Asymptotic Gluing of Shear-Free Hyperboloidal Initial Data Sets

Abstract

We present a procedure for asymptotic gluing of hyperboloidal initial data sets for the Einstein field equations that preserves the shear-free condition. Our construction is modeled on the gluing construction in Isenberg et al. (Ann Henri Poincaré 11(5):881–927, 2010), but with significant modifications that incorporate the shear-free condition. We rely on the special Hölder spaces, and the corresponding theory for elliptic operators on weakly asymptotically hyperbolic manifolds, introduced by the authors in Allen et al. (Commun Anal Geom 26(1):1–61, 2018) and applied to the Einstein constraint equations in Allen et al. (Class Quantum Grav 33(11):115015, 2016).

Appendix A. Uniform Invertibility for Elliptic Operators

In this appendix, we study the vector Laplace operator $L_{\lambda _\varepsilon }$ defined in (1.5) and the linearized Lichnerowicz operator

$$\begin{aligned} {\mathcal {P}}_\varepsilon [1] = \Delta _{\lambda _\varepsilon } - \frac{1}{8}\left( {{\,\mathrm{R}\,}}[\lambda _\varepsilon ] + 7 |\sigma _\varepsilon |^2_{\lambda _\varepsilon } + 30 \right) \end{aligned}$$

(A.1)

given by (7.14) in the special case $\theta =1$. We obtain uniform invertibility of these operators in the following sense.

Proposition A.1

Let $\lambda _\varepsilon$ be the metrics constructed in (5.1). For each $\delta \in [0,3)$, there exists a constant $C_\delta$, independent of $\varepsilon$, such that:

• (a) $L_{\lambda _\varepsilon }:C^{k,\alpha }_{\delta }(M_\varepsilon )\rightarrow C^{k-2,\alpha }_{\delta }(M_\varepsilon )$ is invertible with

  $$\begin{aligned} \Vert X\Vert _{C^{k,\alpha }_{\delta }(M_\varepsilon ;{\tilde{\rho }}_\varepsilon )} \le C_\delta \Vert L_{\lambda _\varepsilon } X\Vert _{C^{k-2,\alpha }_{\delta }(M_\varepsilon ;{\tilde{\rho }}_\varepsilon )} \end{aligned}$$

  for all vector fields $X\in C^{k,\alpha }_{\delta }(M_\varepsilon )$,

• (b) ${\mathcal {P}}_\varepsilon [1]:C^{k,\alpha }_{\delta }(M_\varepsilon )\rightarrow C^{k-2,\alpha }_{\delta }(M_\varepsilon )$ is invertible with

  $$\begin{aligned} \Vert u \Vert _{C^{k,\alpha }_{\delta }(M_\varepsilon ;{\tilde{\rho }}_\varepsilon )} \le C_\delta \Vert {\mathcal {P}}_\varepsilon [1] u \Vert _{C^{k-2,\alpha }_{\delta }(M_\varepsilon ;{\tilde{\rho }}_\varepsilon )} \end{aligned}$$

  for all functions $u\in C^{k,\alpha }_{\delta }(M_\varepsilon )$.

Theorem 1.6 of [2] implies that $L_{\lambda _\varepsilon }$ and ${\mathcal {P}}_\varepsilon [1]$ are Fredholm of index zero; see also [11]. Thus, Proposition A.1 is an immediate consequence of the elliptic regularity estimates in Proposition 5.5 and of the following lemma, which shows the operators have trivial kernel in the relevant spaces, and is proven in Section A.3.

Lemma A.2

For each $\delta \in [0,3)$ there exists $C_\delta$, independent of $\varepsilon$, such that:

• (a) $\Vert X\Vert _{C^0_\delta (M_\varepsilon ;{\tilde{\rho }}_\varepsilon )} \le C_\delta \Vert L_{\lambda _\varepsilon } X\Vert _{C^0_\delta (M_\varepsilon ;{\tilde{\rho }}_\varepsilon )}$ for all $X\in C^{2,\alpha }_\delta (M_\varepsilon )$,

• (b) $\Vert u \Vert _{C^0_\delta (M_\varepsilon ;{\tilde{\rho }}_\varepsilon )} \le C_\delta \Vert {\mathcal {P}}_\varepsilon [1] u \Vert _{C^0_\delta (M_\varepsilon ;{\tilde{\rho }}_\varepsilon )}$ for all $u\in C^{2,\alpha }_\delta (M_\varepsilon )$.

Prior to proving Lemma A.2, we introduce a general framework for blowup analysis and establish some results concerning the kernels of model operators.

1.1 A.1 Exhaustions of Weighted Riemannian Manifolds

Let $(M _*, g _*)$ be a Riemannian manifold. We say that a sequence of Riemannian manifolds $({\varvec{M}}_{\varvec{j}}, {\varvec{g}}_{\varvec{j}})$ forms an exhaustion of $({\varvec{M}} _*, {\varvec{g}} _*)$ if

• $M_j$ are non-empty precompact open subsets of $M _*$,

• $M_1\subseteq \overline{M}_1\subseteq M_2\subseteq \overline{M}_2 \subseteq M_3 \subseteq \cdots$ ,

• $\bigcup _{j=1}^\infty M_j=M _*$, and

• $\Vert g_j-g _*\Vert _{C^2(K, g_*)}\rightarrow 0$ on each precompact set $K\subset M _*$.

If in addition we have continuous functions $w_j:M_j\rightarrow (0,\infty )$ and $w _*:M _*\rightarrow (0,\infty )$ such that $\Vert w_j-w_*\Vert _{C^0(K)}\rightarrow 0$ on each precompact set $K\subset M _*$, then we say that $({\varvec{M}}_{\varvec{j}}, {\varvec{g}}_{\varvec{j}}, {\varvec{w}}_{\varvec{j}})$ forms an exhaustion of $({\varvec{M}} _*, {\varvec{g}} _*, {\varvec{w}} _*)$.

We now give a definition of convergence for linear differential operators. Let $(M_j, g_j)$ be an exhaustion of $(M _*, g _*)$. Consider a second-order linear differential operator $P_*$ acting on sections of some tensor bundle over $M_*$, and operators $P_j$ acting on the restriction of that bundle to $M_j$. We write $P_j=A_j\nabla ^2+B_j\nabla +C_j$ where $A_j, B_j, C_j$ are appropriate bundle homomorphisms and where $\nabla$ is the connection associated with $g_j$. Similarly, we write $P_*=A _*\nabla ^2+B _*\nabla +C _*$. We say that ${\varvec{P}}_{\varvec{j}}$ converges to ${\varvec{P}}_*$, and write $P_j\rightarrow P_*$, if

$$\begin{aligned} \Vert A_j-A _* \Vert _{C^2(K, g_*)}+ \Vert B_j-B _* \Vert _{C^1(K, g_*)}+ \Vert C_j-C _* \Vert _{C^0(K, g_*)}\rightarrow 0 \end{aligned}$$

on each precompact $K\subset M _*$.

Clearly, if $P_j\rightarrow P _*$, then for each precompact K and each smooth tensor field $\eta$ we have

$$\begin{aligned} \Vert P_j \eta -P _* \eta \Vert _{C^0(K, g_*)}\rightarrow 0 \quad \text { and }\quad \Vert P_j^\dagger \eta -P _*^\dagger \eta \Vert _{C^0(K, g_*)}\rightarrow 0, \end{aligned}$$

where $P_j^\dagger$ and $P_*^\dagger$ denote the formal adjoints of $P_j$ and $P_*$, respectively. If in addition the operators $P_j$ and $P _*$ are elliptic, then the constants in the interior elliptic regularity estimates can be chosen independently of (sufficiently large) j. Finally, the reader should notice that if $(M_j, g_j)$ is an exhaustion of $(M _*, g _*)$, then any family of second-order geometric operators $P_j=P[g_j]$ and $P _*=P[g _*]$ satisfies $P_j\rightarrow P _*$.

