# A Positive Energy Theorem for Fourth-Order Gravity

R. Avalos Federal University of Ceará, Mathematics Department, Fortaleza,
Ceará, Brazil  P. Laurain Institut Mathématique de Jussieu, Université de
Paris, Bâtiment Sophie Germain, Case 7052, 75205 Paris Cédex 13, France & DMA,
Ecole normale supérieure, CNRS, PSL Research University, 75005 Paris.  J. H.
Lira∗

###### Abstract

In this paper we prove a positive energy theorem related to fourth-order
gravitational theories, which is a higher-order analogue of the classical ADM
positive energy theorem of general relativity. We will also show that, in
parallel to the corresponding situation in general relativity, this result
intersects several important problems in geometric analysis. For instance, it
underlies positive mass theorems associated to the Paneitz operator, playing a
similar role in the positive $Q$-curvature conformal prescription problem as
the Schoen-Yau positive energy theorem does for the Yamabe problem. Several
other links to $Q$-curvature analysis and rigidity phenomena are established.

##  1 Introduction

In this paper we will analyse the properties of a recently proposed energy
associated to higher-order gravitational theories in the stationary limit, see
[3]. Specifically, in parallel work, we have analysed gravitational theories
described on globally hyperbolic space-times $(V\doteq
M\times\mathbb{R},\bar{g})$ by an action functional of the form

$\displaystyle S(\bar{g})=\int_{V}\left(\alpha
R^{2}_{\bar{g}}+\beta\langle\mathrm{Ric}_{\bar{g}},\mathrm{Ric}_{\bar{g}}\rangle\right)dV_{\bar{g}},$
(1)

where $\alpha$ and $\beta$ are free parameters in this variational setting.
Let us highlight that the study of these kinds of higher-order gravitational
action functional is well-motivated within contemporary theoretical physics as
they appear in connection with effective field theories of gravity [6, 16,
39], as well as in the context of inflationary cosmology [44] and certain
approaches to quantum gravity, such as conformal gravity [31, 32].

In order to make sense of the above functional, we can assume that the class
of metrics considered above are such that $R^{2}_{\bar{g}}$ and
$\langle\mathrm{Ric}_{\bar{g}},\mathrm{Ric}_{\bar{g}}\rangle$ are integrable.
Then, the functional $\bar{g}\mapsto S(\bar{g})$ is well-defined and we have
an $L^{2}$-gradient for this functional, given by a divergence-free tensor
field $A_{\bar{g}}\in\Gamma(T^{0}_{2}V)$, which is explicitly given by

$\displaystyle\begin{split}A_{\bar{g}}&=\beta\Box_{\bar{g}}{\mathrm{Ric}_{\bar{g}}}+(\frac{1}{2}\beta+2\alpha)\Box_{\bar{g}}R\>{\bar{g}}-(2\alpha+\beta)\bar{\nabla}^{2}R_{\bar{g}}-2\beta{\mathrm{Ric}_{\bar{g}}}_{\cdot}\mathrm{Riem}_{\bar{g}}\\\
&+2\alpha R_{\bar{g}}\mathrm{Ric}_{\bar{g}}-\frac{1}{2}\alpha
R^{2}_{\bar{g}}{\bar{g}}-\frac{1}{2}\beta\langle\mathrm{Ric}_{\bar{g}},\mathrm{Ric}_{\bar{g}}\rangle_{\bar{g}}{\bar{g}},\end{split}$
(2)

where above we denoted
${{\mathrm{Ric}_{\bar{g}}}_{\cdot}\mathrm{Riem}_{\bar{g}}}_{ij}\doteq{\mathrm{Ric}_{\bar{g}}}^{kl}{\mathrm{Riem}_{\bar{g}}}{}_{kijl}$.
In a parallel situation to what is well-known in the context of general
relativity (GR), we have shown that there is a canonical notion of energy,
which we denote by $\mathcal{E}_{\alpha,\beta}(\bar{g})$, associated to
asymptotically Euclidean (AE) solutions of the space-time field equations
$A_{\bar{g}}=0$ which arise as perturbations of solutions $\bar{g}_{0}$ which
possess a time-like Killing field. Although the analysis of such an energy
could be quite involved in general, we are able to identify some particular
choices of the parameters $\alpha$ and $\beta$ for which its analysis is
tractable. More important, we establish positivity and rigidity results
results for the energy in those cases. These results are intimately connected
with the existence of metrics with positive constant $Q$-curvature

Our aim here is to analyse the particular case of stationary solutions of the
fourth-order field equations parameterized by $2\alpha+\beta=0$. Recall that
globally hyperbolic stationary space-times are manifolds of the form
$V=M\times\mathbb{R}$ endowed with a Lorentzian metric that can be written as

$\displaystyle\bar{g}=-N^{2}dt^{2}+\tilde{g},$ (3)

where $N:M\mapsto\mathbb{R}^{+}$ is the lapse function and
$\tilde{g}\in\Gamma(T^{0}_{2}M)$ restricts to a Riemannian metric $g$ on each
$t=constant$ hypersurface. In this setting, the appropriate notion of energy
associated to the action $S$ and the corresponding field equations
$A_{\bar{g}}=0$ becomes

$\displaystyle\begin{split}\mathcal{E}_{\alpha}(g)\doteq-\alpha\lim_{r\rightarrow\infty}\int_{S^{n-1}_{r}}\left(\partial_{j}\partial_{i}\partial_{i}g_{aa}-\partial_{j}\partial_{u}\partial_{i}g_{ui}\right)\nu^{j}d\omega_{r}\end{split}.$
(4)

In this expression, we are assuming that $(M,g)$ is an AE manifold. This means
that for a given compact set $K$ the asymptotic region $M\backslash K$ is
diffeomorphic to $\mathbb{R}^{n}\backslash\overline{B_{1}(0)}$ and it is
foliated by topological $(n-1)$-dimensional spheres $S^{n-1}_{r}$, whose
Euclidean volume element is denoted in the expression above by $d\omega_{r}$.
There, $\nu$ stands for the Euclidean unit normal field to these spheres.

The nature of (4) as a conserved quantity in the context of higher-order
gravitational theories make $\mathcal{E}_{\alpha}(g)$ a very good fourth-order
analogue to the Arnowitt, Deser and Misner (ADM) energy in the context of GR.
Let us recall that the total energy of an isolated gravitational system in GR,
whose initial data is modelled as an AE manifold, is given by (see [1])

$\displaystyle E_{ADM}(g)$
$\displaystyle=c(n)\lim_{r\rightarrow\infty}\int_{S^{n-1}_{r}}\left(\partial_{i}g_{ji}-\partial_{j}g_{ii}\right)\nu^{j}d\omega_{r},$
(5)

where $c(n)$ stands for a dimensional constant (see [4] for the detailed
analytical properties of (5)). This energy $E_{ADM}$ has had a huge impact
both within GR and in geometric analysis. Most notoriously, it was the work of
R. Schoen in [40] that elucidated the role that the ADM energy plays in the
final resolution of the Yamabe problem (see also [28] for a review on this
topic). In particular, in order to solve the Yamabe problem in the positive
Yamabe class, in dimensions 3,4 and 5 or in locally conformally flat
manifolds, Schoen noticed that it was enough to prove that an appropriate
constant appearing in the expansion of the Green function $G_{L_{g}}$
associated to the conformal Laplacian $L_{g}$ was non-negative and that the
zero case implied rigidity with the round sphere. Furthermore, it was pointed
out that this constant was precisely the ADM energy of the AE manifold
obtained via an stereographic projection. Therefore, the proof of the positive
energy theorem in GR actually underlies the proof of the Yamabe problem in
these cases. This beautiful relation shows what a fundamental result the
positive energy theorem actually is within geometric analysis, being a
cornerstone in the resolution of the Yamabe problem. Since then, the ADM
energy has influenced many other constructions within geometric analysis and
mathematical GR, which are not necessarily concerned with the Yamabe problem.
For instance, rigidity phenomena associated to positive scalar curvature [7,
9, 10], isoperimetric problems on AE manifolds [5, 17, 25], geometric
foliations and center of mass constructions [17, 24, 25, 37] and even gluing
constructions [10] (for nice reviews of many of these topics, see [8, 27]).

In view of the above paragraph, we consider that the analysis of the positive
energy theorem associated to (4) stands as a highly well-motivated problem
both for the development of a program devoted to the mathematical analysis of
fourth order theories of gravity, as well as a tool which can potentially play
a fundamental role in several fourth order geometric problems. Thus, the main
objective of this paper is the establishment of such a positive energy
theorem, and, afterwards, we will show how it underlies some fundamental
problems in geometric analysis.

Since we will be concerned with positivity issues related to
$\mathcal{E}_{\alpha}(g)$, we must actually fix the sign of $\alpha$ a priori.
For reasons that will become apparent through this paper, we will fix
$\alpha=-1$ and define $\mathcal{E}(g)\doteq\mathcal{E}_{-1}(g)$. Let us
notice that before embarking on the proof of the positive energy theorem, we
should first analyse under what conditions $\mathcal{E}(g)$ is actually well-
defined and prove that it is actually an intrinsic geometric object within a
suitable class of AE-manifolds. This will be done in Proposition 1,
Proposition 2 and Theorem 1. Once this is done, we will first explore
appropriate geometric conditions that could in principle provide a rigidity
statement in the critical cases. Explicitly, such conditions will involve a
$Q$-curvature condition of the form $Q_{g}\geq 0$ (see Proposition 3 and
Theorem 4). Let us recall that given a Riemannian manifold $(M^{n},g)$, $n\geq
3$, its $Q$-curvature is defined by111See Appendix B for details on our
conventions on $Q$-curvature.

$\displaystyle
Q_{g}\doteq-\frac{1}{2(n-1)}\Delta_{g}R_{g}-\frac{2}{(n-2)^{2}}|\mathrm{Ric}_{g}|^{2}_{g}+\frac{n^{3}-4n^{2}+16n-16}{8(n-1)^{2}(n-2)^{2}}R_{g}^{2}.$

In this context, we will establish the following theorem:

######  Theorem A (Positive Energy).

Let $(M^{n},g)$ be an $n$-dimensional AE manifold, with $n\geq 3$, which
satisfies the decaying conditions: (i) $g_{ij}-\delta_{ij}=O_{4}(r^{-\tau})$,
with $\tau>\max\\{0,\frac{n-4}{2}\\}$, in some coordinate system associated to
an structure of infinity; (ii) $Q_{g}\in L^{1}(M,dV_{g})$, and such that
$Y([g])>0$ and $Q_{g}\geq 0$. Then, the fourth order energy $\mathcal{E}(g)$
is non-negative and $\mathcal{E}(g)=0$ if and only if $(M,g)$ is isometric to
$(\mathbb{R}^{n},\delta)$.

The proof of this statement will be the content of Theorem 2 (see also
theorems 3 and 4). Let us notice that some hypotheses in the above theorem can
be relaxed while keeping important results. In particular, the positivity and
rigidity statements are somehow decoupled. This implies that under hypotheses
(i)-(ii) and $R_{g}>0$ at infinity, it follows that $\mathcal{E}(g)\geq 0$
(see Theorem 3), while under hypotheses (i)-(ii), $Q_{g}\geq 0$ and $g$ Yamabe
positive, if $\mathcal{E}(g)=0$, then the rigidity statement follows (see
Theorem 4). Furthermore, let us notice that the borderline case of $n=4$ is
special, since it follows that any 4-dimensional AE manifold satisfies the
decaying conditions (i)-(ii) and, in fact, under these conditions
$\mathcal{E}(g)=0$. These last comments also apply to the case $n=3$, which
can be checked explicitly quite straightforwardly. This imposes restrictions
on the positive curvature conditions that these manifolds can admit, providing
us with the following corollaries.

######  Corollary.

Any n-dimensional AE-Riemannian manifold $(M^{n},g)$, with $n\in\\{3,4\\}$,
such that $Q_{g}\geq 0$ and $Y([g])>0$, is isometric to
$(\mathbb{R}^{n},\delta)$.

Let us notice that Yamabe positive AE manifolds in low dimensions (specially
three) are very natural objects for instance in GR, since maximal vacuum
initial data for isolated systems belong to this class of manifolds.
Therefore, the above corollary, in some sense, can be used to separate the
trivial solutions via an appeal to their $Q$-curvature.

The following result, which in particular gives a conformally invariant
statement, is also a direct consequence of the analysis related to the above
positive energy theorem for $\mathcal{E}(g)$ (see Proposition 5 for more
details).

######  Corollary.

Let $(M,g)$ be an asymptotically Euclidean four manifold with
$\kappa_{g}=\int_{M}Q_{g}\,dv_{g}\geq 0$ and $Y([g])>0$, then $(M,g)$ is
conformal to the euclidean $\mathbb{R}^{4}$.

After presenting the above results concerning the positive energy theorem
associated to $\mathcal{E}(g)$, we will make contact with important problems
in geometric analysis. In particular, our aim is to show that that there is a
very clear parallel in the role played by the ADM positive energy theorems of
Schoen-Yau [41, 42, 43] in the resolution of the Yamabe problem, to the role
Theorem A plays in the resolution of the positive $Q$-curvature Yamabe-type
problems which are known up to date. This problem concerns finding a conformal
deformation of a closed Riemannian manifold $(M,g)$ so that the resulting
metric has constant $Q$-curvature. There has been great progress in this
program in recent times and the analysis depends on whether $n\geq 5$ or
$n=3,4$. In particular, for the most updated resolutions of this problem to
our knowledge in dimension four see [15, 18, 11] and references therein, while
for $n\geq 5$, with different degrees of generality, let us draw the reader’s
attention to [23, 20, 38]. Although we will not be concerned with the three
dimensional case, let us point out that the problem has been addressed in this
case in [21, 22].

Let us now focus on the results of [23]. There, the authors prove that given a
closed manifold $(M,g)$ of dimension $n\geq 5$ which is Yamabe non-negative
and satisfies $Q_{g}\geq 0$ not identically zero,222In this cases we will say
that $Q_{g}$ is semi-positive. there is a conformal deformation to constant
positive $Q$-curvature. In particular, this is done in Theorem 4 therein where
in the cases of dimensions $n=5,6,7$ or $n\geq 5$ and locally conformally
flat, there is a parallel to Schoen’s resolution of the Yamabe problem. That
is, the problem can be reduced to showing that a certain coefficient in the
Green function expansion of the Paneitz operator near a pole is non-negative.
In analogy to Schoen’s ideas, this constant has been called the mass of the
Paneitz operator (whenever defined) and its positivity has been analysed first
by Humbert-Raulot in the locally conformally flat case [26], then by Gursky-
Malchiodi who incorporated the cases $5\leq n\leq 7$ without this last
restriction, and finally by Hang-Yang [21], who weakened the hypotheses on the
scalar curvature imposed in [20] and achieved the final form of this results
which was applied in Theorem 4 of [23]. Along these lines, let us also draw
the reader’s attention to an unpublished work by B. Michel [34], who also
analysed these kinds of positive mass theorems associated to the Green
function of the Paneitz operator previously and obtained related results
through similar methods to those of [26, 20, 21].

In Section 2.2 we will show that exactly in dimensions $5\leq n\leq 7$ or if
$(M,g)$ has a point $p$ around which it is conformally flat, then the mass of
the Green function $G_{P_{g}}$ associated to the Paneitz operator $P_{g}$ is
positively proportional to the energy $\mathcal{E}(\hat{g})$ of the
asymptotically euclidean manifold
$(\hat{M}=M\backslash\\{p\\},\hat{g}=G^{\frac{4}{n-4}}_{P_{g}}g)$ obtained via
a stereographic projection. Then, we show that the following $Q$-curvature
positive mass theorem follows from Theorem A:

######  Theorem B ($Q$-curvature Positive Mass [20, 23, 26]).

Let $(M,g)$ be a closed $n$-dimensional Riemannian manifold, with $5\leq n\leq
7$ or $n\geq 8$ and locally conformally flat around some point $p\in M$. If
$Y([g])\geq 0$ and $(M,g)$ admits a conformal metric with semi-positive
$Q$-curvature, then the mass of $G_{P_{g}}$ at $p$ is non-negative and
vanishes if and only if $(M,g)$ is conformal to the standard sphere.

The above theorem is exactly the positive mass theorem used by Hang-Yang in
[23] to solve the $Q$-curvature prescription problem in dimensions $5\leq
n\leq 7$ or $n\geq 5$ and $M$ locally conformally flat around a point. This
highlights the potential parallel of energy $\mathcal{E}(g)$ in the analysis
of fourth-order problems to the role of the ADM energy in classical second
order geometric problems.

Finally, along the lines of the remarks of the previous paragraphs, we will
analyse the critical four-dimensional case and provide an independent proof a
Theorem B in [18] appealing to the techniques derived in this paper.
Concretely, the following theorem follows from our analysis:

######  Theorem C (Gursky).

Let $(M^{4},g)$ a $4$-dimensional manifold with $Y([g])\geq 0$, then
$\kappa_{g}\leq 16\pi^{2}$ with equality holding iff $(M^{4},g)$ is conformal
to the standard sphere.

We would like to highlight that, in this case, our techniques make contact
with the positive mass theorem of 2-dimensional manifolds, as presented in
[27].

Finally, let us comment that, with the aim of delivering a self-contained
presentation but trying to avoid a long introduction to preliminary results
concerning analysis on AE-manifolds, $Q$-curvature analysis and constructions
concerning conformal normal coordinates, which are things very well-known for
experts in each of these fields, we have compiled the main results which will
be used in this paper is the Appendices A,B and C, where detailed references
can be found.

It has been brought to our attention by Marc Herzlich that one of his
students, Benoît Michel, has previously analysed a similar notion of energy,
see [34].

Acknowledgments: The authors would like to thank the CAPES-COFECUB, CAPES-PNPD
and ANR (ANR- 18-CE40-002) for their financial support. Also, we would like
thank the comments, suggestions and critics made by an anonymous referee to
this paper, which have helped us improve the presentation and content of the
paper.

