A Positive Energy Theorem for Fourth-Order Gravity

Home , Gerhard Huisken, Yamabe problem

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g¯ = −N 2dt2 +˜g, (3)

R+ 0 where N : M 7→ is the lapse function andg ˜ ∈ Γ(T2 M) restricts to a Riemannian metric g on each t = cte hypersurface. In this setting, the appropriate notion of energy associated to the action S and the corresponding ﬁeld equations Ag¯ = 0 becomes

. j Eα(g) = −α lim (∂j∂i∂igaa − ∂j∂u∂igui) ν dωr. (4) r→∞ n−1 ZSr In this expression, we are assuming that (M,g) is an AE manifold. This means that for a n given compact set K the asymptotic region M\K is diﬀeomorphic to R \B1(0) and it is n−1 foliated by topological (n−1)-dimensional spheres Sr , whose Euclidean volume element is denoted in the expression above by dωr. There, ν 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 Eα,β(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])

j EADM (g)= c(n) lim (∂igji − ∂jgii) ν dωr, (5) r→∞ n−1 ZSr where c(n) stands for a dimensional constant (see [4] for the detailed analytical properties of (5)). This energy EADM has had a huge impact both within GR and in geometric analysis. Most notoriously, it was the work of R. Schoen in [38] that elucidated the role that the ADM energy plays in the ﬁnal resolution of the Yamabe problem (see also [27] 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 GLg associated to the conformal Laplacian Lg 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

2 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, 24], geometric foliations and center of mass constructions [17, 23, 24, 35] and even gluing constructions [10] (for nice reviews of many of these topics, see [8, 26]). 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 devel- opment 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 establishing 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 Eα(g), we must actually fix the sing of α a priori. For reasons that will become apparent through this paper, . we will fix α = −1 and define E(g) = E−1(g). Let us notice that before embarking on the proof of the positive energy theorem, we should first analyse under what conditions 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 Qg ≥ 0 (see Proposition 3 and Theorem 4). Let us recall that given a Riemannian manifold (M n,g), n ≥ 3, its Q-curvature is defined by1 . 1 2 n3 − 4n2 + 16n − 16 Q = − ∆ R − |Ric |2 + R2. g 2(n − 1) g g (n − 2)2 g g 8(n − 1)2(n − 2)2 g 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 ≥ −τ n−4 3, which satisfies the decaying conditions: (i) gij − δij = O4(r ), with τ > max{0, 2 }, 1 in some coordinate system associated to an structure of infinity; (ii) Qg ∈ L (M,dVg), and such that Y ([g]) > 0 and Qg ≥ 0. Then, the fourth order energy E(g) is non-negative and E(g)=0 if and only if (M,g) is isometric to (Rn, ·). 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 Rg > 0 at infinity, it follows that E(g) ≥ 0 (see Theorem 3), while under hypotheses (i)-(ii), Qg ≥ 0 and g Yamabe positive, if 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 E(g) = 0. 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 ∈{3, 4}, such n that Qg ≥ 0 and Y ([g]) > 0, is isometric to (R , δ).

1See Appendix B for details on our conventions on Q-curvature.

3 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 E(g) (see Proposition 5 for more details).

Corollary. Let (M,g) be an asymptotically Euclidean four manifold with κg = M Qg dvg ≥ 0 and Y ([g]) ≥ 0, then (M,g) is isometric to the euclidean R4. R After presenting the above results concerning the positive energy theorem associated to 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 [39, 40, 41] in the resolution of the Yam- abe 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 ≥ 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 ≥ 5, with different degrees of generality, let us draw the reader’s attention to [22, 19, 36]. 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 [20, 21]. Let us now focus on the results of [22]. There, the authors prove that given a closed manifold (M,g) of dimension n ≥ 5 which is Yamabe non-negative and satisfies Qg ≥ 0 not identically zero,2 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 ≥ 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 [25], then by Gursky-Malchiodi who incorporated the cases 5 ≤ n ≤ 7 without this last restriction, and finally by Hang-Yang [20], who weakened the hypotheses on the scalar curvature imposed in [19] and achieved the final form of this results which was applied in Theorem 4 of [22]. In Section 2.2 we will show that exactly in dimensions 5 ≤ n ≤ 7 or if (M,g) has a point p around which it is conformally flat, then the mass of the Green function GPg associated to the Paneitz operator Pg is positively proportional to the energy E(ˆg) of the 4 ˆ n−4 asymptotically euclidean manifold (M = M\{p}, gˆ = GPg 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 [19, 22, 25]). Let (M,g) be a closed n-dimensional Riemannian manifold, with 5 ≤ n ≤ 7 or n ≥ 8 and locally conformally flat around some point p ∈ M. If Y ([g]) ≥ 0 and (M,g) admits a conformal metric with semi-positive

2 In this cases we will say that Qg is semi-positive.

4 Q-curvature, then the mass of GPg 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 in by Hang-Yang in [22] to solve the Q-curvature prescription problem in dimensions 5 ≤ n ≤ 7 or n ≥ 5 and M locally conformally ﬂat around a point. This highlights the potential parallel of energy 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]) ≥ 0, then 2 4 κg ≤ 16π with equality holding iﬀ (M ,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 [26]. 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-know 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, Benoˆıt Michel, has previously analysed a similar notion of energy, see [33].

Acknowledgments: The authors would like to thank the CAPES-COFECUB, CAPES- PNPD and ANR (ANR- 18-CE40-002) for their ﬁnancial support.

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.

ij • Given a Riemannian manifold (M,g), we will denote by ∆gu = g ∇i∇ju the negative Laplacian.

• Given a Riemannian manifold (M,g) we denote by ∇ its Riemannian connection and deﬁne the curvature tensor by

R(V, W )Z = ∇V ∇W Z −∇W ∇V Z −∇[V,W ]Z, V,W,Z ∈ Γ(T M),

i n i i and we label its components in some coordinate system {x }i=1 via Rjkl = dx (R(∂k, ∂l)∂j). k Then, the Ricci tensor is deﬁned locally via the contraction Rjl = Rjkl.

5 • Given a tensor ﬁeld T , indices of ∇kT that result of covariant diﬀerentiation will be separated by a comma. That is, if Tij are the components of a 2-tensor, then we 2 denote the components of ∇ T by Tij,kl. Given Σ ֒→ M an embedded compact hypersuface in (M,g), we will denote by ν • the outward pointing g-unit normal to Σ.

Given Σ ֒→ Rn an embedded compact hypersuface, we will denote by νe Euclidean • outward pointing unit normal to Σ.

n−1 n ωn−1 will denote the volume of the unit sphere S ֒→ R and dωn−1 its canonical • volume measure.

• ∞ R∗ let (M,g) a compact manifold, the conformal class of g is {ug ; u ∈ C (M, +)}, denoted [g].

• let (M,g) a compact manifold, ist Yamabe ivariant is deﬁned as

M Rg˜ dvg˜ Y ([g]) = inf n−2 . g˜ vol(R M) 2

k n τ • Given a function f ∈ C (R ), we say that f = Ok(|x| ) for some k ≥ 0 and τ ∈ R if for any multi-index α with |α| ≤ k, the functions ∂αf = O(|x|τ−|α|) either as |x|→∞ or |x|→ 0, depending on the context.

2 1 Deﬁnition 1 (Weighted spaces). Let σ = (1+ r ) 2 and δ ∈ R, we set

• p Rn p p −δp−n Lδ ( ) = u ∈ Lloc | Rn |u| σ dx < +∞ , we equip this set with the norm kukp = |u|pσ−δp−n dx. p,δ Rn R • ∞ Rn R ∞ −δ Lδ ( )= u ∈ Lloc | supRn |u|σ < +∞ , we equip this set with the norm kuk∞,δ = sup |u|σ−δ. Rn

• k,p Rn k,p k j Wδ ( ) = u ∈ Wloc | j=0 kd ukp,δ−j < +∞ , we equip this set with the norm k j kukk,p,δ = jn=0 kd ukp,δ−Pj. o Then we introduceP the notion of asymptotically ﬂat manifolds.

k,q Deﬁnition 2 (AE manifolds). A (M,g) complete Riemannian manifold, with g ∈ Wloc (M) for some k ≥ 1 and q > n is said asymptotically euclidean (with one end) if there exists n K ⊂ M compact and φ : M \ K → R \ B1(0) a diﬀeomorphism such that

• φ∗(g) is a uniformly positive deﬁned metric, i.e. there exists λ> 1 such that 1 |ξ|2 ≤ g (x)ξiξj ≤ λ|ξ|2 ∀x ∈ Rn \ B ∀ξ ∈ Rn. λ ij 1 • k,q Rn φ∗(g)ij − δij ∈ W−τ ( \ B(0, 1)) for some τ > 0 called the decreasing rate.