Proposition A.3

Let $(M_j, g_j, w_j)$ be an exhaustion of $(M _*, g _*, w _*)$, and let $P_j$ and $P _*$ be second-order elliptic linear differential operators on $(M_j, g_j)$ and $(M _*, g _*)$ with $P_j\rightarrow P _*$. Suppose also that there exists points $q_j\in M_1$ converging to $q_*\in M_1$ with respect to $g_*$, a sequence of tensor fields $u_j\in C^2(M_j)$, and constants $c,C>0$ such that:

• (a) for all j we have $\big (w_j^{-1}|u_j|_{g_j}\big )\Big |_{q_j}\ge c$;

• (b) for all j we have $\displaystyle {\sup _{M_j}}\, w_j^{-1}|u_j|_{g_j}\le C$;

• (c) we have $\displaystyle {\sup _{M_j}}\, w_j^{-1}|P_ju_j|_{g_j}\rightarrow 0$ as $j\rightarrow \infty$.

Then, there is a nonzero tensor field $u_*\in C^{0}(M_*, g_*)$ and a subsequence $\{u_{j_n}\}$ such that

• $u_{j_n}\rightarrow u_*$ uniformly on compact sets;

• $\displaystyle {\sup _{M}}\, w _*^{-1}|u_*|_{g _*}<\infty$;

• $P_* u_*=0$ in the weak sense.

Proof

Fix $p>\dim (M_*)$ so that the Sobolev space $H^{1,p}(M_j, g_*)$ embeds continuously into $C^0(M_j, g_*)$ for each j.

We now describe a process for extracting a subsequence of $\{u_j\}$ that we use iteratively in order to produce the desired subsequence via a diagonal argument. Given the sequence $\{u_j\}$ and the sets $M_1\subset {\overline{M}}_1\subset M_2$, we extract a subsequence $u_{j_n,1}$ as follows. Our assumptions imply that for sufficiently large j we have

$$\begin{aligned} |u_j|_{g _j}\le 2Cw_*, \quad |P_j u_j|_{g_j}\le Cw_* \quad \text { on }M_2. \end{aligned}$$

As the volumes ${{\,\mathrm{vol}\,}}_{g_j}(M_2)$ are uniformly bounded for $j>2$, we have that the Sobolev norms $\Vert u_j\Vert _{H^{0,p}(M_2,g_j)}$ and $\Vert P_ju_j\Vert _{H^{0,p}(M_2,g_j)}$ are bounded uniformly. Since the assumption $P_j\rightarrow P_*$ implies

$$\begin{aligned} \Vert u_j\Vert _{H^{2,p}(M_1,g_j)}\le C'\left( \Vert P_ju_j\Vert _{H^{0,p}(M_2,g_j)}+\Vert u_j\Vert _{H^{0,p}(M_2,g_j)}\right) \end{aligned}$$

for some constant $C'$ independent of j, we have that $\Vert u_j\Vert _{H^{2,p}(M_1,g_j)}$ are bounded, and thus so are $\Vert u_j\Vert _{H^{2,p}(M_1,g_*)}$. Applying Rellich’s lemma yields a subsequence $\{u_{j_n,1}\}$ that converges in $H^{1,p}(M_1, g_*)$ to some function $u_1$. Since p has been chosen such that $H^{1,p}(M_1,g_*)\subset C^0(M_1)$, it follows that we have uniform pointwise convergence

$$\begin{aligned} u_{j_n, 1}\rightarrow u_{1}\quad \text { in } C^0(M_1,g_*). \end{aligned}$$

Furthermore, assumptions (a) and (b) imply that

$$\begin{aligned} |u_{1}(q_*)|_{g_*}\ge \frac{c}{2}, \quad |u_{1}|_{g_*}\le 2C w_* \quad \text { on } M_1. \end{aligned}$$

The process that produces the subsequence $\{u_{j_n,1}\}$ from the sequence $\{u_j\}$ and the sets $M_1 \subset {\overline{M}}_1 \subset M_2$ is now applied iteratively. For example, applying this process to the sequence $\{u_{j_n, 1}\}$ and the sets $M_2 \subset \overline{M}_2 \subset M_3$ gives rise to the subsequence $\{ u_{j_n, 2}\}$ of $\{u_{j_n,1}\}$ that converges in $C^0(M_2, g_*)$ to some limit $u_2$. Since $u_{j_n,1} \rightarrow u_1$ in $C^0(M_1, g_*)$, we see that the function $u_2$ is a continuous extension of $u_1$ to the domain $M_2$. Furthermore, we have that

$$\begin{aligned} |u_{2}(q_*)|_{g_*}\ge \frac{c}{2}, \quad |u_{2}|_{g_*}\le 2C w_* \quad \text { on } M_2. \end{aligned}$$

Repeating this process inductively, we obtain subsequences $\{u_{j_n,l}\}$ of $\{u_j\}$ and limiting functions $u_l\in C^0(M_l)$ such that

$$\begin{aligned} u_{j_n, l} \rightarrow u_l \quad \text { in }C^0(M_l, g_*) \end{aligned}$$

as $n\rightarrow \infty$. Consequently, the diagonal sequence $\{u_{j_n,n}\}$ is uniformly convergent on every compact subset of $M_*$ to a limit $u_*\in C^0(M_*, g_*)$. Furthermore, we have

$$\begin{aligned} |u_{*}(q_*)|_{g_*}\ge \frac{c}{2}, \quad \text { and }\quad |u_*|_{g_*}\le 2C w_* \quad \text { on }M_*. \end{aligned}$$

For the remainder of the proof, we denote the subsequence $\{u_{j_n, n}\}$ by $\{u_{j_n}\}$.

We now show that $P_*u_*=0$ weakly. Consider a smooth tensor field $\eta$ supported on some $\Omega \subseteq \overline{\Omega }\subseteq M_*$, where ${\overline{\Omega }}$ is compact. Since $P_{j_n}\rightarrow P_*$ and $g_{j_n}\rightarrow g_*$, we have

$$\begin{aligned} \left| \int _{M_*} \langle P_*^\dagger \eta , u_*\rangle _{g_*} \mathrm{d}V_{g_*}\right|&=\lim _{n\rightarrow \infty } \left| \int _{\Omega } \langle P_{j_n}^\dagger \eta , u_{j_n}\rangle _{g_{j_n}} \mathrm{d}V_{g_{j_n}}\right| \\&\le \lim _{n\rightarrow \infty } \int _{\Omega } \left| \langle \eta , P_{j_n} u_{j_n}\rangle _{g_{j_n}}\right| \mathrm{d}V_{g_{j_n}} \\&\le \Vert \eta \Vert _{C^0(\Omega , g_*)} \mathrm {Vol}_{g_*}(\Omega ) \cdot \lim _{n\rightarrow \infty } \Vert P_{j_n}u_{j_n}\Vert _{C^0(\Omega , g_{j_n})}. \end{aligned}$$

