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Do uniruled six-manifolds contain Sol Lagrangian 0 submanifolds? 1 0 2 Frédéric Mangolte Jean-Yves Welschinger n a January 16, 2010 J 7 1 Abstract ] We prove using symplectic field theory that if the suspension of a G hyperbolicdiffeomorphismofthetwo-torusLagrangianembedsinaclosed S uniruled symplectic six-manifold, then its image contains the boundary . of a symplectic disc with vanishing Maslov index. This prevents such h t a Lagrangian submanifold to be monotone, for instance the real locus a of a smooth real Fano manifold. It also prevents any Sol manifold to m beinthereallocusof anorientablerealDelPezzofibrationoveracurve, [ confirminganexpectationofJ.Kollár. Finally,itconstraintsHamiltonian diffeomorphisms of uniruled symplectic four-manifolds. 1 v 7 Introduction 2 9 2 Complex projective uniruled manifolds play a special rôle in algebraic geome- . try, these are the manifolds of special type in the sense of Mori. What can be 1 0 the topology of the real locus of such a manifold when defined over R? This 0 natural question has a symplectic counterpart. What can be the topology of 1 Lagrangian submanifolds of such uniruled manifolds? Uniruled manifolds of : v dimension two are rational or ruled surfaces. Comessatti proved in [4] that no i orientablecomponentofthereallocusofsuchasurfacecanhavenegativeEuler X characteristic. Actually, closed symplectic four-manifolds with b+ = 1 cannot r 2 a contain any orientable Lagrangian submanifold with negative Euler characteristic. By the way, it is provedin [30] that even the unit cotangent bundle of an orientable hyperbolic surface does not embed as a hypersurface of contact type of a uniruled symplectic four-manifold. In complex dimension three, a great piece of work was done by Kollár [18, 19, 20, 21] in order to carry out Mori’s minimal model program(MMP) over R for uniruled manifolds. Roughly, the upshot [17] is that up to connected sums with RP3 or S2 S1 and modulo a finite number of closedthree-manifolds, the × orientable real uniruled three-manifolds are Seifert fibered spaces or connected sums of lens spaces. This result however depends on two expectations. The first one is that closed hyperbolic manifolds cannot appear. The second one is that closed Sol manifolds cannot appear. Quickly, this first expectation was confirmed by Viterbo and Eliashberg ([28, 15, 5]). Namely, a closed uniruled symplectic manifold of dimension greater than four cannot contain a closed Lagrangiansubmanifold with negative curvature. The proof of Eliashberg uses 1 symplecticfieldtheory(SFT),whichappearstobeaverypowerfultooltotackle this question. The aim of this paper is to prove the second one, using SFT as well, at least as far as the precise expectation of Kollár is concerned. We unfortunately could not prove such a general result as Viterbo-Eliashbeg’s one, but proved the following (see Theorem 2.1). Let L (X,ω) be a closed Lagrangian sub- ⊂ manifold homeomorphic to the suspension of a hyperbolic diffeomorphism of the two-torus, where (X,ω) is a closed symplectic uniruled six-manifold. Then X contains a symplectic disc of vanishing Maslov index and with boundary on L, non-trivial in H (L;Q). This prevents L from being monotone, for instance 1 the real locus of a smooth Fano manifold. It also actually prevents any Sol manifold to be in the real locus of a projective three-manifold fibered over a curve with rational fibers, at least provided this real locus be orientable, see Corollary 2.1 and the discussion which follows. This was the actual problem raised by