##  2 Preliminaries

In this section we will collect some definitions and notational conventions
which will be used along the paper.

Some notational conventions:

  * •

$(M^{n},g)$ will denote an $n$-dimensional Riemannian manifold.

  * •

Given a Riemannian manifold $(M,g)$, we will denote by
$\Delta_{g}u=g^{ij}\nabla_{i}\nabla_{j}u$ the negative Laplacian.

  * •

Given a Riemannian manifold $(M,g)$ we denote by $\nabla$ its Riemannian
connection and define the curvature tensor by

$\displaystyle
R(V,W)Z=\nabla_{V}\nabla_{W}Z-\nabla_{W}\nabla_{V}Z-\nabla_{[V,W]}Z,\;\;V,W,Z\in\Gamma(TM),$

and we label its components in some coordinate system $\\{x^{i}\\}_{i=1}^{n}$
via $R^{i}_{jkl}=dx^{i}\left(R(\partial_{k},\partial_{l})\partial_{j}\right)$.
Then, the Ricci tensor is defined locally via the contraction
$R_{jl}=R^{k}_{jkl}$.

  * •

Given a tensor field $T$, indices of $\nabla^{k}T$ resulting from covariant
differentiation will be separated by a comma. That is, if $T_{ij}$ are the
components of a $2$-tensor, then we denote the components of $\nabla^{2}T$ by
$T_{ij,kl}$.

  * •

Given $\Sigma\hookrightarrow M$ an embedded compact hypersuface in $(M,g)$, we
will denote by $\nu$ the outward pointing $g$-unit normal to $\Sigma$.

  * •

Given $\Sigma\hookrightarrow\mathbb{R}^{n}$ an embedded compact hypersuface,
we will denote by $\nu^{e}$ Euclidean outward pointing unit normal to
$\Sigma$.

  * •

$\omega_{n-1}$ will denote the volume of the unit sphere
$\mathbb{S}^{n-1}\hookrightarrow\mathbb{R}^{n}$ and $d\omega_{n-1}$ is its
canonical volume measure.

  * •

Given a compact manifold $(M,g)$, the conformal class of $g$ is
$\\{ug\;;\;u\in C^{\infty}(M,\mathbb{R}_{+}^{*})\\}$, denoted $[g]$.

  * •

Given a compact manifold $(M,g)$, its Yamabe invariant is defined as

$Y([g])=\inf_{\tilde{g}}\frac{\int_{M}R_{\tilde{g}}\,dv_{\tilde{g}}}{\mathrm{vol}(M)^{\frac{n-2}{2}}}.$

  * •

Given a function $f\in C^{k}(\mathbb{R}^{n})$, we say that
$f=O_{k}(|x|^{\tau})$ for some $k\geq 0$ and $\tau\in\mathbb{R}$ if for any
multi-index $\alpha$ with $|\alpha|\leq k$, the functions
$\partial^{\alpha}f=O(|x|^{\tau-|\alpha|})$ either as $|x|\rightarrow\infty$
or $|x|\rightarrow 0$, depending on the context.

######  Definition 1 (Weighted spaces).

Given $x\in\mathbb{R}^{n}$, let us define $r(x)\doteq|x|$,
$\sigma(x)\doteq(1+r^{2}(x))^{\frac{1}{2}}$ and consider
$\delta\in\mathbb{R}$. Then, we set

  * •

$L^{p}_{\delta}(\mathbb{R}^{n})=\left\\{u\in
L^{p}_{loc}\,|\,\int_{\mathbb{R}^{n}}|u|^{p}\sigma^{-\delta
p-n}\,dx<+\infty\right\\}$, we equip this set with the norm
$\|u\|_{p,\delta}^{p}=\int_{\mathbb{R}^{n}}|u|^{p}\sigma^{-\delta p-n}\,dx$.

  * •

$L^{\infty}_{\delta}(\mathbb{R}^{n})=\left\\{u\in
L^{\infty}_{loc}\,|\,\sup_{\mathbb{R}^{n}}|u|\sigma^{-\delta}<+\infty\right\\}$,
we equip this set with the norm
$\displaystyle\|u\|_{\infty,\delta}=\sup_{\mathbb{R}^{n}}|u|\sigma^{-\delta}$.

  * •

$W^{k,p}_{\delta}(\mathbb{R}^{n})=\left\\{u\in
W^{k,p}_{loc}\,|\,\sum_{j=0}^{k}\|d^{j}u\|_{p,\delta-j}<+\infty\right\\}$, we
equip this set with the norm
$\|u\|_{k,p,\delta}=\sum_{j=0}^{k}\|d^{j}u\|_{p,\delta-j}$.

In particular, by Sobolev embedding, when $kp>n$ if $u\in
W^{k,p}_{\delta}(\mathbb{R}^{n})$ then $u=O(r^{\delta})$ at infinity. See [4]
for more details about weighted spaces.

Then we introduce the notion of asymptotically flat manifolds.

######  Definition 2 (AE manifolds).

A complete Riemannian manifold $(M,g)$ with $g\in W^{k,q}_{loc}(M)$ for some
$k\geq 1$ and $q>n$ is said to be asymptotically Euclidean (with one end) if
there exists a compact set $K\subset M$ and a diffeomorphism $\phi:M\setminus
K\rightarrow\mathbb{R}^{n}\setminus\overline{B_{1}(0)}$ such that

  * •

$\phi_{*}(g)$ is a uniformly positive defined metric, i.e. there exists
$\lambda>1$ such that

$\frac{1}{\lambda}|\xi|^{2}\leq
g_{ij}(x)\xi^{i}\xi^{j}\leq\lambda|\xi|^{2}\;\forall
x\in\mathbb{R}^{n}\setminus B_{1}\;\forall\xi\in\mathbb{R}^{n}.$

  * •

$\phi_{*}(g)_{ij}-\delta_{ij}\in W^{k,q}_{-\tau}(\mathbb{R}^{n}\setminus
B(0,1))$

for some $\tau>0$ called the decreasing rate.

Important remark:  In this chart we defined $\sigma$ and we remark that the
definition of $L^{q}_{\delta}$ is independent of the chart. But
$W^{k,q}_{\delta}$ depends on the chart $\phi$, since the partial derivatives
will depend on the choice of coordinates. It will be denoted
$W^{k,q}_{\delta}(\phi)$. In the rest of the section, we will call such a
chart a structure at infinity. Once the structure at infinity is chosen, we
naturally extend the definition of $W^{k,q}_{\delta}(\mathbb{R}^{n}\setminus
B(0,1))$ to $W^{k,q}_{\delta}(M)$ on the compact part, noticing that all
choices of charts in the compact region define equivalent norms.

In the following definition, which will be used in the core of this paper, we
will restrict to AE manifold which posses more regularity than that of the
general definition given above.

######  Definition 3.

We will say that a (smooth) AE manifold $(M,g)$ is of order $\tau>0$ with
respect to some structure at infinity $\Phi:M\backslash
K\mapsto\mathbb{R}^{n}\backslash\bar{B}$, if, in such coordinates,
$\Phi_{*}g_{ij}-\delta_{ij}=O_{4}(|x|^{-\tau})$.

Of course we can define asymptotically flat structures with multiple ends. But
since analysis phenomena are determined by the behaviour at infinity, it is
very easy to isolate each end and to consider that there is only one.

##  3 The positive energy theorem

The main result of this section will be a positive energy theorem related to
(4) with its corresponding rigidity statement. But, before this, we will begin
by analysing geometric conditions under which (4) is well-defined. In the
following proposition, we will establish such geometric criteria. Let us first
set

$\tau_{n}=\left\\{\begin{array}[]{ll}0&\hbox{ if }n=3,4\\\
\frac{n}{2}-2&\hbox{ if }n\geq 5\end{array}\right..$

######  Proposition 1.

Let $(M^{n},g)$ be an AE manifold of dimension $n\geq 3$ satisfying the
following conditions

  1. i)

There are end rectangular coordinates, given by a structure of infinity
$\Phi$, where $g_{ij}=\delta_{ij}+O_{4}(r^{-\tau})$, where $\tau>\tau_{n}$;

  2. ii)

The $Q$-curvature of $g$ is in $L^{1}(M,dV_{g})$.

Then, given an exhaustion of $M$ by compact sets $\Omega_{k}$ such that
$S_{k}\doteq\Phi(\partial\Omega_{k})$ are smooth connected $(n-1)$-dimensional
manifolds without boundary in $\mathbb{R}^{n}$ satisfying

$\displaystyle\begin{split}R_{k}\doteq\inf\\{|x|:x\in
S_{k}\\}\xrightarrow[k\rightarrow\infty]{}\infty,\\\
R_{k}^{n-1}\mathrm{area}(S_{k})\text{ is bounded as
}k\rightarrow\infty,\end{split}$ (6)

the limit

$\displaystyle\mathcal{E}^{(\Phi)}(g)=\lim_{k\rightarrow\infty}\int_{S_{k}}\left(\partial_{j}\partial_{i}\partial_{i}g_{aa}-\partial_{j}\partial_{u}\partial_{i}g_{ui}\right)\nu^{j}dS,$
(7)

exists and is independent of the sequence of $\\{S_{k}\\}$ used to compute it.

###### Proof.

From the decaying conditions and the familiar expression

$\displaystyle R_{g}$
$\displaystyle=\partial_{i}\partial_{j}g_{ij}-\partial_{j}\partial_{j}g_{ii}+O_{2}((g-\delta)\partial^{2}g)+O_{2}(\left(\partial
g\right)^{2}),$
$\displaystyle=\partial_{i}\partial_{j}g_{ij}-\partial_{j}\partial_{j}g_{ii}+O_{2}(r^{-2\tau-2})$

we get

$\displaystyle\Delta_{g}R_{g}$
$\displaystyle=g^{ab}\nabla_{a}\nabla_{b}R_{g}=g^{ab}\left(\partial_{a}\partial_{b}R_{g}+\Gamma^{c}_{ab}\partial_{c}R_{g}\right),$
$\displaystyle=g^{ab}\partial_{a}\partial_{b}\Big{(}\partial_{i}\partial_{j}g_{ij}-\partial_{j}\partial_{j}g_{ii}\Big{)}+g^{ab}\partial_{a}\partial_{b}\Big{(}O_{2}(r^{-2\tau-2})\Big{)}+g^{ab}\Gamma^{c}_{ab}\partial_{c}\left(\partial_{i}\partial_{j}g_{ij}-\partial_{j}\partial_{j}g_{jj}\right)$
$\displaystyle+g^{ab}\Gamma^{c}_{ab}\partial_{c}\left(O_{2}\left(r^{-2\tau-2}\right)\right),$
$\displaystyle=g^{ab}\partial_{a}\partial_{b}\Big{(}\partial_{i}\partial_{j}g_{ij}-\partial_{j}\partial_{j}g_{ii}\Big{)}+\left(g^{ab}-\delta^{ab}\right)O(r^{-2\tau-4})+O(r^{-2\tau-4})$
$\displaystyle+O_{3}\left((g-\delta)\partial
g\right))O_{1}(r^{-\tau-3})+O_{3}(\partial
g)O_{1}(r^{-\tau-3})+O_{3}\left((g-\delta)\partial
g\right))O_{1}(r^{-2\tau-3})$ $\displaystyle+O_{3}(\partial
g)O_{1}(r^{-2\tau-3}),$
$\displaystyle=\partial_{a}\partial_{a}\Big{(}\partial_{i}\partial_{j}g_{ij}-\partial_{j}\partial_{j}g_{ii}\Big{)}+O(r^{-2\tau-4}).$

In particular, denoting by $D_{k}$ the annular region between $S_{k+1}$ and
$S_{k}$, we get that

$\displaystyle\int_{D_{k}}\partial_{j}\partial_{j}\left(\partial_{u}\partial_{i}g_{ui}-\partial_{u}\partial_{u}g_{ii}\right)dV_{e}$
$\displaystyle=\int_{S_{k+1}}(\partial_{j}\partial_{u}\partial_{i}g_{ui}-\partial_{j}\partial_{u}\partial_{u}g_{ii})\nu^{e}_{j}dS^{n-1}$
$\displaystyle-\int_{S_{k}}(\partial_{j}\partial_{u}\partial_{i}g_{ui}-\partial_{j}\partial_{u}\partial_{u}g_{ii})\nu^{e}_{j}dS^{n-1},$

where, following our convention, $\nu^{e}$ denotes the outward pointing
Euclidean unit normal to each hypersurface. Therefore, if the left-hand side
is integrable over $\mathbb{R}^{n}\backslash K$, then the above boundary
integrals form a Cauchy sequence as $r\rightarrow\infty$ and therefore (7) is
well defined. But from the above computations, we see that this can be reduced
to $\Delta_{g}R_{g}\in L^{1}(M)$ and $2(\tau+2)>n$, that is $\tau>\tau_{n}$.
In particular, notice that under our decaying conditions
$R_{g},\mathrm{Ric}_{g}=O(r^{-\tau-2})$ near infinity, implying that

$\displaystyle Q_{g}=-\frac{1}{2(n-1)}\Delta_{g}R_{g}+O(r^{-2\tau-4}),$

therefore the leading order is carried on the first term and
$\Delta_{g}R_{g}\in L^{1}(M)$ can be replaced by $Q_{g}\in L^{1}(M)$. ∎

Now, we intend to show that (4) is a geometric object, independent of the
structure of infinity we use. With this in mind, let us start by rewriting (4)
in more geometric fashion, which will be proved to be more useful for its
analysis.

######  Proposition 2.

Let $(M,g)$ be an AE manifold which satisfies the decaying conditions i) and
ii) of Proposition 1. Then, we can rewrite the energy (4) as

$\displaystyle\mathcal{E}(g)$
$\displaystyle=-\lim_{r\rightarrow\infty}\int_{S^{n-1}_{1}}\partial_{r}R_{g}r^{n-1}d\omega_{n-1}.$
(8)

###### Proof.

Let us denote by $\nu$ the outward $g$-unit normal to
$S^{n-1}_{r}\hookrightarrow\mathbb{R}^{n}$, with $r$ sufficiently large, and
by $\nu^{e}$ the outward euclidean unit normal to the same sphere. Then, it
follows that $|\nu^{e}-\nu|_{g}=O(r^{-\tau})$. Thus, we find that

$\displaystyle g(\nabla R_{g},\nu)$ $\displaystyle=g(\nabla
R_{g},\nu^{e})+g(\nabla R_{g},\nu-\nu^{e}),$
$\displaystyle=\partial_{i}R_{g}\nu^{e}_{i}+\underbrace{(g-\delta)_{ij}\nabla^{i}R_{g}(\nu^{e})^{j}}_{O_{1}(r^{-2\tau-3})}+\underbrace{\partial_{i}R_{g}(\nu-\nu^{e})_{i}}_{O_{1}(r^{-2\tau-3})}+\underbrace{(g-\delta)_{ij}\nabla^{i}R_{g}(\nu-\nu^{e})^{j}}_{O_{1}(r^{-3\tau-3})}.$

Also, we already know that
$R_{g}=\partial_{i}\partial_{j}g_{ij}-\partial_{j}\partial_{j}g_{ii}+O_{2}(r^{-2\tau-2})$,
which implies

$\displaystyle g(\nabla R_{g},\nu)$
$\displaystyle=\partial_{i}\left(\partial_{k}\partial_{j}g_{kj}-\partial_{j}\partial_{j}g_{kk}\right)\frac{x^{i}}{r}+O_{1}(r^{-2\tau-3})$

Thus,

$\displaystyle\int_{S^{n-1}_{1}}g(\nabla R_{g},\nu)r^{n-1}d\omega_{n-1}$
$\displaystyle=\int_{S^{n-1}_{1}}\partial_{i}\left(\partial_{k}\partial_{j}g_{kj}-\partial_{j}\partial_{j}g_{kk}\right)\frac{x^{i}}{r}r^{n-1}d\omega_{n-1}+O_{1}(r^{n-2\tau-4}).$

Now, from our decaying condition $\tau>\tau_{n}$, we see that $n-2\tau-4<0$,
which implies that in the limit $r\rightarrow\infty$ the last term vanishes.
Therefore, under these conditions

$\displaystyle\lim_{r\rightarrow\infty}\int_{S^{n-1}_{1}}g(\nabla
R_{g},\nu)r^{n-1}d\omega_{n-1}$
$\displaystyle=\lim_{r\rightarrow\infty}\int_{S^{n-1}_{1}}\partial_{i}\left(\partial_{k}\partial_{j}g_{kj}-\partial_{j}\partial_{j}g_{kk}\right)\nu^{e}_{i}r^{n-1}d\omega_{n-1}.$

That is,

$\displaystyle\begin{split}\mathcal{E}(g)&=-\lim_{r\rightarrow\infty}\int_{S^{n-1}_{1}}g(\nabla
R_{g},\nu)r^{n-1}d\omega_{n-1},\\\
&=-\lim_{r\rightarrow\infty}\int_{S^{n-1}_{1}}\nabla
R_{g}\cdot\nu^{e}r^{n-1}d\omega_{n-1},\\\
&=-\lim_{r\rightarrow\infty}\int_{S^{n-1}_{1}}\partial_{r}R_{g}r^{n-1}d\omega_{n-1}.\end{split}$
(9)

∎

With the aid of the above two results, we can establish the following theorem,
which establishes that $\mathcal{E}(g)$ is a geometric object within a
suitable class of AE manifolds.

