# On Carlotto-Schoen-type scalar-curvature gluings111Preprint UWThPh 2016-19

Piotr T. Chruściel, Erwann Delay

###### Abstract

We carry out a Carlotto-Schoen-type gluing with interpolating scalar curvature
on cone-like sets, or deformations thereof, in the category of smooth
Riemannian asymptotically Euclidean metrics.

##  1 Introduction

In an outstanding paper [5] Carlotto and Schoen have constructed non-trivial
asymptotically Euclidean scalar-flat metrics which are Minkowskian outside of
a solid cone. In this note we show how to generalise the construction there in
several directions:

  1. 1.

Rather than gluing an asymptotically Euclidean metric to a flat one, any two
asymptotically Euclidean metrics $g_{1}$ and $g_{2}$ are glued together.

  2. 2.

In the spirit of [11], the gluings at zero-scalar curvature are replaced by
gluings where the scalar curvature of the final metric equals

$\chi R(g_{1})+(1-\chi)R(g_{2})$

with a cut-off function $\chi$.

  3. 3.

We provide a detailed proof with exponential weights near the boundary.

  4. 4.

The geometry of the gluing region is allowed to be more general than the
interface between two cones.

A special case of our main Theorem 3.1 below provides the following variant of
the Riemannian-geometry version of the main theorem of [5]:

######  Theorem 1.1.

Let $n\geq 3$. Given two smooth $n$-dimensional asymptotically Euclidean
Riemannian metrics $g_{1}$ and $g_{2}$ there exist cones and smooth
asymptotically Euclidean metrics $g$ which coincide with $g_{1}$ outside of
the cones and with $g_{2}$ inside slightly smaller cones, see Figure 1.1, with
the scalar curvature $R(g)$ between $R(g_{1})$ and $R(g_{2})$ in the
intermediate region.

Figure 1.1: If the second metric is flat and both metrics are scalar-flat,
then the new metric is flat inside the smaller cone, scalar-flat everywhere,
and coincides with the first one outside the larger cone. Both cones extend to
infinity, and their tips are located very far in the asymptotically Euclidean
region. Right picture: Iterating the construction, one can embed any finite
number of distinct initial data sets into Minkowskian data, or paste-in
Minkowskian data inside several cones into a given data set.

In particular if $R(g_{1})=R(g_{2})=0$, then $R(g)=0$.

In Section 6 we show how Theorem 1.1 follows from our main Theorem 3.1 below.
The key to the proof of this last theorem is a weighted Poincaré inequality
involving radially-scaled exponential weights, proved in Proposition 5.6
below. The rest of the proof is a verification of the hypotheses of the rather
general results proved in [7].

As already mentioned, Theorem 3.1 below allows more general cones than the
ones in [5], as described at the beginning of Section 3. In fact, our
arguments apply to a large class of deformed cones as well, e.g.
“logarithmically-rotated ones”, cf. Theorem 4.1 and the discussion in Section
4.

##  2 Definitions, notations and conventions

We use the summation convention throughout, indices are lowered with $g_{ij}$
and raised with its inverse $g^{ij}$.

We will have to frequently control the asymptotic behavior of the objects at
hand. Given a tensor field $T$ and a function $f$, we will write

$T=O(f),$

when there exists a constant $C$ such that the $g$-norm of $T$ is dominated by
$Cf$.

A metric $g$ on a manifold $M$ will be said to be asymptotically Euclidean
(AE) if $M$ contains a region, diffeomorphic to the complement of a ball in
$\mathbb{R}^{n}$, on which the metric $g$ approaches the Euclidean metric
$\delta$ as one recedes to infinity.

Let $\phi$ and $\psi$ be two smooth strictly positive functions on an
$n$-dimensional manifold $M$. The function $\psi$ will be used to control the
growth of the fields involved near boundaries or in the asymptotic regions,
while $\phi$ will control how the growth is affected by derivations. For
$k\in\mathbb{N}$ let $H^{k}_{\phi,\psi}(g)$ be the space of
$H^{k}_{\textrm{\scriptsize loc}}$ functions or tensor fields such that the
norm222The reader is referred to [2, 1, 12] for a discussion of Sobolev spaces
on Riemannian manifolds.

$\|u\|_{H^{k}_{\phi,\psi}(g)}:=(\int_{M}(\sum_{i=0}^{k}\phi^{2i}|\nabla^{(i)}u|^{2}_{g})\psi^{2}d\mu_{g})^{\frac{1}{2}}$
(2.1)

is finite, where $\nabla^{(i)}$ stands for the tensor
$\underbrace{\nabla...\nabla}_{i\mbox{ \scriptsize times}}u$, with $\nabla$ —
the Levi-Civita covariant derivative of $g$. We assume throughout that the
metric is at least $W^{1,\infty}_{\textrm{\scriptsize loc}}$; higher
differentiability will be usually indicated whenever needed.

For $k\in\mathbb{N}$ we denote by $\mathring{H}^{k}_{\phi,\psi}$ the closure
in $H^{k}_{\phi,\psi}$ of the space of $H^{k}$ functions or tensors which are
compactly (up to a negligible set) supported in $M$, with the norm induced
from $H^{k}_{\phi,\psi}$. The $\mathring{H}^{k}_{\phi,\psi}$’s are Hilbert
spaces with the obvious scalar product associated with the norm (2.1). We will
also use the following notation

$\quad\mathring{H}^{k}:=\mathring{H}^{k}_{1,1}\,,\quad
L^{2}_{\psi}:=\mathring{H}^{0}_{1,\psi}=H^{0}_{1,\psi}\,,$

so that $L^{2}\equiv\mathring{H}^{0}:=\mathring{H}^{0}_{1,1}$.

For $\phi$ and $\varphi$ — smooth strictly positive functions on M, and for
$k\in\mathbb{N}$ and $\alpha\in[0,1]$, we define $C^{k}_{\phi,\varphi}$ to be
the space of $C^{k}$ functions or tensor fields for which the norm

$\begin{array}[]{l}\|u\|_{C^{k}_{\phi,\varphi}(g)}=\sup_{x\in
M}\sum_{i=0}^{k}\|\varphi\phi^{i}\nabla^{(i)}u(x)\|_{g}\end{array}$

is finite.

Let $g$, $g_{0}$ be two Riemannian metrics and let $W$ be a space of two-
covariant symmetric tensors. We will write $g\in M_{g_{0}+W}$ if $g-g_{0}\in
W$. An important example is provided by metrics $g\in
M_{\delta+C^{k+4}_{r,r^{\epsilon}}}$, where $\delta$ denotes the Euclidean
metric. Such metrics are asymptotically Euclidean, with the difference
$g-\delta$ decaying as $O(r^{-\epsilon})$, and with derivatives of order
$1\leq\ell\leq k+4$ decaying as $O(r^{-\epsilon-\ell})$.

##  3 Gluing on sets which are scale-invariant at large distances

Let $S(p,R)\subset\mathbb{R}^{n}$ denote a sphere of radius $R$ centred at
$p$.

Let $\Omega\subset\mathbb{R}^{n}$ be a domain with smooth boundary (thus
$\Omega$ is open and connected, and
$\partial\Omega=\overline{\Omega}\setminus\Omega$ is a smooth manifold). We
further assume that $\partial\Omega$ has _exactly two_ connected components,
and that there exists $R_{0}\geq 1$ such that

$\Omega_{S}:=\Omega\cap S(0,R_{0})$ (3.1)

also has again exactly two connected components, with

$\Omega\setminus B(0,R_{0})=\\{\lambda p\ |\ p\in\Omega_{S}\,,\lambda\geq
1\\}\,.$ (3.2)

When (3.2) holds, we say that $\Omega$ is _invariant under scaling at large
distances_.