It follows from our assumptions that $\sup _\Omega w_{j_n}^{-1}|P_{j_n}u_{j_n}|_{g_{j_n}}\rightarrow 0$. As the functions $w_{j_n}$ converge uniformly to the positive function $w_*$ on the precompact set $\Omega$, they are uniformly bounded from above and below on $\Omega$. Thus, $\Vert P_{j_n}u_{j_n}\Vert _{C^0(\Omega , g_{j_n})}\rightarrow 0$ and hence

$$\begin{aligned} \int _{M_*} \langle P_*^\dagger \eta , u_*\rangle _{g_*} \mathrm{d}V_{g_*}=0. \end{aligned}$$

Therefore, $P_* u_*=0$ weakly. $\square$

1.2 A.2 Invertibility of Model Operators

Our blowup analysis uses the mapping properties of elliptic geometric operators defined using one of two model CMCSF hyperboloidal initial data sets: the data assumed in the main theorem, given by $(g,\Sigma )$ on M, which serves as a model away from the gluing region, and the data given by $(\breve{g},0)$ on ${\mathbb {H}}^3$, which serves as a model in the gluing region. In the first case, our aim is to establish the injectivity of the vector Laplace operator $L_g$ and of the operator ${\mathcal {P}}_0$ given by

$$\begin{aligned} {\mathcal {P}}_0u&= \Delta _gu -\frac{1}{8}\left( {{\,\mathrm{R}\,}}[g] + 7|\Sigma |^2_g + 30\right) u\\&= \Delta _g u - (3+|\Sigma |^2_g), \end{aligned}$$

where we have used (1.2). The operator $\mathcal P_0$ serves as a model for the linearization $\mathcal P_\varepsilon [1]$ of the Lichnerowicz operator about the function 1; see (7.14). In the second case, we establish injectivity of the analogous operators defined by $(\breve{g}, 0)$.

First, we consider the case of the data assumed in the main theorem.

Proposition A.4

Let $(g,\Sigma )$ be initial data on M as in Theorem 1.1 and suppose $|1-\delta |<2$.

• (a) If a continuous vector field X on M satisfies $|X|_g\le C\rho ^\delta$ for some constant C and if $L_gX =0$, then $X=0$.

• (b) If a continuous function u on M satisfies $|u|\le C\rho ^\delta$ for some constant C and if ${\mathcal {P}}_0u=0$, then $u=0$.

Proof

For the first claim, we note that $L_g$ is an elliptic geometric operator. Thus, from the elliptic regularity results in [2, Lemma 5.1] we have $X\in C^{k,\alpha }_\delta (M)$. From Proposition 6.3 of [3], we have that

$$\begin{aligned} L_g :C^{k,\alpha }_\delta (M) \rightarrow C^{k-2,\alpha }_\delta (M) \end{aligned}$$

is invertible, and thus $X=0$.

For the second claim, we note that $\Delta _g -3$ is an elliptic geometric operator. Since $|\Sigma |^2_g\in C^{k-1,\alpha }_2(M)$, adding $- |\Sigma |^2_g u$ to the lower-order term does not affect the arguments leading to elliptic regularity results for $\mathcal P_0$; see [11, Lemma 4.8] and [2, Lemma 5.1]. (Note that the sign convention for the Laplacian $\Delta _g$ in [11] is the opposite of the one used here.) Thus, $u\in C^{k,\alpha }_\delta (M)$. Since Proposition 6.5 of [3] implies that

$$\begin{aligned} {\mathcal {P}}_0 :C^{k,\alpha }_\delta (M) \rightarrow C^{k-2,\alpha }_\delta (M) \end{aligned}$$

is invertible, we conclude that $u=0$. $\square$

We now turn to the model of hyperbolic space. As in section 2.1, we use the coordinates (x, y) on the half-space model of hyperbolic space and write $r^2 = |x|^2 + y^2$. Recall also the function $\breve{\rho }$ defined in (2.6) and the function F described in Proposition 2.2. It is established in [11, Theorem 5.9] that any self-adjoint elliptic geometric operator $\breve{P}$ on hyperbolic space is an isomorphism

$$\begin{aligned} \breve{P}:C^{k,\alpha }_\delta ({\mathbb {H}}^3) \rightarrow C^{k-2,\alpha }_\delta ({\mathbb {H}}^3) \end{aligned}$$

provided $|\delta -1|<R$, where R is the indicial radius of the operator $\breve{P}$. In particular, this applies to the vector Laplace operator $L_{\breve{g}}$ and to the operator $\Delta _{\breve{g}} - 3$ for $|\delta -1|<2$.

The isomorphism property of $\breve{P}$, together with interior elliptic regularity, implies that any continuous tensor field $v\in \ker \breve{P}$ with $|v|_{\breve{g}}\le C \breve{\rho }^\delta$ must in fact vanish. In our blowup analysis, we require a slight strengthening of this statement that makes use of the functions y and yF on the half-space model of hyperbolic space. The argument we present is a generalization of the proof of Proposition 13 and Corollary 14 in [10].

Proposition A.5

Let $\breve{P}$ be a self-adjoint elliptic geometric operator on ${\mathbb {H}}$ with indicial radius $R>0$. Suppose that for some $\delta$ satisfying $|1-\delta |<R$, there exists $v\in \ker \breve{P}$ satisfying either $|v|_{\breve{g}} \le C(yF)^\delta$ or $|v|_{\breve{g}} \le Cy^\delta$ . Then, v is identically zero.

Proof

We argue by contradiction and consider first the case that there exists a nonzero tensor field $v\in \ker \breve{P}$ and constants $C,\delta \in {\mathbb {R}}$ such that $|\delta -1|<R$ and $|v|_{\breve{g}}\le C (yF)^\delta$. Let $r_0>0$ be such that v does not vanish identically on the set where $r<r_0$. From Proposition 2.2, we have

$$\begin{aligned}&|v|_{\breve{g}} \le C y^\delta \quad \text { where }r\ge r_0, \end{aligned}$$

(A.2)

$$\begin{aligned}&|v|_{\breve{g}} \le C \frac{y^\delta }{r^{2\delta }} \quad \text { where }r\le 3r_0, \end{aligned}$$

(A.3)

where here, and in the following, the value of C may vary from line to line.

Let $\varphi _0:{\mathbb {R}}^2 \rightarrow [0,1]$ be a smooth cutoff function supported on $|x|\le 2r_0$, where $x = (x^1,x^2)$ are Cartesian coordinates on ${\mathbb {R}}^2$, and define ${\tilde{v}}$ on ${\mathbb {H}}$ by

$$\begin{aligned} \tilde{v}(x,y)=\int _{{\mathbb {R}}^2} v(x-\xi ,y)\varphi _0(\xi )\,\mathrm{d}\xi . \end{aligned}$$

Since v does not vanish on $r<2r_0$ by assumption, one can always choose $\varphi _0$ so that $\tilde{v}$ is not identically zero. Differentiation under the integral sign shows that $\breve{P} \tilde{v}=0$.