6 Important remark: In this chart we defined σ and we remark that the definition q k,q of Lδ is independent of the chart. But Wδ depends on the chart φ, since the partial k,q derivatives will depend on the choice of coordinates. It will be denoted Wδ (φ). In the rest of the section, we will call such a chart a structure at infinity. Once the structure k,q Rn k,q at infinity is chosen we naturally extend the definition of Wδ ( \ B(0, 1)) to Wδ (M) since on the compact part all choices of chart defined the same structure.

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 τ > 0 with n respect to some structure at infinity Φ: M\K 7→ R \B¯, if, in such coordinates, gij −δij = −τ O4(|x| ). 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

0 if n =3, 4 τ = . n n − 2 if n ≥ 5 2 Proposition 1. Let (M n,g) be an AE manifold of dimension n ≥ 3 satisfying the following conditions

i) There are end rectangular coordinates, given by a structure of inﬁnity Φ, where −τ gij = δij + O4(r ), where τ > τn;

1 ii) The Q-curvature of g is in L (M,dVg). . Then, given an exhaustion of M by compact sets Ωk such that Sk = Φ(∂Ωk) are smooth connected (n − 1)-dimensional manifolds without boundary in Rn satisfying . Rk = inf{|x| : x ∈ Sk} −−−→ ∞, k→∞ (6) n−1 Rk area(Sk) is bounded as k →∞, the limit

(Φ) j E (g) = lim (∂j∂i∂igaa − ∂j∂u∂igui) ν dS, (7) k→∞ ZSk exists and is independent of the sequence of {Sk} used to compute it.

7 Proof. From the decaying conditions and the familiar expression 2 2 Rg = ∂i∂jgij − ∂j∂jgii + O2((g − δ)∂ g)+ O2((∂g) ), −2τ−2 = ∂i∂jgij − ∂j∂jgii + O2(r ) we get ab ab c ∆gRg = g ∇a∇bRg = g (∂a∂bRg +Γab∂cRg) , ab ab −2τ−2 c = g ∂a∂b ∂i∂jgij − ∂j∂jgii + g ∂a∂b O2(r ) +Γab∂c (∂i∂jgij − ∂j∂jgjj) c −2τ−2 +Γab∂c O2 r , ab ab ab −2τ−4 −2τ−4 = g ∂a∂b ∂i∂jgij −∂j∂jgii + g − δ O(r )+ O(r ) −τ−3 −τ−3 −2τ−3 + O3 ((g −δ)∂g))O1(r )+ O3(∂g)O1(r )+ O3 ((g − δ)∂g))O1(r ) −2τ−3 + O3(∂g)O1(r ), −2τ−4 = ∂a∂a ∂i∂jgij − ∂j∂jgii + O(r ). In particular, denoting by Dk the annular region between Sk+1 and Sk, we get that

n−1 ∂j∂j (∂u∂igui − ∂u∂ugii) dVe = (∂j∂u∂igui − ∂j∂u∂ugii)νjdS ZDk ZSk+1 n−1 − (∂j∂u∂igui − ∂j∂u∂ugii)νjdS . ZSk Therefore, if the left-hand side is integrable over Rn\K, then the above boundary integrals form a Cauchy sequence as r →∞ and therefore (7) is well defined. But from the above 1 computations, we see that this can be reduced to ∆gRg ∈ L (M) and 2(τ + 2) > n, that −τ−2 is τ > τn. In particular, notice that under our decaying conditions Rg, Ricg = O(r ) near infinity, implying that 1 Q = − ∆ R + O(r−2τ−4), g 2(n − 1) g g 1 therefore the leading order is carried on the first term and ∆gRg ∈ L (M) can be replaced 1 by Qg ∈ L (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 prove 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

n−1 E(g)= − lim ∂rRgr dωn−1. (8) − r→∞ Sn 1 Z 1 n−1 Rn Proof. Let us denote by ν the outward g-unit normal to Sr ֒→ , with r suﬃciently large, and by νe the outward euclidean unit normal to the same sphere. Then, it follows e −τ that |ν − ν|g = O(r ). Thus, we ﬁnd that e e g(∇Rg, ν)= g(∇Rg, ν )+ g(∇Rg, ν − ν ), e i e j e i e j = ∂iRgνi +(g − δ)ij∇ Rg(ν ) + ∂iRg(ν − ν )i +(g − δ)ij∇ Rg(ν − ν ) .

−2τ−3 −2τ−3 −3τ−3 O1(r ) O1(r ) O1(r ) | {z } | {z } | {z } 8 −2τ−2 Also, we already know that Rg = ∂i∂jgij − ∂j∂jgii + O2(r ), which implies xi g(∇R , ν)= ∂ (∂ ∂ g − ∂ ∂ g ) + O (r−2τ−3) g i k j kj j j kk r 1 Thus,

i n−1 x n−1 n−2τ−4 g(∇Rg, ν)r dωn−1 = ∂i (∂k∂jgkj − ∂j∂jgkk) r dωn−1 + O1(r ). − − Sn 1 Sn 1 r Z 1 Z 1

Now, from our decaying condition τ > τn, we see that n − 2τ − 4 < 0, which implies that in the limit r →∞ the last term vanishes. Therefore, under these conditions

n−1 e n−1 lim g(∇Rg, ν)r dωn−1 = lim ∂i (∂k∂jgkj − ∂j∂jgkk) ν r dωn−1. − − i r→∞ Sn 1 r→∞ Sn 1 Z 1 Z 1 That is,

n−1 E(g)= − lim g(∇Rg, ν)r dωn−1, − r→∞ Sn 1 Z 1 e n−1 = − lim ∇Rg · ν r dωn−1, − (9) r→∞ Sn 1 Z 1 n−1 = − lim ∂rRgr dωn−1. − r→∞ Sn 1 Z 1

With the aid of the above two results, we can establish the following theorem, which establishes that E(g) is a geometric object within a suitable class of AE manifolds. Theorem 1. Let (φ, x) and (ψ,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 τ1, τ2 . (φ) (ψ) respectively, satisfying τ = min{τ1, τ2} > τn. Then, the energies E (g) and E (g) are well-defined and equal. Moreover, the following coordinate independent identity follows:

E(g)= − ∆gRgdVg. (10) ZM Proof. Appealing to Proposition 1, let us compute the energy Eφ(g), associated to the structure of inﬁnity given by (φ, x) using a sequence of spheres near inﬁnity of radii ∞ {rk}k=1. From the above lemma, we have that

(φ) E (g)= − lim ∇RgydVe, − |x|→∞ Sn 1 Z 1 n−1 where dVe = r dr ∧ dωn−1 stands for the canonical Euclidean volume form. Appealing to the decaying conditions of g, we know that det(g)=1+ O(r−τ ) and therefore

n−τ−4 ∇RgydVg = ∇RgydVpe + O(r ). Since n − τ − 4 < 0 under our hypotheses, the above implies that

(φ) E (g)= − lim ∇RgydVg. r→∞ n−1 ZSr

9 Finally, since divg(∇Rg)dVg = d(∇RgydVg), we ﬁnd that

(φ) E (g)= − lim ∆gRgdVg = − ∆gRgdVg, r→∞ ZDr ZM n−1 where Dr ֒→ M is the inner region in M with ∂Dr = Sr . 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 (φ, x) and claim that for any structure of infinity of M were g satisfies conditions (i)-(ii) of Proposition 1, it holds that

E(g)= − ∆gRgdVg. ZM

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 ≥ 3, which satisfies the decaying conditions i) and ii) of Proposition 1 and such that ∆gRg ≤ 0. Then, the fourth order energy E(g) is non-negative and 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 reach (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 [19] will be used to insure that Rg > 0 as soon as we are not flat. Proposition 3. Let (M n,g) be an AE Riemannian manifold with n ≥ 3 satisfying (1) Qg ≥ 0 and (2) Rg ≥ 0. Then, either Rg > 0 or g is flat.