######  Theorem 1.

Let $(\phi,x)$ and $(\psi,y)$ be two structures of infinity for the AE
manifold $(M,g)$ which satisfies the conditions i) and ii) of the above
Proposition with decay rates $\tau_{1},\tau_{2}$ respectively, satisfying
$\tau\doteq\min\\{\tau_{1},\tau_{2}\\}>\tau_{n}$. Then, the energies
$\mathcal{E}^{(\phi)}(g)$ and $\mathcal{E}^{(\psi)}(g)$ are well-defined and
equal. Moreover, the following coordinate independent identity follows:

$\displaystyle\mathcal{E}(g)=-\int_{M}\Delta_{g}R_{g}dV_{g}.$ (10)

###### Proof.

Appealing to Proposition 1, let us compute the energy $\mathcal{E}^{\phi}(g)$,
associated to the structure of infinity given by $(\phi,x)$ using a sequence
of spheres near infinity of radii $\\{r_{k}\\}_{k=1}^{\infty}$. From the above
proposition, we have that

$\displaystyle\mathcal{E}^{(\phi)}(g)=-\lim_{|x|\rightarrow\infty}\int_{S^{n-1}_{1}}\nabla
R_{g}\lrcorner dV_{e},$

where $dV_{e}=r^{n-1}dr\wedge d\omega_{n-1}$ stands for the canonical
Euclidean volume form. Appealing to the decaying conditions of $g$, we know
that $\sqrt{\mathrm{det}(g)}=1+O(r^{-\tau})$ and therefore

$\displaystyle\int_{S^{n-1}_{1}}\nabla R_{g}\lrcorner
dV_{g}=\int_{S^{n-1}_{1}}\nabla R_{g}\lrcorner dV_{e}+O(r^{n-2\tau-4}).$

Since $n-2\tau-4<0$ under our hypotheses, the above implies that

$\displaystyle\mathcal{E}^{(\phi)}(g)=-\lim_{r\rightarrow\infty}\int_{S^{n-1}_{r}}\nabla
R_{g}\lrcorner dV_{g}.$

Finally, since $\mathrm{div}_{g}(\nabla R_{g})dV_{g}=d(\nabla R_{g}\lrcorner
dV_{g})$, we find that

$\displaystyle\mathcal{E}^{(\phi)}(g)=-\lim_{r\rightarrow\infty}\int_{D_{r}}\Delta_{g}R_{g}dV_{g}=-\int_{M}\Delta_{g}R_{g}dV_{g},$

where $D_{r}\hookrightarrow M$ is the inner region in $M$ with $\partial
D_{r}=S_{r}^{n-1}$. In particular, the right-hand side of the above expression
is independent of the coordinates used near infinity. Thus we can drop the
reference to $(\phi,x)$ and claim that for any structure of infinity of $M$
were $g$ satisfies conditions (i)-(ii) of Proposition 1, it holds that

$\displaystyle\mathcal{E}(g)=-\int_{M}\Delta_{g}R_{g}dV_{g}.$

∎

######  Remark 1.

Let us highlight that the form for $\mathcal{E}$ given by (10) allows us to
make contact with the ideas of [35], where the author introduced a general
method to produce _asymptotic charges_ from a Riemannian functional on a
complete manifold with a model asymptotic structure. In particular, (10) is an
example of such a functional. Let us nevertheless stress that the definition
of energy we are interested in is the one (a priori) defined by equation (4),
which is the quantity appearing as a conserved charge for the fourth order
gravitational theories described in the introduction. In this sense, the
connection between (4) and (10) is only revealed after the work done above
through Propositions 1, 2 and Theorem 1, after which the conditions necessary
to achieve asymptotic invariance have already been shown explicitly. Such
conditions correspond to a somewhat simpler and explicit formulation of the
corresponding abstract conditions presented in [35], which must be checked in
a case by case basis.

Besides establishing sufficient conditions for the energy to be a well-defined
intrinsic geometric object, the above theorem provides us with an easy
positive energy corollary.

######  Corollary 1.

Let $(M^{n},g)$ be an $n$-dimensional AE manifold, with $n\geq 3$, which
satisfies the decaying conditions i) and ii) of Proposition 1 and such that
$\Delta_{g}R_{g}\leq 0$. Then, the fourth order energy $\mathcal{E}(g)$ is
non-negative and $\mathcal{E}(g)=0$ if and only if $(M,g)$ is scalar flat.

The positivity statement in the above corollary is self-evident. On the other
hand, the rigidity statement follows from the injectivity of the Laplacian
under our hypotheses (see (A.1)).

Let us notice that the kind of rigidity that we get from the above corollary
is quite weak. In fact, although it is well-known that topological
obstructions exist, the set of AE manifolds with zero scalar curvature is in
general quite large (see, for instance, the discussion after Theorem 2).
Therefore, in order to get a stronger rigidity statement, we will need to
impose so other geometric condition. We will explore this in the next
proposition and in the main theorem given below. The following proposition,
which is an adaptation of a result in [20] will be used to insure that
$R_{g}>0$ as long as $g$ is not flat.

######  Proposition 3.

Let $(M^{n},g)$ be an AE Riemannian manifold with $n\geq 3$ satisfying (1)
$Q_{g}\geq 0$ and (2) $R_{g}\geq 0$. Then, either $R_{g}>0$ or $g$ must be
flat.

###### Proof.

If $Q_{g}\geq 0$, then we find that

$\displaystyle\Delta_{g}R_{g}-c_{1}(n)R^{2}_{g}\leq-
c_{2}(n)|\mathrm{Ric}_{g}|^{2}_{g}\leq 0,$

for some constants $c_{1}(n),c_{2}(n)>0$. From the the condition $R_{g}\geq 0$
we can appeal to Lemma 4 in [33] to conclude that if $R_{g}$ vanishes at a
single point, then it must be identically zero. Thus, we already see that
either $R_{g}>0$ or $R_{g}\equiv 0$. In the second case, we find that $0\leq
Q_{g}=-c_{2}(n)|\mathrm{Ric}_{g}|^{2}_{g}\leq 0$, and thus
$\mathrm{Ric}_{g}\equiv 0$, which implies (from asymptotic flatness) that $g$
is flat. Indeed, since $\mathrm{Ric}_{g}\geq 0$, then by Bishop-Gromov theorem
insures that $r\mapsto\frac{vol(B_{g}(p,r))}{\omega_{r}r^{n}}$ is non-
increasing, hence it must be constant by asymptotic flatness, finally the
equality case of Bishop-Gromov theorem implies that $g$ must be flat. ∎

We will now present the main result of this paper. We should highlight that,
in particular, the proof of the rigidity statement in the following theorem
follows ideas close to the proofs of the positive mass theorems associated to
the Paneitz operator of [26, 20, 21].

######  Theorem 2.

Let $(M^{n},g)$ be an $n$-dimensional AE manifold, with $n\geq 3$, which
satisfies the decaying conditions i) and ii) of Proposition 1 and such that
$Q_{g},R_{g}\geq 0$. Then, the fourth order energy $\mathcal{E}(g)$ is non-
negative and $\mathcal{E}(g)=0$ if and only if $Q_{g}\equiv 0$ and $(M,g)$ is
isometric to $(\mathbb{R}^{n},\cdot)$.

###### Proof.

Let us first notice that we need only work with the case $Q_{g}\geq 0$ and
$R_{g}>0$, since in the remaining case the result follows from Proposition 3.
Thus, in what follows we assume that $Q_{g}\geq 0$ and $R_{g}>0$.

Proof of positivity

Let us start by noticing that under our hypotheses we have
$R_{g}=O_{2}(r^{-\tau-2})$ with $\tau>\tau_{n}$. Then, the energy exists and
is independent of the sequence of spheres used to compute it. Furthermore,
these decaying conditions imply that (8) also holds. We can rewrite

$\displaystyle\int_{S^{n-1}_{1}}\partial_{r}R_{g}r^{n-1}d\omega_{n-1}$
$\displaystyle=\frac{\partial}{\partial
r}\left(r^{n-1}\int_{S^{n-1}_{1}}R_{g}d\omega_{n-1}\right)-(n-1)r^{n-2}\int_{S^{n-1}_{1}}R_{g}d\omega_{n-1}.$

Define

$\displaystyle h(r)\doteq r^{n-2}\int_{S^{n-1}_{1}}R_{g}d\omega_{n-1}.$ (11)

Then,

$\displaystyle\int_{S^{n-1}_{1}}\partial_{r}R_{g}r^{n-1}d\omega_{n-1}$
$\displaystyle=\frac{\partial}{\partial r}\left(rh(r)\right)-(n-1)h(r).$ (12)

By assumption we have that $h>0$ and from our decreasing assumption that there
is some $r_{0}$ sufficiently large, so that $R_{g}(x)\leq Cr^{-\tau-2}(x)$ for
all $x\in M$ such that $r(x)>r_{0}$. In particular, this last condition
implies that $h(r)\leq C\omega_{n-1}r^{-\tau+n-4}$ for all $r>r_{0}$.

Suppose that there is some $r^{*}>r_{0}$ such that (12) is positive at
$r^{*}$. Then, by continuity, there is some interval
$(r^{*}-\epsilon,r^{*}+\epsilon)$ so that

$\displaystyle rh^{\prime}(r)>(n-2)h(r),$

which implies

$\displaystyle\log\left(\frac{h(r)}{h(r^{*})}\right)>(n-2)\log\left(\frac{r}{r^{*}}\right),\text{
for any }r\in(r^{*},r^{*}+\epsilon).$

That is $h(r)>\frac{h(r^{*})}{{r^{*}}^{n-2}}r^{n-2}$ for all
$r\in(r^{*},r^{*}+\epsilon)$. But notice that also $h(r)\leq
C\omega_{n-1}r^{-\tau+n-4}$ for all such $r$, which shows that

$\displaystyle\frac{h(r^{*})}{{r^{*}}^{n-2}}r^{n-2}<h(r)\leq
C\omega_{n-1}r^{-\tau+n-4}\text{ for any }r\in(r^{*},r^{*}+\epsilon).$

Since $r^{*}$ and $C$ are fixed, the above inequalities are clearly falsified
for large enough $r$, since $r^{n-2}$ grows faster that $r^{n-4-\tau}$. This,
in turn, implies that the interval $I=[r^{*},r^{*}+\epsilon)$ where (12) is
positive cannot be extended indefinitely, since that would force us into the
above inequalities being valid for arbitrary large $r>r^{*}$, presenting a
contradiction. Therefore, this implies that there must be some
$\epsilon_{max}$ where (12) is non-positive at $r^{*}+\epsilon_{max}$. That
is, there must be some $r_{1}>r^{*}$ where

$\displaystyle
r_{1}^{n-1}\int_{S^{n-1}_{1}}\partial_{r}R_{g}(r_{1},\theta)d\omega_{n-1}(\theta)$
$\displaystyle=\left(\frac{\partial}{\partial
r}\left(rh(r)\right)-(n-1)h(r)\right)|_{r=r_{1}}\leq 0.$ (13)

The idea is now to repeat this procedure so as to select a sequence of spheres
$\\{S^{n-1}_{r_{i}}\\}_{i=1}^{\infty}$ along which (12) is manifestly non-
positive. The fact that such radii can be chosen under our hypotheses is a
consequence of the above argument. Then

$\displaystyle\mathcal{E}(g)_{j}=-r_{j}^{n-1}\int_{S_{r_{j}}}\partial_{r}R_{g}d\omega_{n-1}\geq
0\text{ for all }j,$

which implies

$\displaystyle\mathcal{E}(g)=-\lim_{j\rightarrow\infty}\int_{S_{r_{j}}}\partial_{r}R_{g}r^{n-1}d\omega_{n-1}\geq
0.$ (14)

Proof of rigidity

Now, the idea is to analyze whether $\mathcal{E}(g)=0$ implies $g$-flatness.
First we look for a conformal metric $\tilde{g}=u^{\frac{4}{n-2}}g$ such that
$R_{\tilde{g}}\equiv 0$, that is,

$\displaystyle L_{g}(u)=\Delta_{g}u-c_{n}R_{g}u=0,$ (15)

where $c_{n}=\frac{n-2}{4(n-1)}$. Setting $u=1+\phi$, this is equivalent to
solve

$\displaystyle\Delta_{g}\phi-c_{n}R_{g}\phi=c_{n}R_{g},$ (16)

We know that $R_{g}=O_{2}(r^{-\tau-2})$, this implies that $R_{g}\in
L^{p}_{-\delta-2}$ for any $\delta<\tau$ and any $p$. Thanks to corollary A.1,
we know that $L_{g}:W^{2,p}_{\rho}\mapsto L^{p}_{\rho-2}$ is an isomorphism
for $2-n<\rho<0$. Thus, choosing $p>n/2$, we find a (smooth) solution $\phi\in
W^{2,p}_{-\delta}$ for any $0<\delta<\sigma\doteq\min\\{\tau,n-2\\}$.333In the
case $\tau>n-2$, we can achieve $\delta=\sigma$.. By our choice of $p$, $\phi$
goes to zero at infinity, then, the fact that $R_{g}>0$ and the maximum
principle, see lemma 2 and 4 [33], insure that $u=1+\phi>0$.

I) $n\not=4$

Then, let $\Phi\doteq u^{-\frac{n-4}{n-2}}$; rewrite
$g=\Phi^{\frac{4}{n-4}}\tilde{g}$ and notice that transformation rule for the
Paneitz operator gives us

$\displaystyle\frac{n-4}{2}\Phi^{\frac{n+4}{n-4}}Q_{g}=P_{\tilde{g}}\Phi.$

Also, since $R_{\tilde{g}}=0$, we have that
$Q_{\tilde{g}}=-\frac{2}{(n-2)^{2}}|\mathrm{Ric}_{\tilde{g}}|^{2}_{\tilde{g}}$,
and thus

$\displaystyle
P_{\tilde{g}}\Phi=\Delta^{2}_{\tilde{g}}\Phi+\mathrm{div}_{\tilde{g}}\left(4S_{\tilde{g}}(\nabla\Phi,\cdot)\right)-\frac{(n-4)}{(n-2)^{2}}|\mathrm{Ric}_{\tilde{g}}|^{2}_{\tilde{g}}\Phi,$
(17)

where, $S_{\tilde{g}}=\frac{1}{n-2}\mathrm{Ric}_{\tilde{g}}$ since
$R_{\tilde{g}}=0$. Therefore, denoting by $D_{r}$ the bounded region in $M$
whose boundary is given by the sphere $S_{r}$ in the end of $M$, we see that

$\displaystyle\int_{D_{r}}\frac{n-4}{2}\Phi^{\frac{n+4}{n-4}}Q_{g}+\frac{(n-4)}{(n-2)^{2}}|\mathrm{Ric}_{\tilde{g}}|^{2}_{\tilde{g}}\Phi\,dv_{\tilde{g}}=\int_{S_{r}}\tilde{g}(\tilde{\nabla}\Delta_{\tilde{g}}\Phi,\tilde{\nu})d\omega_{\tilde{g}}+\frac{4}{n-2}\int_{S_{r}}\mathrm{Ric}_{\tilde{g}}(\nabla\Phi,\tilde{\nu})d\omega_{\tilde{g}}$
(18)

Let us now estimate the terms in the right-hand side.

Estimates on $\mathrm{Ric}_{\tilde{g}}(\nabla\Phi,\tilde{\nu})$

From the above analysis we find that $u=1+O_{4}(r^{-\delta})$ for any
$\delta<\sigma=\min\\{\tau,n-2\\}$, and therefore

$\displaystyle\tilde{g}_{ij}$
$\displaystyle=(1+O_{4}(r^{-\delta}))(\delta_{ij}+O_{4}(r^{-\tau})),$
$\displaystyle=\delta_{ij}+O_{4}(r^{-\delta}).$

Then, $\mathrm{Ric}_{\tilde{g}}=O_{2}(r^{-\delta-2})$ and also

$\displaystyle\nabla\Phi=-\frac{n-4}{n-2}u^{-\frac{n-4}{n-2}-1}\nabla
u=O_{3}(r^{-\delta-1}).$

Which implies

$\displaystyle\mathrm{Ric}_{\tilde{g}}(\nabla\Phi,\tilde{\nu})$
$\displaystyle=\mathrm{Ric}_{\tilde{g}}(\nabla\Phi,\nu^{e})+{\mathrm{Ric}_{\tilde{g}}}_{ij}\nabla^{i}\Phi(\nu^{e}-\tilde{\nu})^{j}=O_{2}(r^{-2\delta-3}),$
$\displaystyle\mathrm{Ric}_{\tilde{g}}(\nabla\Phi,\tilde{\nu})r^{n-1}$
$\displaystyle=O_{2}(r^{n-2\delta-4}).$

Since $\sigma>\frac{n}{2}-2$, then, we can choose $\delta$ such that

$\displaystyle|\mathrm{Ric}_{\tilde{g}}(\nabla\Phi,\tilde{\nu})|r^{n-1}\rightarrow
0\hbox{ as }r\rightarrow+\infty.$

Estimates on $\Delta_{\tilde{g}}\Phi$

Let us rewrite

$\displaystyle\tilde{\nabla}_{i}\Phi$
$\displaystyle=-\frac{n-4}{n-2}u^{-2\frac{n-3}{n-2}}\tilde{\nabla}_{i}u,$
$\displaystyle\tilde{\nabla}_{j}\tilde{\nabla}_{i}\Phi$
$\displaystyle=-\frac{n-4}{n-2}\left(u^{-2\frac{n-3}{n-2}}\tilde{\nabla}_{j}\tilde{\nabla}_{i}u-2\frac{n-3}{n-2}u^{-\frac{3n-8}{n-2}}\tilde{\nabla}_{j}u\tilde{\nabla}_{i}u\right),$

from which we get

$\displaystyle\Delta_{\tilde{g}}\Phi$
$\displaystyle=-\frac{n-4}{n-2}u^{-2\frac{n-3}{n-2}}\Delta_{\tilde{g}}u+2\frac{(n-4)(n-3)}{(n-2)^{2}}u^{-\frac{3n-8}{n-2}}|\tilde{\nabla}u|^{2}_{\tilde{g}}.$

Now, we can compute that

$\displaystyle\Delta_{\tilde{g}}u$
$\displaystyle=\tilde{g}^{ij}\tilde{\nabla}_{j}\nabla_{i}u=\tilde{g}^{ij}\left(\nabla_{j}\nabla_{i}u-(\Gamma^{k}_{ij}(\tilde{g})-\Gamma^{k}_{ij}(g))\nabla_{k}u\right),$
$\displaystyle=\Delta_{g}u+(\tilde{g}^{ij}-{g}^{ij})\nabla_{j}\nabla_{i}u-\tilde{g}^{ij}(\Gamma^{k}_{ij}(\tilde{g})-\Gamma^{k}_{ij}(g))\nabla_{k}u$

and we can estimate
$\tilde{g}^{ij}-g^{ij}=O_{4}(r^{-\delta})-O_{4}(r^{-\tau})=O_{4}(r^{-\delta})$
and
$\Gamma^{k}_{ij}(\tilde{g})-\Gamma^{k}_{ij}(g)=O_{3}(r^{-\delta-1})-O_{3}(r^{-\tau-1})=O_{3}(r^{-\delta-1})$,
since $\sigma=\min\\{\tau,n-2\\}$, and also $\nabla u=O_{3}(r^{-\delta-1})$.
Therefore

$\displaystyle\begin{split}\tilde{g}^{ij}(\Gamma^{k}_{ij}(\tilde{g})-\Gamma^{k}_{ij}(g))\nabla_{k}u&=O_{3}(r^{-2\delta-2}),\\\
(\tilde{g}^{ij}-{g}^{ij})\nabla_{j}\nabla_{i}u&=O_{2}(r^{-2\delta-2}).\end{split}$
(19)

On the other hand, from (15) we find that444Notice that, actually,
$R_{g}=O_{2}(r^{-\tau-2})$ which is stronger than they the decay we are using,
but for our purposes keeping just one weight parameter given by $\delta$ is
enough to achieve the required estimates.