We claim that

$$\begin{aligned} |\tilde{v}|_{\breve{g}}\le C\left( y^{2-\delta }+y^{\delta }\right) . \end{aligned}$$

(A.4)

On the region where $r\ge 3r_0$, the estimate (A.2) implies (A.4) and thus we focus attention on the region where $r\le 3r_0$. There, (A.3) implies that

$$\begin{aligned} |\tilde{v}|_{\breve{g}}\le C \int _{|\xi |\le 2r_0} \frac{y^{\delta }}{(y^2+|x-\xi |^2)^{\delta }} \mathrm{d}\xi . \end{aligned}$$

We now use the change of variables $\xi =x-y\zeta$ and observe that $r\le 3r_0$ and $|\xi |\le 2r_0$ implies $|\zeta |\le \tfrac{5r_0}{y}$. Thus, using polar coordinates yields

$$\begin{aligned} \begin{aligned} |\tilde{v}|_{\breve{g}}&\le C y^{2-\delta }\int _{|x-y\zeta |\le 2r_0}\frac{\mathrm{d}\zeta }{(1+|\zeta |^2)^\delta } \\&\le C y^{2-\delta }\int _0^{5r_0/y} \frac{t}{(1+t^2)^\delta }\,\mathrm{d}t. \end{aligned} \end{aligned}$$

It follows from

$$\begin{aligned} \frac{1}{(1+t^2)^\delta }\le {\left\{ \begin{array}{ll} C &{} \text { for }t\le 1,\\ C t^{-2\delta } &{} \text { for }t\ge 1 \end{array}\right. } \end{aligned}$$

that

$$\begin{aligned} \int _0^{5r_0/y} \frac{t}{(1+t^2)^\delta }\,\mathrm{d}t\le C( 1+y^{2\delta -2}). \end{aligned}$$

This completes the proof of (A.4).

We now define $u={\mathcal {I}}^*\tilde{v}$, where ${\mathcal {I}}$ is the inversion operator defined in (2.3). Note that $\breve{P}u=0$ due to ${\mathcal {I}}$-invariance of $\breve{g}$, and consequently the ${\mathcal {I}}$-invariance of $\breve{P}$. Choose $r_1>0$ so that $u\ne 0$ on $r\le r_1$. Choose also a smooth function $\varphi _1 :{\mathbb {R}}^2\rightarrow [0,1]$ supported on $|x|<2r_1$ such that the tensor field

$$\begin{aligned} \tilde{u}(x,y)=\int _{{\mathbb {R}}^2} u(x-\xi ,y)\varphi _1(\xi )\,\mathrm{d}\xi \end{aligned}$$

is nonzero. Differentiation under the integral sign shows that $\breve{P}\tilde{u}=0$.

We claim that

$$\begin{aligned} |{\widetilde{u}}|_{\breve{g}}\lesssim \breve{\rho }^{2-\delta }+\breve{\rho }^\delta . \end{aligned}$$

(A.5)

To this end, observe that (A.4) implies

$$\begin{aligned} |u|_{\breve{g}}\lesssim \left( \frac{y}{r^2}\right) ^{2-\delta }+\left( \frac{y}{r^2}\right) ^{\delta }. \end{aligned}$$

The same change of variables argument involved in the proof of (A.4) shows that

$$\begin{aligned} |{\widetilde{u}}|_{\breve{g}} \le C y^{2-\delta }+y^{\delta } \quad \text { on }\quad r\le 3r_1. \end{aligned}$$

In the region where $r\le 3r_1$, we have $C^{-1} y\le \breve{\rho }\le C y$, which implies the estimate (A.5) in that region.

In the region where $r\ge 3r_1$ and $|\xi |\le 2r_1$, we have

$$\begin{aligned} C^{-1}r^2 \le y^2+|x-\xi |^2 Cr^2. \end{aligned}$$

The estimate (A.5) now follows from the fact that $C^{-1}\breve{\rho }\le \frac{y}{r^2}\le C \breve{\rho }$ where $r\ge 3r_1$.

Since $\breve{\rho }\le C$, the estimate (A.5) implies that $|{\tilde{u}}|_{\breve{g}} \le C \breve{\rho }^\nu$ where $\nu = \min {(2-\delta , \delta )}$. Thus, ${\tilde{u}}\in \ker \breve{P}$ and ${\tilde{u}}\in C^0_\nu ({\mathbb {H}}^3)$, where $|1-\nu |<R$. The isomorphism property of $\breve{P}$ implies that ${\tilde{u}} =0$, which is the desired contradiction.

Suppose now that $\breve{P}v=0$ and that $|v|_{\breve{g}} \le C y^\delta$. If $\delta <0$, then the fact that $y\ge C\breve{\rho }$ implies $|v|_{\breve{g}} \le C \breve{\rho }^\delta$ and hence, the isomorphism property of $\breve{P}$ implies $v=0$. If $\delta \ge 0$, then the fact that $F^\delta \ge C$ implies that $|v|_{\breve{g}} \le C (yF)^\delta$ and thus $v=0$ by the previous argument. $\square$

1.3 A.3 Proof of Lemma A.2

We now establish Lemma A.2. We present the argument for the estimate

$$\begin{aligned} \Vert u\Vert _{C^0_\delta (M_\varepsilon ;{\tilde{\rho }}_\varepsilon )}\lesssim \Vert \mathcal P_\varepsilon [1]u\Vert _{C^0_\delta (M_\varepsilon ;{\tilde{\rho }}_\varepsilon )}; \end{aligned}$$

(A.6)

the estimate for the vector Laplace operator follows from analogous reasoning.

We argue by contradiction and assume that (A.6) does not hold. Thus, there exists $\delta \in [0,3)$ and a sequence $\varepsilon _j\rightarrow 0$, together with functions $u_j\in C^{2,\alpha }_\delta (M_{\varepsilon _j})$, such that

$$\begin{aligned} \Vert u_j\Vert _{C^0_\delta (M_{\varepsilon _j};{\tilde{\rho }}_{\varepsilon _j})} =1 \end{aligned}$$

(A.7)

and

$$\begin{aligned} \Vert \mathcal P_{\varepsilon _j}[1]u_j\Vert _{C^0_\delta (M_{\varepsilon _j};{\tilde{\rho }}_{\varepsilon _j})} \rightarrow 0. \end{aligned}$$

(A.8)

Hence, there exist points $q_j\in M\setminus (U_{1,\varepsilon _j} \cup U_{2,\varepsilon _j})$, where we recall (2.10), such that at the point $\pi _{\varepsilon _j}(q_j)\in M_{\varepsilon _j}$ we have

$$\begin{aligned} \big ({\tilde{\rho }}_{\varepsilon _j}^{-\delta } |u_j|\big )\Big |_{\pi _{\varepsilon _j}(q_j)} \ge \frac{1}{2} . \end{aligned}$$

(A.9)

Passing to a subsequence if necessary, we may assume that $q_j \rightarrow q\in {\overline{M}}$. We now consider several cases, depending on the location of $q\in {\overline{M}}$, obtaining a contradiction in each case.

1.3.1 Case 1: $q\in M$

In this case we define $(M_j, g_j, w_j)$ by setting

$$\begin{aligned} M_j = \{p\in M\setminus (U_{1,\varepsilon _j} \cup U_{2,\varepsilon _j}):\rho _{\varepsilon _j}(\pi _{\varepsilon _j}(p))> \varepsilon _j\}, \end{aligned}$$

$g_j = \pi _{\varepsilon _j}^*\lambda _{\varepsilon _j}$, and $w_j = \pi _{\varepsilon _j}^*{\tilde{\rho }}_{\varepsilon _j}^\delta = \pi _{\varepsilon _j}^*({\rho _{\varepsilon _j}}/{\omega _{\varepsilon _j}} )^\delta ;$ see (4.7).

Let $\psi :(0,\infty ) \rightarrow (0,1]$ be the smooth cutoff function used in (4.5), and define the function $\omega _*:{\overline{M}}\rightarrow (0,\infty )$ by setting $\omega =1$ outside the domain of the preferred background coordinates $\Theta _i = (\theta , \rho )$ and by requiring that $\omega _* = \psi (|(\theta ,\rho )|)$ in each background coordinate chart. Set $w_* = (\rho /\omega _*)^\delta$ on M.