We let $x:\overline{\Omega}\to\mathbb{R}$ be any smooth defining function for
$\partial\Omega$ which has been chosen so that

$x(\lambda p)=\lambda x(p)$ for $\lambda\geq 1$ and for $p,\lambda
p\in\Omega\setminus B(0,R_{0})$. (3.3)

Equivalently, for $p\in\Omega_{S}$ and $\lambda$ larger than one, we require
$x(\lambda p)=\lambda x_{S}(p)$, where $x_{S}$ is a defining function for
$\partial\Omega_{S}$ within $S(0,R_{0})$.

By definition of $\Omega$ there exists a constant $c$ such that the distance
function $d(p)$ from a point $p\in\Omega$ to $\partial\Omega$ is smooth for
all $d(p)\leq cr(p)$. The function $x$ can be chosen to be equal to $d$ in
that region.

A simple example, considered in [5], satisfying the above is the following:
let $0<\theta_{1}<\theta_{2}<\pi$ and let $\Omega_{\theta_{1},\theta_{2}}$ any
domain with smooth boundary such that

$\Omega_{\theta_{1},\theta_{2}}\setminus
B(0,R_{0})=(\mathring{C}_{\theta_{2}}\setminus C_{\theta_{1}})\setminus
B(0,R_{0})\,,$ (3.4)

where $C_{\theta}$ is a closed solid cone with aperture $\theta>0$; compare
the left Figure 1.1. Here, and elsewhere, $\mathring{U}$ denotes the interior
of a set $U$. In this example $\Omega_{S}:=\Omega_{\theta_{1},\theta_{2}}\cap
S(0,R_{0})$ is connected, with a boundary which has two connected components.

Some other examples can be found in Figure 3.1

Figure 3.1: Left figure: A possible set $\Omega$, located between the two
surfaces. The metric coincides with $g_{1}$ above the first surface, with
$g_{2}$ below the second one, and is scalar-flat everywhere if both $g_{1}$
and $g_{2}$ were. Right figure: A scale-invariant thickening of the displayed
hypersurface provides a set $\Omega$ so that $\mathbb{R}^{3}\setminus\Omega$
has two components. The metric will coincide with $g_{1}$ in one component,
and with $g_{2}$ in the other.

Figure 3.2: After iterating the gluing a finite number of times (first two
pictures), or an infinite number of times (last picture) one can obtain
aesthetically pleasing sets $\Omega_{S}$. The infinite iteration might require
going to larger and larger distances for each individual gluing. Images by Jos
Leys, with kind permission of the author.

We will denote by $r$ a smooth positive function which coincides with
$|\vec{x}|$ for $|\vec{x}|\geq 1$.

For $\beta,s,\mu\in\mathbb{R}$ we define

$\phi=\left(\frac{x}{r}\right)^{2}r=\frac{x^{2}}{r}\,,\quad\psi={r}^{-n/2-\beta}\left(\frac{x}{r}\right)^{\sigma}e^{-sr/x}=:{r}^{\mu}{x}^{\sigma}e^{-sr/x}$
(3.5)

on $\Omega$. The factor $r^{-\beta}$ is directly related to the large-distance
radial behavior of the solutions, while the exponential factor with $s>0$ will
force the solution to vanish at all orders at $\partial\Omega$ (compare (3.12)
below).

We will make a gluing-by-interpolation of scalar curvature. The main interest
is that of scalar-flat metrics, which then remain scalar-flat, or for metrics
with positive scalar curvature, which then remains positive. Since the current
problem is related to the construction of initial data sets for Einstein
equations, in some general-relativistic matter models, such as Vlasov or dust,
an interpolation of scalar curvature is of direct interest.

In order to carry out the interpolation, recall that $\partial\Omega$ has
exactly two connected components. We denote by $\chi$ a smooth function with
the following properties:

  1. 1.

$0\leq\chi\leq 1$;

  2. 2.

$\chi$ equals one in a neighborhood of one of the components and equals zero
in a neighborhood of the other component;

  3. 3.

on $\Omega\setminus B(0,R_{0})$ the function $\chi$ is required to depend only
upon $x/r$.

Starting with two AE metrics $g$ and $\hat{g}$ we define

$\displaystyle g_{\chi}:=\chi\hat{g}+(1-\chi)g\,,$ (3.6) $\displaystyle
R_{\chi}:=\chi R(\hat{g})+(1-\chi)R(g)\,,\quad\delta
R_{\chi}:=R_{\chi}-R\big{(}g_{\chi}\big{)}\,.$ (3.7)

We denote by $P_{g}$ the linearisation of the scalar-curvature operator at a
metric $g$, and will write $P$ for $P_{g}$ when $g$ is obvious in the context.
Its formal adjoint $P_{g}^{*}$ reads

$P_{g}^{*}(f)=\nabla\nabla f-\Delta fg-f\mbox{\rm Ric}(g)\,,$ (3.8)

where $\Delta=\nabla^{i}\nabla_{i}$.

Letting $\Omega$, $x$ and $r$ be as just described, with $\psi$, $\phi$ given
by (3.5), we have:

######  Theorem 3.1.

Let $\epsilon>0$, $k>n/2$, $\beta\in[-(n-2),0)$, and
${\tilde{\beta}}<\min(\beta,-\epsilon)$. Suppose that $g\in
M_{\delta+C^{k+4}_{r,r^{\epsilon}}}\cap C^{\infty}$. For all real numbers
$\sigma$ and $s>0$ and

all smooth metrics ${\hat{g}{}}$ close enough to $g$ in
$C^{k+4}_{r,r^{-{\tilde{\beta}}}}(\Omega)$

there exists on $\Omega$ a unique smooth two-covariant symmetric tensor field
$h$ of the form

$h=\phi^{4}\psi^{2}P^{*}_{g_{\chi}}(\delta
N)\in\psi^{2}\phi^{2}\mathring{H}^{k+2}_{\phi,\psi}(g)$ (3.9)

such that the metric $g_{\chi}+h$ solves

$R\left[g_{\chi}+h\right]=\chi R(\hat{g})+(1-\chi)R(g)\,.$ (3.10)

The tensor field $h$ vanishes at $\partial\Omega$ and can be
$C^{\infty}$-extended by zero across $\partial\Omega$, leading to a smooth
asymptotically Euclidean metric $g_{\chi}+h$.

######  Remark 3.2.

Some comments about the asymptotic behaviour of the metrics involved are in
order. The requirement $g\in M_{\delta+C^{k+4}_{r,r^{\epsilon}}}\cap
C^{\infty}$ (equivalently, $g-\delta\in{C^{k+4}_{r,r^{\epsilon}}}\cap
C^{\infty}$) guarantees that $g$ is asymptotically Euclidean, with $g$
approaching the Euclidean metric as $O(r^{-\epsilon})$. Note that $\epsilon$
can be arbitrary small. Next, $g-\hat{g}$ is required to fall off as
$r^{\tilde{\beta}}$, so that $g_{\chi}$ will approach $\delta$ as
$O(r^{\min(-\epsilon,{\tilde{\beta}})})$. The final metric $g_{\chi}+h$ will a
priori approach $\delta$ somewhat slower. Indeed, the proof below shows that
there exists a constant $C$ such that

$\displaystyle\|h\|_{\psi^{2}\phi^{2}\mathring{H}^{k+2}_{\phi,\psi}(g_{\chi})}\leq
C\left\|\delta
R_{\chi}\right\|_{\psi^{2}\mathring{H}^{k}_{\phi,\psi}(g_{\chi})}\,.$ (3.11)

(The right-hand side is finite and small only if the weight $-\beta$ is larger
then $\max(-{\tilde{\beta}},\epsilon)$, which explains the hypothesis.)

