A sphere theorem on locally conformally flat even-dimensional manifolds

Transcription

1 A sphere heorem on locally conformally fla even-dimensional manifol Giovanni CATINO a, Zindine DJADLI b and Cheikh Birahim NDIAYE c a Universià di Pisa - Diparimeno di aemaica Largo Bruno Ponecorvo, 5 I-5617 Pisa - Ialy b Insiu Fourier - UR rue des ahs F-3840 Sain arin d Hères Cedex - France c SISSA Via Beiru, -4 I Triese - Ialy absrac. In his paper, we prove ha a closed even-dimensional manifold which is locally conformally fla wih posiive scalar curvaure, posiive Euler characerisic and which saisfies some addiional condiion on is curvaure is diffeomorphic o he sphere or projecive space. Key Wor: geomery of n-manifol, rigidiy, conformal geomery, fully non-linear equaion AS subjec classificaion: 53C4, 53C0, 53C1, 53C5 1 Inroducion A compac surface Σ, g) wih posiive scalar curvaure mus have Euler-Poincaré characerisic χσ) > 0 by he Gauss-Bonne formula. Then, by he classificaion of surfaces, Σ is eiher up o diffeomorphisms) S or RP, and hen one can conclude using he classical uniformizaion heorem ha he meric g is conformal o he canonical meric of consan curvaure. Since his resul uses he Gauss-Bonne formula in a fundamenal manner say as a bridge beween opological and geomerical informaions), he siuaion in higher dimension should be more complicaed. Neverheless, due o. Gursky and independenly by E. Hebey and. Vaugon in he four dimensional case, and remembering ha surfaces are locally conformally fla, one can prove he following Theorem 1.1 Gursky, [13] and Hebey Vaugon, [15] and [16]). Le, g) be a closed, locally conformally fla, n-dimensional Riemannian manifold, n = 4 or 6, wih non-negaive scalar curvaure. Then χ). Furhermore, χ) = if and only if, g) is conformally diffeomorphic o he sandard sphere, and χ) = 1 if and only if, g) is conformally diffeomorphic o he sandard real projecive space. 1 addresses: caino@mail.dm.unipi.i, Zindine.Djadli@ujf-grenoble.fr, ndiaye@sissa.i 1

2 As poined ou by Gursky, his resul is no rue for higher dimensions. Take for example he produc of S 4 equipped wih he canonical meric and a four dimensional hyperbolic space form. So, in view of generalizing he classificaion resul of Gursky in higher dimensions, one have o add some addiional condiion on he geomery of he manifold. In order o sae such a resul we need o inroduce some noaions. Consider, g), a compac, smooh, n-dimensional Riemannian manifold wihou boundary. Given a secion A of he bundle of symmeric wo ensors, we can use he meric o raise an index and view A as a ensor of ype 1, 1), or equivalenly as a secion of EndT ). This allows us o define σ k g 1 A) he k-h elemenary funcion of he eigenvalues of g 1 A, namely, if we denoe by λ 1,..., λ n hese eigenvalues σ k g 1 A) = λ i1... λ ik. A g = 1 n 1 i 1< <i k n In his paper we choose he ensor here is a real number) Ric g n 1) R gg where Ric g and R g denoe he Ricci and he scalar curvaure of ) g respecively. Noe ha for = 1, A 1 g is he classical Schouen ensor A 1 g = 1 n Ric g 1 n 1) R gg see [1]). Hence, wih our noaions, σ k g 1 A g) denoes he k-h elemenary symmeric funcion of he eigenvalues of g 1 A g. We call Γ + k he cone defined by ), Γ + k := {λ 1,..., λ n ) R n /σ 1 λ 1,..., λ n ) > 0,..., σ k λ 1,..., λ n ) > 0}. We say ha for some, A g is in Γ + k, if he eigenvalues of A g are in Γ + k. A Gauss-Bonne ype formula was proved by Viaclovsky in [18] for locally conformally fla manifol, which relaes he classical Gauss-Bonne-Chern o he inegral of he n -h elemenary funcion of he eigenvalues of he Schouen ensor. ore precisely we have 1 π) n 1) n )!!) g 1 A g ) dv g = χ), where χ) denoes he Euler characerisic of. Our main resul is he following: Theorem 1.. Le, g) be a closed, locally conformally fla, n dimensional Riemannian manifold, n 8 even, wih posiive scalar curvaure and wih posiive Euler Poincaré characerisic. There exiss a consan 0 = 0 n, diam, g), Rm ) < 1 such ha, if A g Γ + n, for some [ 0, 1], hen here exiss a meric g conformal o g such ha A 1 g Γ+ n. In paricular, g) has non negaive Ricci curvaure Ric g 0). Noe ha, since he manifold is assumed o be locally conformally fla, we can also wrie ha 0 = 0 n, diam, g), Ric ). Using a classificaion resul for compac, locally conformally fla manifol wih nonnegaive Ricci curvaure and a vanishing resul, we can prove he following classificaion resul Theorem 1.3. Le, g) be a closed, locally conformally fla, n dimensional Riemannian manifold, n 8 even, wih posiive scalar curvaure and wih posiive Euler Poincaré characerisic. There exiss a consan 0 = 0 n, diam, g), Ric ) < 1 such ha if A g Γ + n, for some [ 0, 1], hen is diffeomorphic o eiher S n or RP n.