We claim that $(M_j, g_j, w_j)$ forms an exhaustion of $(M, g, w_*)$. The convergence of the metrics is immediate from the fact that $\iota _\varepsilon ^*\lambda _\varepsilon = g$ on $E_c$; see Proposition 5.6. To see the convergence of the weight functions, we recall from Sect. 4.3 that in preferred background coordinates $\Theta _i = (\theta , \rho )$ we have $\pi _\varepsilon ^*\omega _\varepsilon = \psi (|(\theta ,\rho )| + \varepsilon ^2|(\theta ,\rho )|^{-1})$. Thus, $\pi _{\varepsilon _j}^*\omega _{\varepsilon _j} \rightarrow \omega$ uniformly on every precompact subset of M. As $\pi _\varepsilon ^*\rho _\varepsilon = \rho$ on $E_c$, we see that $w_j\rightarrow w_*$ uniformly on precompact sets as well.

Let $v_j = \pi _{\varepsilon _j}^* u_j$. As $q_*\in M_j$ for sufficiently large j, we may pass to a subsequence to ensure that $q_j\in M_1$ for all j, and hence that $q_*\in M_1$. From (A.9) and (A.7), we have

$$\begin{aligned} \big (w_j^{-1} |v_j|\big )\Big |_{q_j}\ge \frac{1}{2} \quad \text { and }\quad \sup _{M_j} w_j^{-1}|u_j|\le 1. \end{aligned}$$

Furthermore, setting $P_j = \pi _{\varepsilon _j}^*\mathcal P_{\varepsilon _j}[1]$, we have $\sup _{M_j} w_j^{-1}|P_j v_j| \rightarrow 0$.

The convergence of $\iota _\varepsilon ^*|\sigma _\varepsilon |^2_{\lambda _\varepsilon }$ to $|\Sigma |^2_g$ in the exterior region given by (6.11) implies that ${\mathcal {P}}_{\varepsilon _j}[1]\rightarrow {\mathcal {P}}_0$ as described in §A.1. Thus, from Proposition A.3, there exists a nonzero function $v_*\in C^0(M)$ such that $|v_*|\le C (\rho /\omega _*)^\delta$ and $\mathcal P_0 v_* = 0$. Note that $\rho \le C \omega _* \le C.$ Thus, since $\delta \ge 0$, we have $|v_*|\le C$. But Proposition A.4 implies that the only continuous and bounded function in the kernel of ${\mathcal {P}}_0$ is the zero function, contradicting that $v_*$ is nonzero.

1.3.2 Case 2: $q\in \partial {\overline{M}} \setminus \{p_1, p_2\}$

Let $\Theta = (\theta ,\rho )$ be background coordinates on M centered at q as introduced in Sect. 2.2. After an affine change of coordinates, we can arrange that at q we have ${{\bar{g}}}_{ij}\mathrm{d}\Theta ^i\mathrm{d}\Theta ^j = \delta _{ij}\mathrm{d}\Theta ^i\mathrm{d}\Theta ^j$. For j sufficiently large, $q_j$ is contained in the domain Z(q) of $\Theta$; denote $\Theta (q_j)$ by $({\hat{\theta }}_j, {\hat{\rho }}_j)$. Let $r_*>0$ be such that neither $p_1$ nor $p_2$ is contained in that part of Z(q) where $|(\theta ,\rho )|\le r_*$. Without loss of generality, we may assume that $|{\hat{\theta }}_j|< r_*/2$.

Set $M_j = \{ (x,y)\in {\mathbb {H}} :|(x,y)| < r_*/4{\hat{\rho }}_j, y>{\hat{\rho }}_j/2\}$ and use the background coordinates $(\theta ,\rho )$ about q to define $\Phi _j:M_j \rightarrow M$ by

$$\begin{aligned} \Phi _j:(x,y) \mapsto (\theta ,\rho ) = ({\hat{\theta }}_j + {\hat{\rho }}_j x, {\hat{\rho }}_j y). \end{aligned}$$

Note that $\Phi _j(0,1) = q_j$ and that for sufficiently small c (and hence, in view of (4.2), sufficiently small $\varepsilon _j$) the image of $\Phi _j$ is contained in the exterior region $E_c$. Thus, we may define $T_j :M_j \rightarrow M_{\varepsilon _j}$ by $T_j = \iota _{\varepsilon _j}\circ \Phi _j$.

Set $g_j = T_j^*\lambda _{\varepsilon _j}$. Let ${\hat{\omega }}_j = \omega _{\varepsilon _j}(\iota _{\varepsilon _j}(q_j))$ and, recalling from (4.7) that ${\tilde{\rho }}_\varepsilon = \rho _\varepsilon /\omega _\varepsilon$, define

$$\begin{aligned} w_j = {\tilde{\rho }}_{\varepsilon _j}(\iota _{\varepsilon _j}(q_j))^{-\delta } T_j^*{\tilde{\rho }}_{\varepsilon _j}^\delta = \left( \frac{{\hat{\rho }}_j}{{\hat{\omega }}_j}\right) ^{-\delta }T_j^*{\tilde{\rho }}_{\varepsilon _j}^\delta . \end{aligned}$$

(A.10)

We claim that $(M_j, g_j, w_j)$ forms an exhaustion of $({\mathbb {H}}, \breve{g}, y^\delta )$. Since ${\hat{\rho }}_j\rightarrow 0$, we have $\bigcup _{j} M_j = {\mathbb {H}}$. To see that $g_j\rightarrow \breve{g}$, recall from Proposition 5.6 that $\iota _\varepsilon ^*\lambda _\varepsilon = g$ on $E_c$, and thus, $g_j$ is simply the pullback of g by $\Phi _j$. In coordinates $\Theta = (\theta ,\rho )$, we write $g = \rho ^{-2} {\overline{g}}_{ij}(\theta ,\rho ) \mathrm{d}\Theta ^i \mathrm{d}\Theta ^j$. Thus, in coordinates $X = (x,y)$ we have $g_j = \Phi _j^* g= y^{-2} {\overline{g}}_{ij} ({\hat{\theta }}_j + {\hat{\rho }}_jx, {\hat{\rho }}_j y) \mathrm{d}X^i \mathrm{d}X^j$. Thus, on any precompact set $K\subset {\mathbb {H}}$ we have $g_j \rightarrow y^{-2} {\overline{g}}_{ij}(q)\mathrm{d}X^i\mathrm{d}X^j = y^{-2} g_\text {E} = \breve{g}$ uniformly. Finally, to see the convergence of the weight function, note that ${\hat{\rho }}_j^{-1}T_j^*\rho _{\varepsilon _j} = y$ and that on any precompact $K\subset {\mathbb {H}}$ we have ${\hat{\omega }}_j^{-1} T_j^*\omega _{\varepsilon _j} \rightarrow 1$ uniformly. Thus, the claim is verified.

Define the functions $v_j$ on $M_j$ by $v_j = ({\hat{\rho }}_j/{\hat{\omega }}_j)^{-\delta }T_j^*u_j$. Using (A.10) we see that $w_j^{-1} |v_j| = T_j^* ({\tilde{\rho }}_{\varepsilon _j}^{-\delta } |u_j|)$. Thus, from (A.9) and (A.7) we have

$$\begin{aligned} \big (w_j^{-1}|v_j| \big )\Big |_{(0,1)} \ge \frac{1}{2} \quad \text { and }\quad \sup _{M_j} w_j^{-1} |v_j| \le 1; \end{aligned}$$

hence, assumptions (a) and (b) of Proposition A.3 hold.