$\displaystyle\Delta_{g}u=c_{n}R_{g}+c_{n}\underbrace{R_{g}}_{O_{2}(r^{-\delta-2})}\underbrace{\phi}_{O_{4}(r^{-\delta})}=c_{n}R_{g}+O_{2}(r^{-2\delta-2}).$
(20)

Putting together (19)-(20), we finally find that

$\displaystyle\Delta_{\tilde{g}}u$
$\displaystyle=c_{n}R_{g}+O_{2}(r^{-2\delta-2}),$ (21)

which, in turn, implies

$\displaystyle\begin{split}\Delta_{\tilde{g}}\Phi&=-\frac{n-4}{4(n-1)}u^{-2\frac{n-3}{n-2}}R_{g}+O_{2}(r^{-2\delta-2}),\\\
\nabla(\Delta_{\tilde{g}}\Phi)&=-\frac{n-4}{4(n-1)}u^{-2\frac{n-3}{n-2}}\nabla
R_{g}+O_{1}(r^{-2\delta-3}).\end{split}$ (22)

Therefore

$\displaystyle\tilde{g}(\tilde{\nabla}\Delta_{\tilde{g}}\Phi,\tilde{\nu})$
$\displaystyle=\tilde{g}(\tilde{\nabla}\Delta_{\tilde{g}}\Phi,\nu^{e})+\tilde{g}(\tilde{\nabla}\Delta_{\tilde{g}}\Phi,\tilde{\nu}-\nu^{e}),$
$\displaystyle=\nabla_{i}\Delta_{\tilde{g}}\Phi\frac{x^{i}}{r}+\underbrace{(\tilde{g}_{ij}-\delta_{ij})\nabla^{i}(\Delta_{\tilde{g}}\Phi)}_{O_{1}(r^{-2\delta-3})}\frac{x^{j}}{r}+\underbrace{\tilde{g}(\tilde{\nabla}\Delta_{\tilde{g}}\Phi,\tilde{\nu}-\nu^{e})}_{O_{1}(r^{-2\delta-3})},$
$\displaystyle=-\frac{n-4}{4(n-1)}u^{-2\frac{n-3}{n-2}}\nabla_{i}R_{g}\frac{x^{i}}{r}+O_{1}(r^{-2\delta-3}),$

implying

$\displaystyle\tilde{g}(\tilde{\nabla}\Delta_{\tilde{g}}\Phi,\tilde{\nu})r^{n-1}$
$\displaystyle=-\frac{n-4}{4(n-1)}\underbrace{u^{-2\frac{n-3}{n-2}}}_{\xrightarrow[r\rightarrow\infty]{}1}\partial_{r}R_{g}r^{n-1}+\underbrace{O_{1}(r^{n-2\delta-4})}_{\xrightarrow[r\rightarrow\infty]{}0},$

where we have used, as above, that $\delta>\frac{n}{2}-2$, implying we can
choose $\delta>0$ such that $n-2\delta-4<0$. From all this and
$d\omega_{\tilde{g}}=(1+O(r^{-\delta}))d\omega_{r}$, we find that

$\displaystyle\begin{split}\lim_{r\rightarrow\infty}\int_{S^{n-1}_{r}}\tilde{g}(\tilde{\nabla}\Delta_{\tilde{g}}\Phi,\tilde{\nu})d\omega_{r}&=-\frac{n-4}{4(n-1)}\lim_{r\rightarrow\infty}\left(\int_{S^{n-1}_{1}}\partial_{r}R_{g}r^{n-1}d\omega_{n-1}+\int_{S^{n-1}_{1}}\underbrace{\partial_{r}R_{g}O(r^{-\delta})r^{n-1}}_{O(r^{n-2\delta-4})}d\omega_{n-1}\right),\\\
&=\frac{n-4}{4(n-1)}\mathcal{E}(g),\end{split}$

Putting together the above analysis, from (18), we find that

$\displaystyle\int_{M}\frac{n-4}{2}\Phi^{\frac{n+4}{n-4}}Q_{g}+\frac{(n-4)}{(n-2)^{2}}|\mathrm{Ric}_{\tilde{g}}|^{2}_{\tilde{g}}\Phi\,dv_{\tilde{g}}=\frac{n-4}{4(n-1)}\mathcal{E}(g).$
(23)

Thus, if $\mathcal{E}(g)=0$, then $Q_{g}\equiv 0$ and
$\mathrm{Ric}_{\tilde{g}}\equiv 0$, which implies, through results such as
those exposed in the proof of Proposition 3 that $\tilde{g}$ is flat and
$M\cong\mathbb{R}^{n}$. Therefore, since $\tilde{g}=u^{\frac{4}{n-2}}g$, then
$g$ is conformally-flat.

II) $n=4$

Now, fix $\Phi=-\ln(u)$ and rewrite $\tilde{g}=e^{-2\Phi}g$. Then, using the
transformation rule (B.7) we find

$\displaystyle 0\leq Q_{g}=e^{-4\Phi}(P_{\tilde{g}}\Phi+Q_{\tilde{g}}).$ (24)

Since $R_{\tilde{g}}=0$, then
$Q_{\tilde{g}}=-\frac{1}{2}|\mathrm{Ric}_{\tilde{g}}|^{2}_{\tilde{g}}$, which
implies

$\displaystyle
0\leq\int_{D_{r}}\left(Q_{g}e^{4\Phi}+\frac{1}{2}|\mathrm{Ric}_{\tilde{g}}|^{2}_{\tilde{g}}\right)dV_{\tilde{g}}=\int_{S_{r}}\tilde{g}(\tilde{\nabla}\Delta_{\tilde{g}}\Phi,\tilde{\nu})d\omega_{\tilde{g}}+2\int_{S_{r}}\mathrm{Ric}_{\tilde{g}}(\nabla\Phi,\tilde{\nu})d\omega_{\tilde{g}}$
(25)

Estimates on $\mathrm{Ric}_{\tilde{g}}(\nabla\Phi,\tilde{\nu})$

We clearly have $u=1+O_{4}(r^{-\delta})$ for some $\delta<\sigma$. Then, from
$\nabla\Phi=-\frac{1}{u}\nabla u=O_{3}(r^{-\delta-1})$, we find that

$\displaystyle\begin{split}\mathrm{Ric}_{\tilde{g}}(\nabla\Phi,\tilde{\nu})r^{3}&=\underbrace{\mathrm{Ric}_{\tilde{g}}(\nabla\Phi,\nu_{e})}_{O_{3}(r^{-2\delta-3})}r^{3}+\underbrace{\mathrm{Ric}_{\tilde{g}}(\nabla\Phi,\tilde{\nu}-\nu_{e})}_{O_{3}(r^{-3\delta-3})}r^{3}=O_{3}(r^{-2\delta}).\end{split}$
(26)

Combining this with $d\omega_{\tilde{g}}=(1+O(r^{-\delta}))d\omega_{r}$ we get

$\displaystyle\int_{S_{r}}\mathrm{Ric}_{\tilde{g}}(\nabla\Phi,\tilde{\nu})d\omega_{\tilde{g}}=o(1).$
(27)

Estimates on $\tilde{g}(\nabla\Delta_{\tilde{g}}\Phi,\tilde{\nu})$

Straightforwardly we see that

$\displaystyle\begin{split}\tilde{\nabla}_{j}\tilde{\nabla}_{i}\Phi&=-u^{-1}\tilde{\nabla}_{j}\tilde{\nabla}_{i}u+u^{-2}\tilde{\nabla}_{j}u\tilde{\nabla}_{i}u=-u^{-1}\tilde{\nabla}_{j}\tilde{\nabla}_{i}u+O_{3}(r^{-2\delta-2}),\\\
\tilde{\nabla}_{j}\tilde{\nabla}_{i}u&=\nabla_{j}\nabla_{i}u-\Delta\Gamma^{k}_{ji}\nabla_{k}u=\nabla_{j}\nabla_{i}u+O_{3}(r^{-2\delta-2}),\end{split}$
(28)

where $\Delta\Gamma^{k}_{ji}=\Gamma^{k}_{ij}(\tilde{g})-\Gamma^{k}_{ij}(g)$.
Thus, we see that

$\displaystyle\Delta_{\tilde{g}}\Phi$
$\displaystyle=-u^{-1}\Delta_{g}u-u^{-1}\underbrace{(\tilde{g}^{ij}-g^{ij})\nabla_{j}\nabla_{i}u}_{O_{3}(r^{-2\delta-2})}+O_{3}(r^{-2\delta-2})=-u^{-1}\Delta_{g}u+O_{3}(r^{-2\delta-2}),$
$\displaystyle=-c_{n}R_{g}+O_{3}(r^{-2\delta-2}),$

thus,

$\displaystyle\nabla\Delta_{\tilde{g}}\Phi$ $\displaystyle=-c_{n}\nabla
R_{g}+O_{2}(r^{-2\delta-3}),$
$\displaystyle\tilde{g}(\nabla\Delta_{\tilde{g}}\Phi,\tilde{\nu})$
$\displaystyle=-c_{n}\tilde{g}(\nabla
R_{g},\nu_{e})-c_{n}\underbrace{\tilde{g}(\nabla
R_{g},\tilde{\nu}-\nu_{e})}_{O_{1}(r^{-2\delta-3})}+O_{2}(r^{-2\delta-3}),$

implying

$\displaystyle\int_{S_{r}}\tilde{g}(\nabla\Delta_{\tilde{g}}\Phi,\tilde{\nu})d\omega_{\tilde{g}}$
$\displaystyle=-c_{n}\int_{S_{r}}\tilde{g}(\nabla
R_{g},\nu_{e})d\omega_{\tilde{g}}+O(r^{-2\delta}),$
$\displaystyle=-c_{n}\int_{S_{r}}\tilde{g}(\nabla
R_{g},\nu_{e})d\omega_{r}-\int_{S_{r}}\underbrace{\tilde{g}(\nabla
R_{g},\nu_{e})}_{O(r^{-\tau-3})}O(r^{-\delta})d\omega_{r}+O(r^{-2\delta}),$
$\displaystyle=-c_{n}\int_{S_{r}}\tilde{g}(\nabla
R_{g},\nu_{e})d\omega_{r}+o(1)$ $\displaystyle=-c_{n}\int_{S_{r}}g(\nabla
R_{g},\nu_{e})d\omega_{r}-c_{n}\int_{S_{r}}(\underbrace{u^{2}-1}_{o(1)})\underbrace{g(\nabla
R_{g},\nu_{e})}_{O(r^{-\delta-3)}}d\omega_{r}+o(1)$
$\displaystyle=-c_{n}\int_{S_{r}}g(\nabla R_{g},\nu_{e})d\omega_{r}+o(1).$

Therefore, we find that

$\displaystyle\lim_{r\rightarrow\infty}\int_{S_{r}}\tilde{g}(\nabla\Delta_{\tilde{g}}\Phi,\tilde{\nu})d\omega_{\tilde{g}}$
$\displaystyle=c_{n}\mathcal{E}(g).$ (29)

Finally, putting together (25),(27) and (29), we find that

$\displaystyle
0\leq\int_{M}\left(Q_{g}e^{4\Phi}+\frac{1}{2}|\mathrm{Ric}_{\tilde{g}}|^{2}_{\tilde{g}}\right)dV_{\tilde{g}}=c_{n}\mathcal{E}(g),$
(30)

which implies that if $\mathcal{E}(g)=0$, then $\mathrm{Ric}_{\tilde{g}}\equiv
0$ and therefore $g$ is conformally-flat with $Q_{g}\equiv 0$.

Finally, $g=v^{\frac{4}{n-4}}\delta$ (or $g=e^{2v}\delta$ if $n=4$), with
$\Delta^{2}v=0$ and $\displaystyle\lim_{r\rightarrow\infty}v=1$ (or
$\displaystyle\lim_{r\rightarrow\infty}v=0$), hence, by maximum principle, we
get that $v\equiv 1$ (or $v\equiv 0$) which achieves the proof of the theorem.
∎

It is worth noticing that, looking more carefully to the above proof, we can
weaken the hypotheses of the previous theorem and still get interesting
results. Indeed, in the positivity part, we only use the fact that the scalar
curvature is positive at infinity, while for the rigidity statement, we only
use the fact that we can make the scalar curvature flat via a conformal
transformation, which is ensured by assuming that $Y([g])>0$ by theorem 5.1 of
[14], and that $Q_{g}$ is non-negative. Furthermore, under this last
condition, equations (23) and (30) also provide proofs of positivity. All
these observations give us the following results.

######  Theorem 3.

Let $(M^{n},g)$ be an $n$-dimensional AE manifold, with $n\geq 3$, which
satisfies the decaying conditions i) and ii) of Proposition 1 and such that
$R_{g}>0$ on $M\setminus K$, for some compact set $K$ then the fourth order
energy $\mathcal{E}(g)\geq 0$.

######  Theorem 4.

Let $(M^{n},g)$ be an $n$-dimensional AE manifold, with $n\geq 3$, which
satisfies the decaying conditions i) and ii) of Proposition 1 and such that
$Q_{g}\geq 0$ and $Y([g])>0$, then $\mathcal{E}(g)\geq 0$ with equality
holding if and only if $(M,g)$ is isometric to $(\mathbb{R}^{n},\cdot)$.

Let us now notice that the $Q$-curvature assumption in Theorem 4 cannot be
weakened while keeping the rest of the hypotheses. This can be seen as
follows: Consider $(\mathbb{R}^{n},\delta)$, with $n\geq 3$, and let
$\\{h_{k}\\}_{k=1}^{\infty}\in C^{\infty}(U)$, be a sequence of smooth
compactly supported symmetric second rank tensor fields,
$\bar{U}\subset\subset M$, with $||h_{k}||<\epsilon$, for some fixed small
$\epsilon>0$, such that $h_{k}\xrightarrow[k\rightarrow\infty]{}0$. Then,
consider the sequence of metrics $g_{k}=\delta+h_{k}$, where the construction
of the $h_{k}$ can be made so that $g_{k}$ are not conformally flat.
Furthermore, this construction can be fit so that $\\{g_{k}\\}_{k=1}^{\infty}$
are all Yamabe positive.555For an explicit construction of this see Example 1
in [2] Therefore, there are conformal factors $\\{u_{k}=1+\varphi_{k}\\}$,
with $\varphi_{k}\in H^{s}_{-\delta}$, with $\delta<n-2$ arbitrary, such that
$\Delta_{g_{k}}u_{k}-R_{g_{k}}u_{k}=0$.666In fact, from Theorem 1.17 in [4],
it follows that  $\displaystyle u_{k}=1+\frac{a_{k}}{r^{n-2}}+o(r^{-(n-2)}),$
where $a_{k}$ are constants. This means that
$R_{u_{k}^{\frac{4}{n-2}}g_{k}}=0$ and thus
$Q_{u_{k}^{\frac{4}{n-2}}g_{k}}=-c_{2}(n)|\mathrm{Ric}_{u_{k}^{\frac{4}{n-2}}g_{k}}|^{2}_{u_{k}^{\frac{4}{n-2}}g_{k}}<0$.
Furthermore, all the metrics $u_{k}^{\frac{4}{n-2}}g_{k}$ satisfy the decaying
conditions of Proposition 1 and are, by construction, Yamabe positive.
Nevertheless, from their asymptotics
$u_{k}^{\frac{4}{n-2}}{g_{k}}_{ij}=(1+\frac{a_{k,n}}{r^{n-2}})\delta_{ij}+o_{k}(r^{-(n-2)})$
we see that $\mathcal{E}(g_{k})=0$ for all $k$, although none of the $g_{k}$
are conformally flat. Finally, by considering $k$ sufficiently large, we can
make $-\frac{1}{k}<Q_{u^{\frac{4}{n-2}}_{k}g_{k}}<0$, which shows that there
are Yamabe positive AE metrics satisfying conditions (i)-(ii) of Proposition 1
which have negative $Q$-curvature which is as small as we want, zero energy
and are not conformally flat.

In dimensions $n=3,4$, under the above hypotheses, we fall into a curious
situation, since any such $n$-dimensional AE-manifold has $\mathcal{E}(g)=0$
as a consequence of the fall-off conditions, since $\nabla
R_{g}=0(r^{-\tau-3})$ decreases faster than the volume of the $S_{r}^{n-1}$.

######  Corollary 2.

Any n-dimensional AE-Riemannian manifold $(M^{n},g)$, with $n\in\\{3,4\\}$,
such that $Q_{g}\geq 0$ and $Y([g])>0$, is isometric to
$(\mathbb{R}^{n},\delta)$.