Proof. If Qg ≥ 0, then we ﬁnd that

2 2 ∆gRg − c1(n)Rg ≤ −c2(n)|Ricg|g ≤ 0, for some constants c1(n),c2(n) > 0. From the the condition Rg ≥ 0 we can appeal to Lemma 4 in [32] to conclude that if Rg vanishes at a single point, then it must be identically zero. Thus, we already see that either Rg > 0 or Rg ≡ 0. In the second case, we find 2 that 0 ≤ Qg = −c2(n)|Ricg|g ≤ 0, and thus Ricg ≡ 0, which implies (from asymptotic flatness) that g is flat. Indeed, since Ricg ≥ 0, then by Bishop-Gromov theorem insures vol(Bg (p,r)) that r 7→ n is non-increasing, hence it must be constant by asymptotic flatness, ωrr finally the equality case of Bishop-Gromov theorem implies that we 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 [25, 19, 20].

10 Theorem 2. Let (M n,g) be an n-dimensional AE manifold, with n ≥ 3, which satisﬁes the decaying conditions i) and ii) of Proposition 1 and such that Qg, Rg ≥ 0. Then, the fourth order energy E(g) is non-negative and E(g)=0 if and only if Qg ≡ 0 and (M,g) is conformal to (Rn, ·).

Proof. Let us ﬁrst notice that we need only work with the case Qg ≥ 0 and Rg > 0, since in the remaining case the result follows from Proposition 3. Thus, in what follows we assume that Qg ≥ 0 and Rg > 0. Proof of positivity −τ−2 Let us start by noticing that under our hypotheses we have Rg = O2(r ) with τ > τn. 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

n−1 ∂ n−1 n−2 ∂rRgr dωn−1 = r Rgdωn−1 − (n − 1)r Rgdωn−1. − − − Sn 1 ∂r Sn 1 Sn 1 Z 1 Z 1 ! Z 1 Deﬁne

. n−2 h(r) = r Rgdωn−1. (11) − Sn 1 Z 1 Then,

n−1 ∂ ∂rRgr dωn−1 = (rh(r)) − (n − 1)h(r). (12) − Sn 1 ∂r Z 1 By assumption we have that h> 0 and from our decreasing assumption that there is −τ−2 some r0 suﬃciently large, so that Rg(x) ≤ Cr (x) for all x ∈ M such that r(x) >r0. −τ+n−4 In particular, this last condition implies that h(r) ≤ Cωn−1r for all r>r0. ∗ ∗ Suppose that there is some r >r0 such that (12) is positive at r . Then, by continuity, there is some interval (r∗ − ǫ, r∗ + ǫ) so that

rh′(r) > (n − 2)h(r), which implies

h(r) r log > (n − 2) log , for any r ∈ [r∗,r∗ + ǫ). h(r∗) r∗ ∗ h(r ) n−2 ∗ ∗ −τ+n−4 That is h(r) > r∗n−2 r for all r ∈ [r ,r +ǫ). But notice that also h(r) ≤ Cωn−1r for all such r, which shows that

h(r∗) rn−2 < h(r) ≤ Cω r−τ+n−4 for any r ∈ [r∗,r∗ + ǫ). r∗n−2 n−1

∗ Since r is ﬁxed, the above implies that there must be some ǫmax where (12) is non-positive ∗ ∗ around r + ǫmax. That is, there must be some r1 >r where

n−1 ∂ r ∂rRg(r1, θ)dωn−1(θ)= (rh(r)) − (n − 1)h(r) ≤ 0. (13) 1 − Sn 1 ∂r Z 1

11 n−1 ∞ The idea is now to repeat this procedure so as to select a sequence of spheres {Sri }i=1 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

E n−1 (g)j = −rj ∂rRgdωn−1 ≥ 0 for all j, S Z rj which implies

n−1 E(g)= − lim ∂rRgr dωn−1 ≥ 0. (14) j→∞ S Z rj Proof of rigidity Now, the idea is to analyze whether E(g) implies g-ﬂatness. First we look for a 4 − conformal metricg ˜ = u n 2 g such that Rg˜ ≡ 0, that is,

Lg(u) = ∆gu − cnRgu =0, (15)

n−2 where cn = 4(n−1) . Setting u =1+ φ, this is equivalent to solve

∆gφ − cnRgφ = cnRg, (16)

−τ−2 k We know that Rg = O2(r ), this implies that Rg ∈ H−δ−2 for any δ < τ and any k. s s−2 n Thanks to corroraly A.1, we know that Lg : Hρ 7→ Hρ−2 is an isomorphism for s > 2 s . and 2 − n<ρ< 0. Thus we ﬁnd a (smooth) solution φ ∈ H−δ for any 0 <δ<σ = min{τ, n − 2}.3. I) n =46

− . − n 4 4 Then, let Φ = u n−2 ; rewrite g = Φ n−4 g˜ and notice that transformation rule for the Paneitz operator gives us

n − 4 n+4 Φ n−4 Q = P Φ. 2 g g˜

2 2 Also, since Rg˜ = 0, we have that Qg˜ = − (n−2)2 |Ricg˜|g˜, and thus

(n − 4) P Φ = ∆2Φ + div (4S (∇Φ, ·)) − |Ric |2Φ, (17) g˜ g˜ g˜ g˜ (n − 2)2 g˜ g˜

1 where, Sg˜ = n−2 Ricg˜ since Rg˜ = 0. Therefore, denoting by Dr the bounded region in M whose boundary is given by the sphere Sr in the end of M, we see that

n − 4 n+4 (n − 4) 2 4 Φ n−4 Q + |Ric | Φ dv = g˜(∇˜ ∆ Φ, ν˜)dω + Ric (∇Φ, ν˜)dω 2 g (n − 2)2 g˜ g˜ g˜ g˜ g˜ n − 2 g˜ g˜ ZDr ZSr ZSr (18)

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

Estimates on Ricg˜(∇Φ, ν˜)

3In the case τ>n − 2, we can achieve δ = σ.

12 −δ From the above analysis we ﬁnd that u =1+ O4(r ) for any δ < σ = min{τ, n − 2}, and therefore

−δ −τ g˜ij = (1+ O4(r ))(δij + O4(r )), −δ = δij + O4(r ).

−δ−2 Then, Ricg˜ = O2(r ) and also

− n − 4 − n 4 −1 −δ−1 ∇Φ= − u n−2 ∇u = O (r ). n − 2 3 Which implies

e i e j −2δ−3 Ricg˜(∇Φ, ν˜) = Ricg˜(∇Φ, ν ) + Ricg˜ij∇ Φ(ν − ν˜) = O2(r ), n−1 n−2δ−4 Ricg˜(∇Φ, ν˜)r = O2(r ).

n Since σ > 2 − 2, then, we can choose δ such that n−1 |Ricg˜(∇Φ, ν˜)|r → 0 as r → +∞.

Estimates on ∆g˜Φ Let us rewrite

− n − 4 −2 n 3 ∇˜ Φ= − u n−2 ∇˜ u, i n − 2 i − − n − 4 −2 n 3 n − 3 − 3n 8 ∇˜ ∇˜ Φ= − u n−2 ∇˜ ∇˜ u − 2 u n−2 ∇˜ u∇˜ u , j i n − 2 j i n − 2 j i from which we get

− − n − 4 −2 n 3 (n − 4)(n − 3) − 3n 8 2 ∆ Φ= − u n−2 ∆ u +2 u n−2 |∇˜ u| . g˜ n − 2 g˜ (n − 2)2 g˜ Now, we can compute that ij ˜ ij k k ∆g˜u =g ˜ ∇j∇iu =g ˜ ∇j∇iu − (Γij(˜g) − Γij(g))∇ku , ij ij ij k k = ∆gu +(˜g − g )∇j∇iu − g˜ (Γij(˜g) − Γij(g))∇ku ij ij −δ −τ −δ k k and we can estimateg ˜ − g = O4(r ) − O4(r ) = O4(r ) and Γij(˜g) − Γij(g) = −δ−1 −τ−1 −δ−1 −δ−1 O3(r ) − O3(r )= O3(r ), since σ = min{τ, n − 2}, and also ∇u = O3(r ). Therefore ij k k −2δ−2 g˜ (Γij(˜g) − Γij(g))∇ku = O3(r ), ij ij −2δ−2 (19) (˜g − g )∇j∇iu = O2(r ). On the other hand, from (15) we ﬁnd that

−2δ−2 ∆gu = cnRg + cn Rg φ = cnRg + O2(r ). (20)

−δ−2 −δ O2(r ) O4(r ) Putting together (19)-(20), we ﬁnally|{z} ﬁnd that|{z}

−2δ−2 ∆g˜u = cnRg + O2(r ), (21)