Define the differential operator $P_j$ on $M_j$ by $P_j = T_j^*{\mathcal {P}}_{\varepsilon _j}[1]$. We claim that $P_j\rightarrow \Delta _{\breve{g}} - 3$. To see this, note first that since $\iota _\varepsilon ^*\lambda _\varepsilon = g$ in the exterior region $E_c$, applying the constraint equations (1.2) to (A.1) yields

$$\begin{aligned} \begin{aligned} \iota _\varepsilon ^*{\mathcal {P}}_{\varepsilon _j}[1]&= \Delta _g - 3 - \frac{1}{8}|\Sigma |^2_g - \frac{7}{8}\iota _\varepsilon ^*|\sigma _\varepsilon |^2_{\lambda _\varepsilon } \\&= \Delta _{g} - 3 - |\Sigma |^2_{g} -\frac{7}{8}\left( \iota _{\varepsilon _j}^*|\sigma _{\varepsilon _j}|^2_{\lambda _{\varepsilon _j}} - |\Sigma |^2_g\right) . \end{aligned} \end{aligned}$$

(A.11)

By the hypotheses of Theorem 1.1, we have that $|{\overline{\Sigma }}|^2_{{\overline{g}}}$ is bounded, and thus for some constant C we have $\Phi _j^*|\Sigma |^2_{g} =\Phi _j^*(\rho ^2 |{\overline{\Sigma }}|^2_{{\overline{g}}})\le C({\hat{\rho }}_j y)^2$, which tends to zero uniformly on any precompact set. Furthermore, from (6.11), we have

$$\begin{aligned} \Phi _j^*\left| \iota _{\varepsilon _j}^*|\sigma _{\varepsilon _j}|^2_{\lambda _{\varepsilon _j}} - |\Sigma |^2_g \right| \le C ({\hat{\rho }}_j y)^2 \varepsilon _j, \end{aligned}$$

which also tends to zero. Finally, since $g_j = \Phi _j^*g\rightarrow \breve{g}$ we have $\Delta _g\rightarrow \Delta _{\breve{g}}$, which establishes the claim.

From (A.8), we have $w_j^{-1}|P_j v_j| \rightarrow 0$. Applying Proposition A.3, we obtain a continuous, nonzero function $v_*$ on ${\mathbb {H}}$ such that $|v_*| \le C y^\delta$ and $\Delta _{\breve{g}} v_* - 3v_*=0$. This, however, contradicts Proposition A.5.

1.3.3 Case 3: $q\in \{p_1, p_2\}$

In this case, we may assume, without loss of generality, that $q = p_1$ and, by passing to a subsequence if necessary, that the points $q_j$ are contained in the domain $Z(p_1)$ of the preferred background coordinates $(\theta ,\rho )$ centered about $p_1$. Let $({\hat{\theta }}_j,{\hat{\rho }}_j)$ be the background coordinates of $q_j$. Since $q_j\in M\setminus (U_{1,\varepsilon _j} \cup U_{2,\varepsilon _j})$ we have $|({\hat{\theta }}_j,{\hat{\rho }}_j)| \ge \varepsilon _j$. Since $|({\hat{\theta }}_j,{\hat{\rho }}_j)| \rightarrow 0$, we may assume that $|{\hat{\theta }}_j|+{\hat{\rho }}_j<1/8$.

Below, we consider three sub-cases, depending on the nature of the convergence $({\hat{\theta }}_j,{\hat{\rho }}_j)\rightarrow (0,0)$. In each case, we define nested precompact subsets $M_j \subset {\mathbb {H}}$ and maps $T_j:M_j \rightarrow M_{\varepsilon _j}$. We arrange $T_j$ so that the preferred background coordinate expression for $T_j(x,y)$ satisfies

$$\begin{aligned} 8\varepsilon _j^2< |T_j(x,y)| < \frac{1}{8}, \end{aligned}$$

(A.12)

which ensures that $g_j = T_j^*\lambda _{\varepsilon _j}$ is well defined. We then show that $(M_j, g_j)$ forms an exhaustion of $({\mathbb {H}}, \breve{g})$, and that $P_j = T_j^*P_{\varepsilon _j}[1]\rightarrow \Delta _{\breve{g}} - 3$. Finally, in each case we construct a sequence of functions $v_j$ and weights $w_j$ satisfying the hypotheses of Proposition A.3. We thus obtain a nonzero limiting function $v_*$, from which we obtain a contradiction via Proposition A.5.

Case 3(a): Both $|{\hat{\theta }}_j|/{\hat{\rho }}_j$ and $|({\hat{\theta }}_j,{\hat{\rho }}_j)|/\varepsilon _j$ are bounded above Thus, there exists $C>1$ such that $|{\hat{\theta }}_j|\le C{\hat{\rho }}_j$ and $|({\hat{\theta }}_j,{\hat{\rho }}_j)|\le C\varepsilon _j$ for all j. Thus, ${\hat{\rho }}_j \le |({\hat{\theta }}_j,{\hat{\rho }}_j)| \le |{\hat{\theta }}_j| + {\hat{\rho }}_j \le 2C{\hat{\rho }}_j$. Furthermore, since $\varepsilon _j\le |({\hat{\theta }}_j,{\hat{\rho }}_j)|$, we have $\varepsilon _j \le 2C \hat{\rho }_j$ and ${\hat{\rho }}_j \le |({\hat{\theta }}_j, {\hat{\rho }}_j)| \le C \varepsilon _j$. Combining these yields

$$\begin{aligned} \frac{1}{2C}{\hat{\rho }}_j \le |({\hat{\theta }}_j,{\hat{\rho }}_j)| \le 2C {\hat{\rho }}_j \quad \text { and }\quad \frac{1}{2C}\varepsilon _j\le {\hat{\rho }}_j \le 2C\varepsilon _j. \end{aligned}$$

(A.13)

Let

$$\begin{aligned} M_j = \left\{ (x,y)\in {\mathbb {H}} :|(x,y)|< \frac{1}{8\varepsilon _j}, y>8\varepsilon _j \right\} \end{aligned}$$

and define $T_j:M_j \rightarrow M_{\varepsilon _j}$ by setting $T_j(x,y) = \Psi _{\varepsilon _j}(x,y) = (\varepsilon _j x, \varepsilon _j y)$. For each $(x,y)\in M_j$ we have, for sufficiently large j, that

$$\begin{aligned} 8\varepsilon _j^2<\varepsilon _j y \le |(\varepsilon _j x, \varepsilon _j y)| < \frac{1}{8} \end{aligned}$$

and thus (A.12) holds. Let

$$\begin{aligned} w = \left( \frac{yF}{2(r+ 1/r)}\right) ^\delta . \end{aligned}$$

Thus, from (4.3) and (4.6), we have $w = T_j^*{\tilde{\rho }}_{\varepsilon _j}^\delta$. From Proposition 5.6, we have $g_j = \Psi _{\varepsilon _j}^*\lambda _{\varepsilon _j} \rightarrow \breve{g}$ on precompact sets of ${\mathbb {H}}$, and thus $(M_j, g_j, w)$ is an exhaustion of $({\mathbb {H}}, \breve{g}, w)$. Furthermore, it follows from (6.12) that $T_j^* |\sigma _{\varepsilon _j}|^2_{\lambda _{\varepsilon _j}} \rightarrow 0$ uniformly on precompact subsets of ${\mathbb {H}}$; hence $P_j = T_j^* \mathcal P_{\varepsilon _j}[1] \rightarrow \Delta _{\breve{g}} - 3$.