### Rigidity in critical cases

The aim of this section is to comment further on the especial case that occurs
in dimension four. In particular, the fact that the energy $\mathcal{E}(g)$ is
always zero has some topological consequences as a corollary. It is worth to
contrast this with the second order case associated to the ADM energy. In this
last case, the critical case is given in dimension two, where asymptotic
flatness is too strong a condition in order to detect any meaningful
information concerning the ADM energy (for details, see Chapter 3 in [27]).
Let us start by briefly commenting on this case as a warm-up.

Let $(M,g)$ be an asymptotically flat surface with $\tau>0$ and non-negative
integrable Gaussian curvature. Let us consider a chart a infinity, and recall
the definition of the ADM energy

$m(g)=\lim_{R\rightarrow+\infty}\int_{\partial
B_{R}}(\partial_{i}g_{ij}-\partial_{j}g_{ii})\nu^{j}\,ds,$

where $\nu$ is the outer normal. Because of the decreasing assumptions, we
easily see that the mass must vanish. Let us write the Gauss-Bonnet formula
which gives

$2\chi(M)=\int_{M\setminus B(0,R)^{c}}K\,d\sigma+\int_{\partial
B(0,R)}k_{g}\,ds.$

Here we write $\chi(M)$ instead of $\chi(M\setminus B(0,R)^{c})$ since they
are equal for $R$ large. Then in polar coordinate we have,

$\displaystyle k_{g}$
$\displaystyle=-\frac{1}{r^{2}}g\left(\nabla_{\partial_{\theta}}\partial_{\theta},\partial_{r}\right)=-\frac{1}{r^{2}}\Gamma^{r}_{\theta\theta}(g)+O\left(\frac{1}{r^{1+\tau}}\right)=-\frac{1}{r^{2}}\Gamma^{r}_{\theta\theta}(\delta)+O\left(\frac{1}{r^{1+\tau}}\right)$
$\displaystyle=\frac{1}{r}+O\left(\frac{1}{r^{1+\tau}}\right),$

hence passing to the limit we obtain that

$2\pi\chi(M)-2\pi=\int_{M}K\,d\sigma.$

But we know that

$\chi(M)=2-2g-r,$

where $g$ is the genus of $M$ and $r$ its number of ends. Then we necessary
get that $g=0$, $r=1$ and $K\equiv 0$. Hence we recover the following well-
known proposition, see chapter 3 [27],

######  Proposition 4.

Let $(M,g)$ an asymptotically flat surface with non-negative Gauss curvature.
Then, $(M,g)$ is the Euclidean plane.

Let us turn to the $4$-dimensional case. Let $(M,g)$ be a 4-dimensional AE
manifold with $\tau>0$ with only one end such that $|W_{g}|^{2}$ is
integrable. From Corollary 2, we know that the mass is necessarily vanishing

$\displaystyle\mathcal{E}(g)=-\lim_{R\rightarrow\infty}\int_{S_{R}}\partial_{r}R_{g}r^{3}d\omega_{3}=0.$
(31)

Along the lines of the $2$-dimensional case, let us try to get a similar proof
of the rigidity result. First, we apply the Gauss-Bonnet-Chern formula with
boundary (see [12] for instance)

$32\pi^{2}\chi(M)=\int_{K_{R}}|W|^{2}+16\sigma_{2}(S_{g})\,dv+8\int_{\partial
B(0,R)}B_{g}\,dv$

where $K_{R}=\left(M\setminus\overline{B(0,R)}\right)^{c}$,
$S_{g}=\frac{1}{2}\left(\mathrm{Ric}-\frac{1}{6}R_{g}g\right)$ is the Schouten
tensor and
$B_{g}=\frac{1}{2}R_{g}H-\mathrm{Ric}(\nu,\nu)H-R_{\gamma\alpha\beta\gamma}\mathrm{II}^{\alpha\beta}+\frac{1}{3}H^{3}-H|\mathrm{II}|^{2}+\frac{2}{3}tr(\mathrm{II}^{3})$,
where $\mathrm{II}$ and $H$ are respectively the second form and the mean
curvature of a hypersurface $\Sigma\hookrightarrow M$,
$H=\mathrm{tr}_{\Sigma}\mathrm{II}$, taken with respect to the inner unit
normal to $\Sigma$, the Greek indices are indices tangent to $\Sigma$, and
$\mathrm{II}^{3}$ is defined in local coordinates by
${\mathrm{II}^{3}}^{i}_{j}=\mathrm{II}^{i}_{k}\mathrm{II}^{k}_{l}\mathrm{II}^{l}_{j}$.
Moreover, in dimension $4$, we have

$Q_{g}=-\frac{1}{6}\Delta_{g}R_{g}+4\sigma_{2}(S_{g}),$

and thanks to our decreasing assumption, we get

$B_{g}=\frac{2}{r^{3}}+O\left(\frac{1}{r^{3+\tau}}\right)$

which gives, using once more our decreasing assumption, that

$\displaystyle\begin{split}32\pi^{2}(\chi(M)-1)&=\int_{K_{R}}|W|^{2}+4Q_{g}+\frac{2}{3}\Delta_{g}R_{g}\,dv_{g}+O\left(\frac{1}{R^{\tau}}\right)\\\
&=\int_{K_{R}}|W|^{2}+4Q_{g}\,dv_{g}+O\left(\frac{1}{R^{\tau}}\right)\end{split}$
(32)

Hence passing to the limit, we get

$32\pi^{2}(\chi(M)-1)=\int_{M}|W|^{2}+4Q_{g}\,dv_{g}$

Let us set

$\kappa_{g}=\int_{M}Q_{g}\,dv_{g}.$

One can use the above expression to recover Corollary 2 in the four
dimensional case as consequence of the conformal invariance of $\kappa_{g}$.
Also, we can easily deduce the following curvature-topology proposition. This
proposition is not really new but we just would like to put them in
perspective with our rigidity result

######  Proposition 5.

Let $(M,g)$ be an AE 4-manifold with $\kappa_{g}\geq 0$, then $\chi(M)\geq 1$
with equality if $(M,g)$ is locally conformally flat with $\kappa_{g}=0$.
In particular, if $\kappa_{g}\geq 0$ and $Y([g])>0$ or the second betti number
vanishes, then $\kappa_{g}=0$ and $(M,g)$ is conformal to the euclidean
$\mathbb{R}^{4}$.

###  3.1 The Q-curvature positive mass theorem

In this section we will make contact with a series of recent results
associated with the positive mass theorem for the Paneitz operator, namely
[20, 21, 26]. In particular, we will show that these results follow from
Theorem 4.

Let us consider a closed manifold $(M^{n},g)$ with $n\geq 5$. If, the Panietz
operator is coercive, that is to say

$\inf_{u\in H^{2}(M)\,;\,\|u\|_{2}=1}\int_{M}P_{g}(u)u\,dv_{g}>0,$

then it admits a Green functions $G_{P_{g}}$. This is in particular guaranteed
if $R_{g}\geq 0$ and $Q_{g}$ is semi-positive, see proposition B of [20], or
more generally by the fact that

$Y_{4}([g])=\inf_{u\in
H^{2}(M)\,;\,\|u\|_{2^{\\#}}=1}\int_{M}P_{g}(u)u\,dv_{g}>0,$

where $2^{\\#}=\frac{2n}{n-4}$. This last infimum has the advantage to be
conformally invariant and it plays a similar role to the Yamabe invariant for
the $Q$-curvature. Through this section, we will assume that $Y_{4}([g])>0$
and therefore $G_{P_{g}}$ exists for every $g\in[g]$.

Nevertheless, contrary to the conformal Laplacian, nothing here guarantees
that $G_{P_{g}}$ is positive. This will be one of our assumptions, which is in
particular satisfied if $Y([g])\geq 0$ and $Q_{\tilde{g}}$ is semi-positive
for some conformal metric $\tilde{g}$, due to lemma 3.2 in [23].

######  Remark 2.

In fact Hang and Yang assume $Y([g])>0$, but if $Y([g])=0$ and there is some
$\tilde{g}\in[g]$ with $Q_{\tilde{g}}$ semi-positive, then there exists
$\bar{g}=u^{\frac{4}{n-2}}\tilde{g}$ such that $R_{\bar{g}}=0$ which implies
that $Q_{\bar{g}}\leq 0$ which contradicts the fact that

$\int_{M}Q_{\tilde{g}}v\,dv_{\tilde{g}}=\int_{M}v^{\frac{n+4}{n-4}}Q_{\bar{g}}\,dv_{\tilde{g}},$

with $v=u^{\frac{n-4}{n-2}}$. Hence the existence of a conformal metric with
semi-positive $Q$-curvature forces the Yamabe invariant to be positive.

In fact, if $Y([g])\geq 0$ , thanks to, theorem 1.1 of [23], the existence of
a positive $G_{P_{g}}$ is equivalent to the existence of a conformal metric
$\tilde{g}$ with semi-positive $Q$-curvature.

Let now us focus on the expansion of the Green function around a singularity.
If we assume that $5\leq n\leq 7$ or $g$ is locally conformally flat, then, in
conformal normal coordinates $\\{x^{i}\\}$ for the conformal metric
$\tilde{g}$, the Green function $G_{P}$ of $P_{\tilde{g}}$ admits an expansion
of the form (see Proposition 2.5 in [20])

$\displaystyle G_{P}(p,x)=\frac{\gamma_{n}}{r^{n-4}}+\alpha+O_{4}(r),$ (33)

where $r(x)\doteq d_{\tilde{g}}(p,x)$,
$\gamma_{n}\doteq\frac{1}{2(n-2)(n-4)\omega_{n-1}}$ and $\alpha$ is a constant
called the mass.

######  Remark 3.

As remarked in [23], under the condition $\mathrm{Ker}(P_{g})=0$ (which is
itself conformally invariant), the sign of the Green function is a conformal
invariant since,

$G_{P_{u^{\frac{4}{n-4}}g}}(p,q)=u(p)^{-1}u(q)^{-1}G_{P_{g}}(p,q).$

Hence, the sign of the mass is also a conformal invariant.

This terminology arises in analogy to the Yamabe problem, where the Green
function $G_{L_{g}}$ of the conformal Laplacian is involved and, in important
cases, it admits a similar expansion to (33). In the case of the conformal
Laplacian, it was an observation of R. Schoen that the constant which appears
in the place of $\alpha$ is precisely the ADM mass of the AE-manifold obtained
by via the stereographic projection
$(M\backslash\\{p\\},G^{\frac{4}{n-2}}_{L}g)$. This showed that the resolution
of the positive mass conjecture in general relativity would amount to
completing the resolution of the Yamabe problem, which concerned the cases
$Y(g)>0$ and dimensions $n=3,4,5$ or $M$ locally conformally flat (see [40]
and [28]).

The positive mass theorem presented in [20] as Theorem 2.9 states that under
the conditions described above which provide us with the expansion (33), the
mass satisfies $\alpha\geq 0$ with equality holding if and only if $(M,g)$ is
conformal to the round sphere. This came about as a generalisation of [26],
where this result was proven in the conformally flat case, and, in turn, all
this was generalised in [21], where the condition $R_{g}\geq 0$ was replaced
by $Y([g])>0$. Furthermore, in Theorem 1.4 of [23], this positive mass theorem
is used to solve the $Y_{4}([g])>0$ $Q$-curvature prescription problem in very
much the same spirit as the usual positive mass theorem was used by Schoen in
[40]. We intend to show that all these results are special cases of the
positive energy theorem presented above.

Finally, let us notice that, for instance, in the $Q$-curvature prescription
problem it is the sign of the mass that is actually important rather than its
precise value. This becomes especially useful when it is combined with the
additional observation that the sign of the mass is itself a conformal
invariant, which was highligted in Remark 3. Thus, being concerned with a
conformal problem, given $(M^{n},[g])$, we will always consider a choice of
$g$ satisfying the conformal normal coordinate properties (C.1),(C.3),(C.4).
Notice that from the discussion presented in that appendix, this can always be
achieved by first going to a related conformal metric.

In order to prepare for the main statements of this section, let us first
establish a couple of preliminary results.

######  Proposition 6.

Let $(M^{n},g)$ be a closed manifold satisfying $n\geq 5$ and whose Panietz
operator admits a positive Green function $G_{P}$ with an expansion as (33)
around some point $p\in M$. Then, the manifold $(\hat{M}\doteq
M\backslash\\{p\\},\hat{g}\doteq G_{P}(p,\cdot)^{\frac{4}{n-4}}g)$ is an
asymptotically flat manifold of order $\tau=1$ if $n=5$ and $\tau=2$ if $n>5$.
Furthermore, either if $5\leq n\leq 7$ or $g$ is flat around $p$, then
$\mathcal{E}(\hat{g})=8(n-1)(n-2)\omega_{n-1}\gamma_{n}\alpha$.

###### Proof.

Let us fix the conformal normal coordinates $\\{x^{i}\\}$ where (33) holds.
Let us appeal to the conformal normal coordinate construction of order $N\geq
4$ described Appendix B so that (C.3)-(C.4) hold, and start by considering the
following expansion near $p$

$\displaystyle
g_{ij}=\delta_{ij}+\frac{1}{3}R_{iklj}(p)x^{k}x^{l}+\frac{1}{6}R_{iklj,a}(p)x^{k}x^{l}x^{a}+O(r^{4}),$
(34)

Now, consider the inverted coordinates
$z^{i}=\gamma_{n}^{\frac{2}{n-4}}\frac{x^{i}}{r^{2}}$ in a neighborhood of $p$
and define $\rho^{2}\doteq|z|^{2}$, so that
$\rho^{2}=\gamma_{n}^{\frac{4}{n-4}}r^{-2}$ and
$x^{i}=\gamma_{n}^{\frac{2}{n-4}}\frac{z^{i}}{\rho^{2}}$. Then, it holds that
$\gamma_{n}r^{-(n-4)}=\gamma_{n}^{-1}\rho^{n-4}$ and

$\displaystyle\frac{\partial}{\partial
z^{i}}=\gamma_{n}^{\frac{2}{n-4}}\rho^{-2}\left(\delta^{j}_{i}-2\rho^{-2}z^{i}z^{j}\right)\frac{\partial}{\partial
x^{j}},$ (35)

which, appealing to (33), implies

$\displaystyle\hat{g}_{ij}(z)$
$\displaystyle=\left(\gamma^{-1}_{n}\rho^{n-4}+\alpha+O_{4}(\rho^{-1})\right)^{\frac{4}{n-4}}g(\partial_{z^{i}},\partial_{z^{j}}),$
$\displaystyle=\left(1+\frac{\gamma_{n}\alpha}{\rho^{n-4}}+O_{4}(\rho^{3-n})\right)^{\frac{4}{n-4}}\left(\delta^{i}_{k}-2\rho^{-2}z^{i}z^{k}\right)\left(\delta^{j}_{l}-2\rho^{-2}z^{j}z^{l}\right)g_{kl}\left(\gamma_{n}^{\frac{2}{n-4}}\rho^{-2}z\right).$

Invoking (34), this implies

$\displaystyle\hat{g}_{ij}(z)$
$\displaystyle=\left(1+\frac{\gamma_{n}\alpha}{\rho^{n-4}}+O_{4}(\rho^{3-n})\right)^{\frac{4}{n-4}}h_{ij}(z),$

where

$\displaystyle h_{ij}(z)$
$\displaystyle=\left(\delta_{k}^{i}\delta_{l}^{j}-2\frac{z^{i}z^{k}}{\rho^{2}}\delta_{l}^{j}-2\frac{z^{j}z^{l}}{\rho^{2}}\delta_{k}^{i}+4\frac{z^{i}z^{j}z^{k}z^{l}}{\rho^{4}}\right)\left(\delta_{kl}+\frac{\gamma_{n}^{\frac{4}{n-4}}}{3}R_{kmnl}(p)\frac{z^{m}z^{n}}{\rho^{4}}\right.$
$\displaystyle+\left.\frac{\gamma^{\frac{6}{n-4}}_{n}}{6}R_{kmnl,p}(p)\frac{z^{m}z^{n}z^{p}}{\rho^{6}}+O_{4}\left(\frac{1}{\rho^{4}}\right)\right),$
$\displaystyle=\delta_{ij}+H^{2}_{ij}(z)+H^{3}_{ij}(z)+O_{4}\left(\frac{1}{\rho^{4}}\right)$

where

$\displaystyle\gamma^{-\frac{4}{n-4}}_{n}H^{2}_{ij}$
$\displaystyle=\frac{1}{3}R_{imnj}(p)\frac{z^{m}z^{n}}{\rho^{4}}-\frac{2}{3}\underbrace{R_{kmnj}(p)z^{k}z^{m}}_{=0}\frac{z^{n}z^{i}}{\rho^{6}}-\frac{2}{3}\underbrace{R_{imnl}(p)z^{n}z^{l}}_{=0}\frac{z^{m}z^{j}}{\rho^{6}}$
$\displaystyle+\frac{4}{3}\underbrace{R_{kmnl}(p)z^{k}z^{m}}_{=0}\frac{z^{n}z^{l}z^{i}z^{j}}{\rho^{8}},$
$\displaystyle=\frac{1}{3}R_{imnj}(p)\frac{z^{m}z^{n}}{\rho^{4}}$

and

$\displaystyle\gamma^{-\frac{6}{n-4}}_{n}H^{3}_{ij}$
$\displaystyle=\frac{1}{6}R_{imnj,a}(p)\frac{z^{m}z^{n}z^{a}}{\rho^{6}}-\frac{1}{3}\underbrace{R_{kmnj,a}(p)z^{k}z^{m}}_{=0}\frac{z^{n}z^{i}z^{a}}{\rho^{8}}-\frac{1}{3}\underbrace{R_{imnl,a}(p)z^{n}z^{l}}_{=0}\frac{z^{m}z^{j}z^{a}}{\rho^{8}}$
$\displaystyle+\frac{2}{3}\underbrace{R_{kmnl,a}(p)z^{k}z^{m}}_{=0}\frac{z^{n}z^{l}z^{i}z^{j}z^{a}}{\rho^{10}},$
$\displaystyle=\frac{1}{6}R_{imnj,a}(p)\frac{z^{m}z^{n}z^{a}}{\rho^{6}}.$