3 Remark 1.4. The assumpion ha here exiss a consan 0 = 0 n, diam, g), Ric ) < 1 such ha A g Γ + n, for some [ 0, 1], by he Guan Viaclovsky Wang inequaliy see [1]), does no imply ha he meric g has non negaive Ricci curvaure. This would be rue, if one could ge 0 = 1. We need o poin ou ha here is anoher resul in he same direcion due o Guan-Lin-Wang [10], where hey proved, as a corollary of a more general resul, ha if, g) is a locally conformally fla manifold of even dimension, wih posiive Euler-Poincaré characerisic and wih A 1 g Γ n 1, hen is diffeomorphic o eiher S n or RP n. Also in his case, here isn a relaion beween heir and our assumpion. Ellipiciy Following [14], we will discuss he ellipiciy properies of equaion 1). Definiion.1. Le λ 1,, λ n ) R n. We view he k-elemenary symmeric funcion as a funcion on R n : σ k λ 1,, λ n ) = λ i1 λ ik, and we define Γ + k = 1 j k 1 i 1< <i k n {σ j λ 1,, λ n ) > 0} R n, For a symmeric linear ransformaion A : V V, where V is an n-dimensional inner produc space, he noaion A Γ + will mean ha he eigenvalues of A lie in he corresponding se. We noe ha his noaion also makes sense for a symmeric -ensor on a Riemannian manifold. If A Γ + n, le σ /n n A) = { A)}/n. Definiion.. Le A : V V, where V is an n-dimensional inner produc space. The n 1)-Newon ransformaion associaed wih A is T n 1) A) := Also, for R we define he linear ransformaion We have he following: Lemma.3. i) Γ + n ii) If A Γ + n, hen T n n 1 1) j σ j A)A j. j=0 L A) := T n 1) A) + 1 n σ 1T n 1) A)) I. is an open convex cone wih verex a he origin. 1 A) is posiive definie. Hence for all 1, L A) is posiive definie., hen ρ [0, 1], ρa + 1 ρ)b Γ + n, iii) If A and B are symmeric linear ransformaions, A, B Γ + n and σ n n Lemma.4. If A : R HomV, V ), hen ρa + 1 ρ)b) ρσ n A) + 1 ρ)σ n B). d A)s) = i,j T n 1) A) ij s) d A) ijs), i.e, he n 1)-Newon ransformaion is wha arises from differeniaion of. 3