Equation (3.11) implies (compare (3.24) below)

$h=o(r^{-n-\beta+2}(x/r)^{\sigma+4-n}e^{-sr/x})=o(r^{-n-\beta+2})\,.$ (3.12)

Since $\beta<0$, (3.12) shows that the decay of the metric to the flat-one is
_most likely_ slower than the Schwarzschildean decay rate $O(r^{-(n-2)})$.

######  Remark 3.3.

A well defined ADM energy-momentum requires the metric to approach the flat-
one as $o(r^{-(n-2)/2})$. Comparing with (3.12), this will be satisfied if in
Theorem 3.1 we require instead

$-\frac{n-2}{2}\leq\beta\leq 0\,.$ (3.13)

######  Remark 3.4.

An analogous result holds if $g$ is not necessarily smooth: If $g\in
M_{\delta+C^{k+4}_{r,r^{\epsilon}}}$, then the construction can be modified as
in [8] so that the final metric $g_{\chi}+h$ will be of $C^{k+4}$
differentiability class.

###### Proof.

We start by showing that Theorem 3.6 of [7] with $K\equiv 0\equiv Y$ applies,
with $g_{0}$ taken to be the Euclidean metric. This requires, first, checking
the “scaling property” of the functions $\phi$ and $\psi$. This is a routine
exercise, using scaling in $r$ and [7, Appendix B]. Next, note that with a
flat metric $g_{0}$ Equation (3.8) reads

$P_{g_{0}}^{*}(f)=\nabla\nabla f-\Delta fg_{0}\,.$ (3.14)

Since $\beta\neq 0$, the property [7, Equation (3.4)] needed in Theorem 3.6
follows directly from Proposition 5.6 below used twice, first for $u$ and then
for $\nabla u$. Remark that the integrals involving $u$ in the right-hand side
of (5.22) will vanish if the support of $u$ lies in the region $|\vec{x}|\geq
R$.

Now, the operator $\pi_{{\mathcal{K}}_{0}^{\perp}}$ appearing in the statement
of [7, Theorem 3.6] projects on the space ${\mathcal{K}}_{0}^{\perp}$ of
_static KIDs_ , namely the kernel of $P^{*}$. We claim that under our
hypotheses this space is trivial, and so the operator ${L}_{\phi,\psi}$ of [7,
Theorem 3.6] is an isomorphism:

Indeed, it is well-known that static KIDs that are $o(1)$ along some cone
$C_{\varepsilon}$ have to vanish everywhere. (This is essentially [10,
Proposition 2.1], together with unique continuation for solutions of the KID
equation.) So it suffices to show that functions $\delta
N\in\mathring{H}^{k+4}_{\phi,\psi}(g)$ are $o(1)$ in some cone contained in
$\Omega$. For this let us choose a cone $C_{\varepsilon}\subset\Omega$ on
which $x/r$ is bounded away from zero. Then the restriction $\delta
N|_{C_{\varepsilon}}$ is in the space

$\,\,\,\,\mathring{\\!\\!\\!\\!\mycal
H}_{k}^{\beta}(C_{\varepsilon}):=\mathring{H}^{k}_{r,{r}^{-n/2-\beta}}(C_{\varepsilon})\,.$
(3.15)

We note the inclusions [3]

$C^{\beta^{\prime}}_{k}(C_{\varepsilon})\subset\,\,\,\,\mathring{\\!\\!\\!\\!\mycal
H}_{k}^{\beta}(C_{\varepsilon})\,,\ \beta^{\prime}<\beta\,,\ \mbox{ and }\
\quad\,\,\,\,\mathring{\\!\\!\\!\\!\mycal
H}_{k}^{\beta}(C_{\varepsilon})\subset C^{\beta}_{\lfloor
k-n/2\rfloor}(C_{\varepsilon})\,,\ k>n/2\,.$ (3.16)

In fact [3]

$f\in\,\,\,\,\mathring{\\!\\!\\!\\!\mycal H}_{k}^{\beta}(C_{\varepsilon})\,,\
k>n/2\quad\Longrightarrow\quad f=o(r^{\beta})\,.$ (3.17)

So, the requirement that there are no KIDs in the space under consideration
will be satisfied for

$\beta\leq 0\,.$ (3.18)

(A more careful inspection shows (compare (3.24) below) that

$\delta N\in\mathring{H}^{k}_{\phi,\psi}\,,\
k>n/2\quad\Longrightarrow\quad\delta
N=o\Big{(}r^{\beta}\big{(}\frac{x}{r}\big{)}^{-\sigma-n}e^{sr/x}\Big{)}\,,$
(3.19)

but a possible blow-up of $\delta N$ when $x$ tends to zero is irrelevant as
long as $\delta N$ is forced to go to zero when receding to infinity away from
$\partial\Omega$.)

Summarising, [7, Theorem 3.6] applies and shows that the map $L_{\phi,\psi}$
there is an isomorphism.

We now want to use the inverse function theorem as in [7, Theorem 3.7] to
solve the equation (3.10); equivalently

$R\left[g_{\chi}+h\right]-R(g_{\chi})=\delta R_{\chi}\,.$ (3.20)

This requires checking differentiability of the map defined there. This will
follow by standard arguments if the metrics of the form $g_{\chi}+h$, with $h$
given by (3.9), are asymptotically Euclidean. Now, given a perturbation
$\delta R$ of the Ricci scalar on $\Omega$, the linearized perturbed metric
$h$ is obtained from the solution $\delta N$ of the equation

$\psi^{2}L_{\phi,\psi}\delta N\equiv P\underbrace{\phi^{4}\psi^{2}P^{*}\delta
N}_{=:h}=\delta R\in\psi^{2}\mathring{H}^{k}_{\phi,\psi}(g)\,.$ (3.21)

In cones $C_{\varepsilon}$ staying away from the boundary we have $\delta
N\in\,\,\,\,\mathring{\\!\\!\\!\\!\mycal H}_{k+4}^{\beta}(C_{\varepsilon})$,
and since on $C_{\varepsilon}$ we have $\psi\sim r^{-n/2-\beta}$ and $\phi\sim
r$ we obtain

$\displaystyle h$ $\displaystyle=$ $\displaystyle\phi^{4}\psi^{2}P^{*}\delta
N=r^{-n-2\beta+4}o(r^{\beta-2})=o(r^{-\beta-n+2})\,.$ (3.22)

We conclude that $h$ will decay to zero in such cones provided that

$-(n-2)\leq\beta\,.$ (3.23)

A more careful treatment, without invoking interior cones, uses the weighted
Sobolev embedding

$\displaystyle h\in\psi^{2}\phi^{2}\mathring{H}^{k+2}_{\phi,\psi}(g_{\chi})$
$\displaystyle\subset$ $\displaystyle
r^{-n-2\beta+2}(x/r)^{2\sigma+4}e^{-2sr/x}C^{k+2-\lfloor
n/2\rfloor+\alpha}_{x^{2}/r,r^{-\beta}(x/r)^{\sigma+n}e^{-sr/x}}$ (3.24)
$\displaystyle=$ $\displaystyle C^{k+2-\lfloor
n/2\rfloor+\alpha}_{r^{n+\beta-2}(x/r)^{-\sigma-4+n}e^{sr/x}}$

where $\alpha$ is any number in $(0,1)$ when $n$ is even, and $\alpha=1/2$
when $n$ is odd. (The inclusion above is obtained by a calculation as in [7,
Equation (B.4)] where, using the notation there, in the fifth line the
elliptic regularity estimate is replaced by the Sobolev inequality on
$B(0,1/4)$ for the function $u\circ\varphi_{p}$.) This guarantees
differentiability of the constraints map, as required for applicability of the
inverse function theorem.

