative index is defined as

\begin{equation*}\operatorname{rel-ind}(\delta_1, \delta_2) = \operatorname{ind}(\delta_1) - \operatorname{ind}(\delta_2).\end{equation*}

We use duality once more (i.e. the operator L_g is formally self-adjoint in L ²) to see so long as $0 \lt \delta \lt \min_{1\leq i \leq k} \gamma_1(\varepsilon_i)$ we have

(26)\begin{eqnarray} \operatorname{rel-ind} (\delta, -\delta) & = & \operatorname{ind}(\delta) - \operatorname{ind} (-\delta) = 2 \operatorname{ind}(\delta) \end{eqnarray}

\begin{eqnarray*} & = & 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. \end{eqnarray*}

Thus it suffices to show that $\operatorname{rel-ind} (\delta, -\delta) = 2k$ for an appropriate choice of δ. We choose $0 \lt \delta \lt \min_{1 \leq i \leq k} \gamma_1(\varepsilon_i)$.

We compute this relative index theorem using the Melrose’s relative index theorem (see Theorem 6.5 of [Reference Melrose11]). We first decompose

\begin{equation*}\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 ).\end{equation*}

We can now write L_g as the sum of restrictions

\begin{equation*}L_g = \left. L_g \right |_{\Omega^c} + \sum_{i=1}^k \left. L_g \right |_{\mathbf{B}_r(p_i) \backslash \{p_i\}} \end{equation*}

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

\begin{equation*}\partial \Omega^c = \bigcup_{i=1}^k \partial \mathbf{B}_r(p_i),\end{equation*}

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

\begin{equation*}\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}).\end{equation*}

This is where we use Melrose’s machinery, as developed in Chapters 4,5 and 6 of [Reference Melrose11]. To do so we introduce the Fourier-Laplace transform

(27)\begin{equation} \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)\end{equation}

and the twisted operator

\begin{equation*}\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})\end{equation*}

defined by

(28)\begin{equation} \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).\end{equation}

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 ζ is in the half-space $\{\zeta : \mathrm{Im} (\zeta) \lt -\delta T_{\varepsilon_i}\}$. 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 ζ. This allows us to use the analytic Fredholm theorem (see Section 5.2 of [Reference Melrose11]) to conclude that $\widetilde{\mathcal{L}_{\varepsilon_i}}$ is Fredholm so long as ζ 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 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 [Reference Ratzkin12] and the proof of Proposition 4.15 of [Reference Mazzeo, Pollack and Uhlenbeck10]. 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

\begin{eqnarray*} 2 \dim (\mathcal{B}_g) & = & \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})) \\ && - \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})) \\ & = & \sum_{i=1}^k 2 = 2k, \end{eqnarray*}

as we claimed.