4 Proof. The proof of his lemma is a consequence of an easy compuaion. See Gursky-Viaclovsky [14] Proposiion.5 Ellipiciy propery). Le u C ) be a soluion of equaion 1) for some 1 and le g = e u g. Assume ha A g Γ+ n. Then he linearized operaor a u, L : C,α ) C α ), is ellipic and inverible 0 < α < 1). Proof. Define he operaor F [u, g u, gu] = g 1 A g) fx) n e nu, so ha soluions of he equaion 1) are exacly he zeroes of F. Define he funcion u s = u + sϕ, hen he linearizaion a u of he oeraor F is defined by L ϕ) = d F [u s, g u s, g u s ] = d s=0 g 1 A g) ) s=0 d n fx) e nu s ) s=0 ). From Lemma.4 we have d g 1 A g) ) ) = T n s=0 1 g 1 A g) d ij We compue A g ) ij )) s=0. d A g )ij )) s=0 = gϕ) ij + 1 n gϕ)g ij ) g u g ϕ g ij + du dϕ. Easly we have also d n fx) e nu s ) s=0 ) = nfx) n e nu ϕ. Puing all ogheer, we conclude L ϕ) = T n 1 g 1 A g) ij gϕ) ij + 1 ) n gϕ)g ij nfx) n e nu ϕ + where he las erms denoe addiional ones wich are linear in g ϕ. The firs erm of he linearizaion is exacly he one defined in.1, i.e. L A g) ij = T n 1 A g) ij + 1 n T n 1 A g) pp δ ij. So finally, we have L ϕ) = L A g) ij gϕ) ij nfx) n e nu ϕ + Since A g Γ+ n, by Lemma.3, we have ha he ensor L A g ) is posiive definie. So he linearized operaor a any soluion u mus be ellipic. Noe also ha, by he previous formula, he operaor is of he form L ϕ) = Eϕ) cx)ϕ, where Eϕ) is a second order linear ellipic operaor and cx) is a sricly posiive funcion on, since cx) = nfx) n e nu and fx) > 0. This allows us o inver his operaor beween he Hölder spaces C,α ) and C α ). 4

5 For he proof of Theorem 1. and Theorem 1.3, we will be concerned wih he following equaion for a conformal meric g = e u g: 1) g 1 A g) ) /n = fe u, where f is a posiive funcion on. Le σ 1 g 1 A 1 g) be he race of A 1 g wih respec o he meric g. We have he following formula for he ransformaion of A g under his conformal change of meric: ) A g = A g + gu + 1 n gu)g + du du gu gg. Since A g = A 1 g + 1 n σ 1g 1 A 1 g)g, his formula follows easily from he sandard formula for he ransformaion of he Schouen ensor see [18]): 3) A 1 g = A 1 g + gu + du du 1 gu gg. Using his formula we may wrie 1) wih respec o he background meric g g 1 A g + gu + 1 n gu)g + du du )) /n gu gg = fx)e u. 3 Upper bound and higher order esimae Throughou he sequel,, g) will be a closed n-dimensional Riemannian manifold n even) wih posiive scalar curvaure and locally conformally fla. Since R g > 0, here exiss δ > such ha A δ g Γ + n. for example we can ake δ such ha A δ g is posiive definie, i.e. Ric g n 1) R gg > 0 on ). Noe ha δ only depen on Ric. For [δ, 1], consider he pah of equaions in he sequel we use he noaion A u := A g for g given by g = e u g) 4) σ /n n g 1 A u ) = fe u, where f = σ /n n g 1 A δ g) > 0. Noe ha u 0 is a soluion of 4) for = δ. Proposiion 3.1 Upper bound). Le u C ) be a soluion of 4) for some [δ, 1], wih A u Γ + n. Then u δ, where δ depen only on Ric. Proof. From Newon s inequaliy we have δ σ n n C n σ 1, for some C n > 0. So for all x fe u C n σ 1 g 1 A u ). Le p be a maximum of u, hen using ), since he gradien erms vanish a p and u )p) 0, fp)e up) C n σ 1 g 1 A u )p) = C n σ 1 g 1 A n n g)p) + C n u )p) n C n σ 1 g 1 A g)p) C n σ 1 g 1 A δ g)p). Since is compac, hen u δ, for some δ depending only on Ric. 5