Let $({{\hat{x}}}_j,{{\hat{y}}}_j) = ({\hat{\theta }}_j/\varepsilon _j, \hat{\rho }_j/\varepsilon _j) \in M_j$ so that $T_j ({{\hat{x}}}_j, {{\hat{y}}}_j) = ({\hat{\theta }}_j, {\hat{\rho }}_j)$. From (A.13), the sequence $({{\hat{x}}}_j, {{\hat{y}}}_j)$ is bounded, and ${{\hat{y}}}_j$ is bounded away from zero. Thus, by passing to a subsequence we have $({{\hat{x}}}_j, {{\hat{y}}}_j) \rightarrow ({{\hat{x}}}_*, {{\hat{y}}}_*)$ with ${{\hat{y}}}_*>0$.

Set $v_j = T_j^* u_j$. By assumption we have

$$\begin{aligned} \big (w^{-1}|v_j|\big )\Big |_{({{\hat{x}}}_j, {{\hat{y}}}_j)} = \big ({\tilde{\rho }}_{\varepsilon _j}^{-\delta } |u_j|\big )\Big |_{\pi _{\varepsilon _j}(q_j)} \ge \frac{1}{2} \end{aligned}$$

and

$$\begin{aligned} \sup _{M_j} w^{-1}|v_j| =\sup _{M_j} T_j^*\big ({\tilde{\rho }}_{\varepsilon _j}^{-\delta } |u_j|\big ) \le 1. \end{aligned}$$

Furthermore,

$$\begin{aligned} \sup _{M_j}w^{-1}|P_j v_j| = \sup _{M_j}T_j^*\big ({\tilde{\rho }}_{\varepsilon _j}^{-\delta } |P_{\varepsilon _j}[1]u_j|\big ) \le \Vert \mathcal P_{\varepsilon _j}[1]u_j\Vert _{C^0_\delta (M_{\varepsilon _j};{\tilde{\rho }}_{\varepsilon _j})}\rightarrow 0. \end{aligned}$$

Thus, the hypotheses of Proposition A.3 are satisfied, and there exists a nonzero function $v_*$ on ${\mathbb {H}}$ with $\Delta _{\breve{g}}v_* - 3v_* =0$ and

$$\begin{aligned} |v_*| \le w = \left( \frac{yF}{2(r+ 1/r)}\right) ^\delta . \end{aligned}$$

As we are assuming $\delta \ge 0$, this implies $|v_*| \le (yF)^\delta$ and the desired contradiction is obtained from Proposition A.5.

Case 3(b): $|{\hat{\theta }}_j|/{\hat{\rho }}_j$ is bounded above, But $|({\hat{\theta }}_j,{\hat{\rho }}_j)|/\varepsilon _j$ is not bounded above

In this case we may, after passing to a subsequence if necessary, suppose that $|{\hat{\theta }}_j|\le C{\hat{\rho }}_j$ for some constant $C>1$ and that ${{\hat{r}}}_j = |({\hat{\theta }}_j,{\hat{\rho }}_j)|/\varepsilon _j \rightarrow \infty$. Thus,

$$\begin{aligned} {\hat{\rho }}_j \le \varepsilon _j {{\hat{r}}}_j =|({\hat{\theta }}_j,{\hat{\rho }}_j)| \le 2C {\hat{\rho }}_j \quad \text { and }\quad \frac{{\hat{\rho }}_j}{\varepsilon _j}\rightarrow \infty . \end{aligned}$$

(A.14)

Let

$$\begin{aligned} M_j = \left\{ (x,y) \in {\mathbb {H}} :|(x,y)|<\frac{1}{16C{\hat{\rho }}_j}, y> \frac{8\varepsilon _j}{{\hat{\rho }}_j} \right\} . \end{aligned}$$

For sufficiently large j, we have $\varepsilon _j / {\hat{\rho }}_j < 8C$ and thus $M_j\subset A_{\varepsilon _j}$. Hence, we may define $T_j :M_j \rightarrow M_{\varepsilon _j}$ by $T_j(x,y) = \Psi _{\varepsilon _j}({{\hat{r}}}_j x, {{\hat{r}}}_j y)$. In preferred background coordinates $(\theta ,\rho )$ about $p_1$ we have $T_j(x,y) = \left( \varepsilon _j{{\hat{r}}}_jx, \varepsilon _j{{\hat{r}}}_jy\right)$ and thus from (A.14) we see that

$$\begin{aligned} 8\varepsilon _j^2< 8 \varepsilon _j< {\hat{\rho }}_j y \le \varepsilon _j{{\hat{r}}}_j y \le |T_j(x,y)| \le \varepsilon _j{{\hat{r}}}_j |(x,y)| < \frac{1}{8}; \end{aligned}$$

thus (A.12) holds.

Let $({{\hat{x}}}_j, {{\hat{y}}}_j) = (\varepsilon _j {{\hat{r}}}_j)^{-1} ({\hat{\theta }}_j, {\hat{\rho }}_j)$ so that $T_j({{\hat{x}}}_j, {{\hat{y}}}_j) = \pi _{\varepsilon _j}(q_j)$. By construction we have $|({{\hat{x}}}_j, {{\hat{y}}}_j)| = 1$ and it follows from (A.14) that ${{\hat{y}}}_j \ge 1/2C$. Thus we may pass to a subsequence such that $({{\hat{x}}}_j, {{\hat{y}}}_j) \rightarrow ({{\hat{x}}}_*, {{\hat{y}}}_*)$ with $|({{\hat{x}}}_*, {{\hat{y}}}_*)| = 1$ and $y_*>0$.

Setting $g_j = T_j^*\lambda _{\varepsilon _j}$ and $w_j = T_j^*{\tilde{\rho }}_{\varepsilon _j}^{\delta }$, we claim that $(M_j, g_j, w_j)$ forms an exhaustion of $({\mathbb {H}}, \breve{g}, (y/2r)^\delta )$, where as usual we write $r = |(x,y)|$. To see this, first note that Proposition 5.6 implies that $\Psi _{\varepsilon _j}^*\lambda _{\varepsilon _j} \rightarrow \breve{g}$ uniformly on precompact sets. Dilation by ${{\hat{r}}}_j$ is an isometry of hyperbolic space that preserves unweighted norms; see (2.15). Thus $g_j\rightarrow \breve{g}$ uniformly on precompact sets. Next observe from (4.7), while using (4.3) and (4.6), that

$$\begin{aligned} T_j^*{\tilde{\rho }}_{\varepsilon _j} = \frac{y F({{\hat{r}}}_j r)}{2\left( r + {1}/{{{\hat{r}}}_j^2 r}\right) }. \end{aligned}$$

The hypotheses that define this case include ${{\hat{r}}}_j \rightarrow \infty$, while Proposition 2.2 states that $F({{\hat{r}}}_j r) = 1$ if ${{\hat{r}}}_j r >2$. Thus, we see that $T_j^*{\tilde{\rho }}_{\varepsilon _j} \rightarrow y/2r$ uniformly on precompact subsets of ${\mathbb {H}}$ and the claim is established.