Smoothness of solutions is standard: away from $\partial\Omega$ this follows
from elliptic regularity, while near $\partial\Omega$ this is guaranteed by
the exponential decay of $h$ and its derivatives at $\partial\Omega$. The
proof is now complete.

##  4 Beyond scale-invariant sets

Let $(\mycal S,g)$ be an asymptotically Euclidean manifold and let
$\Omega\subset\mycal S$, $r$ and $x$ be as described at the beginning of
Section 3. Let $g_{0}$ be any smooth metric on $\mycal S$ which coincides with
the Euclidean metric at all large distances in the asymptotically Euclidean
end. We will use the metric $g_{0}$ in the definitions of the functional
spaces.

Suppose that $\Psi:\mycal S\to\mycal S$ is a smooth diffeomorphism such that

$\Psi\in C^{k+5}_{r,r^{-1}}$, with $\Psi^{*}g$ uniformly equivalent to $g$.
(4.1)

The symbol $\chi$ will denote a cut-off function which equals the function
$\chi$ defined shortly before (3.6) _composed with $\Psi$_.

We have:

######  Theorem 4.1.

Under (4.1) and the remaining hypotheses of Theorem 3.1, the conclusions of
Theorem 3.1, as well as Remarks 3.2-3.4, hold with $\Omega$ replaced by
$\Psi(\Omega)$.

######  Remark 4.2.

As in our remaining analysis, the requirement of closedness of $g$ and
$\hat{g}$ can be achieved by translating $\Omega$ sufficiently far to the
asymptotic region, or by scaling as in Section 6).

###### Proof.

We note that there are no KIDs on $\Psi(\Omega)$ which tend to zero when
staying away from the boundary of $\Psi(\Omega)$. This can be proved by
calculations similar to those of [4, Proposition 2.2], where integration over
the rays

$\gamma_{p}=\\{rp\,,\ r\in[1,\infty)\\}\,,\ p\in S(R_{0})\,,$

is replaced by integration over $\Psi(\gamma_{p})$.

The result follows now by applying the following result to the metrics
$\Psi^{*}g$ and $\Psi^{*}\hat{g}$ on $\Omega$:

######  Theorem 4.3.

Let $\epsilon>0$, $k>n/2$, $\beta\in[-(n-2),0)$, and
${\tilde{\beta}}<\min(\beta,-\epsilon)$. Let $g_{0}$ be any smooth metric on
$\mycal S$ which coincides with the Euclidean metric in the asymptotic region.
Suppose that $g\in C^{k+4}_{r,1}\cap C^{\infty}$ is uniformly equivalent to
$g_{0}$ and has no static KIDs on $\Omega$. For all real numbers $\sigma$ and
$s>0$ and

all smooth metrics ${\hat{g}{}}$ close enough to $g$ in
$C^{k+4}_{r,r^{-{\tilde{\beta}}}}(\Omega)$

there exists on $\Omega$ a unique smooth two-covariant symmetric tensor field
$h$ of the form

$h=\phi^{4}\psi^{2}P^{*}_{g_{\chi}}(\delta
N)\in\psi^{2}\phi^{2}\mathring{H}^{k+2}_{\phi,\psi}(g)$ (4.2)

such that the metric $g_{\chi}+h$ solves

$R\left[g_{\chi}+h\right]=\chi R(\hat{g})+(1-\chi)R(g)\,.$ (4.3)

The tensor field $h$ vanishes at $\partial\Omega$ and can be
$C^{\infty}$-extended by zero across $\partial\Omega$, leading to a smooth
metric $g_{\chi}+h$ which approaches $g$ as $r$ recedes to infinity in the
asymptotic region.

###### Proof.

Theorem 4.3 follows from the inverse function theorem applied to the map

$r_{\chi}:\mathring{H}^{k+4}_{\phi,\psi}(g)\ni\delta
N\mapsto\psi^{-2}[(R(g_{\chi}+\phi^{4}\psi^{2}P^{*}_{g_{\chi}}(\delta
N))-R(g_{\chi})]\in\mathring{H}^{k}_{\phi,\psi}(g)\,,$ (4.4)

exactly as in the proof of Theorem 3.1. Recall that the map $r_{\chi}$ will
have $\psi^{-2}\delta R_{\chi}$ in its range when $g_{\chi}$ is close to $g$
in $C^{k+4}_{\phi,1}$ because the inverse, $Dr_{\chi}(0)^{-1}$, of the
linearised map $Dr_{\chi}(0)$ has a bound uniform over a neighborhood of
$g_{\chi}$ in this topology, as follows in part from the uniformity of the
constant in the weighted Poincaré inequality of Section 5.

######  Remark 4.4.

An alternative proof of Theorem 4.1 proceeds by checking that Propositions 5.4
and 5.6 below hold with $\Omega$ replaced by $\Psi(\Omega)$, $r$ replaced by
$r_{\Psi}:=r\circ\Psi^{-1}$, and $x$ replaced by $x_{\Psi}:=x\circ\Psi^{-1}$.
Since we have already seen (see the proof of Theorem 4.1) that there are no
KIDs vanishing at infinity on $\Psi(\Omega)$, the proof of Theorem 3.1 applies
verbatim.

To verify the propositions, one starts by noting that there exists a constant
$C_{1}$ such that we have

$C^{-1}_{1}r\leq r_{\Psi}\leq C_{1}r\,.$ (4.5)

This can be established by standard considerations (cf., e.g., the proof of
[6, Equation (30)]). Hence, decay rates in $r_{\Psi}$ are equivalent to decay
rates in $r$.

