We prove that under suitable assumptions, the constant term in the Green
function of the Paneitz-Branson operator on a compact Riemannian manifold
$(M,g)$ is positive unless $(M,g)$ is conformally diffeomophic to the standard
sphere. The proof is inspired by the positive mass theorem on spin manifolds by
Ammann-Humbert.Comment: 7 page

B. Ammann

E. Hebey

Emmanuel Humbert

J. Qing

J.M. Lee

R. Schoen

S.M. Paneitz

Simon Raulot

T.P. Branson

English

arXiv

7 pagesInternational audienceWe prove that under suitable assumptions, the constant term in the Green function of the Paneitz-Branson operator on a compact Riemannian manifold $(M,g)$ is positive unless $(M,g)$ is conformally diffeomophic to the standard sphere. The proof is inspired by the positive mass theorem on spin manifolds by Ammann-Humbert

Humbert, Emmanuel

Raulot, Simon

INRIA a CCSD electronic archive server

HAL Id: hal-00297324https://hal.archives-ouvertes.fr/hal-00297324Submitted on 15 Jul 2008HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestinée au dépôt et à la diffusion de documentsscientifiques de niveau recherche, publiés ou non,émanant des établissements d’enseignement et derecherche français ou étrangers, des laboratoirespublics ou privés.Positive mass theorem for the Paneitz-Branson operatorEmmanuel Humbert, Simon RaulotTo cite this version:Emmanuel Humbert, Simon Raulot. Positive mass theorem for the Paneitz-Branson operator. Cal-culus of Variations and Partial Differential Equations, Springer Verlag, 2009, 36 (4), pp.525-531.￿10.1007/s00526-009-0241-6￿. ￿hal-00297324￿POSITIVE MASS THEOREM FOR THE PANEITZ-BRANSONOPERATOREMMANUEL HUMBERT AND SIMON RAULOTAbstract. We prove that under suitable assumptions, the constant term inthe Green function of the Paneitz-Branson operator on a compact Riemannianmanifold (M, g) is positive unless (M, g) is conformally diffeomophic to thestandard sphere. The proof is inspired by the positive mass theorem on spinmanifolds by Ammann-Humbert [AH03].1. IntroductionLet (M, g) be a compact Riemannian manifold of dimension n ≥ 4. We denoteby Qg the Q-curvature for the metric g defined byQg :=n2 − 48n(n − 1)2S2g −2(n − 2)2|Eg|2 +12(n − 1)∆gSg,where ∆g = −divg∇ is the Laplace-Beltrami operator, Sg stands for the scalarcurvature of g, |Eg| denotes the g-norm of the Einstein tensor Eg := Ricg −Sgngand Ricg is the Ricci curvature of g. The Paneitz-Branson operator introducedfor n = 4 by Paneitz in [Pa83] and whose definition was generalized in dimensiongreater than 5 by Branson [Br87], is defined for all u ∈ C∞(M) byPgu := ∆2gu − divg (Agdu) +n − 42QguwhereAg :=(n − 2)2 + 42(n − 1)(n − 2)Sgg −4n − 2Ricg.This operator is closely related to the problem of prescribing Q-curvature in aconformal class as well as the Yamabe operator (see (5) below) is related to theproblem of prescribing the scalar curvature in a conformal class. It is a conformallycovariant operator in the sense that if g′ = e2fg is conformal to g, then for allv ∈ C∞(M),Pg′(e−n−42fv) = e−n+42fPg(v).In particular, if n ≥ 5, and if we set u = en−42f so that g′ = u4n−4 g, we get for allv ∈ C∞(M)Pg′ (u−1v) = u−n+4n−4 Pg(v). (1)From now on, we make the following assumptions:(a) g is conformally flat;(b) n ≥ 5;Key words and phrases. Paneitz-Branson operator, positive mass theorem.12 EMMANUEL HUMBERT AND SIMON RAULOT(c) the Yamabe invariant is positive (see for instance [Au98] or [He97]) i.e. g isconformal to a metric g′ for which the scalar curvature is positive.