Putting together all the above, we find that

$\displaystyle\hat{g}_{ij}(z)$
$\displaystyle=\left(1+\frac{4}{n-4}\frac{\gamma_{n}\alpha}{\rho^{n-4}}\right)\delta_{ij}+\frac{\gamma^{\frac{4}{n-4}}_{n}}{3}R_{iabj}(p)\frac{z^{a}z^{b}}{\rho^{4}}+\frac{\gamma^{\frac{6}{n-4}}_{n}}{6}R_{iabj,c}(p)\frac{z^{a}z^{b}z^{c}}{\rho^{6}}$
$\displaystyle+O_{4}(\rho^{-(n-3)})+O_{4}(\rho^{-4}),$

which establishes the asymptotic flatness condition claimed in the
proposition. Notice that if $5\leq n\leq 7$, then, from Proposition 1, we know
that $\mathcal{E}(\hat{g})$ is well-defined and the above expansion implies

$\displaystyle\partial_{a}\hat{g}_{ij}$
$\displaystyle=-4\gamma_{n}\alpha\frac{z^{a}}{\rho^{n-2}}\delta_{ij}+\partial_{a}H^{2}_{ij}+\partial_{a}H^{3}_{ij}+O_{3}\left(\rho^{-(n-2)}\right),$
$\displaystyle\partial_{ba}\hat{g}_{ij}$
$\displaystyle=-4\gamma_{n}\alpha\left(\frac{\delta_{ab}}{\rho^{n-2}}-(n-2)\frac{z^{b}z^{a}}{\rho^{n}}\right)\delta_{ij}+\partial_{ba}H^{2}_{ij}+\partial_{ba}H^{3}_{ij}+O_{2}\left(\rho^{-(n-1)}\right).$

Appealing to (C.3)-(C.4), it is not difficult to explicitly compute that
$\partial_{aa}H^{2}_{ii}$ and $\partial_{aa}H^{3}_{ii}$ both vanish. This
implies

$\displaystyle\partial_{aa}\hat{g}_{ii}$
$\displaystyle=-8n\gamma_{n}\alpha\rho^{-(n-2)}+O_{2}(\rho^{-(n-1)}).$ (36)

Similarly, appealing to (C.3)-(C.4), it also holds that
$\partial_{ij}H^{2}_{ij}=0$ and $\partial_{ij}H^{3}_{ij}=0$, and therefore we
find that

$\displaystyle\partial_{ij}\hat{g}_{ij}$
$\displaystyle=-8\gamma_{n}\alpha\rho^{-(n-2)}+O_{2}(\rho^{-(n-1)}).$ (37)

Thus, putting together (36)-(37), we see that

$\displaystyle\left(\partial_{caa}\hat{g}_{ii}(z)-\partial_{cij}\hat{g}_{ij}\right)\frac{z^{c}}{\rho}$
$\displaystyle=8\gamma_{n}\alpha(n-1)(n-2)\rho^{-(n-1)}+O_{1}(\rho^{-n}),$
(38)

implying that

$\displaystyle\mathcal{E}(\hat{g})$
$\displaystyle=\lim_{\rho\rightarrow\infty}\int_{S_{\rho}}\left(\partial_{caa}\hat{g}_{ii}(z)-\partial_{cij}\hat{g}_{ij}(z)\right)\frac{z^{c}}{\rho}d\omega_{\rho}=\omega_{n-1}8\gamma_{n}\alpha(n-1)(n-2).$
(39)

Finally, in $g$ is flat near $p$, without restriction on the dimension, we
know that the expansion (33) holds in rectangular coordinates around $p$,
where $g_{ij}(x)=\delta_{ij}$. Thus, the same computations as above show that

$\displaystyle\hat{g}_{ij}(z)$
$\displaystyle=\left(1+\frac{4}{n-4}\frac{\gamma_{n}\alpha}{\rho^{n-4}}\right)\delta_{ij}+O_{4}(\rho^{-(n-3)}).$

In this case the order of decay is improved for $n\geq 7$ and in particular we
know that $\mathcal{E}(\hat{g})$ is well-defined. From the above expression,
the same computations that led us to (39) prove that
$\mathcal{E}(\hat{g})=8\omega_{n-1}(n-1)(n-2)\gamma_{n}\alpha$. ∎

In order to apply the positive energy theorem to establish that $\alpha\geq
0$, we first need to check that $\hat{g}$ satisfies its hypotheses. With this
in mind, consider the following proposition.

######  Proposition 7.

Consider a closed Riemannian manifold $(M^{n},g)$, with $n\geq 5$, which
admits a conformal metric with positive $Q$-curvature such that $Y([g])\geq
0$. Then, there exists a conformal metric $\tilde{g}$ such that the
asymptotically flat manifold
$(\hat{M}=M\backslash\\{p\\},\hat{g}=G_{P_{\tilde{g}}}^{\frac{4}{n-4}}\tilde{g})$
satisfies $Y([\hat{g}])>0$ and $Q_{\hat{g}}\equiv 0$.

###### Proof.

Thanks to Remark 2, $Y([g])>0$ and $G_{P_{\tilde{g}}}$ exists and is positive
for every $\tilde{g}\in[g]$. Then, thanks to proposition 6, we only have to
prove that $Y([\hat{g}])>0$. Since $Y([g])>0$, let $G_{L_{\tilde{g}}}$ be the
Green’s function of the conformal Laplacian. We trivially get that
$\bar{g}=\left(\frac{G^{2}_{L}}{G_{P}^{\frac{2(n-2)}{n-4}}}\right)^{\frac{2}{n-2}}\hat{g}$
is scalar flat. ∎

Equipped with the above two propositions, we can recover the following
theorem, originally proved by Hang-Yang in [23], and which is an extension of
results of Gursky-Malchiodi [20] and Humbert-Raulot [26].

######  Theorem 5.

Let $(M,g)$ be a closed $n$-dimensional Riemannian manifold, with $5\leq n\leq
7$ or $n\geq 8$ and locally conformally flat around some point $p\in M$. If
$Y([g])\geq 0$ and $(M,g)$ admits a conformal metric with semi-positive
$Q$-curvature, then the mass of $G_{P}$ at $p$ is non-negative and vanishes if
and only if $(M,g)$ is conformal to the standard sphere.

It is important to note that here the hypothesis are conformally invariant. In
dimension $n\geq 6$, thanks to Corollary 1.1 [19] and remark 2, they are
equivalent to $Y([g])>0$ and $Y_{4}^{*}([g])>0$, where

$Y_{4}^{*}([g])=\frac{n-4}{2}\inf_{\tilde{g}\in[g],R_{\tilde{g}}>0}\frac{\int_{M}Q_{\tilde{g}}\,dv_{\tilde{g}}}{\mathrm{Vol}_{\tilde{g}}(M)}.$

Of course $Y_{4}^{*}([g])$ is also conformally invariant.

###### Proof.

Under these conditions, we know that the Green function $G_{P}$ exists and is
positive for every element in $[g]$. From Proposition 6 we also know that
there is a conformal metric $\tilde{g}$ such that the manifold
$(\hat{M}=M\backslash\\{p\\},\hat{g}=G_{P_{\tilde{g}}}^{\frac{4}{n-4}}\tilde{g})$
is AE, satisfies the decay assumptions of Theorem 4 and, furthermore the
energy is positively proportional to the mass of $G_{P_{\tilde{g}}}$. Finally,
from Proposition 7, we know that $Y([\hat{g}])>0$ and $Q_{\hat{g}}\equiv 0$.
Therefore, $(\hat{M},\hat{g})$ satisfies all the hypotheses of Theorem 4, and
thus the non-negativity follows directly from $\mathcal{E}(\hat{g})$ being
positively proportional to $\alpha$. Finally, if $\alpha=0$, we find that
$\hat{M}$ is isometric to $\mathbb{R}^{n}$. Being $M$ the one point
compactification of $\hat{M}$, we see that $M\cong S^{n}$ and $g$ is conformal
to the round metric.

∎

###  3.2 The 4-dimensional case

Finally, we would like to end by briefly discussing some peculiarities
concerning the analysis of the four dimensional case. In particular, a large
part of what we have done above for the $Q$-curvature positive mass theorem
can be translated perfectly well to $n=4$. Nevertheless, in view of Corollary
2 it should be clear that strong restrictions should appear in the treatment
of this special case. Although the origins of some of these restriction may be
evident to experts in $Q$-curvature analysis, the aim in what follows is to
make explicit the subtle differences of this case and where the
particularities arise. In this spirit, let us start by presenting the
following lemma which combines results of [36]-[29].

######  Lemma 1.

Let $(M^{4},g)$ be a closed Riemannian manifold satisfying
$\mathrm{Ker}(P_{g})=\mathbb{R}$ and $\kappa_{g}>0$. Then, given $p\in M$,
there exists a Green’s function $G_{p}\doteq G_{P_{g}}(p,\cdot)\in
C^{\infty}(M\backslash\\{p\\})$ with a pole at $p$, unique up to an additive
constant, which satisfies

$\displaystyle P_{g}G_{p}+Q_{g}=\kappa_{g}\delta_{p}$ (40)

as distributions. Furthermore, near $p$, in $g$-normal coordinates
$\\{x^{i}\\}_{i=1}^{4}$ the following expansion holds

$\displaystyle G_{p}$
$\displaystyle=\frac{\kappa_{g}}{16\pi^{2}}\ln(r^{-2})+\frac{\kappa_{g}}{16\pi^{2}}S_{0}+\frac{\kappa_{g}}{16\pi^{2}}a_{i}x^{i}+\frac{\kappa_{g}}{16\pi^{2}}b_{ij}x^{i}x^{j}+o_{4}(r^{2}).$
(41)

for some constants $S_{0},a_{i},b_{ij}$.

###### Proof.

From Lemma 2.1 in [36] we know that if $\mathrm{Ker}(P_{g})=\mathbb{R}$, then
the Paneitz operator $P_{g}$ admits a Green function $G_{P}$, which is to say
that for every $u\in C^{4}(M)$ it holds that

$\displaystyle u(x)-\bar{u}=\int_{M}G_{P}(x,y)P_{g}(u(y))dV_{g}(y),$ (42)

where $\bar{u}\doteq\mathrm{vol_{g}}(M)^{-1}\int_{M}udV_{g}$. Also, since
$\mathrm{Ker}(P_{g})=\mathbb{R}$, we know that there is a smooth function $U$
satisfying

$\displaystyle P_{g}U=-(Q_{g}-\bar{Q}_{g}).$

Define $G_{p}\doteq\kappa_{g}G_{P}+U$, so that

$\displaystyle\langle P_{g}G_{p},u\rangle$
$\displaystyle=\kappa_{g}u(p)-\kappa_{g}\bar{u}+\langle P_{g}U,u\rangle,$
$\displaystyle=\kappa_{g}u_{p}-\kappa_{g}\bar{u}-\langle
Q_{g},u\rangle+\langle\frac{\kappa_{g}}{\mathrm{vol}_{g}(M)},u\rangle=\kappa_{g}u_{p}-\langle
Q_{g},u\rangle.$

Thus, we see that

$\displaystyle\langle P_{g}G_{p}+Q_{g},u\rangle=\kappa_{g}u(p)\;\;\forall u\in
C^{\infty}(M),$ (43)

which is to say that $P_{g}G_{p}+Q_{g}=\kappa_{g}\delta_{p}$. The uniqueness
claim follows since two different such functions $G_{p}$ and $\tilde{G}_{p}$
must satisfy $P_{g}(G_{p}-\tilde{G}_{p})=0$, implying $G_{p}=\tilde{G}_{p}+c$.
If we rewrite (40) as
$P_{g}\left(\frac{16\pi^{2}}{k_{g}}G_{p}\right)+\frac{16\pi^{2}}{k_{g}}Q_{g}=16\pi^{2}\delta_{p}$
we can appeal quite straightforwardly to the computations of the Appendix in
[29] to conclude that, in $g$-normal coordinates, the Green function has the
following expansion near $p$:

$\displaystyle\frac{16\pi^{2}}{\kappa_{g}}G_{p}$
$\displaystyle=-2\ln(r)+S_{0}+a_{i}x^{i}+b_{ij}x^{i}x^{j}+o_{4}(r^{2}).$ (44)

which proves (41). ∎

Let us now consider the same setting as in the above lemma and consider the
inverted coordinates $z^{i}=\frac{x^{i}}{r^{2}}$ and define
$(\hat{M}=M\backslash\\{p\\},\hat{g}=e^{2G_{p}}g)$. Then,

$\displaystyle
Q_{\hat{g}}(q)=e^{-4G_{p}(q)}\left(P_{g}G_{p}(q)+Q_{g}(q)\right)=0,\text{ for
any }q\in\hat{M}.$

and

$\displaystyle\begin{split}\hat{g}_{ij}(z)&=e^{\ln(\rho^{4\frac{\kappa_{g}}{16\pi^{2}}})+\frac{\kappa_{g}}{8\pi^{2}}S_{0}+\frac{\kappa_{g}}{8\pi^{2}}a_{i}\frac{z^{i}}{\rho^{2}}+\frac{\kappa_{g}}{8\pi^{2}}b_{ij}\frac{z^{i}z^{j}}{\rho^{4}}+O_{4}(\rho^{-3})}g(\partial_{z^{i}},\partial_{z^{j}}),\\\
&=\rho^{4\frac{\kappa_{g}}{16\pi^{2}}}e^{\frac{\kappa_{g}}{8\pi^{2}}S_{0}}e^{\frac{\kappa_{g}}{8\pi^{2}}a_{i}\frac{z^{i}}{\rho^{2}}+\frac{\kappa_{g}}{8\pi^{2}}b_{ij}\frac{z^{i}z^{j}}{\rho^{4}}+O_{4}(\rho^{-3})}\rho^{-4}\left(\delta^{i}_{k}-2\rho^{-2}z^{i}z^{k}\right)\left(\delta^{j}_{l}-2\rho^{-2}z^{j}z^{l}\right)g_{kl}(\rho^{-2}z),\\\
&=\rho^{4\Delta\kappa_{g}}e^{\frac{\kappa_{g}}{8\pi^{2}}S_{0}}e^{\frac{\kappa_{g}}{8\pi^{2}}a_{i}\frac{z^{i}}{\rho^{2}}+\frac{\kappa_{g}}{8\pi^{2}}b_{ij}\frac{z^{i}z^{j}}{\rho^{4}}+O_{4}(\rho^{-3})}\left(\delta_{ij}+O_{4}(\rho^{-2})\right),\\\
&=\rho^{4\Delta\kappa_{g}}e^{\frac{\kappa_{g}}{8\pi^{2}}S_{0}}\left(1+\frac{\kappa_{g}}{8\pi^{2}}a_{i}\frac{z^{i}}{\rho^{2}}+O_{4}(\rho^{-2})\right)\left(\delta_{ij}+O_{4}(\rho^{-2})\right),\end{split}$
(45)

where we have defined $\Delta\kappa_{g}\doteq\frac{\kappa_{g}}{16\pi^{2}}-1$.
Let us notice that the above inversion gives an AE gemtric $\hat{g}$ iff
$\Delta\kappa_{g}=0$, which is to say $\kappa_{g}=16\pi^{2}$. In particular,
in this case, via a coordinate change of the form
$\bar{z}^{i}=e^{S_{0}}z^{i}$, we find that

$\displaystyle\hat{g}_{ij}(\bar{z})$
$\displaystyle=\left(1+2e^{S_{0}}a_{k}\frac{\bar{z}^{k}}{\bar{\rho}^{2}}+O_{4}(\bar{\rho}^{-2})\right)\left(\delta_{ij}+O_{4}(\bar{\rho}^{-2})\right),$
(46)

where $\bar{\rho}=|\bar{z}|=e^{S_{0}}\rho$. That is,
$\rho^{-k}=e^{kS_{0}}\bar{\rho}^{-k}$. We see that $(\hat{M},\hat{g})$ is an
AE-manifold or order $\tau=1$. Therefore, since by construction
$Q_{\hat{g}}\equiv 0$, if $Y(g)>0$, in view of Corollary 2, we must conclude
that $(M,g)$ is conformal to the round sphere. That is, the above computations
together with Corollary 2 imply the following Corollary, which is not new,
since it concerns the rigidity statement involved in Theorem B in [18].

######  Corollary 3.

Let $(M,g)$ be a closed 4-dimensional manifold which satisfies
$\mathrm{Ker}(P_{g})=\mathbb{R}$ and $\kappa_{g}=16\pi^{2}$. If $Y(g)>0$, then
$(M,g)$ is conformal to the round sphere.

Finally, let us highlight that the techniques developed so far allow us to
actually get more than the previous corollary. In fact, we can recover the
full statement of Theorem B in [18] by an independent and simple proof.

######  Theorem 6 (Gursky).

Let $(M^{4},g)$ a $4$-dimensional manifold with $Y([g])\geq 0$, then
$\kappa_{g}\leq 16\pi^{2}$ with equality holding iff $(M^{4},g)$ is conformal
to the standard sphere.