Let $u_{\Psi}:=u\circ\Psi$, and let us denote by $\mu_{g}$ the Riemannian
measure associated with a metric $g$. We then have for functions $u$ compactly
supported in $\Psi(\Omega)$, using change-of-variables, Proposition 5.6, and a
trace theorem,

$\displaystyle\int_{\Psi(\Omega)}x_{\Psi}^{2\sigma}r_{\Psi}^{2\mu}e^{-sr_{\Psi}/x_{\Psi}}|u|^{2}\,d\mu_{g}=\int_{\Omega}x^{2\sigma}r^{2\mu}e^{-sr/x}|u_{\Psi}|^{2}\,d\mu_{\Psi^{*}g}$
(4.6) $\displaystyle\leq$ $\displaystyle
C\int_{\Omega}x^{2\sigma}r^{2\mu}e^{-sr/x}|u_{\Psi}|^{2}\,d\mu_{g}$
$\displaystyle\leq$ $\displaystyle C^{2}\bigg{(}\int_{\\{x\geq
x_{0},\,r=R\\}}|u_{\Psi}|^{2}+\int_{\\{x\geq x_{0},\,r\leq
R\\}}|u_{\Psi}|^{2}\,d\mu_{g}$
$\displaystyle\phantom{C\bigg{(}}+\int_{\Omega}x^{2\sigma+4}r^{2\mu-2}e^{-sr/x}|\nabla
u_{\Psi}|^{2}_{g}\,d\mu_{g}\bigg{)}$ $\displaystyle\leq$ $\displaystyle
C^{3}\bigg{(}\int_{\\{x\geq x_{0},\,r\leq
R\\}}|u_{\Psi}|^{2}\,d\mu_{\Psi^{*}g}+\int_{\Omega}x^{2\sigma+4}r^{2\mu-2}e^{-sr/x}|\nabla
u_{\Psi}|^{2}_{\Psi^{*}g}\,d\mu_{\Psi^{*}g}\bigg{)}$ $\displaystyle=$
$\displaystyle C^{3}\bigg{(}\int_{\Psi(\\{x\geq x_{0},\,r\leq
R\\})}|u|^{2}\,d\mu_{g}+\int_{\Psi(\Omega)}x_{\Psi}^{2\sigma+4}r_{\Psi}^{2\mu-2}e^{-sr_{\Psi}/x_{\Psi}}|\nabla
u|^{2}_{g}\,d\mu_{g}\bigg{)}\,,\phantom{xxx}$

as claimed, assuming of course $s\neq 0$ and $\sigma+\mu+n/2\neq 0$.

The result for tensor fields $u$ easily follows (cf., e.g., [9, Remark 4.7]).

There is likewise an equivalent of (5.9) in the current setting. $\Box$

A trivial example of maps $\Psi$ which satisfy the above is linear maps of
$\mathbb{R}^{n}$. This, however, does not lead to a new family of sets
$\Omega$ as compared to Theorem 3.1. Another example is provided by maps which
are asymptotic to linear maps, which again does not lead to any additional
essential changes at large distances.

A non-trivial example is provided by _logarithmic rotations_ , whose action on
cones or tennis-ball curves can be seen in Figure 4.1, and which we define as
follows:

Figure 4.1: Left figure: A logarithmic rotation applied to two cones; $\Omega$
is the region between the cones. Right figure: A logarithmic rotation applied
to the “tennis-ball curve” (namely, the curve along which the halves of a
tennis-ball are joined together). A scale-invariant thickening of the surface
displayed provides the set $\Omega$.

Let $R(t)$ denote a rotation by angle $t$ around the $z$-axis. For
$\alpha\in\mathbb{R}$ consider the smooth map $\Psi_{\alpha}$ defined as

$\mathbb{R}^{3}\setminus\overline{B(1)}\ni\vec{x}\mapsto
R(\alpha\ln|\vec{x}|)\vec{x}\in\mathbb{R}^{3}\setminus\overline{B(1)}\,.$

In spherical coordinates on $\mathbb{R}^{3}$ the map $\Psi_{\alpha}$ takes the
form

$(r,\theta,\varphi)\mapsto(r,\theta,\varphi+\alpha\ln r)\,.$

This leads to the following form of the pull-back $\Psi_{\alpha}^{*}\delta$ of
the Euclidean metric $\delta$:

$\displaystyle\Psi_{\alpha}^{*}\delta$ $\displaystyle=$ $\displaystyle
dr^{2}+r^{2}\Big{(}d\theta^{2}+\sin^{2}\theta\big{(}d\varphi+\frac{\alpha
dr}{r}\big{)}^{2}\Big{)}$ (4.7) $\displaystyle=$
$\displaystyle(1+\alpha^{2}\sin^{2}\theta)dr^{2}+r^{2}\Big{(}d\theta^{2}+\sin^{2}\theta
d\varphi^{2}\Big{)}+2\alpha rdrd\varphi\,.$

Using

$2rdrd\varphi\leq dr^{2}+r^{2}d\varphi^{2}\leq\delta\,$

we find for $|\alpha|<1$

$(1-|\alpha|)\delta\leq\Psi_{\alpha}^{*}\delta\leq(1+|\alpha|+\alpha^{2})\delta\leq
3\delta\,,$ (4.8)

as desired.

We note that while the image of a scale-invariant set under a logarithmic
rotation is not scale-invariant, it is invariant under a discrete scaling,
which would have been enough for a direct analysis in Section 5 anyway.

Composing the above rotations along different axes leads to further non-
trivial examples on $\mathbb{R}^{3}$.

The above generalises in an obvious way to higher dimensions by writing
$\mathbb{R}^{n}=\mathbb{R}^{3}\times\mathbb{R}^{n-2}$, and letting $R(t)$ be
the identity on the $\mathbb{R}^{n-3}$ factor.

A similar calculation applies to the map

$\mathbb{R}^{3}\ni\vec{x}\mapsto
R\Big{(}\alpha\ln\sqrt{x^{2}+y^{2}+1}\Big{)}\vec{x}\in\mathbb{R}^{3}\,,$ (4.9)

leading again to the estimate (4.8) for $|\alpha|<1$.

On $\mathbb{R}^{2n}$, further non-trivial maps satisfying (4.1) can be
obtained by making independent logarithmic rotations (4.9) on each
$\mathbb{R}^{2}$ factor, etc.

##  5 Weighted Poincaré inequalities

We will need the following [7, Proposition C.2]:

######  Proposition 5.1.

Let $u$ be a $C^{1}$ compactly supported tensor field on a Riemannian manifold
$(M,g)$, and let $w,v$ be two $C^{2}$ functions defined on a neighborhood of
the support of $u$. Then

$\int_{M}e^{2v}|\nabla u|^{2}\geq\int_{M}e^{2v}\left[\Delta v+\Delta w+|\nabla
v|^{2}-|\nabla w|^{2}\right]|u|^{2}\,.$ (5.1)

Throughout this section we assume that the metric $g$ approaches the Euclidean
metric as one recedes to infinity, with no decay rate imposed. The first
derivatives are required to fall-off as $o(1/r)$. Wherever second derivatives
of the metric arise in a calculation, they are required to fall-off as
$o(1/r^{2})$, etc.

We let $r$ be any smooth positive function on $\Omega$ bounded away from zero
which coincides with the coordinate radius on $\Omega\setminus B(R_{0})$.

Our first goal is to prove Proposition 5.4 below. As a first step, we prove:

######  Lemma 5.2.

Let $\Omega$ be as above and let $\sigma,\mu\in\mathbb{R}$ be such that
$\sigma\neq-1/2$. There exist constants $c,C>0$ such that for all $C^{2}$
tensor fields $u$ compactly supported in

$\Omega^{\prime}:=\Omega\cap\\{0<x<cr\\}$

it holds that

$\int_{\Omega}x^{2\sigma+2}r^{2\mu}|\nabla u|^{2}\geq
C\int_{\Omega}x^{2\sigma}r^{2\mu}|u|^{2}\,.$ (5.2)

######  Remark 5.3.

Once the inequality has been established for a metric $g$ as explained above,
it is immediate that (5.2) also holds, when $u$ is a function, for any metric
which is uniformly equivalent to $g$. But then it holds for tensors for all
differentiable metrics uniformly equivalent to $g$ (no decay conditions on the
derivatives needed) by a standard argument, cf. e.g. [9, Remark 4.7].