13 which, in turn, implies

− n − 4 −2 n 3 −2δ−2 ∆ Φ= − u n−2 R + O (r ), g˜ 4(n − 1) g 2 (22) − n − 4 −2 n 3 −2δ−3 ∇(∆ Φ) = − u n−2 ∇R + O (r ). g˜ 4(n − 1) g 1

Therefore

e e g˜(∇˜ ∆g˜Φ, ν˜)=˜g(∇˜ ∆g˜Φ, ν )+˜g(∇˜ ∆g˜Φ, ν˜ − ν ), xi xj = ∇ ∆ Φ +(˜g − δ )∇i(∆ Φ) +˜g(∇˜ ∆ Φ, ν˜ − νe), i g˜ r ij ij g˜ r g˜ −2δ−3 −2δ−3 O1(r ) O1(r )

− i n − 4 −2 n 3 x −2δ−3 = − u | n−2 ∇ R{z + O}(r |), {z } 4(n − 1) i g r 1 implying

− n−1 n − 4 −2 n 3 n−1 n−2δ−4 g˜(∇˜ ∆ Φ, ν˜)r = − u n−2 ∂ R r + O (r ), g˜ 4(n − 1) r g 1 1 −−−→→∞ −−−→0 r r→∞

n | {z } | {z } where we have used, as above, that δ > 2 − 2, implying we can choose δ > 0 such that −δ n − 2δ − 4 < 0. From all this and dωg˜ = (1+ O(r ))dωr, we ﬁnd that

n − 4 n−1 −δ n−1 lim g˜(∇˜ ∆g˜Φ, ν˜)dωr = − lim ∂rRgr dω + ∂rRgO(r )r dω , r→∞ n−1 r→∞  n−1 n−1  Sr 4(n − 1) Sr Sr Z Z Z n−2δ−4  O(r )  n − 4   = E(g), | {z } 4(n − 1)

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

n − 4 n+4 (n − 4) 2 n − 4 Φ n−4 Q + |Ric | Φ dv = E(g). (23) 2 g (n − 2)2 g˜ g˜ g˜ 4(n − 1) ZM

Thus, if E(g) = 0, then Qg ≡ 0 and Ricg˜ ≡ 0, which implies, through results such as those exposed in the proof of Proposition 3 thatg ˜ is flat and M ∼= Rn. Therefore, since 4 g˜ = u n−4 g, then g is conformally-flat. II) n =4 Now, fix Φ = − ln(u) and rewriteg ˜ = e−2Φg. Then, using the transformation rule (B.7) we find

−4Φ 0 ≤ Qg = e (PgΦ+ Qg˜). (24)

1 2 Since Rg¯ = 0, then Qg˜ = − 2 |Ricg˜|g˜, which implies 1 0 ≤ Q e4Φ + |Ric |2 dV = g˜(∇˜ ∆ Φ, ν˜)dω +2 Ric (∇Φ, ν˜)dω (25) g 2 g˜ g˜ g˜ g˜ g˜ g˜ g˜ ZDr ZSr ZSr

14 Estimates on Ricg˜(∇Φ, ν˜) −δ 1 We clearly have u = 1+ O4(r ) for some δ < σ. Then, from ∇Φ = − u ∇u = −δ−1 O3(r ), we ﬁnd that 3 3 3 −2δ Ricg˜(∇Φ, ν˜)r = Ricg˜(∇Φ, νe) r + Ricg˜(∇Φ, ν˜ − νe) r = O3(r ). (26) −2δ−3 −3δ−3 O3(r ) O3(r ) −δ Combining this with dωg˜ = (1+| O{z(r ))}dωr we| get {z }

Ricg˜(∇Φ, ν˜)dωg˜ = o(1). (27) ZSr

Estimates on g˜(∇∆g˜Φ, ν˜) Straightforwardly we see that −1 −2 −1 −2δ−2 ∇˜ j∇˜ iΦ= −u ∇˜ j∇˜ iu + u ∇˜ ju∇˜ iu = −u ∇˜ j∇˜ iu + O3(r ), (28) ˜ ˜ k −2δ−2 ∇j∇iu = ∇j∇iu − ∆Γji∇ku = ∇j∇iu + O3(r ), k k k where ∆Γji =Γij(˜g) − Γij(g). Thus, we see that −1 −1 ij ij −2δ−2 −1 −2δ−2 ∆g˜Φ= −u ∆gu − u (˜g − g )∇j∇iu +O3(r )= −u ∆gu + O3(r ),

−2δ−2 O3(r ) −1 −2δ−2 = −cnu Rg + O3(|r ){z, } thus, −1 2 −2δ−3 −1 −2δ−3 ∇∆g˜Φ= −cnu ∇Rg + cnu ∇uRg + O2(r )= −u ∇Rg + O1(r ), −1 −1 −2δ−3 g˜(∇∆g˜Φ, ν˜)= −cnu g˜(∇Rg, νe) − cnu g˜(∇Rg, ν˜ − νe) +O1(r ),

−2δ−3 O1(r ) implying | {z }

−1 −2δ g˜(∇∆g˜Φ, ν˜)dωg˜ = −cn u g˜(∇Rg, νe)dωg˜ + O(r ), ZSr ZSr −1 −1 −δ −2δ = −cn u g˜(∇Rg, νe)dωr − u g˜(∇Rg, νe) O(r )dωr + O(r ), Sr Sr Z Z O(r−τ−3)

−1 = −cn u g˜(∇Rg, νe)dωr + o(1). | {z } ZSr Therefore, we ﬁnd that

lim g˜(∇∆g˜Φ, ν˜)dωg˜ = cnE(g). (29) r→∞ ZSr Finally, putting together (25),(27) and (29), we ﬁnd that 1 0 ≤ Q e4Φ + |Ric |2 dV = c E(g), (30) g 2 g˜ g˜ g˜ n ZM which implies that if E(g) = 0, then Ricg˜ ≡ 0 and therefore g is conformally-ﬂat with Qg ≡ 0. 4 v Finally, g = v n−4 δ (or g = e2 δ if n = 4), with ∆2v = 0 and lim v = 1 (or lim v = 0), r→∞ r→∞ hence, by maximum principle, we get that v ≡ 1 (or v ≡ 0) which achieves the proof of the theorem.

15 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 Qg is non-negative. Furthermore, under this last condition, equations (23) and (23) 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 ≥ 3, which satisﬁes the decaying conditions i) and ii) of Proposition 1 and such that Rg > 0 on M \ K, for some compact set K then the fourth order energy E(g) ≥ 0.

Theorem 4. Let (M n,g) be an n-dimensional AE manifold, with n ≥ 3, which satisﬁes the decaying conditions i) and ii) of Proposition 1 and such that Qg ≥ 0 and Y ([g]) > 0, then E(g) ≥ 0 with equality holding if and only if (M,g) is isometric to (Rn, ·).

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 Rn ∞ ∞ ( , δ), with n ≥ 3, and let {hk}k=1 ∈ C (U), be a sequence of smooth compactly supported symmetric second rank tensor fields, U¯ ⊂⊂ M, with ||hk|| < ǫ, for some fixed small ǫ > 0, such that hk −−−→ 0. Then, consider the sequence of metrics k→∞ gk = δ + hk, where the construction of the hk can be made so that gk are not con- ∞ formally flat. Furthermore, this construction can be fit so that {gk}k=1 are all Yamabe 4 s positive. Therefore, there are conformal factors {uk = 1+ ϕk}, with ϕk ∈ H−δ, with 5 δ < n − 2 arbitrary, such that ∆ u − R u = 0. This means that R 4 = 0 and gk k gk k n−2 uk gk 4 2 n−2 4 4 thus Q − = −c2(n)|Ric − | 4 < 0. Furthermore, all the metrics uk gk sat- n 2 n 2 n−2 u gk u gk k k uk gk isfy the decaying conditions of Proposition 1 and are, by construction, Yamabe positive. 4 n−2 ak,n −(n−2) Nevertheless, from their asymptotics uk gkij = (1+ rn−2 )δij + ok(r ) we see that E(gk) = 0 for all k, although none of the gk are conformally flat. Finally, by considering 1 k sufficiently large, we can make − < Q 4 < 0, which shows that there are Yamabe k n−2 uk gk 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 E(g) = 0 as a consequence of the fall-oﬀ conditions.

Corollary 2. Any n-dimensional AE-Riemannian manifold (M n,g), with n ∈ {3, 4}, n such that Qg ≥ 0 and Y ([g]) > 0, is isometric to (R , δ). 4For an explicit construction of this see Example 1 in [2] 5In fact, from Theorem 1.17 in [4], it follows that

a − − u =1+ k + o(r (n 2)), k rn−2 where ak are constants.