###### Proof.

Since we need only pay attention to the cases $\kappa_{g}>0$, let us start by
noticing that, due to Theorem A in [18], under our hypotheses
$\mathrm{Ker}(P_{g})=\mathbb{R}$. Furthermore, the case $Y([g])=0$ is also
trivial, since in this case $\kappa_{g}\leq 0$, thus we will assume
$Y([g])>0$. Since our hypotheses are conformally invariant, let us start
assuming that $g$ has been picked within $[g]$ so as to be the metric of a
conformal normal coordinate system. We can now appeal to Lemma 1 and the
expansion (45) to construct the manifold
$(\hat{M}=M\backslash\\{p\\},\hat{g})$. Alhtough in general this manifold will
not be AE, it holds that $Q_{\hat{g}}\equiv 0$. Also, since $Y([g])>0$, we now
that the Green function $G_{L}$ of the conformal Laplacian (with a pole at
$p\in M$) exists, it is positive and therefore we can construct the metric
$\tilde{g}\doteq G^{2}_{L}g$ on $\hat{M}$. In inverted normal coordinates
around $p$, from [28], we know that

$\displaystyle\tilde{g}(z)(\partial_{z^{i}},\partial_{z^{j}})=\delta_{ij}+O_{4}(\rho^{-2}),$

which shows that $(\hat{M},\tilde{g})$ is AE of order $\tau=2$. Furthermore,
$\hat{g}$ and $\tilde{g}$ are related via

$\displaystyle\hat{g}=e^{-\ln(G^{2}_{L})}e^{2G_{p}}\tilde{g}=e^{2(G_{p}-\ln(G_{L}))}\tilde{g}$

and therefore, from the conformal covariance associate to the Paneitz
operator, we see that

$\displaystyle 0=e^{4\Phi}Q_{\hat{g}}=P_{\tilde{g}}\Phi+Q_{\tilde{g}},$ (47)

where we have defined $\Phi\doteq G_{P}-\ln(G_{L})$. Following the proof
theorem 2, we know that
$Q_{\tilde{g}}=-\frac{1}{2}|\mathrm{Ric}_{\tilde{g}}|^{2}_{\tilde{g}}$, and
therefore we find that, for $\rho$ large enough,

$\displaystyle\int_{D_{\rho}}\frac{1}{2}|\mathrm{Ric}_{\tilde{g}}|^{2}_{\tilde{g}}dV_{\tilde{g}}=\int_{S_{\rho}}\tilde{g}(\tilde{\nabla}\Delta_{\tilde{g}}\Phi,\tilde{\nu})d\tilde{\omega}+2\int_{S_{\rho}}\mathrm{Ric}_{\tilde{g}}(\tilde{\nabla}\Phi,\tilde{\nu})d\tilde{\omega}.$
(48)

Now, the main difference with respect to theorem 2 is that we cannot estimate
the derivatives of $\Phi$ in the same way, since it does not solve the same
equation as in that proof. Nevertheless, in this case, we have an explicit
expression for $\Phi$, at least asymptotically. Thus, let us notice that for
sufficiently large $\rho$ the following holds

$\displaystyle\Phi$
$\displaystyle=\ln(\rho^{2\alpha})+S_{0}+O_{4}(\rho^{-1})-\ln(\rho^{2}+A+O_{4}(\rho^{-1})),$
$\displaystyle=\ln(\rho^{2\alpha})+S_{0}-\ln(\rho^{2})-\ln(1+A\rho^{-2}+O_{4}(\rho^{-3}))+O_{4}(\rho^{-1}),$
$\displaystyle=\ln(\rho^{2(\alpha-1)})+S_{0}-\ln(1+A\rho^{-2}+O_{4}(\rho^{-3}))+O_{4}(\rho^{-1}),$

where above we have defined $\alpha\doteq\frac{\kappa_{g}}{16\pi^{2}}$ and
both $A$ and $S_{0}$ are constants. From all this we can directly compute that

$\displaystyle\tilde{\nabla}_{i}\Phi$
$\displaystyle=2(\alpha-1)\frac{z^{i}}{\rho^{2}}+O_{3}(\rho^{-2}),$
$\displaystyle\Delta_{\tilde{g}}\Phi$
$\displaystyle=\frac{4(\alpha-1)}{\rho^{2}}+O_{2}(\rho^{-3}),$
$\displaystyle\tilde{\nabla}_{i}\Delta_{\tilde{g}}\Phi$
$\displaystyle=-\frac{8(\alpha-1)}{\rho^{3}}\frac{z^{i}}{\rho}+O_{2}(\rho^{-4}).$

From the above expression, it follows that

$\displaystyle\mathrm{Ric}_{\tilde{g}}(\tilde{\nabla}\Phi,\tilde{\nu})$
$\displaystyle=o(\rho^{-3}),$
$\displaystyle\tilde{g}(\tilde{\nabla}\Delta_{\tilde{g}}\Phi,\tilde{\nu})$
$\displaystyle=\frac{8(1-\alpha)}{\rho^{3}}+O_{1}(\rho^{-4}),$

implying that

$\displaystyle\int_{D_{\rho}}\frac{1}{2}|\mathrm{Ric}_{\tilde{g}}|^{2}_{\tilde{g}}dV_{\tilde{g}}=8\omega_{3}(1-\alpha)+o(1).$
(49)

Finally, passing to the limit as $r$ goes to infinity, we find that

$\displaystyle\int_{\hat{M}}|\mathrm{Ric}_{\tilde{g}}|^{2}_{\tilde{g}}dV_{\tilde{g}}=8\omega_{3}(1-\alpha)\geq
0.$ (50)

This implies that $\alpha\leq 1$ and the equality case has already been dealt
with in the previous corollary, which establishes the theorem.

∎

###  3.3 The 3-dimensional case

In this section, we briefly explain how to recover the following theorem due
to Hang and Yang [22].

######  Theorem 7 (Proposition 2.4 [22]).

Assume the Yamabe invariant $Y([g])>0$, $\ker P_{g}=0$. If there is some $p\in
M$ such that $G_{p}\doteq G_{P_{g}}(p,\cdot)<0$ on $M\backslash\\{p\\}$, then
$G_{p}(p)<0$ except when $(M,g)$ is conformally equivalent to the standard
$S^{3}$.

In this case the proof runs along the same lines as in the previous theorem.
In particular, its hypotheses and conclusions are conformally invariant. Thus,
we can assume that $g$ is the metric of a conformal normal coordinate system.
Then, let $\hat{M}=M\backslash\\{p\\}$ and define $\hat{g}\doteq G^{-4}_{p}g$
and $\tilde{g}=G^{4}_{L}g$ on this manifold, where, as in the proof of the
previous theorem, $G_{L}$ denotes the Green function of the conformal
Laplacian with a pole at $p$. Let us recall the following expansions, valid in
conformal normal coordinates around $p$ (see, [22] and [28])

$\displaystyle G_{p}$ $\displaystyle=A+O_{4}(r),$ $\displaystyle G_{L}$
$\displaystyle=\frac{1}{r}+\alpha+O_{4}(r),$

where $A$ and $\alpha$ are constants. After going to inverted coordinates
$z=\frac{x}{r}$, with $\rho=r^{-1}$, we find that

$\displaystyle\tilde{g}_{ij}(z)=\left(1+\frac{\alpha}{\rho}\right)\delta_{ij}+O(\rho^{-2}),$

and, clearly, on $\hat{g}=\Phi^{-4}\tilde{g}$, with $\Phi\doteq G_{P}G_{L}$
and $Q_{\hat{g}}\equiv 0$. Therefore, from the conformal covariance of the
Paneitz operator, we find that $P_{\tilde{g}}\Phi\equiv 0$, which translates
to

$\displaystyle
0=\Delta^{2}_{\tilde{g}}\Phi+4\mathrm{div}_{\tilde{g}}\left(\mathrm{Ric}_{\tilde{g}}(\nabla\Phi,\cdot)\right)+|\mathrm{Ric}_{\tilde{g}}|^{2}_{\tilde{g}}\Phi.$

That is

$\displaystyle\int_{D_{\rho}}|\mathrm{Ric}_{\tilde{g}}|^{2}_{\tilde{g}}\Phi
dV_{\tilde{g}}=-\int_{S_{\rho}}\tilde{g}(\tilde{\nabla}\Delta_{\tilde{g}}\Phi,\tilde{\nu})d\tilde{\omega}_{\rho}-4\int_{S_{\rho}}\mathrm{Ric}_{\tilde{g}}(\nabla\Phi,\tilde{\nu})d\tilde{\omega}_{\rho}.$
(51)

As in the previous theorem, we now can compute explicitly the terms in the
right-hand side. That is,

$\displaystyle\Phi$
$\displaystyle=\left(A+O_{4}(\rho^{-1})\right)\left(\rho+\alpha+O_{4}(\rho^{-1})\right)=A\rho+O_{4}(\rho^{0})$
$\displaystyle\nabla_{i}\Phi$
$\displaystyle=A\frac{z^{i}}{\rho}+O_{3}(\rho^{-1}),$
$\displaystyle\Delta_{\tilde{g}}\Phi$
$\displaystyle=\frac{2A}{\rho}+O_{2}(\rho^{-2}),$
$\displaystyle\nabla_{i}\Delta_{\tilde{g}}\Phi$
$\displaystyle=-2A\frac{z^{i}}{\rho^{3}}+O_{1}(\rho^{-3}),$

which, together with $\mathrm{Ric}_{\tilde{g}}=O_{2}(\rho^{-3})$ implies that

$\displaystyle\int_{D_{\rho}}|\mathrm{Ric}_{\tilde{g}}|^{2}_{\tilde{g}}\Phi
dV_{\tilde{g}}=8\pi A+O(\rho^{-1}).$

Passing to the limit and remembering that $\Phi<0$, we find that $A\leq 0$
with equality holding iff $\mathrm{Ric}_{\tilde{g}}\equiv 0$. That is, if
$\hat{M}\cong\mathbb{R}^{3}$ and $\tilde{g}=\delta$, which implies the final
result.

##  Appendix A Appendix: Some analytic results concerning AE manifolds

In this appendix we will collect some facts results concerning AE manifolds
which are used in the core of the paper. Most of these results are well-known
for experts. We include them for the sake completeness and to deliver a self-
contained presentation. For detailed proofs and discussions on these topics,
we refer the reader to Bartnik [4], Lee-Parker [28].

Let us start with the following fundamental theorem regarding the properties
of the Laplacian on AE manifolds.

######  Theorem A.1.

Let $(M,g)$ an asymptotically Euclidean manifold with a structure at infinity
$\phi:M\setminus K\rightarrow\mathbb{R}^{n}\setminus B_{1}$ with decay rate
$\tau$. If $\delta\not\in(\mathbb{Z}\setminus\\{-1,\cdots,3-n\\})$ and $q>1$
then

$\Delta_{g}:W_{\delta}^{2,q}(\phi)\rightarrow L^{q}_{\delta-2}$

is Fredholm. Moreover

$\left\\{\begin{array}[]{l}\hbox{ if }\delta>2-n\hbox{ then }\Delta_{g}\hbox{
is surjective,}\\\ \hbox{ if }2-n<\delta<0\hbox{ then }\Delta_{g}\hbox{ is
bijective,}\\\ \hbox{ if }\delta<0\hbox{ then }\Delta_{g}\hbox{ is
injective.}\end{array}\right.$ (A.1)

######  Remark A.1.

The decay rates in the set $\mathbb{Z}\setminus\\{-1,\cdots,3-n\\}$ are called
exceptional and we say that $\delta$ is non-exceptional if
$\delta\not\in\mathbb{Z}\setminus\\{-1,\cdots,3-n\\}$.

Let us now analyse the relation between different potential structures of
infinity. In particular, the following theorem concerns the existence of
harmonic coordinates.

######  Theorem A.2.

Let $(M,g)$ be an AE manifold, with $g\in W^{k,q}_{loc}$, $k\geq 1$, $q>n$,
and $(\Phi,x):M\backslash K\mapsto
E_{R}\doteq\mathbb{R}^{n}\backslash\overline{B_{R}(0)}$ where $K\subset\subset
M$, $R\geq 1$ is a structure of infinity of order $\tau>0$ with $1-\tau$ non-
exceptional, and fix $1<\eta<2$. There are functions $y^{i}\in
W^{k,q}_{\eta}$, $i=1,\cdots,n$, such that $\Delta_{g}y^{i}=0$ and
$(x^{i}-y^{i})\in W^{k,q}_{1-\tau^{*}}(E_{R})$ for
$\tau^{*}\doteq\min\\{\tau,n-2\\}$, which implies

$\displaystyle\begin{split}|x^{i}-y^{i}|&=o_{k}(r^{1-\tau^{*}}),\\\
|g(\partial_{x^{i}},\partial_{x^{j}})-g(\partial_{y^{i}},\partial_{y^{j}})|&=o_{k}(r^{-\tau^{*}}).\end{split}$
(A.2)

Furthermore, the set of functions $\\{1,y^{i}\\}$ is a basis for
$H_{1}=\\{u\in W^{k,q}_{\eta}:\Delta_{g}u=0\\}$.

###### Proof.

Let us first extend the functions $x^{i}$ smoothly to all of $M$. Then, near
infinity, $\Delta_{g}x^{i}=g^{kl}\Gamma^{i}_{kl}\doteq\Gamma^{i}\in
W^{k-1,q}_{-1-\tau}$. Notice that if $1-\tau>2-n$, then Theorem A.1 implies
the existence of $v^{i}\in W^{k+1,q}_{1-\tau}$ solving
$\Delta_{g}v^{i}=\Gamma^{i}$. Now, if $1-\tau<2-n$ (the equality case is an
exceptional case), we cannot, a priori, improve the decay given by $r^{2-n}$.
This actually follows from Theorem 1.17 in [4]. Therefore, in any case, we
know that there are solutions $v^{i}\in W^{k+1,q}_{1-\tau^{*}}$ to
$\Delta_{g}v^{i}=\Gamma^{i}$, where $\tau^{*}=\min\\{\tau,n-2\\}$. Therefore,
there is some $v^{i}\in W^{k+1,q}_{1-\tau^{*}}$ satisfying
$\Delta_{g}(x^{i}-v^{i})=0$. Let us then define $y^{i}\doteq x^{i}-v^{i}$,
which implies the first estimate in (A.2), and furthermore, since
$\frac{\partial y^{i}}{\partial x^{j}}=\delta^{i}_{j}+o(r^{-\tau^{*}})$, we
see that near infinity $\\{y^{i}\\}$ are coordinates which are asymptotically
Cartesian. This, in turn, implies the second estimate in (A.2). The final
claim concerning $H_{1}$ follows since $\\{1,y^{i}\\}$ spans an
$(n+1)$-dimensional subspace of $H_{1}$, but from Proposition 2.2 in [4]
$\mathrm{dim}(H_{1})=n+1$. ∎

######  Remark A.2.

Let us highlight that the statement of the above theorem is slightly different
than the well-known Theorem 3.1 in [4]. The difference relies in the fact
that, in [4], it does not seem to be explicitly stated that, a priori, the
decay rate $r^{2-n}$ cannot be improved, regardless of how large $\tau$ may
be. Nevertheless, as can be seen in the above proof, the above mild correction
comes about from results contained in the same reference, as is also clear
from a careful examination of the proof of Theorem 3.1 in [4]. Similar
comments apply to the following theorem, which can be found in [4] as
Corollary 3.2, and presents the relation between two different structures of
infinity.

######  Theorem A.3.

Let $(M,g)$ an asymptotically flat manifold, $g\in W^{k,q}_{loc}$, $k\geq 1$
and $q>n$, with two structures at infinity $\phi,\psi:M\setminus
K\rightarrow\mathbb{R}^{n}\setminus\overline{B_{1}(0)}$ with decay rates
$\tau_{\phi}$ and $\tau_{\psi}$, where of each of these weights satisfies that
$1-\tau$ is non-exceptional. There exists $(O,a)\in O(n)\times\mathbb{R}^{n}$
such that

$x^{i}-(O^{i}_{j}z^{j}+a^{i})\in W^{k,q}_{1-\tau}(\mathbb{R}^{n})$

which implies

$\displaystyle|x^{i}-(O^{i}_{j}z^{j}+a^{i})|=o_{k}(r^{1-\tau}),$ (A.3)

where $\tau\doteq\min\\{\tau_{\phi},\tau_{\psi},n-2\\}$, $x=\phi^{-1}$ and
$z=\psi^{-1}$.

###### Proof.

Let $y^{i}$ and $w^{i}$ be the harmonic coordinates constructed in the
previous theorem associated to $\phi$ and $\psi$ respectively. Then, since
$H_{1}$ is intrinsic to $M$ and $\\{1,y^{i}\\}$ and $\\{1,w^{i}\\}$ are bases
for this space, we get that

$\displaystyle w^{i}=A^{i}_{j}y^{j}+a^{i},$

where we $A\in\mathrm{GL}(n,\mathbb{R})$ a priori, but actually, from the
construction of the previous theorem, we know that
$z^{i}=A^{i}_{j}x^{j}+a^{i}-A^{i}_{j}v^{j}+\bar{v}^{i}$, with $v^{i}\in
W^{k,q}_{1-\tau^{*}_{\phi}}$ and $\bar{v}^{i}\in W^{k,q}_{1-\tau^{*}_{\psi}}$,
where $\tau^{*}$ was defined in the previous theorem. Since these last two
systems are Cartesian, then $A\in O(n)$ and we explicitly see that
$z^{i}-A^{i}_{j}x^{j}-a^{i}\in W^{k,q}_{1-\tau}$. ∎

The above estimate is the best possible and it is determined by the Ricci
curvature as shown by the next theorem which corresponds to Proposition 3.3 in
[4], see also [13].

######  Theorem A.4.

Let $(M,g)$ an asymptotically flat manifold with a structure at infinity
$\phi:M\setminus K\rightarrow\mathbb{R}^{n}\setminus B(0,1)$ with decay rate
$\tau$ such that $(\phi_{*}g-\delta)\in
W^{2,q}_{-\tau}(\mathbb{R}^{n}\setminus B(0,1))$ for $q>n$ and such that

$Ric_{g}\in L^{q}_{-2-\eta}(M)\hbox{ for some }\eta>\tau\hbox{ and
}\eta\not\in(\mathbb{Z}\setminus\\{-1,\cdots,3-n\\}).$

Then there exists a structure at infinity $\Theta:M\setminus
K^{\prime}\rightarrow\mathbb{R}^{n}\setminus B_{1}$ such that
$(\Theta_{*}g-\delta)\in W^{2,q}_{-\eta}(\mathbb{R}^{n}\setminus B_{1})$.