(d) the operator Pg is positive.Under Assumptions (a) to (d), it is well known that the Green’s function Gg ofPg exists, is unique and smooth on M \ {p}. By the conformal convariance of thePaneitz-Branson operator, if g′ = u4n−4 g is conformal to g, thenGg′(x, y) =Gg(x, y)u(x)u(y).Now, let p ∈ M . By (1), up to a conformal change of metric, we can assume(a’) g is flat around p.Then, it is known that we have the following expansion when x is close to p,Gg(x, p) =12(n − 2)(n − 4)ωn−1dg(x, p)n−4+ A + αp(x) (2)where ωn−1 stands for the volume of the (n−1)-dimensional sphere, A ∈ R, αp is asmooth function defined around p and satisfying αp(p) = 0. By analogy to the caseof the conformal Laplacian (see again [Au98, He97]), the number A is called themass of the Paneitz-Branson operator. If g′ = u4n−4 g is another metric conformalto g and flat around p, then the mass A′ corresponding to the metric g′ is given byA′ =Au(p)2.Hence, the mass A depends on the choice of the metric in the conformal class, butnot its sign. This is the reason why in the statement of Theorem 1.1 below, we donot need to assume (a′).We also make the following assumption(e) Gg > 0 on M \ {p}.For interesting results concerning Assumptions (d) and (e), the reader may referto Grunau-Robert [GR07].The main result of the paper is the following:Theorem 1.1. Under assumptions (a) to (e), the mass A satisfiesA ≥ 0with equality if and only if (M, g) is conformally diffeomorphic to the sphere.Theorem 1.1 has been already proven with the additional assumption that thePoincaré exponent is small enough (see [QR06a, QR06b]). In this case, Qing andRaske proved also the positivity of the Green’s function of Gg.Our proof is inspired from the positive mass theorem on spin manifolds by Ammann-Humbert in [AH03] (see also Raulot [Ra07]). The difficulty here is to overcomePOSITIVE MASS THEOREM FOR THE PANEITZ-BRANSON OPERATOR 3the fact that on non-spin manifolds, there is no equivalent of the Schrödinger-Lichnerowicz Formula.Hebey and Robert proved the nice following result which is an analogue for geo-metric equations of order 4 of a hard problem concerning the Yamabe Equation:Theorem (Hebey, Robert; [HR04]). Let (M, g) be a conformally flat compactmanifold of dimension n ≥ 5. Assume g has a positive Yamabe invariant, that Pgis positive as well as its Green function and that the mass of Pg is positive. Then,the geometric equationPgu = un+4n−4is compact.In particular, together with Theorem 1.1, we get rid of the positivity of the mass.Acknoledgements We want to thank Emmanuel Hebey and Frédéric Robert whogave us many helpful informations and references on the subject.2. Proof of Theorem 1.1In the whole proof, we can work with Assumption (a′) which does not restrict thegenerality as explained above. To avoid complicated formulas, we setH(x) = 2(n − 2)(n − 4)ωn−1Gg(x, p).By Relation (2), H satisfies the following expansion near pH(x) =1dg(x, p)n−4+ B + α(x) (3)where B = 2(n− 2)(n− 4)ωn−1A and where α = 2(n− 2)(n− 4)ωn−1αp is smootharound p and satisfies α(p) = 0. Theorem 1.1 is equivalent to show that B ≥ 0 withequality if and only if (M, g) is conformally diffeomorphic to the standard sphere.For any metric g, letLg :=4(n − 1)n − 2∆g + Sgbe the Yamabe operator. We recall some well known facts about Lg. The readermay refer to [Au98, He97] for further informations. First, as well as Pg, Lg isconformally covariant. If g′ = u4n−2 g is conformal to g thenLg′(u−1 · ) = u−n+2n−2 Lg( · ) (4)It follows that the scalar curvatures Sg and Sg′ are related by the following equationLgu = Sg′un+2n−2 . (5)By Assumptions (a′) and (b), the Green’s function Λg of Lg exists, is unique,smooth and positive on M \ {p}. Setting Γ(x) = 4(n − 1)ωn−1Λg(x, p) to simplifyformulas, we have when x is close