16 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 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 [26]). Let us start by briefly commenting on this case as a warm-up. Let (M,g) be an asymptotically flat surface with τ > 0 and non-negative integrable Gaussian curvature. Let us consider a chart a infinity, and recall the definition of the ADM energy j m(g) = lim (∂igij − ∂jgii)ν ds, R→+∞ Z∂BR where ν 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

2χ(M)= K dσ + kg ds. c ZM\B(0,R) Z∂B(0,R) Here we write χ(M) instead of χ(M \ B(0, R)c) since they are equal for R large. Then in polar coordinate we have, 1 1 1 1 1 k = − g (∇ ∂ , ∂ )= − Γr (g)+ O = − Γr (δ)+ O g r2 ∂θ θ r r2 θθ r1+τ r2 θθ r1+τ 1 1 = + O , r r1+τ hence passing to the limit we obtain that

2πχ(M) − 2π = K dσ. ZM But we know that χ(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 ≡ 0. Hence we recover the following well-known proposition, see chapter 3 [26], 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 2 with τ > 0 with only one end such that |Wg| is integrable. From Corollary 2, we know that the mass is necessarily vanishing

3 E(g)= − lim ∂rRgr dω3 =0. (31) R→∞ ZSR 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) 2 2 32π χ(M)= |W | + 16σ2(Sg) dv +8 Bg dv ZKR Z∂B(0,R) 17 c 1 1 where KR = M \ B(0, R) , Sg = 2 Ric − 6 Rgg is the Schouten tensor and Bg = 1 αβ 1 3 2 2 3 2 RgH −Ric(ν, ν)H −Rγαβγ II + 3 H −H|II| + 3 tr(II ), where II and H are respectively the second form and the mean curvature of a hypersurface Σ ֒→ M, H = trΣII, taken with respect to the inner unit normal to Σ, the Greek indices are indices tangent to Σ, 3 3i i k l and II is deﬁned in local coordinates by II j = IIkIIl IIj. Moreover, in dimension 4, we have 1 Q = − ∆ R +4σ (S ), g 6 g g 2 g and thanks to our decreasing assumption, we get 2 1 B = + O g r3 r3+τ which gives, using once more our decreasing assumption, that

2 1 32π2(χ(M) − 1) = |W |2 +4Q + ∆ R dv + O (32) g 3 g g g Rτ ZKR 1 = |W |2 +4Q dv + O (33) g g Rτ ZKR Hence passing to the limit, we get

2 2 32π (χ(M) − 1) = |W | +4Qg dvg ZM Let us set

κg = Qg dvg. ZM 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 κg ≥ 0 then χ(M) ≥ 1 with equality if (M,g) is locally conformally ﬂat with vanishing Q-curvature. In particular if κg ≥ 0 and Y ([g]) ≥ 0 or the seccond betti number vanishes, then Qg ≡ 0 and (M,g) is conformal to the euclidean R4.

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 [19, 20, 25]. In particular, we will show that these results follow from Theorem 4. Let us consider a closed manifold (M n,g) with n ≥ 5. If, the Panietz operator is coercive, that is to say

inf Pg(u)u dvg > 0, u∈H2(M); kuk =1 2 ZM then it admits a Green functions GPg . This is in particular guaranteed if Rg ≥ 0 and Qg is semi-positive, see proposition B of [19], or more generally by the fact that

Y4([g]) = inf Pg(u)u dvg > 0, u∈H2(M); kuk =1 2# ZM 18 # 2 where 2 = 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 Y4([g]) > 0 and therefore GPg exists for every g ∈ [g].

Nevertheless, contrary to the conformal Laplacian, nothing here guarantees that GPg is positive. This will be one of our assumptions, which is in particular satisﬁed if Y ([g]) ≥ 0 and Qg˜ is semi-positive for some conformal metricg ˜, due to lemma 3.2 in [22]. Remark 1. In fact Hang and Yang assume Y ([g]) > 0, but if Y ([g]) = 0 and there is 4 − some g˜ ∈ [g] with Qg˜ semi-positive, then there exists g¯ = u n 2 g˜ such that Rg¯ = 0 which implies that Qg¯ ≤ 0 which contradicts the fact that

n+4 − Qg˜v dvg˜ = v n 4 Qg¯ dvg˜, ZM ZM n−4 with v = u 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]) ≥ 0 , thanks to, theorem 1.1 of [22], the existence of a positive GPg is equivalent to the existence of a conformal metric 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 ≤ n ≤ 7 or g is locally conformally ﬂat, then, in conformal normal i coordinates {x } for the conformal metricg ˜, the Green function GP of Pg˜ admits an expansion of the form (see Proposition 2.5 in [19]) γ G (p, x)= n + α + O (r), (34) P rn−4 4

. . 1 where r(x) = dg(p, x), γn = and α is a constant called the mass. ˜ (n−2)(n−4)ωn−1

Remark 2. As remarked in [22], under the condition Ker(Pg)=0 (which is itself conformally invariant), the sign of the Green function is a conformal invariant since,

−1 −1 GP 4 (p, q)= u(p) u(q) GPg (p, q). u n−4 g Hence, the sign of the mass is also a conformal invariant. This terminology arises in analogy to the Yamabe problem, where the Green function

GLg of the conformal Laplacian is involved and, in important cases, it admits a similar expansion to (34). In the case of the conformal Laplacian, it was an observation of R. Schoen that the constant which appears in the place of α is precisely the ADM mass of the 4 n−2 AE-manifold obtained by via the stereographic projection (M\{p},GL 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 [38] and [27]). The positive mass theorem presented in [19] as Theorem 2.9 states that under the conditions described above which provide us with the expansion (34), the mass satisfies α ≥ 0 with equality holding if and only if (M,g) is conformal to the round sphere. This came about as a generalisation of [25], where this result was proven in the conformally flat case, and, in turn, all this was generalised in [20], where the condition Rg ≥ 0 was replaced by Y ([g]) > 0. Furthermore, in Theorem 1.4 of [22], this positive mass theorem is used to solve the Y4([g]) > 0 Q-curvature prescription problem in very much the same

19 spirit as the usual positive mass theorem was used by Schoen in [38]. 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 2. 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 ≥ 5 and whose Panietz operator admits a positive Green function GP with an expansion as (34) around some 4 . . − point p ∈ M. Then, the manifold (Mˆ = M\{p}, gˆ = GP (p, ·) n 4 g) is an asymptotically flat manifold of order τ =1 if n =5 and τ =2 if n> 5. Furthermore, either if 5 ≤ n ≤ 7 or g is flat around p, then E(ˆg)=8(n − 1)(n − 2)ωn−1γnα. Proof. Let us fix the conformal normal coordinates {xi} where (34) holds. Let us appeal to the conformal normal coordinate construction of order N ≥ 4 described Appendix B so that (C.3)-(C.4) hold, and start by considering the following expansion near p 1 1 g = δ + R (p)xkxl + R (p)xkxlxa + O(r4), (35) ij ij 3 iklj 6 iklj,a 2 i n−4 xi Now, consider the inverted coordinates z = γn r2 in a neighborhood of p and define 4 2 2 . 2 2 n−4 −2 i n−4 zi −(n−4) −1 n−4 ρ = |z| , so that ρ = γn r and x = γn ρ2 . Then, it holds that γnr = γn ρ and 2 ∂ n−4 −2 j −2 i j ∂ = γn ρ δ − 2ρ z z , (36) ∂zi i ∂xj which, appealing to (34), implies 4 −1 n−4 −1 n−4 gîj(z)= γn ρ + α + O4(ρ ) g(∂zi , ∂zj ), 4 n−4 γnα j = 1+ + O (ρ3−n) δi − 2ρ−2zizk δ − 2ρ−2zjzl g (ρ−2z). ρn−4 4 k l kl Invoking (35), this implies 4 γ α n−4 gˆ (z)= 1+ n + O (ρ3−n) h (z), ij ρn−4 4 ij where 4 i k j l i j k l n−4 m n z z z z z z z z γn z z h (z)= δi δj − 2 δj − 2 δi +4 δ + R (p) ij k l ρ2 l ρ2 k ρ4  kl 3 kmnl ρ4 6  n−4 m n p γn z z z 1 + R (p) + O , 6 kmnl,p ρ6 4 ρ4  1  = δ + H2 (z)+ H3 (z)+ O ij ij ij 4 ρ4