Set $v_j = T_j^* u_j$ and $P_j = T_j^*{\mathcal {P}}_{\varepsilon _j}[1]$. We now verify that the hypotheses of Proposition A.3 are satisfied. The assumption (A.9) implies that

$$\begin{aligned} \big (w_j^{-1} |v_j|\big )\Big |_{({{\hat{x}}}_j, {{\hat{y}}}_j)} \ge \frac{1}{2} \end{aligned}$$

and thus hypothesis (a) holds. As the assumptions (A.7) and (A.8) imply that hypotheses (b) and (c) hold, it remains to establish the convergence of the operators $P_j$. Proposition 6.8 implies that $T_j^* |\sigma _{\varepsilon _j}|^2_{\lambda _{\varepsilon _j}}\rightarrow 0$ uniformly on precompact subsets of ${\mathbb {H}}$. Thus, the convergence $g_j \rightarrow \breve{g}$ implies that $P_j \rightarrow \Delta _{\breve{g}} - 3$.

We now invoke Proposition A.3 to conclude that there exists a nonzero continuous function $v_*$ on ${\mathbb {H}}$ such that $|v_*|\le C(y/r)^\delta \le Cy^0$ and $\Delta _{\breve{g}} v_* - 3 v_* =0$. Consequently, Proposition A.5 yields a contradiction.

1.3.4 Case 3(c): $|{\hat{\theta }}_j|/{\hat{\rho }}_j$ is not Bounded Above

Passing to a subsequence we may assume that $|{\hat{\theta }}_j|/{\hat{\rho }}_j \rightarrow \infty$. We may further assume that ${\hat{\rho }}_j / |{\hat{\theta }}_j| < 1/2$; when combined with the fact that $\varepsilon _j \le |({\hat{\theta }}_j, {\hat{\rho }}_j)|$, we find that $|{\hat{\theta }}_j|\ge \varepsilon _j/2$.

Let $({{\hat{x}}}_j, {{\hat{y}}}_j) = \varepsilon _j^{-1}({\hat{\theta }}_j, {\hat{\rho }}_j)$ so that $\Psi _{\varepsilon _j}({{\hat{x}}}_j, {{\hat{y}}}_j) = \pi _{\varepsilon _j}(q_j)$. Set

$$\begin{aligned} M_j = \left\{ (x,y)\in {\mathbb {H}} :|(x,y)|< \frac{|{\hat{\theta }}_j|}{2{\hat{\rho }}_j}, y > \varepsilon _j \right\} . \end{aligned}$$

For $(x,y)\in M_j$ we may use $|{\hat{\theta }}_j| < 1/8$ to conclude that

$$\begin{aligned} |({{\hat{x}}}_j + {{\hat{y}}}_j x, {{\hat{y}}}_j y)| \le \frac{|{\hat{\theta }}_j|}{\varepsilon _j} + \frac{{\hat{\rho }}_j}{\varepsilon _j}|(x,y)|< 2\frac{|{\hat{\theta }}_j|}{\varepsilon _j} <\frac{1}{8\varepsilon _j} \end{aligned}$$

(A.15)

and use ${\hat{\rho }}_j |x| < |{\hat{\theta }}_j|/2$ to obtain

$$\begin{aligned} |({{\hat{x}}}_j + {{\hat{y}}}_j x, {{\hat{y}}}_j y)| \ge |{{\hat{x}}}_j + {{\hat{y}}}_j x| = \frac{1}{\varepsilon _j} |{\hat{\theta }}_j + {\hat{\rho }}_j x| \ge \frac{|{\hat{\theta }}_j|}{2\varepsilon _j}> \frac{1}{4} > 8 \varepsilon _j. \end{aligned}$$

(A.16)

Thus, the map $\Phi _j :M_j \rightarrow A_{8\varepsilon _j}$ given by $\Phi _j (x,y) = ({{\hat{x}}}_j + {{\hat{y}}}_j x, {{\hat{y}}}_j y)$ is well defined, and the map $T_j= \Psi _{\varepsilon _j}\circ \Phi _j:M_j \rightarrow M_{\varepsilon _j}$ satisfies (A.12). In preferred background coordinates $(\theta ,\rho )$ about $p_1$, we have $T_j (x,y) = (\varepsilon _j{{\hat{x}}}_j + \varepsilon _j{{\hat{y}}}_j x, \varepsilon _j{{\hat{y}}}_j y) = ({\hat{\theta }}_j + {\hat{\rho }}_j x, {\hat{\rho }}_j y)$ and thus $T_j(0,1) = \pi _{\varepsilon _j}(q_j)$.

We now estimate $T_j^*{\tilde{\rho }}_{\varepsilon _j}$ using (4.8), which implies that

$$\begin{aligned} T_j^*{\tilde{\rho }}_{\varepsilon _j} = \frac{{{\hat{y}}}_j y F(|\Phi _j(x,y)|)}{2\left( |\Phi _j(x,y)| + \frac{1}{|\Phi _j(x,y)|} \right) }. \end{aligned}$$

The estimate (A.16) implies that $|\Phi _j(x,y)| > 1/4$ and thus from Proposition 2.2 we have $F(|\Phi _j(x,y)|)$ uniformly bounded above and below. The lower bound $|\Phi _j(x,y)| > 1/4$ furthermore implies that

$$\begin{aligned} |\Phi _j(x,y)| \le |\Phi _j(x,y)| + \frac{1}{|\Phi _j(x,y)|} \le 17 |\Phi _j(x,y)|. \end{aligned}$$

As (A.15) and (A.16) imply that

$$\begin{aligned} \frac{|{\hat{\theta }}_j|}{2\varepsilon _j} \le |\Phi _j(x,y)| \le 2\frac{|{\hat{\theta }}_j|}{\varepsilon _j}, \end{aligned}$$

and as ${{\hat{y}}}_j = {\hat{\rho }}_j / \varepsilon _j$, we find that

$$\begin{aligned} \frac{1}{C }\frac{{\hat{\rho }}_j}{|{\hat{\theta }}_j|} y \le T_j^* {\tilde{\rho }}_{\varepsilon _j} \le C \frac{{\hat{\rho }}_j}{|{\hat{\theta }}_j|} y \end{aligned}$$

(A.17)

for some constant C.

Let $g_j = T_j^*\lambda _{\varepsilon _j}$ and set $w_j = y^\delta$. From Proposition 5.6, we see that $g_j \rightarrow \breve{g}$ on precompact sets and thus $(M_j, g_j, w_j)$ forms an exhaustion of $({\mathbb {H}}, \breve{g}, y^\delta )$. We seek to apply Proposition A.3 to the functions

$$\begin{aligned} v_j = \left( \frac{{\hat{\rho }}_j}{|{\hat{\theta }}_j|}\right) ^{-\delta } T_j^* u_j \end{aligned}$$

and operators $P_j = T_j^*{\mathcal {P}}_{\varepsilon _j}[1]$. Proposition 6.8 implies that $T_j^* |\sigma _{\varepsilon _j}|^2_{\lambda _{\varepsilon _j}}\rightarrow 0$ uniformly on precompact subsets of ${\mathbb {H}}$. Thus, the convergence $g_j \rightarrow \breve{g}$ implies that $P_j \rightarrow \Delta _{\breve{g}} - 3$. Thus by applying (A.17) to (A.7), (A.8), and (A.9) we have that the hypotheses of Proposition A.3 are satisfied. The result is a nonzero function $v_*$ satisfying both $|v_*| \le Cy^\delta$ and $\Delta _{\breve{g}} v - 3v=0$. This, however, is in contradiction to Proposition A.5.

With all cases exhausted, the proof of Lemma A.2 is complete.