to pΓ(x) =1dg(x, p)n−2+ C + β(x) (6)4 EMMANUEL HUMBERT AND SIMON RAULOTwhere by C ∈ R, β is a smooth function defined around p and satisfies β(0) = 0.We define a new metric g′ := Γ4n−2 g conformal to g on M0 := M \ {p}. Then, by(5)Sg′ = Γ−n+2n−2 Lg(Γ) ≡ 0 (7)on M0. We set H′ = Γ−n−4n−2 H . By conformal covariance of the Paneitz-Bransonoperator (1) and since PgH = 0 on M0, we have Pg′H′ ≡ 0 on M0. Define for allǫ > 0 small enough, Mǫ := M \Bg(p, ǫ) where Bg(p, ǫ) stands for the ball of centerp and radius ǫ with respect to the metric g. We have∫MǫPg′H′dvg′ = 0. (8)By Relation (7) and from the definition of Pg we havePg′H′ = ∆2g′H′ − divg′(4n − 2Ricg′dH′)−n − 4(n − 2)2|Eg′ |2H ′.Set Sǫ := ∂Mǫ = ∂Bg(p, ǫ) be the (n−1)-dimensional sphere of center p and radiusǫ. We let dsg′ (resp. dsg) be the volume element induced by g′ (resp. g) on Sǫ.Integrating by part the above relation, we obtain∫MǫPg′H′dvg′ = −I +4n − 2II −12∫Mǫ|Eg′ |2H ′dvg′ (9)where{I =∫Sǫ∂ν′∆g′H′dsg′II =∫SǫRicg′(gradg′H ′, ν′)dsg′ .Here, ν′ denotes the unit outer normal vector on Sǫ = ∂Mǫ with respect to themetric g′.2.1. Computation of I. First, we notice that the scalar curvatures Sg and Sg′vanish on Sǫ. For g, this comes from Assumption (a′) and for g′, this follows from(7). Consequently, using Formula (4) and∆g′H′ =n − 24(n − 1)Lg′H′=n − 24(n − 1)Γ−n+2n−2 Lg (ΓH′)=n − 24(n − 1)Γ−n+2n−2 Lg(Γ2n−2 H)We obtain∆g′H′ = Γ−n+2n−2 ∆g(Γ2n−2 H). (10)We set r := dg(x, p). From Formulas (3) and (6), we have:Γ2n−2 H =(1rn−2+ C + β(x))2n−2(1rn−4+ B + α(x)).Then, using Taylor formula at p,Γ2n−2 H = r2−n + Br−2 + O(r−1)POSITIVE MASS THEOREM FOR THE PANEITZ-BRANSON OPERATOR 5where in the whole proof, O(rm) denotes a smooth function defined in a neighbor-hood of p and which satisfies|∇kgO(rm)|g ≤ Ckrm−kfor all k ∈ N. Since g is flat around p, we have for radially symmetric functions f ,∆gf(r) = −f′′(r) −n − 1rf ′(r). (11)Hence, this gives that near p,∆gΓ2n−2 H = 2(n − 4)Br−4 + O(r−3)and hence by (10) and (6)∆g′H′ = Γ−n+2n−2 ∆gH = 2(n − 4)Brn−2 + O(rn−1).We then obtain∂∂r(∆g′H′) = 2(n − 2)(n − 4)Brn−3 + O(rn−2). (12)On Sǫ, r ≡ ǫ. In addition,ν′ = −Γ−2n−2∂∂r= −(ǫ2 + o(ǫ2))∂∂r(13)anddsg′ = Γ2n−1n−2 dsg = Γ2n−1n−2 ǫn−1ds =(ǫ1−n + o(ǫ1−n))ds. (14)where ds stands for the standard volume element on the unit (n − 1)-sphere. ByFormulas (12), (13) and (14), we obtainI = −2(n− 2)(n − 4)ωn−1B + o(1) (15)2.2. Computation of II. If g′ = e2fg is conformal to g, then the following formulaholds (see [He97] p. 240 or [Au98]):Ricg′ = Ricg − (n − 2)∇2f + (n − 2)∇f ⊗∇f +(∆gf − (n − 2)|∇f |2g)g. (16)In this context, f = 2n−2 log(Γ). By (6), we have near pf =2n − 2log(1rn−2+ O(1))= −2 log(r) + O(rn−2). (17)Let (r, Θ1, · · · , Θn−1) be polar coordinates on Rn. The Christoffel symbols Γrr,Θiof the Euclidean metric in these coordinates identically vanish. This implies thatfor any radially symmetric function h, the mixed terms ∇2rΘih are zero. Since g isflat near p, we deduce that∇2f =2r2dr2 + b + Ō(rn−4)where, as in what follows, we denote by Ō(rm) a 2-form whose norm with respectto g is O(rm) and where b is a 2-form such thatb(· ,∂∂r)≡ 0. (18)6 EMMANUEL HUMBERT AND SIMON RAULOTUsing (11), one also computes that∇f ⊗∇f =4r2dr2 + Ō(rn−4)∆gf =2(n − 2)r2+ O(rn−4)|∇f |2g =4r2+ O(rn−4).Since g is flat near p, Ricg vanishes and g = dr2 + r2σn−1 where σn−1 stands forthe usual metric on the standard sphere Sn−1. We deduce from these computationsthatRicg′ = −(n − 2)b −2(n − 2)r2dr2 +4(n − 2)r2dr2 +(2(n − 2)r2−4(n − 2)r2+ O(rn−4))(dr2 + r2σn−1) + Ō(rn−4)= −(n − 2)b − 2(n − 2)σn−1 + Ō(rn−4).We get from (3), (6) and the definition of H ′ that on Sǫgradg′H ′ = Γ−4n−2 