###### Proof.

We use (5.1) with

$v=\sigma\ln x+\mu\ln r\,.$ (5.3)

Then

$\Delta v+|\nabla v|^{2}=\frac{\sigma}{x^{2}}\bigg{[}(\sigma-1)|\nabla
x|^{2}+{2}{\mu x}\frac{\nabla x\cdot\nabla r}{r}+x\Delta
x\bigg{]}+\frac{\mu}{r^{2}}\bigg{[}(\mu-1){|\nabla r|^{2}}+r\Delta
r\bigg{]}\,.$ (5.4)

Letting

$w=-\frac{1}{2}\ln x+\frac{\nu}{2}\ln r$

one similarly finds

$\Delta w-|\nabla w|^{2}=\frac{1}{4x^{2}}\bigg{[}|\nabla x|^{2}+{2\nu
x}\frac{\nabla x\cdot\nabla r}{r}-2x\Delta
x\bigg{]}+\frac{\nu}{4r^{2}}\bigg{[}-(\nu+2){|\nabla r|^{2}}+2r\Delta
r\bigg{]}\,.$ (5.5)

Given $\sigma\neq 1$, $\mu,\nu\in\mathbb{R}$, $\epsilon>0$, and a compact set
$K$, for small $x$ the dominant contribution comes from the $|\nabla
x|^{2}/x^{2}$ terms. This shows that we can choose $x_{0}>0$ small enough so
that for $0<x<x_{0}$ on $K$ we have

$\Delta v+|\nabla v|^{2}+\Delta w-|\nabla
w|^{2}\geq\frac{(2\sigma-1)^{2}-\epsilon}{4x^{2}}|\nabla x|^{2}\,.$ (5.6)

We consider now $p\in\Omega$ with

$\lambda:=r(p)\geq 2R_{0}\,,$ (5.7)

Then the scaling transformation $\mathbb{R}^{n}\ni\vec{y}\mapsto
2R_{0}\lambda^{-1}\vec{y}$ maps the set

$\Gamma_{\lambda}:=\Omega\cap\\{\lambda/2\leq|\vec{y}|\leq 2\lambda\\}$

to $\Gamma_{1}=\Omega\cap\\{R_{0}\leq|\vec{y}|\leq 4R_{0}\\}$. The associated
pull-back will carry the metric $g$ on $\Gamma_{\theta,\lambda}$ to a metric
$\lambda^{-2}g_{\lambda}$ on $\Gamma_{\theta,1}$, with $g_{\lambda}$
approaching the flat metric on $\Gamma_{1}$ as $\lambda$ tends to infinity. We
can use (5.6) for all rescaled metrics $g_{\lambda}$, with perhaps a smaller
constant $x_{0}$ independent of $\lambda$. Since both sides are invariant
under the transformation $(x,r)\mapsto(ax,ar)$, scaling back to
$\Omega_{\lambda}$ gives there

$\Delta v+|\nabla v|^{2}+\Delta w-|\nabla
w|^{2}\geq\frac{(\sigma-\frac{1}{2})^{2}}{2x^{2}}\,,$ (5.8)

for all $0<x/r<c$ small enough, as desired. $\Box$

We are ready to prove now:

######  Proposition 5.4.

Let $\Omega$ be as above and let $\sigma,\mu\in\mathbb{R}$ be such that
$\sigma\neq-1/2$ and $\sigma+\mu+n/2\neq 0$. There exist constants $x_{0},C,R$
such that for all $C^{2}$ tensor fields $u$ compactly supported in $\Omega$ it
holds that

$\displaystyle\int_{\Omega}x^{2\sigma}r^{2\mu}|u|^{2}$ $\displaystyle\leq$
$\displaystyle C\bigg{(}\int_{\\{x\geq x_{0},\,r=R\\}}|u|^{2}+\int_{\\{x\geq
x_{0},\,r\leq R\\}}|u|^{2}$ (5.9)
$\displaystyle\phantom{C\bigg{(}}+\int_{\Omega}x^{2\sigma+2}r^{2\mu}|\nabla
u|^{2}\bigg{)}\,.$

###### Proof.

We will use a family of cut-off functions $\Xi_{\lambda}$ satisfying

$|\phi\nabla\Xi_{\lambda}|\leq C/\lambda\,,$ (5.10)

for $\lambda$ large. Indeed, let $\chi_{\lambda}(t)=\chi(-\ln(t)/\lambda)$
with $\chi$ a smooth non-decreasing function satisfying $\chi(s)=0$ for $s<1$
and $\chi(s)=1$ for $s\geq 2$. Recall that in the current case $\phi=x$. We
set $\Xi_{\lambda}=\chi_{\lambda}(x/r)$, then

$\displaystyle
x\nabla\Xi_{\lambda}=x\nabla\left(\chi(-\ln(x/r)/\lambda)\right)=\displaystyle\frac{\chi^{\prime}}{\lambda}\left(\frac{x}{r}\nabla
r-\nabla x\right)\,,$

which is bounded on $\Omega$. Hence (5.10) holds.

We note that $\Xi_{\lambda}=0$ if and only if $x\geq e^{-\lambda}r$, and
$\Xi_{\lambda}=1$ if and only if $x\leq e^{-2\lambda}r$. Set

$u=\Xi_{\lambda}u+(1-\Xi_{\lambda})u=:v+w\,,$

thus $v$ vanishes outside $\\{0<x\leq re^{-\lambda}\\}$, and $w$ vanishes for
$x\leq e^{-2\lambda}r$.

Let

$\psi=x^{\sigma}r^{\mu}\,.$

Applying Lemma 5.2 to $v$ we obtain, for all $\lambda$ large enough,

$\int_{\\{0<x\leq re^{-\lambda}\\}}\psi^{2}\phi^{2}|\nabla
u|^{2}+\frac{c_{1}}{\lambda^{2}}\int_{\\{0<x\leq
re^{-\lambda}\,,\nabla\Xi_{\lambda}\neq 0\\}}\psi^{2}|u|^{2}\geq
c_{2}\int_{\\{v\neq 0\\}}\psi^{2}|v|^{2}\,.$ (5.11)

On each half-ray we have the classical Poincaré inequality for tensor fields
with compact support in $[R,\infty)\subset\mathbb{R}$: if $2\mu+2\sigma+n\neq
0$, then

$\int_{R}^{\infty}r^{2\mu+2\sigma+n-1}|u|^{2}dr\leq
C_{1}|u|^{2}(R)+C_{2}\int_{R}^{\infty}r^{2\mu+2\sigma+n+1}|\partial_{r}u|^{2}dr\,,$
(5.12)

with constants $C_{2}$ depending only upon $2\mu+2\sigma+n$, and $C_{1}$
depending only upon $2\mu+2\sigma+n$ and $R$. (In fact, $C_{1}<0$ when
$2\mu+2\sigma+n>0$, but this will not be needed in our considerations.)