20 where

4 m n n i m j − n−2 2 1 z z 2 k m z z 2 n l z z γn H = R (p) − R (p)z z − R (p)z z ij 3 imnj ρ4 3 kmnj ρ6 3 imnl ρ6 =0 =0 4 znzlzizj + R (p)zkzm | , {z } | {z } 3 kmnl ρ8 =0 1 zmzn = R| imnj({zp) } 3 ρ4 and

6 m n a n i a m j a − n−2 3 1 z z z 1 k m z z z 1 n l z z z γn H = R (p) − R (p)z z − R (p)z z ij 6 imnj,a ρ6 3 kmnj,a ρ8 3 imnl,a ρ8 =0 =0 2 znzlzizjza + R (p)zkzm | , {z } | {z } 3 kmnl,a ρ10 =0 1 zmznza = R| imnj,a{z(p) } . 6 ρ6 Putting together all the above, we ﬁnd that

4 6 n−2 a b n−2 a b c 4 γ α γn z z γn z z z gˆ (z)= 1+ n δ + R (p) + R (p) ij n − 4 ρn−4 ij 3 iabj ρ4 6 iabj,c ρ6 −(n−3) −4 + O4(ρ )+ O4(ρ ), which establishes the asymptotic flatness condition claimed in the proposition. Notice that if 5 ≤ n ≤ 7, then, from Proposition 1, we know that E(ˆg) is well-defined and the above expansion implies za ∂ gˆ = −4γ α δ + ∂ H2 + ∂ H3 + O ρ−(n−2) , a ij n ρn−2 ij a ij a ij 3 δ zbza ∂ gˆ = −4γ α ab − (n − 2) δ + ∂ H2 + ∂ H3 + O ρ−(n−1) . ba ij n ρn−2 ρn ij ba ij ba ij 2 2 3 Appealing to (C.3)-(C.4), it is not difficult to explicitly compute that ∂aaHii and ∂aaHii both vanish. This implies

−(n−2) −(n−1) ∂aagˆii = −8nγnαρ + O2(ρ ). (37)

2 3 Similarly, appealing to (C.3)-(C.4), it also holds that ∂ijHij = 0 and ∂ijHij = 0, and therefore we ﬁnd that

−(n−2) −(n−1) ∂ijgˆij = −8γnαρ + O2(ρ ). (38)

Thus, putting together (37)-(38), we see that zc (∂ gˆ (z) − ∂ gˆ ) =8γ α(n − 1)(n − 2)ρ−(n−1) + O (ρ−n), (39) caa ii cij ij ρ n 1

21 implying that zc E(ˆg) = lim (∂caagîi(z) − ∂cijgîj(z)) dωρ = ωn−18γnα(n − 1)(n − 2). (40) ρ→∞ ρ ZSρ Finally, in g is flat near p, without restriction on the dimension, we know that the expansion (34) holds in rectangular coordinates around p, where gij(x) = δij. Thus, the same computations as above show that

4 γ α gˆ (z)= 1+ n δ + O (ρ−(n−3)). ij n − 4 ρn−4 ij 4 In this case the order of decay is improved for n ≥ 7 and in particular we know that E(ˆg) is well-defined. From the above expression, the same computations that led us to (40) prove that E(ˆg)=8ωn−1(n − 1)(n − 2)γnα. In order to apply the positive energy theorem to establish that α ≥ 0, we first need to check thatg ˆ satisfies its hypotheses. With this in mind, consider the following proposition.

Proposition 7. Consider a closed Riemannian manifold (M n,g), with n ≥ 5, which admits a conformal metric with positive Q-curvature such that Y ([g]) ≥ 0. Then, there exists a conformal metric g˜ such that the asymptotically ﬂat manifold (Mˆ = M\{p}, gˆ = 4 G n−4 g˜) satisﬁes Y ([ˆg]) > 0 and Q ≡ 0. Pg˜ gˆ

Proof. Thanks to Remark 1, Y ([g]) > 0 and GPg˜ exists and is positive for everyg ˜ ∈ [g]. Then, thanks to proposition 6, we only have to prove that Y [ˆg] > 0. Since Y ([g]) >

0, let GLg the Green function of the conformal Laplacian. We trivially get that g¯ = 2 n−2 GL 2(n−2) gˆ is scalar ﬂat. n−4 GP !

Equipped with the above two propositions, we can recover the following theorem, originally proved by Hang-Yang in [22], and which is an extension of results of Gursky- Malchiodi [19] and Humbert-Raulot [25].

Theorem 5. Let (M,g) be a closed n-dimensional Riemannian manifold, with 5 ≤ n ≤ 7 or n ≥ 8 and locally conformally ﬂat around some point p ∈ M. If Y ([g]) ≥ 0 and (M,g) admits a conformal metric with semi-positive Q-curvature, then the mass of GP at p is non-negative and vanishes if and only if (M,g) is conformal to the standard sphere.

Proof. Under these conditions, we know that the Green function GP exists and is positive for every element in [g]. From Proposition 6 we also know that there is a conformal metric 4 g˜ such that the manifold (Mˆ = M\{p}, gˆ = G n−4 g˜) is AE, satisfies the decay assumptions Pg˜ of Theorem 4 and, furthermore the energy is positively proportional to the mass of GPg˜ . Finally, from Proposition 7, we know that Y ([ˆg]) > 0 and Qgˆ ≡ 0. Therefore, (M,ˆ gˆ) satisfies all the hypotheses of Theorem 4, and thus the non-negativity follows directly from E(g) being positively proportional to α. Finally, if α = 0, we find that Mˆ is conformally equivalent to Rn. Being M the one point compactification of Mˆ , we see that M ∼= Sn and g is conformal to the round metric.

22 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. Nev- ertheless, 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 [34]-[28].

4 Lemma 1. Let (M ,g) be a closed Riemannian manifold satisfying Ker(Pg) = R and ∞ κg > 0. Then, given p ∈ M, there exists a Green function Gp ∈ C (M\{p}), unique up to an additive constant, which satisﬁes

PgGp + Qg = κgδp (41)

i 4 as distributions. Furthermore, near p, in g-normal coordinates {x }i=1 the following expansion holds k k k k G = p ln(r−2)+ p S + p a xi + p b xixj + o (r2). (42) p 16π2 16π2 0 16π2 i 16π2 ij 4 for some constants S0, ai, bij.

Proof. From Lemma 2.1 in [34] we know that if Ker(Pg)= R, then the Paneitz operator 4 Pg admits a Green function GP , which is to say that for every u ∈ C (M) it holds that

u(x) − u¯ = GP (x, y)Pg(u(y))dVg(y), (43) ZM . −1 R whereu ¯ = volg(M) M udVg. Also, since Ker(Pg)= , we know that there is a smooth function U satisfying R PgU = −(Qg − Q¯g). . Deﬁne Gp = κgGP + U, so that

hPgGp,ui = κgu(p) − κgu¯ + hPgU, ui, κg = κgup − κgu¯ −hQg,ui + h ,ui = κup −hQg,ui. volg(M) Thus, we see that

∞ hPgGp + Qg,ui = κgu(p) ∀u ∈ C (M), (44) which is to say that PgGp + Qg = κgδp. The uniqueness claim follows since two diﬀerent such functions Gp and G˜p must satisfy Pg(Gp − G˜p) = 0, implying G˜p = G˜p + c. If we rewrite (41) as P 16π2 G + 16π2 Q = 16π2δ we can appeal quite straightforwardly to g kp p kp g p the computations of the Appendix in [28] to conclude that, in g-normal coordinates, the Green function has the following expansion near p:

2 16π i i j 2 Gp = −2 ln(r)+ S0 + aix + bijx x + o4(r ). (45) kp which proves (42).