6 Once we have an upper bound for he soluions of equaion 4), by he work of S. Chen [7], we immediaely ge C 1 and C esimaes: Proposiion 3. C 1 and C esimaes). Le u C 4 ) be a soluion of 4) for some [δ, 1], wih A u Γ +. Then sup g u g + ) gu g C1, where C 1 depens only on n, diam, g) and Rm. Now, by he Yamabe equaion for he conformal deformaion of he scalar curvaure, we have ha R g e u = R g + n 1) g u n 1)n ) g u g). So we obained an uniform esimaes for he scalar curvaure of g, i.e. Proposiion 3.3. Le u C 4 ) be a soluion of 4) for some [δ, 1], wih A u Γ +. Then 0 < R g e u Λ, where Λ is a posiive consan depending only on n diam, g) and Rm. 4 Lower bound Proposiion 4.1 Lower Bound). There exiss a consan 0 = 0 n, diam, g), Rm ) < 1 such ha, if u C ) is a soluion of 4) and if A u Γ + n for some [ 0, 1], hen u δ for some uniform consan δ depending only on diam, g) and Rm. Proof. I s easy o see ha he following formula hol g 1 A g) = g 1 A g ) + C 1 1 ) n σ1 g 1 A g ) n + n 1 for some posiive consans C 1 and c n,i depending only on n and i. i=1 ) n 1 i c n,i σ i g 1 A g )σ 1 g 1 A g )) n i, n Since A u Γ + n, we have σ i gu 1 A u ) > 0 for all 1 i n/. So, ieraing he previous formula, we can easily check ha σ i gu 1 A 1 u ) > C i 1 ) i σ 1 gu 1 A 1 u ) ) i, for some posiive consans C i depending only on n. Hence, by he previous formula, we have g 1 u A u ) g 1 u A 1 u ) + C 1 1 ) n σ1 g 1 u A 1 u ) n C 1 ) n σ1 g 1 u A 1 u ) n. On he oher hand, since u is a soluion of equaion 4), we have g 1 u A u ) = e nu g 1 A u ) = e nu f n, or equivalenly e nu g 1 u A u ) = e nu f n. Inegraing on his wih respec o dv g, we obain C e nu dv g e nu f n dvg = e nu g 1 u A u ) dv g = g 1 u A u ) dv gu g 1 u A 1 u ) dv gu + C 1 1 ) n R n gu dv gu C 1 ) n 6 R n gu dv gu,

7 where C > 0 is chosen so ha f n C recall ha, since f = σ /n n g 1 A δ g), C depen only on Ric ). Using Hölder inequaliy and he definiion of he Yamabe invarian which is posiive), we ge R n gu dv gu Y, [g])) n. oreover, by he resul of Viaclovsky in [18], we have he conformal invariance g 1 u A 1 u ) dv gu = g 1 A 1 g) dv g. Thus we ge C e nu dv g g 1 A 1 g) dv g + C 1 1 ) n n Y, [g])) C 1 ) n Now, by Proposiion 3.3, we have an uniform esimae for he quaniy R n g dv g = R n g e nu dv g. R n gu dv gu. Hence for 0 sufficienely close o 1, since he Euler-Poincaré characerisic of is posiive, we can always assume ha g 1 A 1 g) dv g + C 1 1 ) n n Y, [g])) C 1 ) n R n g dv g = λ > 0, for every [ 0, 1]. This gives max u 1 n log λ Cdiam, g), Ric ). By Proposiion 3., max g u g C 1. This implies he Harnack inequaliy max u min u + Cdiam, g), Rm ), by simply inegraing along a geodesic connecing poins a which u aains is maximum and minimum. Combining hese wo inequaliies, we obain u min u 1 n log λ C =: δ, where C only depen on diam, g) and Rm. This en he proof of he lemma. In he previous secion we prove ha for a soluion u of he equaion 4), wih A u Γ + n, we have a priori C 1 and C esimaes jus depending on he upper bound of he funcion u. Now, if [ 0, 1], where 0 is he one of Proposiion 4.1, since u C ) has a lower bound oo, from Proposiion 3. we have u L ) + g u L ) + gu L ) C, where C depen only on n, diam, g) and Rm. By he works of Krylov [17] and Evans [8] we obain C,α esimaes, i.e Proposiion 4.. Le u C 4 ) be a soluion of 4) for some [ 0, 1], wih A u u C,α ) C, where C is a posiive consan depending only on n, diam, g) and Rm. Γ + n. Then 7