Integrating over the angles with $u$ replaced by $w$, this gives

$\int_{\\{w\neq 0\\}}r^{2\mu+2\sigma+2}|\nabla w|^{2}+\int_{\\{w\neq
0\\}\cap\\{r=R\\}}|w|^{2}\geq C\int_{\\{w\neq 0\\}\cap\\{r\geq
R\\}}r^{2\sigma+2\mu}|w|^{2}\,.$ (5.13)

This implies

$\displaystyle\int_{\\{w\neq 0\\}}r^{2\mu+2\sigma+2}|\nabla
u|^{2}+\frac{c_{3}}{\lambda^{2}}\int_{\\{w\neq 0\,,\nabla\Xi_{\lambda}\neq
0\\}}r^{2\mu+2\sigma}|u|^{2}$ (5.14) $\displaystyle+\int_{\\{w\neq
0\\}\cap\\{r=R\\}}|u|^{2}+\int_{\\{w\neq 0\\}\cap\\{r\leq R\\}}|u|^{2}\geq
c_{4}\int_{\\{w\neq 0\\}}r^{2\sigma+2\mu}|w|^{2}\,.\phantom{xx}$

On the support of $w$ and $\nabla\Xi_{\lambda}$ we have $e^{-2\lambda}r\leq
x\leq e^{-\lambda}r$, hence

$\displaystyle\frac{1}{2}\int x^{2\sigma}r^{2\mu}|u|^{2}\leq\int
x^{2\sigma}r^{2\mu}|v|^{2}+\int x^{2\sigma}r^{2\mu}|w|^{2}$
$\displaystyle\leq$ $\displaystyle\int
x^{2\sigma}r^{2\mu}|v|^{2}+\max(e^{\lambda\sigma}\,,e^{2\lambda\sigma})\int
r^{2\sigma+2\mu}|w|^{2}$ $\displaystyle\leq$
$\displaystyle\frac{C_{3}}{2}\bigg{[}\int_{\\{0<x\leq
re^{-\lambda}\\}}x^{2\sigma+2}r^{2\mu}|\nabla
u|^{2}+\frac{c_{1}}{\lambda^{2}}\int_{\\{0<x\leq
re^{-\lambda}\\}}x^{2\sigma+2}r^{2\mu}|u|^{2}$
$\displaystyle+\max(e^{\lambda\sigma}\,,e^{2\lambda\sigma})\bigg{(}\int_{\\{w\neq
0\\}\cap\\{r=R\\}}|u|^{2}+\int_{\\{w\neq 0\\}\cap\\{r\leq R\\}}|u|^{2}$
$\displaystyle+\int_{\\{x\geq re^{-2\lambda}\\}}r^{2\mu+2\sigma+2}|\nabla
u|^{2}+\frac{c_{3}}{\lambda^{2}}\int_{\\{re^{-2\lambda}\leq x\leq
re^{-\lambda}\\}}r^{2\mu+2\sigma}|u|^{2}\bigg{)}\bigg{]}\,.$

For $\lambda\geq 1$ we can rewrite this as

$\displaystyle\int
x^{2\sigma}r^{2\mu}\left(1-C_{3}\frac{c_{1}}{\lambda^{2}}\right)|u|^{2}$
$\displaystyle\leq$ $\displaystyle
C_{3}\bigg{[}c_{3}\max(e^{\lambda\sigma}\,,e^{2\lambda\sigma})\bigg{(}\int_{\\{re^{-2\lambda}\leq
x\leq re^{-\lambda}\\}}r^{2\mu+2\sigma}|u|^{2}+\int_{\\{w\neq
0\\}\cap\\{r=R\\}}|u|^{2}$ $\displaystyle+\int_{\\{w\neq 0\\}\cap\\{r\leq
R\\}}|u|^{2}+\int_{\\{x\geq re^{-2\lambda}\\}}r^{2\mu+2\sigma+2}|\nabla
u|^{2}\bigg{)}$ $\displaystyle+\int_{\\{0<x\leq
re^{-\lambda}\\}}x^{2\sigma+2}r^{2\mu}|\nabla u|^{2}\bigg{]}\,.$

Choosing $\lambda$ large enough we obtain, keeping in mind that $x<Cr$ on
$\Omega$,

$\displaystyle\int x^{2\sigma}r^{2\mu}|u|^{2}\leq
C_{4}(\lambda)\bigg{(}\int_{\\{re^{-2\lambda}\leq x\leq
re^{-\lambda}\\}}r^{2\mu+2\sigma}|u|^{2}+\int_{\\{w\neq
0\\}\cap\\{r=R\\}}|u|^{2}$ (5.15)
$\displaystyle\phantom{xxxxxxxxxx}+\int_{\\{w\neq 0\\}\cap\\{r\leq
R\\}}|u|^{2}+\int_{M}x^{2\sigma+2}r^{2\mu}|\nabla u|^{2}\bigg{)}\,.$

Integrating (5.12) over the angles gives

$\displaystyle\int_{\\{re^{-2\lambda}\leq x\leq re^{-\lambda}\,,r\geq
R\\}}r^{2\mu+2\sigma}|u|^{2}$ (5.16) $\displaystyle\leq
C(R)\int_{\\{re^{-2\lambda}\leq x\leq re^{-\lambda}\,,r\geq
R\\}}r^{2\mu+2\sigma+2}|\nabla u|^{2}+C_{1}(R)\int_{\\{re^{-2\lambda}\leq
x\leq re^{-\lambda}\,,r=R\\}}|u|^{2}$ $\displaystyle\leq
C^{\prime}(R,\lambda)\int_{\\{re^{-2\lambda}\leq x\leq
re^{-\lambda}\\}}x^{2\sigma+2}r^{2\mu}|\nabla
u|^{2}+C_{1}(R)\int_{\\{re^{-2\lambda}\leq x\leq
re^{-\lambda}\,,r=R\\}}|u|^{2}$ $\displaystyle\leq
C^{\prime}(R,\lambda)\int_{M}x^{2\sigma+2}r^{2\mu}|\nabla
u|^{2}+C_{1}(R)\int_{\\{re^{-2\lambda}\leq x\leq
re^{-\lambda}\,,r=R\\}}|u|^{2}\,.$

Inserting this into the first line of (5.15) gives (5.9) . $\Box$

We note a standard consequence of Proposition 5.4:

######  Corollary 5.5.

Let $\Omega$ be as above and let $\sigma,\mu\in\mathbb{R}$ be such that
$\sigma\neq-1/2$ and $\sigma+\mu+n/2\neq 0$. Let $\mathring{H}$ denote the
completion of $C^{2}$ compactly supported tensor fields in $\Omega$ with
respect to the norm

$\|u\|_{\mathring{H}}^{2}=\int_{\Omega}x^{2\sigma}r^{2\mu}|u|^{2}+\int_{\Omega}x^{2\sigma+2}r^{2\mu}|\nabla
u|^{2}\,.$

Suppose that $\mathring{H}$ contains a closed subspace $X$ transversal to the
space $\\{u\in\mathring{H}:\nabla u=0\\}$. Then there exists a constant $C$
such that for all $u\in X$ we have

$\displaystyle\int_{\Omega}x^{2\sigma}r^{2\mu}|u|^{2}$ $\displaystyle\leq$
$\displaystyle C\int_{\Omega}x^{2\sigma+2}r^{2\mu}|\nabla u|^{2}\,.$ (5.17)

###### Proof.

As already mentioned, the result is standard, we give the proof for
completeness.