Finally, the following statement concerning the conformal Laplacian will be
important in our analysis. It is a variant of theorem 9.2 of [28]

######  Corollary A.1.

Let $(M,g)$ an asymptotically flat manifold of order $\tau>0$ and assume
$R_{g}\geq 0$. If $2-n<\delta<0$ and $q>1$, then

$L_{g}=\Delta_{g}-c_{n}R_{g}:W_{\delta}^{2,q}\rightarrow L^{q}_{\delta-2}$

is an isomorphism.

###### Proof.

Since $L_{g}$ is a compact perturbation of the Laplacian, it is also Fredholm,
hence thanks to its self-adjointness it suffices to proof injectivity to get
the result. Consider $u\in W_{\delta}^{2,q}$ such that
$\Delta_{g}u=c_{n}R_{g}u$, and first notice that by elliptic regularity we can
assume that $u\in W^{2,q}_{\delta}$ with $q>n$. Also, notice that $R_{g}u\in
W^{2,q}_{\delta^{\prime}-2}$ for any $\delta^{\prime}>\delta-\tau$. Since
$\tau>0$, we can pick $\delta^{\prime}$ satisfying
$\max\\{\delta-\tau,2-n\\}<\delta^{\prime}<\delta$ and appeal to Theorem A.1
to obtain $u\in W^{2,q}_{\delta^{\prime}}$. Since we can repeat the argument
as much as necessary, we can assume that $u\in W^{2,q}_{\delta^{\prime}}$ with
$\delta^{\prime}>2-n$ taken arbitrary close to $2-n$. Then, we can multiply
$L_{g}(u)=0$ by $u$ and integrate by parts to obtain

$\int_{M\setminus(\mathbb{R}^{n}\setminus B_{R}(0))}\left(|\nabla
u|_{g}^{2}+c_{n}R_{g}u^{2}\right)dv_{g}=\int_{\partial
B_{R}(0)}u\partial_{\nu}u\;d\sigma,$

Since $u=O_{2}(r^{\delta^{\prime}})$ near infinity, we can estimate $|\nabla
u|_{g}^{2}=O(r^{2\delta^{\prime}-2})$ and
$R_{g}u^{2}=O(r^{2\delta^{\prime}-\tau-2})$ which implies these quantities are
in $L^{1}(M)$ as long as chose $2-n<\delta^{\prime}<\frac{2-n}{2}$.
Furthermore, under these condition one finds
$u\partial_{\nu}u=O(r^{2\delta^{\prime}-1})$ and therefore

$\displaystyle\int_{M\setminus(\mathbb{R}^{n}\setminus B_{R}(0))}\left(|\nabla
u|_{g}^{2}+c_{n}R_{g}u^{2}\right)dv_{g}=O(R^{n-2\delta^{\prime}-2})$

so that we can pass to the limit as $R\rightarrow+\infty$, which gives that
$u\equiv 0$ since
$R_{g}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\geq}0$
and $u\rightarrow 0$ as $|x|\rightarrow\infty$, which concludes the proof. ∎

##  Appendix B Appendix: Conventions on Q-curvature

Let us adopt the following general definition of $Q$-curvature for an
arbitrary Riemannian manifold $(M^{n},g)$ with $n\geq 3$:

$\displaystyle
Q_{g}=-\Delta_{g}\sigma_{1}(S_{g})+4\sigma_{2}(S_{g})+\frac{n-4}{2}\left(\sigma_{1}(S_{g})\right)^{2},$
(B.1)

where
$S_{g}\doteq\frac{1}{n-2}\left(\mathrm{Ric}_{g}-\frac{1}{2(n-1)}R_{g}g\right)$
stands for the Schouten tensor and $\sigma_{k}(S_{g})$ stands for the $k$-th
elementary symmetric function of the eigenvalues of $S_{g}$. In this context,
the Paneitz operator is defined by

$\displaystyle
P_{g}u\doteq\Delta^{2}_{g}u+\mathrm{div}_{g}\left(\left(4S_{g}-(n-2)\sigma_{1}(S_{g})g\right)(\nabla
u,\cdot)\right)+\frac{n-4}{2}Q_{g}u,$ (B.2)

for all $u\in C^{\infty}(M)$. In this context, the following relations hold:

$\displaystyle\begin{split}\sigma_{1}(S_{g})&=\frac{R_{g}}{2(n-1)},\\\
\sigma_{2}(S_{g})&=\frac{1}{2}\left(\frac{n^{2}-n}{4(n-1)^{2}(n-2)^{2}}R^{2}_{g}-\frac{|\mathrm{Ric}_{g}|^{2}_{g}}{(n-2)^{2}}\right),\end{split}$
(B.3)

which implies that

$\displaystyle\begin{split}Q_{g}&=-\frac{1}{2(n-1)}\Delta_{g}R_{g}-\frac{2}{(n-2)^{2}}|\mathrm{Ric}_{g}|^{2}_{g}+\frac{n^{3}-4n^{2}+16n-16}{8(n-1)^{2}(n-2)^{2}}R_{g}^{2},\\\
P_{g}u&=\Delta^{2}u+\mathrm{div}_{g}\left(\left(\frac{4}{n-2}\mathrm{Ric}_{g}-\frac{n^{2}-4n+8}{2(n-1)(n-2)}R_{g}\>g\right)(\nabla
u,\cdot)\right)+\frac{n-4}{2}Q_{g}u.\end{split}$ (B.4)

In particular, for $n\neq 4$, if $\bar{g}=u^{\frac{4}{n-4}}g$, then

$\displaystyle Q_{\bar{g}}=\frac{2}{n-4}u^{-\frac{n+4}{n-4}}P_{g}u.$ (B.5)

In the case of $n=4$ we can apply the above definitions to get

$\displaystyle\begin{split}Q_{g}&=-\frac{1}{6}\Delta_{g}R_{g}-\frac{1}{2}|\mathrm{Ric}_{g}|^{2}_{g}+\frac{1}{6}R_{g}^{2},\\\
P_{g}u&=\Delta^{2}_{g}u+\mathrm{div}_{g}\left(\left(2\mathrm{Ric}_{g}-\frac{2}{3}R_{g}\>g\right)(\nabla
u,\cdot)\right),\end{split}$ (B.6)

and in this case, if $\bar{g}=e^{2u}g$, we have that [30]

$\displaystyle Q_{\bar{g}}=e^{-4u}\left(P_{g}u+Q_{g}\right).$ (B.7)

Now, let us notice that it is also quite standard to redefine the
$Q$-curvature in dimension 4 via (see [20, 29])

$\displaystyle
Q^{(4)}_{g}=\frac{1}{2}Q_{g}=-\frac{1}{12}\Delta_{g}R_{g}-\frac{1}{4}|\mathrm{Ric}_{g}|^{2}_{g}+\frac{1}{12}R_{g}^{2}$
(B.8)

and in this case, it follows that

$\displaystyle 2Q^{(4)}_{\bar{g}}=e^{-4u}\left(P_{g}u+2Q^{(4)}_{g}\right).$
(B.9)

We will not adopt this redefinitions and keep a unified notation via (B.1)
along this paper.

##  Appendix C Appendix: Conformal normal coordinates

In order to deliver a presentation as self-cointained as possible, this
section is meant to summarise some of the results concerning conformal normal
coordinates presented in [28], which which are used in the core of this paper.
The basic construction is given on a smooth Riemannian manifold $(M^{n},g)$
where we intend to expand $g$ around a fixed point $p\in M$. In particular, we
are interested in finding an element $\tilde{g}$ within the conformal class
$[g]$ for which, in $\tilde{g}$-normal coordinates $\\{x^{i}\\}$, the
following expansion holds around $p$:

$\displaystyle\det(\tilde{g})=1+O(r^{N})$ (C.1)

for any chosen $N\geq 2$, where $r=|x|$ (see Theorem 5.1 in [28]). This type
of expansion is achieved by first noticing that, for any Riemannian metric
$g$, in $g$-normal coordinates $\\{x^{i}\\}_{i=1}^{n}$ around $p\in M$, the
following holds

$\displaystyle
g_{ij}(x)=\delta_{ij}+\frac{1}{3}R_{iklj}x^{k}x^{l}+\frac{1}{6}R_{iklj,a}x^{k}x^{l}x^{a}+\left(\frac{1}{20}R_{iklj,ab}+\frac{2}{45}R_{iklc}R_{jabc}\right)x^{k}x^{l}x^{a}x^{b}+O(r^{5}),$

where all the coefficients at evaluated at $p$. From this expression it is
possible to compute $\det(g)$ around $p$ for any such metric, so as to get

$\displaystyle\begin{split}\det(g)(x)&=1-\frac{1}{3}R_{ij}x^{i}x^{j}-\frac{1}{6}R_{ij,k}x^{i}x^{j}x^{k}\\\
&-\left(\frac{1}{20}R_{ij,kl}+\frac{1}{90}R_{aijb}R_{aklb}-\frac{1}{18}R_{ij}R_{kl}\right)x^{i}x^{j}x^{k}x^{l}+O(r^{5}),\end{split}$
(C.2)

where again all the coefficients are evaluated at $p$. Assuming an expanssion
of the form $\det(g)=1+O(r^{N})$ with $N\geq 2$ (the case $N=2$ is valid for
any metric in its own normal coordinates) and appelaing to Theorem 5.2 in
[28], the authors can find a homogeneous polynomial $f\in\mathcal{P}_{N}$ so
that the expansion of $\det(\tilde{g})=1+O(r^{N+1})$ for $\tilde{g}=e^{2f}g$,
which establishes an inductive proof (see the proof of Theorem 5.1). But
notice that this implies that the symmetrization of the coefficients in (C.2)
of order up to $N$ must vanish for $\tilde{g}$. Thus, in the case we do this
construction for $N\geq 4$, we see that this implies

$\displaystyle\begin{split}&\tilde{R}_{ij}(p)=0,\\\
&\tilde{R}_{ij,k}+\tilde{R}_{ki,j}+\tilde{R}_{jk,i}(p)=0.\end{split}$ (C.3)

Putting together the second condition above with the contracted Bianchi
identities, we also see that

$\displaystyle\tilde{R}_{,k}(p)=0.$ (C.4)

## References

  * [1] Richard Arnowitt, Stanley Deser, and Charles W. Misner.  Coordinate invariance and energy expressions in general relativity.  Phys. Rev., 122(3):9971006, 2006.
  * [2] Rodrigo Avalos and Jorge H. Lira.  Reduced thin-sandwich equations on manifolds euclidean at infinity and on closed manifolds: Existence and multiplicity.  J. Math. Phy., 61:122501, 2020.
  * [3] Rodrigo Avalos, Jorge H. Lira, and Nicolas Marque.  Energy in Fourth Order Gravity.  arXiv e-prints, page arXiv:2102.00545, January 2021.
  * [4] Robert Bartnik.  The mass of an asymptotically flat manifold.  Comm. Pure Appl. Math., 39(5):661–693, 1986.
  * [5] Simon Brendle and Michael. Eichmair.  Large outlying stable constant mean curvature spheres in initial data sets.  Invent. Math., 107(3):663–682, 2014.
  * [6] Cliff P. Burgess.  Quantum gravity in everyday life: General relativity as an effective field theory.  Living Rev. Relativity, 7:5, 2004.
  * [7] Alessandro Carlotto.  Rigidity of stable minimal hypersurfaces in asymptotically flat spaces.  Calc. Var. Partial Differ. Equ., 55(3):1–20, 1988.
  * [8] Alessandro Carlotto.  Four Lectures on Asymptotically Flat Riemannian Manifolds.  in: Einstein Equations: Physical and Mathematical Aspects of General Relativity. Eds: S. Cacciatori, Batu Güneysu and S. Pigola. Springer Nature Switzerland AG, 2019.
  * [9] Alessandro Carlotto, Otis Chodosh, and Michael. Eichmair.  Effective versions of the positive mass theorem.  Invent. Math., 206(3):1–20, 2016.
  * [10] Alessandro Carlotto and Richard. Schoen.  Localizing solutions of the einstein constraint equations.  Invent. Math., 205(3):1–20, 2016.
  * [11] Sun-Yung Alice Chang and Paul C. Yang.  Extremal metrics of zeta functional determinants on 4-manifolds.  Ann. of Math., 142:171–212, 1995.
  * [12] Szu-yu Sophie Chen.  Conformal deformation on manifolds with boundary.  Geom. Funct. Anal., 19(4):1029–1064, 2009.
  * [13] Dennis M. DeTurck and Jerry L. Kazdan.  Some regularity theorems in Riemannian geometry.  Ann. Sci. École Norm. Sup. (4), 14(3):249–260, 1981.
  * [14] James Dilts and David Maxwell.  Yamabe classification and prescribed scalar curvature in the asymptotically Euclidean setting.  Comm. Anal. Geom., 26(5):1127–1168, 2018.
  * [15] Zindine Djadli and Andrea Malchiodi.  Existence of conformal metrics with constant $q$-curvature.  Ann. Math., 168:813–858, 2008.
  * [16] John F. Donoghue.  General relativity as an effective field theory: The leading quantum cerrections.  Phys. Rev. D, 50(6):3874–3888, 1994.
  * [17] Michael Eichmair and Jan Metzger.  Large isoperimetric surfaces in initial data sets.  J. Differ. Geom., 94(1):159–186, 2013.
  * [18] Matthew J. Gursky.  The principal eigenvalue of a conformally invariant differential operator, with an application to semilinear elliptic PDE.  Comm. Math. Phys., 207(1):131–143, 1999.
  * [19] Matthew J. Gursky, Fengbo Hang, and Yueh-Ju Lin.  Riemannian manifolds with positive Yamabe invariant and Paneitz operator.  Int. Math. Res. Not. IMRN, (5):1348–1367, 2016.
  * [20] Matthew J. Gursky and Andrea Malchiodi.  A strong maximum principle for the Paneitz operator and a non-local flow for the $Q$-curvature.  J. Eur. Math. Soc. (JEMS), 17(9):2137–2173, 2015.
  * [21] Fengbo Hang and Paul C. Yang.  Sign of Green’s function of Paneitz operators and the $Q$ curvature.  Int. Math. Res. Not. IMRN, (19):9775–9791, 2015.
  * [22] Fengbo Hang and Paul C. Yang.  $Q$ curvature on a class of 3-manifolds.  Comm. Pure Appl. Math., 69(4):734–744, 2016.
  * [23] Fengbo Hang and Paul C. Yang.  Q-curvature on a class of manifolds with dimension at least 5.  Comm. Pure Appl. Math., LXIX:1452–1491, 2016.
  * [24] Luan. H. Huang.  Foliations by stable spheres with constant mean curvature for isolated systems with general asymptotics.  Commun. Math. Phys., 300(2):331–373, 2010.
  * [25] Gerhard Huisken and Shing T. Yau.  Definition of center of mass for isolated physical systems and unique foliations by stable spheres with constant mean curvature.  Invent. Math., 124:281–311, 1996.
  * [26] Emmanuel Humbert and Simon Raulot.  Positive mass theorem for the Paneitz-Branson operator.  Calc. Var. Partial Differential Equations, 36(4):525–531, 2009\.
  * [27] Dan A. Lee.  Geometric relativity, volume 201 of Graduate Studies in Mathematics.  American Mathematical Society, Providence, RI, 2019.
  * [28] John M. Lee and Thomas H. Parker.  The Yamabe problem.  Bull. Amer. Math. Soc. (N.S.), 17(1):37–91, 1987.
  * [29] Jiayu Li, Yuxiang Li, and Pan Liu.  The $Q$-curvature on a 4-dimensional Riemannian manifold $(M,g)$ with $\int_{M}QdV_{g}=8\pi^{2}$.  Adv. Math., 231(3-4):2194–2223, 2012.
  * [30] Yueh-Ju Lin and Wei Yuan.  Deformations of Q-curvature I.  Calc. Var. Partial Differential Equations, 55(4):Art. 101, 29, 2016\.
  * [31] Juan Maldacena.  Einstein gravity from conformal gravity.  arXiv:1105.5632v2.
  * [32] Philip D. Mannheim.  Making the case for conformal gravity.  Found Phys, 42:388–420, 1980.
  * [33] David Maxwell.  Solutions of the Einstein constraint equations with apparent horizon boundaries.  Comm. Math. Phys., 253(3):561–583, 2005.
  * [34] Benoît Michel.  Masse des opérateurs GJMS.  arXiv e-prints, page arXiv:1012.4414, December 2010.
  * [35] Benoît Michel.  Geometric invariance of mass-like asymptotic invariants.  Journal of Mathematical Physics, 52(5):052504, 2011.
  * [36] Cheikh Birahim Ndiaye.  Constant q-curvature metrics in arbitrary dimension.  Journal of Functional Analysis, 251:1–58, 2007.
  * [37] Christopher Nerz.  Foliations by stable spheres with constant mean curvature for isolated systems without asymptotic symmetry.  Calc. Var. Partial Differ. Equ, 54(2):1911–1946, 2015.
  * [38] J. Qing and D. Raske.  On positive solutions to semilinear conformally invariant equations on locally conformally flat manifolds.  Int. Math. Res. Not., 2006(9):94172–94172, 2006.
  * [39] Gerard ’T Hooft and Martinus Veltman.  One loop divergencies in the theory of gravitation.  Ann. Inst. Henri Poincare, A, 20:69–94, 1974.
  * [40] Richard Schoen.  Conformal deformation of a riemannian metric to constant scalar curvature.  J. Diff. Geom, 20:479–595, 1984.
  * [41] Richard Schoen and Shing T. Yau.  On the proof of the positive mass conjecture in general relativity.  Comm. Math. Phys, 65:45–76, 1979.
  * [42] Richard Schoen and Shing T. Yau.  Comm. Math. Phys, 79:231–260, 1981.
  * [43] Richard Schoen and Shing T. Yau.  Conformally flat manifolds, kleinian groups and scalar curvature.  Invent. math., 92:47–71, 1988.
  * [44] Alexei Starobinsky.  A new type of isotropic cosmological models without singularity.  Physics Letters B, 91(1):99–102, 1980.