8 5 Proof of Theorem 1. Le 0 = 0 n, diam, g), Rm ) < 1 be he one of Proposiion 4.1. We assume by hypohesis ha A 0 g Γ + n. The parameer 0 will be he saring poin in order o use he coninuiy mehod. Our 1-parameer family of equaions, for [ 0, 1], is 5) σ /n n g 1 A u ) = fx)e u, wih fx) = σ /n n g 1 A 0 g ) > 0. Define he se S = { [ 0, 1] a soluion u C,α ) of 5) wih A u Clearly, wih our choice of f, u 0 is a soluion for = 0. By assumpion, 0 S, and S. Le S, and u be a soluion. By Proposiion.5, he linearized operaor a u, L : C,α ) C α ), is inverible. The implici funcion heorem ells us ha S is open. From classical ellipic heory, since u C,α ), by a boosrap argumen using classical Schauder esimaes, i follows ha u C ), since f C ). By Proposiion 4. and he classical Ascoli-Arzela s Theorem, we will ge ha S mus be closed, herefore S = [ 0, 1]. The meric g = e u1 g hen saisfies σ k A 1 g ) > 0 for all 1 k n. The inequaliy on he Ricci curvaure in Theorem 1. follows from he following proposiion for k = n see [1] for he proof) Proposiion 5.1. If for some meric g 1 on we have A g1 Γ + k, hen Ricg 1 ) k n kn 1) R g 1 g 1. Γ + n }. 6 Proof of Theorem 1.3 In Theorem 1. we have proved ha if, g) is an even-dimensional, closed, locally conformally fla manifol wih posiive scalar curvaure, posiive Euler-Poincaré characerisic, and close o be n -admissible, hen admis a meric g which is n -admissible, i.e. wih A g Γ + n. In paricular i urns ou ha he Ricci curvaure Ric g ) mus be nonnegaive. By he classificaion heorem for locally conformally fla manifol wih nonnegaive Ricci curvaure for insance, see []), we have ha mus be conformally equivalen o eiher a space form or a finie quoien of a Riemannian S n 1 c) S 1, for some c > 0. oreover, for locally conformally fla manifol which admi k-admissible merics i.e. merics g such ha A g Γ + k ) we have opological resricions see, for example, [6], [9], [11]). In [9] hey proved he following vanishing heorem Proposiion 6.1 Gonzáles, [9]). Le, g) be a closed, locally conformally fla manifold, wih A g Γ + k, k < n/. Then he q-h Bei number b q = 0, for Since A g Γ + n number n k + 1 q n + k Γ + n 1, we can apply his proposiion o he case k = n 1 o ge ha he q-h Bei 1. b q = 0, for q n. Since he Euler characerisic can be defined in erms of he Bei numbers, by Poincaré dualiy, we ge ha χ) = b 1. Then 0 < χ), which forces he manifold o be diffeomorphic o eiher RP n if χ) = 1) or S n if χ) = ). 8

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