gradg(1 + O(rn−4))= O(rn−1)∂∂r+ v (19)is a vector field such that Ricg′(v, ν′) = 0. Observe that by (13) and (18), we haveσn−1( · , ν′) = 0 and b( · , ν′) = 0 on Sǫ. In addition, the estimates (13), (18) thenimply that on SǫRicg′(gradg′H ′, ν′) = Ō(rn−4)(gradg′H ′, ν′)= O(ǫ2n−3).Relation (14) then leads toII = O(ǫn−2) = o(1). (20)2.3. Conclusion. Using (8), (9), (15), (20) and passing to the limit ǫ → 0, weobtain that0 = 2(n − 2)(n − 4)ωn−1B −12∫M\{p}|Eg′ |2H ′dvg′ . (21)Assumption (e) implies that H ′ > 0 and hence B ≥ 0. This proves first part ofTheorem 1.1.Now, assume that B = 0. Then Eg′ ≡ 0 on M \{p}. This implies that (M \{p}, g′)is Einstein and scalar flat hence Ricci flat. Since in addition the Weyl curvature iszero, (M \{p}, g′) turns to be flat (see [He97] p. 123) . It is known that (M \{p}, g′)is asymptotically flat and that its mass satisfies m(g′) = cnC where cn > 0 (seee.g. Lee-Parker [LP87]). Since g′ is flat, m(g′) = 0 so is C and by a positive massTheorem by Schoen-Yau [SY88], (M, g) is conformally diffeomorphic to (Sn, g).Remark 2.1. It is clear from the proof that Assumption (a) can be weakened andreplaced by(a) g is locally flat around a point p and the standard Positive Mass Theorem isvalid on M (i.e. with the notations of Section 2, C ≥ 0 with equality if and onlyPOSITIVE MASS THEOREM FOR THE PANEITZ-BRANSON OPERATOR 7if (M, g) is conformally diffeomorphic to Sn). In particular, by [SY79] and [AH03],this assumption holds if n ∈ {5, 6, 7} or if M is spin.References[AH03] B. Ammann and E. Humbert, Positive mass theorem for the Yamabe problem on spinmanifolds, Geom. and Func. Anal., 15 (2005), No 3, (567–576).[Au98] T. Aubin, Some nonlinear problems in Riemannian geometry, Springer Monographs inMathematics, Springer-Verlag, Berlin, 1998.[Br87] T.P. Branson, Group representations arising from Lorentz conformal geometry, J. Func.Anal. 74 (1987), (199–291).[GR07] H.C. Grunau and F. Robert, Positivity issues of biharmonic Green’s functions underDirichlet boundary conditions, Preprint 2007.[He97] E. Hebey, Introduction à l’analyse non-linéaire sur les variétés, Diderot Éditeur, Artset sciences., 1997.[HR04] E. Hebey and F. Robert, Compactness and global estimates for the geometric Paneitzequation in high dimensions, Electronic Res. Announc. of the AMS, 10 (2004), (135–141).[LP87] J. M. Lee and T. H. Parker. The Yamabe problem, Bull. Am. Math. Soc., New Ser., 1737–91, 1987.[Pa83] S. Paneitz, A quartic conformally covariant differential operator for arbitrary pseudo-Riemannian manifolds, Preprint 1983[QR06a] J. Qing and D. Raske, Compactness for conformal metrics with constant Q-curvatureon locally conformally flat manifolds, Calc. of Variations, 26 (2006), No3, (343–356).[QR06b] J. Qing and D. Raske, On positive solutions to semilinear conformally invariant equa-tions on locally conformally flat manifolds, Int. Math. Res. Not., 6 (2006), Art. ID94172.[SY79] R. Schoen and S.-T. Yau. On the proof of the positive mass conjecture in general rela-tivity, Comm. Math. Phys., 65 (1979), (45–76).[SY88] R. Schoen and S.-T. Yau. Conformally flat manifolds, Kleinian groups and scalar cur-vature, Invent. Math., 92 (1988), (47–71).[Ra07] S. Raulot, Green functions for the Dirac operator under local boundary conditions andapplications, Preprint arXiv:math/0703197v1Institut Élie Cartan, BP 239, Université de Nancy 1, 54506 Vandoeuvre-lès-NancyCedex, FranceE-mail address: humbert@iecn.u-nancy.frInstitut de Mathématiques, Université de Neuchâtel, Rue Emile-Argand 11, 2007Neuchâtel, SuisseE-mail address: simon.raulot@unine.ch

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