23 Let us now consider the same setting as in the above lemma and consider the inverted i i x ˆ 2Gp coordinates z = r2 and deﬁne (M = M\{p}, gˆ = e g). Then,

−4Gp(q) Qgˆ(q)= e (PgGp(q)+ Qg(q))=0, for any q ∈ M.ˆ and

κg 4 κg κg i κg i j − ln(ρ 16π2 )+ S + a z + b z z +O (ρ 3) π2 0 π2 i ρ2 π2 ij ρ4 4 gˆij(z)= e 8 8 8 g(∂zi , ∂zj ), i i j κg κg κg z κg z z −3 4 S ai + bij +O4(ρ ) i i k j j l π2 π2 0 8π2 ρ2 8π2 ρ4 −4 −2 −2 −2 = ρ 16 e 8 e ρ δk − 2ρ z z δl − 2ρ z z gkl(ρ z), κ i κ i j κg g a z + g b z z +O (ρ−3) 4∆κg 2 S0 π2 i ρ2 π2 ij ρ4 4 −2 = ρ e 8π e 8 8 δij + O4(ρ ) ,

κg i 4∆κ S0 κg z −2 −2 = ρ g e 8π2 1+ a + O (ρ ) δ + O (ρ ) , 8π2 i ρ2 4 ij 4 (46)

. κg where we have defined ∆κg = 16π2 − 1. Let us notice that the above inversion gives 2 an AE gemtricg ˆ iff ∆κg = 0, which is to say κg = 16π . In particular, in this case, via a coordinate change of the formz ¯i = eS0 zi, we find that

z¯k gˆ (¯z)= 1+2eS0 a + O (¯ρ−2) δ + O (¯ρ−2) , (47) ij k ρ¯2 4 ij 4 whereρ ¯ = |z¯| = eS0 ρ. That is, ρ−k = ekS0 ρ¯−k. We see that (M,ˆ gˆ) is an AE-manifold or order τ = 1. Therefore, since by construction Qgˆ ≡ 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 Ker(Pg)= R 2 and κg = 16π . 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.

4 Theorem 6 (Gursky). Let (M ,g) a 4-dimensional manifold with Y ([g]) ≥ 0, then κg ≤ 16π2 with equality holding iﬀ (M 4,g) is conformal to the standard sphere.

Proof. Since we need only pay attention to the cases κg > 0, let us start by noticing that, due to Theorem A in [18], under our hypotheses Ker(Pg) = R. Furthermore, the case Y ([g]) = 0 is also trivial, since in this case κg ≤ 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 (46) to construct the manifold (Mˆ = M\{p}, gˆ). Alhtough in general this manifold will not be AE, it holds that Qgˆ ≡ 0. Also, since Y ([g]) > 0, we now that the Green function GL of the conformal Laplacian (with a pole . 2 ˆ at p ∈ M) exists, it is positive and therefore we can construct the metricg ˜ = GLg on M. In inverted normal coordinates around p, from [27], we know that

−2 g˜(z)(∂zi , ∂zj )= δij + O4(ρ ),

24 which shows that (M,ˆ g˜) is AE of order τ = 2. Furthermore,g ˆ andg ˜ are related via

G2 G G G gˆ = e− ln( L)e2 P g˜ = e2( P −ln( L))g˜ and therefore, from the conformal covariance associate to the Paneitz operator, we see that

4Φ 0= e Qgˆ = Pg˜Φ+ Qg˜, (48) . where we have defined Φ = GP − ln(GL). Following the proof theorem 2, we know that 1 2 Qg˜ = − 2 |Ricg˜|g˜, and therefore we find that 1 |Ric |2dV = g˜(∇˜ ∆ Φ, ν˜)dω˜ +2 Ric (∇˜ Φ, ν˜)dω.˜ (49) 2 g˜ g˜ g˜ g˜ g˜ ZDr ZSr ZSr Now, the main difference with respect to theorem 2 is that we cannot estimate the derivatives of Φ 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 Φ, at least asymptotically. Thus, let us notice that for sufficiently large ρ the following holds

2α −1 2 −1 Φ = ln(ρ )+ S0 + O4(ρ ) − ln(ρ + A + O4(ρ )), 2α 2 −2 −3 −1 = ln(ρ )+ S0 − ln(ρ ) − ln(1 + Aρ + O4(ρ )) + O4(ρ ), 2(α−1) −2 −3 −1 = ln(ρ )+ S0 − ln(1 + Aρ + O4(ρ )) + O4(ρ ),

. κg where above we have deﬁned α = 16π2 and both A and S0 are constants. From all this we can directly compute that zi ∇˜ Φ=2(α − 1) + O (ρ−2), i ρ2 3 4(α − 1) ∆ Φ= + O (ρ−4), g˜ ρ2 3 8(α − 1) zi ∇˜ ∆ Φ= − + O (ρ−5). i g˜ ρ3 ρ 2 From the above expression, it follows that

−5 Ricg˜(∇˜ Φ, ν˜)= O(ρ ), 8(1 − α) g˜(∇˜ ∆ Φ, ν˜)= + O (ρ−5), g˜ ρ3 2 implying that 1 |Ric |2dV =8ω (1 − α)+ o(1). (50) 2 g˜ g˜ g˜ 3 ZDr Finally, passing to the limit, we ﬁnd that

2 |Ricg˜|g˜dVg˜ =8ω3(1 − α) ≥ 0. (51) ˆ ZM This implies that α ≤ 1 and the equality case has already been dealt with in the previous corollary, which establishes the theorem.

25 3.3 The 3-dimensional case In this section, we brieﬂy explain how to recover the following theorem due to Hang and Yang [21].

Theorem 7 (Proposition 2.4 [21]). Assume the Yamabe invariant Y ([g]) > 0, ker Pg =0.

If there is some p ∈ M such that GPg (p, ·) < 0 on M\{p}, then GPg (p) < 0 except when (M,g) is conformally equivalent to the standard S3. 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 Mˆ = M\{p} and . −4 4 deﬁneg ˆ = GPg g andg ˜ = GLg g on this manifold. Let us recall the following expansions, valid in conformal normal coordinates around p (see, [21] and [27])

GP = A + O4(r), 1 G = + α + O (r), L r 4

x −1 where A and α are constants. After going to inverted coordinates z = r , with ρ = r , we ﬁnd that α g˜ (z)= 1+ δ + O(ρ−2), ij ρ ij −4 . and, clearly, ong ˆ = Φ g˜, with Φ = GP GL and Qgˆ ≡ 0. Therefore, from the conformal covariance of the Paneitz operator, we ﬁnd that Pg˜Φ ≡ 0, which translates to

2 2 0 = ∆g˜Φ − 4divg˜ (Ricg˜(∇Φ, ·)) + |Ricg˜|g˜Φ. That is

2 ˜ |Ricg˜|g˜ΦdVg˜ = − g˜(∇∆g˜Φ, ν˜)dω˜ρ +4 Ricg˜(∇Φ, ν˜)dω˜ρ. (52) ZDρ ZSρ ZSρ As in the previous theorem, we now can compute explicitly the terms in the right-hand side. That is,

−1 −1 0 Φ= A + O4(ρ ) ρ + α + O4(ρ ) = Aρ + O4(ρ ) zi ∇ Φ= A + O (ρ−1), i ρ 3 2A ∆ Φ= + O (ρ−2), g˜ ρ 2 zi ∇ ∆ Φ= −2A + O (ρ−3), i g˜ ρ−3 1

−3 which, together with Ricg˜ = O2(ρ ) implies that

2 −1 |Ricg˜|g˜ΦdVg˜ =8πA + O(ρ ). ZDρ Passing to the limit and remembering that Φ < 0, we find that A ≤ 0 with equality ∼ 3 holding iff Ricg˜ ≡ 0. That is, if Mˆ = R andg ˜ = δ, which implies the final result.

26 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 [27]. 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 ﬂat manifold with a structure at inﬁnity n φ : M \ K → R \ B1 with decay rate τ. If δ 6∈ (Z \ {−1, ··· , 3 − n}) then

2,q q ∆g : Wδ (φ) → Lδ−2 is Fredholm. Moreover ∆g is

if δ > 2 − n then ∆g is surjective, if 2 − n<δ< 0 then ∆ is bijective, (A.1)  g  if δ < 0 then ∆g is injective. Remark A.1. The decay rates in the set Z \ {−1, ··· , 3 − n} are called exceptional and we say that δ is non-exceptional if δ 6∈ Z \ {−1, ··· , 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.

k,q Theorem A.2. Let (M,g) be an AE manifold, with g ∈ Wloc , k ≥ 1, q > n, and (Φ, x): M\K 7→ ER where K ⊂⊂ M, R ≥ 1 is a structure of inﬁnity of order τ > 0 with i k,q 1 − τ non-exceptional, and ﬁx 1 <η< 2. There are functions y ∈ Wη , i = 1, ··· , n, i i i k,q ∗ . such that ∆gy =0 and (x − y ) ∈ W1−τ ∗ (ER) for τ = min{τ, n − 2}, which implies

i i 1−τ ∗ |x − y | = ok(r ), −τ ∗ (A.2) |g(∂xi , ∂xj ) − g(∂yi , ∂yj )| = ok(r ).