Suppose that (5.17) is wrong, then there exists a sequence of $C^{2}$
compactly supported tensor fields $u_{n}$ such that

$\displaystyle\int_{\Omega}x^{2\sigma}r^{2\mu}|u_{n}|^{2}$ $\displaystyle\geq$
$\displaystyle n\int_{\Omega}x^{2\sigma+2}r^{2\mu}|\nabla u_{n}|^{2}\,.$
(5.18)

We can normalize the sequence $u_{n}$ so that

$\displaystyle\int_{\\{x\geq x_{0},\,r=R\\}}|u_{n}|^{2}+\int_{\\{x\geq
x_{0},\,r\leq R\\}}|u_{n}|^{2}=1\,.$ (5.19)

Equation (5.9) implies

$\displaystyle\int_{\Omega}x^{2\sigma}r^{2\mu}|u_{n}|^{2}$ $\displaystyle\leq$
$\displaystyle
C\bigg{(}1+\frac{1}{n}\int_{\Omega}x^{2\sigma}r^{2\mu}|u_{n}|^{2}\bigg{)}\,.$

Thus, for $n\geq 2C$,

$\displaystyle\int_{\Omega}x^{2\sigma}r^{2\mu}|u_{n}|^{2}$ $\displaystyle\leq$
$\displaystyle 2C\,.$ (5.20)

Equation (5.18) gives now

$\displaystyle\int_{\\{x\geq x_{0},\,r\leq R\\}}|\nabla u_{n}|^{2}\leq
C_{1}\int_{\Omega}x^{2\sigma+2}r^{2\mu}|\nabla
u_{n}|^{2}\leq\frac{2CC_{1}}{n}\,.$ (5.21)

Let $K=\Omega\cap\\{x>x_{0},\,r<R\\}$. Compactness of the embeddings

$W^{1,2}(K)\subset L^{2}(K)\,,\quad W^{1,2}(K)\subset L^{2}(\\{r=R\\}\cap
K)\,,$

implies that $\\{u_{n}\\}$ contains a subsequence, still denoted by
$\\{u_{n}\\}$, which is Cauchy both in $L^{2}(K)$ and in $L^{2}(\\{r=R\\}\cap
K)$. Equation (5.9) applied to $u_{n}-u_{m}$ shows that $\\{u_{n}\\}$ is
Cauchy in $\mathring{H}$. The limit is a non-trivial tensor field satisfying
$\nabla u=0$, which contradicts the fact that zero is the only such tensor
field in $X$. $\Box$

We have the exponentially-weighted version of the above:

######  Proposition 5.6.

Let $\Omega$ be as above and let $s,\sigma,\mu\in\mathbb{R}$ satisfy $s\neq 0$
and $\beta\equiv\sigma+\mu+n/2\neq 0$. There exist constants $x_{0},C,R$ such
that for all $C^{2}$ tensor fields $u$ compactly supported in $\Omega$ it
holds that

$\displaystyle\int_{\Omega}x^{2\sigma}r^{2\mu}e^{-sr/x}|u|^{2}$
$\displaystyle\leq$ $\displaystyle C\bigg{(}\int_{\\{x\geq
x_{0},\,r=R\\}}|u|^{2}+\int_{\\{x\geq x_{0},\,r\leq R\\}}|u|^{2}$ (5.22)
$\displaystyle\phantom{C\bigg{(}}+\int_{\Omega}x^{2\sigma+4}r^{2\mu-2}e^{-sr/x}|\nabla
u|^{2}\bigg{)}\,.$

###### Proof.

The proof is a direct repetition of that of Proposition 5.4, with
$\psi=x^{\sigma}r^{\mu}e^{-sr/(2x)}$, $\phi=x^{2}/r$, and where instead of
Lemma 5.2 the following Lemma is used: $\Box$

######  Lemma 5.7.

Let $\Omega$ be as above and let $s,\mu\in\mathbb{R}$ be such that $s\neq 0$.
There exist constants $c,C>0$ such that for all $C^{2}$ tensor fields $u$
compactly supported in

$\Omega^{\prime}:=\Omega\cap\\{0<x\ <cr\\}$

it holds that

$\int_{M}x^{4+2\sigma}r^{2\mu-2}e^{-sr/x}|\nabla u|^{2}\geq
C\int_{M}x^{2\sigma}r^{2\mu}e^{-sr/x}|u|^{2}\,.$ (5.23)

###### Proof.

We use again (5.1), with $w=0$ and

$v=-\frac{sr}{x}+(2+\sigma)\ln x+(\mu-1)\ln r\,.$ (5.24)

On any compact set one finds, for $0<x/r$ small enough,

$\Delta v+|\nabla
v|^{2}=\big{(}\frac{s^{2}r^{2}}{x^{4}}+O(rx^{-3})+O(x^{-2})+O(r^{-2})\big{)}|\nabla
x|^{2}\,.$ (5.25)

By scaling, for $s\neq 0$ and for $0<x<cr$ with $c$ small enough, this implies

$\Delta v+|\nabla v|^{2}\geq\hat{C}\frac{r^{2}}{x^{4}}\,,$ (5.26)

for some constant $\hat{C}>0$, which together with (5.1) leads to (5.23).
$\Box$

##  6 Applications

In this section we wish to show how to arrange things so that all conditions
of Theorem 3.1 are met.

Consider, then, two smooth metrics $g,\hat{g}\in
M_{\delta+C^{k+4}_{r,r^{\epsilon}}}\cap C^{\infty}$, $i=1,2$, for some
$\epsilon>0$ and $k>n/2$. Let $\Omega$, $\Omega_{S}$ be as in Section 3, cf.
(3.1)-(3.2).

Let $\psi_{\vec{y}}:\mathbb{R}^{n}\to\mathbb{R}^{n}$ denote the translation by
$\vec{y}$,

$\phi_{\vec{y}}(\vec{x}):=\vec{x}+\vec{y}\,.$

When $|\vec{y}|$ tends to infinity the three tensor fields
$\phi_{\vec{y}}^{*}g-\delta$, $\phi_{\vec{y}}^{*}\hat{g}-\delta$, and
$\phi_{\vec{y}}^{*}(g-\hat{g})$ approach zero in
$C^{k+4}_{r,r^{-\epsilon}}(\Omega)$. Thus Theorem 3.1 applies for all
$|\vec{y}|$ large enough, and provides a gluing of $\phi_{\vec{y}}^{*}g$ with
$\phi_{\vec{y}}^{*}\hat{g}$. Taking $\Omega$ to be a translation of the
asymptotic cone as in (3.4) proves Theorem 1.1.

An alternative construction proceeds as follows, assuming that $\Omega\cap
B(0,1)=\emptyset$: For $\lambda\geq 1$ let
$\psi_{\lambda}:\mathbb{R}^{n}\to\mathbb{R}^{n}$ denote the scaling map,
$\psi_{\lambda}(\vec{x}):=\lambda\vec{x}$. Set

$g_{\lambda}=\lambda^{-2}\psi_{\lambda}^{*}g\,,\qquad\hat{g}_{\lambda}=\lambda^{-2}\psi_{\lambda}^{*}\hat{g}\,.$
(6.1)

Then both $g_{\lambda}$ and $\hat{g}_{\lambda}$ tend to $\delta$ in
$C^{k+4}_{r,r^{-\epsilon}}(\Omega)$ as $\lambda$ tends to infinity. Hence
Theorem 3.1 applies for all $\lambda$ large enough. Scaling back the glued
metric provides a gluing of $g$ and $\hat{g}$ along a rescaled set $\Omega$.

Consider, finally, a collection $\Omega_{i}$, $i=1,\ldots,N$, of disjoint sets
satisfying the requirements set forth at the beginning of Section 3, possibly
after translations. The construction just given can be repeated $N$-times to
obtain a gluing of $g$ and $\hat{g}$ across $\cup\Omega_{i}$.

Acknowledgements: Supported in part by the Austrian Research Fund (FWF),
Project P 29517-N16.

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