i k,q Furthermore, the set of functions {1,y } is a basis for H1 = {u ∈ Wη : ∆gu =0}. Proof. Let us first extend the functions xi smoothly to all of M. Then, near infinity, i kl i . i k−1,q ∆gx = g Γkl = Γ ∈ W−1−τ . Notice that if 1 − τ > 2 − n, then Theorem A.1 implies i k+1,q i i the existence of v ∈ W1−τ solving ∆gv = Γ . Now, if 1 − τ < 2 − n (the equality case is an exceptional case), we cannot, a priori, improve the decay given by r2−n. This actually follows from Theorem 1.17 in [4]. Therefore, in any case, we know that there are i k+1,q i i ∗ solutions v ∈ W1−τ ∗ to ∆gv = Γ , where τ = min{τ, n − 2}. Therefore, there is some i k+1,q i i i . i i v ∈ W1−τ ∗ satisfying ∆g(x − v ) = 0. Let us then define y = x − v , which implies ∂yi i −τ ∗ the first estimate in (A.2), and furthermore, since ∂xj = δj + o(r ), we see that near infinity {yi} are coordinates which are asymptotically Cartesian. This, in turn, implies i the second estimate in (A.2). The final claim concerning H1 follows since {1,y } spans an (n + 1)-dimensional subspace of H1, but from Proposition 2.2 in [4] dim(H1)= n + 1.

27 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 r2−n cannot be improved, regardless of how large τ 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. k,q Theorem A.3. Let (M,g) an asymptotically flat manifold, g ∈ Wloc , k ≥ 1 and q > n, n with two structures at infinity φ, ψ : M \ K → R \ B(0, 1) with decay rates τφ and τψ, where of each of these weights satisfies that 1 − τ is non-exceptional. There exists (O, a) ∈ O(n) × Rn such that i i j i k,q Rn x − (Ojz + a ) ∈ W1−τ ( ) which implies

i i j i 1−τ |x − (Ojz + a )| = ok(r ), (A.3)

. −1 −1 where τ = min{τφ, τψ, n − 2}, x = φ and z = ψ . Proof. Let yi and wi be the harmonic coordinates constructed in the previous theorem i associated to φ and ψ respectively. Then, since H1 is intrinsic to M and {1,y } and {1,wi} are bases for this space, we get that

i i j i w = Ajy + a , where we A ∈ GL(n, R) a priori, but actually, from the construction of the previous i i j i i j i i k,q i k,q theorem, we know that z = A x +a +A v −v¯ , with v ∈ W ∗ andv ¯ ∈ W ∗ , where j j 1−τφ 1−τψ τ ∗ was defined in the previous theorem. Since these last two systems are Cartesian, then i i j i k,q A ∈ O(n) and we explicitly see that z − Ajx − a ∈ W1−τ . 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 Rn 2,q Rn φ : M \ K → \ B(0, 1) with decay rate τ such that (φ∗g − δ) ∈ W−τ ( \ B(0, 1)) for q > n and such that q Z Ricg ∈ L−2−η(M) for some η > τ and η 6∈ ( \ {−1, ··· , 3 − n}).

′ n Then there exists a structure at inﬁnity Θ: M \ K → R \ B1 such that (Θ∗g − δ) ∈ 2,q Rn W−η ( \ B1). Finally, the following statement concerning the conformal Laplacian will be important in our analysis. It is a variant of theorem 9.2 of [27] Corollary A.1. Let (M,g) an asymptotically ﬂat manifold of order τ > 0 and assume Rg ≥ 0. If 2 − n<δ< 0 then

2,q q Lg = ∆g − cnRg : Wδ → Lδ−2 is an isomorphism.

28 Proof. Since Lg is a compact perturbation of the Laplacian, it is also Fredholm, hence 2,q thanks to its self-adjointness it suﬃces to proof injectivity to get the result. Let u ∈ Wδ 2,q ′ 2,q such that ∆gu = cnRgu since Rgu ∈ Wδ′ for any δ ≥ δ − τ − 2, hence u ∈ Wδ−τ , since 2,q we can repeat the argument as much as necessary, we can assume that u ∈ Wδ′ with ′ δ < n − 2. Then we can multiply Lg(u) = 0 by u and integrating by part, which gives

2 2 −|∇u| − cnRgu dv = u∂ν u dσ, Rn ZM\( \B(0,R)) Z∂B(0,R) Since u decrease enough, we can pass to the limit as R → +∞, which gives that u ≡ 0 since Rg≥0 and u → 0 as |x|→∞, which concludes the proof.

B Appendix: Conventions on Q-curvature

Let us adopt the following a the general deﬁnition for the Q-curvature an arbitrary Rie- mannian manifold (M n,g) with n ≥ 3: n − 4 Q = −∆ σ (S )+4σ (S )+ (σ (S ))2 , (B.1) g g 1 g 2 g 2 1 g

. 1 1 where Sg = n−2 Ricg − 2(n−1) Rgg stands for the Schouten tensor and σk(Sg) stands for the k-th elementary symmetric function of the eigenvalues of Sg. In this context, the Paneitz operator is deﬁned by . n − 4 P u = ∆2u + div ((4S − (n − 2)σ (S )g)(∇u, ·))+ Q u, (B.2) g g g g 1 g 2 g for all u ∈ C∞(M). In this context, the following relations hold: R σ (S )= g , 1 g 2(n − 1) 2 2 (B.3) 1 n − n |Ricg| σ (S )= R2 − g , 2 g 2 4(n − 1)2(n − 2)2 g (n − 2)2 which implies that 1 2 n3 − 4n2 + 16n − 16 Q = − ∆ R − |Ric |2 + R2, g 2(n − 1) g g (n − 2)2 g g 8(n − 1)2(n − 2)2 g (B.4) 4 n2 − 4n +8 n − 4 P u = ∆2u + div Ric − R g (∇u, ·) + Q u. g g n − 2 g 2(n − 1)(n − 2) g 2 g 4 In particular, for n =6 4, ifg ¯ = u n−4 g, then

2 − n+4 Q = u n−4 P u. (B.5) g¯ n − 4 g In the case of n = 4 we can apply the above deﬁnitions to get 1 1 1 Q = − ∆ R − |Ric |2 + R2, g 6 g g 2 g g 6 g 2 (B.6) P u = ∆2u + div 2Ric − R g (∇u, ·) , g g g g 3 g

29 and in this case, ifg ¯ = e2ug, we have that [29] −4u Qg¯ = e (Pgu + Qg) . (B.7) Now, let us notice that it is also quite standard to redefine the Q-curvature in dimension 4 via (see [19, 28]) 1 1 1 1 Q(4) = Q = − ∆ R − |Ric |2 + R2 (B.8) g 2 G 12 g g 4 g g 12 g and in this case, it follows that (4) −4u (4) 2Qg¯ = e Pgu + Qg . (B.9) We will not adopt this redefinitions and keep a unified notation via (B.1) along this paper.

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 [27], 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 ∈ M. In particular, we are interested in finding an elementg ˜ within the conformal class [g] for which, ing ˜-normal coordinates {xi}, the following expansion holds around p: det(˜g)=1+ O(rN ) (C.1) for any chosen N ≥ 2, where r = |x| (see Theorem 5.1 in [27]). This type of expansion is achieved by first noticing that, for any Riemannian metric g, in g-normal coordinates i n {x }i=1 around p ∈ M, the following holds 1 1 1 2 g (x)= δ + R xkxl + R xkxlxa + R + R R xkxlxaxb + O(r5), ij ij 3 iklj 6 iklj,a 20 iklj,ab 45 iklc jabc 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 1 1 det(g)(x)=1 − R xixj − R xixjxk 3 ij 6 ij,k 1 1 1 (C.2) − R + R R − R R xixjxkxl + O(r5), 20 ij,kl 90 aijb aklb 18 ij kl where again all the coefficients are evaluated at p. Assuming an expanssion of the form det(g)=1+ O(rN ) with N ≥ 2 (the case N = 2 is valid for any metric in its own normal coordinates) and appelaing to Theorem 5.2 in [27], the authors can find a homogeneous N+1 2f polynomial f ∈ PN so that the expansion of det(˜g)=1+ O(r ) forg ˜ = e 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 forg ˜. Thus, in the case we do this construction for N ≥ 4, we see that this implies

R˜ij(p)=0, (C.3) R˜ij,k + R˜ki,j + R˜jk,i(p)=0. Putting together the second condition above with the contracted Bianchi identities, we also see that

R˜,k(p)=0. (C.4)

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