Kollár in [21, Remark 1.4]. Finally, it implies that a Hamiltonian diffeomorphism of a uniruled symplectic four-manifold which preserves some Lagrangian torus cannot restrict to a hyperbolic diffeomorphism of the torus, see Corollary2.3. Our approach,which uses SFT, requires some understanding of the geodesic flow of Sol manifolds, namely the Morse indices of their closed geodesics. The first part of this paper is thus devoted to a study of Sol manifolds and their closed geodesics. The second part is devoted to the proof of our main result. Note that the converse problem remains puzzling. What is the simplest real projective manifold which contains a hyperbolic component? What is the simplest real projectivemanifold which contains a Solcomponent? Recall that every closed orientable three-manifold modeled on any of the six remainingthree-dimensionalgeometriesembeds inthe reallocusofaprojective uniruled manifold [13, 12]. Note also that in the case of the projective space, the absence of orientable Sol Lagrangian submanifolds follows from Theorem 14.1of[6]. Moreover,inthispaperKenjiFukayaremarksthathismethodsmay extend to uniruled manifolds as well. Acknowledgments: This work was supported by the French Agence na- tionale de la recherche, reference ANR-08-BLAN-0291-02. The second author acknowledges Gabriel Paternain for fruitful discussions about Sol manifolds. 1 Sol-geometry 1.1 The group Sol The group R of real numbers acts on the abelian group R2 by R R2 R2 × → (z,(x,y)) (ezx,e zy) . − 7→ The induced semidirect product is denoted by Sol, so that the grouplaw of Sol is given by Sol Sol Sol × → ((α,β,λ),(x,y,z)) eλx+α,e λy+β,z+λ . − 7→ Let K =R2 denote the kernel of the s(cid:0)urjective morphism (x,y,(cid:1)z) Sol z ∼ ∈ 7→ ∈ R, and let e ,e ,e be the elements of Sol of coordinates (1,0,0), (0,1,0), and 1 2 3 2 (0,0,1) respectively. The group K coincides with the derived subgroup of Sol, as shows the relation [e3,xe1+ye2]=(e 1)xe1+(e−1 1)ye2 . − − Denote by ∂ ∂ ∂ X :=ez , Y :=e−z , Z := ∂x ∂y ∂z the left-invariant vector fields of Sol which coincide with ∂ , ∂ , ∂ at the ori- ∂x ∂y ∂z gin. We provide Sol with the Riemannian metric and the orientation making (X,Y,Z) direct orthonormal. The space Sol thus obtained is homogeneous, its isotropygroupisisomorphictothediedralgroupD generatedbytheisometries 4 ρ: (x,y,z) Sol (y, x, z) Sol ∈ 7→ − − ∈ and r : (x,y,z) Sol ( x,y,z) Sol Y ∈ 7→ − ∈ see [26, Lemmas 3.1 and 3.2]. In particular, the isometries of Sol preserve the horizontalfoliation := dz =0 ,andactbyisometriesonitsspaceofleavesR. F { } We denote by P: Isom(Sol) Isom(R) the surjective morphism thus defined. → 1.2 Geodesic flow on Sol GeodesicsofSolhavebeendeterminedin[26],anddividedintothreetypesA,B, andC. GeodesicsoftypeAarethelinesdirectedbythe vectorfieldf := X Y 1 √−2 or f := X+Y; they are contained in the foliation (whose leaves are minimal 2 √2 F surfaces). Geodesics of type B are geodesics contained in the totally geodesic hyperbolic foliations := dy = 0 or := dx = 0 . Among geodesics ′ ′′ H { } H { } of type B, only those directed by the vector field Z will play a rôle in this paper. Geodesics of type C are contained in cylinders whose axes are geodesics of type A. Along such a geodesic, the z-coordinate -corresponding to the axis directed by e - is then bounded between two values, these geodesics of type C 3 will not play a significant rôle in the sequel. The aim of the present paragraph is to calculate the linearization of the geodesic flow along a geodesic of type A or B. DenotebyS Sol:= (q,p) T Sol p =1 theunitarycotangentbundle ∗ ∗ { ∈ | k k } of Sol where the norm is the one induced by the fixed Riemannian metric k·k on Sol. Denote by ξ the contact distribution of S Sol, it is the kernel of the ∗ ∗ restriction of the Liouville form pdq. Likewise, we denote by SSol the unitary tangent bundle of Sol, and by ξ the distribution induced by the identification ♭: SSol ∼ S∗Solgivenbythe metric. Theidentificationis definedinthe basis −→ (e ,e ,e ) by 1 2 3 ♭: TSol T Sol ∗ −→ (x,y,z,x˙,y˙,z˙) x,y,z,e 2zx˙,e2zy˙,z˙ . − 7−→ The Levi-Civita connection gives an ortho(cid:0)gonal direct sum dec(cid:1)omposition ξ =ξ ξ , h v ⊕ 3 where ξ is the space of elements of ξ which are tangent to the fibers of TSol, v while ξ is the orthogonal plane to ξ given by the connection. The planes ξ h v h and ξ are canonically isomorphic; if v is a tangent vector to Sol orthogonalto v a geodesic, we will denote by v its lift to ξ , and by v˙ its lift to ξ , in order to h v distinguish them. 1.2.1 Linearized flow along a geodesic of type A There are two families of geodesics of type A, those given by f = X+Y, and 2 √2 those given by f = X Y . Since these families are exchanged by the isometry 1 √−2 r =ρ2r , we restrict our study to the first family. X Y Let then γ(t) := γ + t X+Y be a geodesic of type A and γ (t) = 0 √2 γ0 ′ (cid:16) (cid:17)| dγ(t)= X+Y . The orthogonalplane to γ (t) in T Sol is generated by dt √2 γ(t) ′ γ(t) (X −Y)|γ(cid:16)(t) and(cid:17)|Z|γ(t). Hence ξ|(γ(t),γ′(t)) =hX −Y,Z,X˙ −Y˙,Z˙i. Let X Y h := − , h :=Z+h˙ , h :=Z+2h˙ , h :=h Z˙ . 1 2 1 3 1 4 1 √2 − Lemma 1.1. Let γ: t R γ +t X+Y Sol be a geodesic of type A, ∈ 7−→ 0 √2 γ0 ∈ where γ Sol. Then the canonical sy(cid:16)mplecti(cid:17)c|form ♭ (dp dq) on the contact 0 ∗ ∈ ∧ distribution ξ along γ is given by dh dh +dh dh . 1 2 3 4 ∧ ∧ Proof. The pull-back of the Liouville form is ♭∗(pdq)(x,y,z,x˙,y˙,z˙) =e−2zx˙dx+e2zy˙dy+z˙dz , | so that the symplectic form writes ♭ (dp dq) ∗ (x,y,z,x˙,y˙,z˙) ∧ | = 2e 2zx˙dz dx+e 2zdx˙ dx + 2e2zy˙dz dy+e2zdy˙ dy − − − ∧ ∧ ∧ ∧ +dz˙ dz ( ) (cid:0) (cid:1) (cid:0) (cid:1) ∧ ∗ =du˙ du+dv˙ dv+dz˙ dz+2(v˙du+u˙dv) dz , ∧ ∧ ∧ ∧ where (u,v) are coordinates in the basis (X Y,X+Y), and (u˙,v˙) are coordi- √−2 √2 nates in the basis (X˙ Y˙ ,X˙+Y˙ ). The restriction of this symplectic form to the √−2 √2 distributionξ alongourgeodesicoftypeAisdu˙ du+dz˙ dz+2du dz,since ∧ ∧ ∧ v˙ 1 and u˙ 0, and this eventually gives dh dh +dh dh . 1 2 3 4 ≡ ≡ ∧ ∧ Proposition 1.1. Let γ: t R γ +t X+Y Sol be a geodesic of ∈ 7−→ 0 √2 γ0 ∈ type A, where γ Sol. The linearization (cid:16)of the(cid:17)g|eodesic flow of Sol along 0 ∈ γ restricted to the contact distribution ξ has the following matrix in the basis (h ,h ,h ,h ): 1 2 3 4 1 t 0 0 0 1 0 0 0 0 cos(√2t) 1 sin(√2t) . −√2 0 0 √2sin(√2t) cos(√2t)      4 Proof. The vector field h is the restriction along γ of a Killing field of Sol. 1 Likewise,thevectorfieldt X Y +Z istherestrictiontoγ ofaKilling √−2 γ(t) |γ(t) field of Sol. We deduce fro(cid:16)m that(cid:17)t|he two first columns of the matrix. Without loss of generality, we can assume that γ =0, so that γ: t R tf (0) Sol. 0 2 ∈ 7→ ∈ A geodesic of type C which is close to γ writes γ (t)=u (t)f (k)+v (t)f (k)+z (t)e . k k 1 k 2 k 3 Then ∂ ∂ ∂f ∂ 2 γ (t) = u (t) f (0)+v (t) + z (t) e k k 1 0 k 3 (cid:18)∂k (cid:19)|k=0 (cid:18)∂k (cid:19)|k=0 ∂k |k=0 (cid:18)∂k (cid:19)|k=0 since u (t) 0 and we consider only the normal part of vector fields. 0 ≡ Now,withthenotationsof[26, 4.4],u (t)=d+µksn(µ(t+τ) K),where, k sinceweassumethatγ (0) 0,eit§herd=µk andτ =0,ord=0a−ndτ = K. k ≡ −µ In the first case,we get ∂ u (t) =√2 1+sin(√2t π) , while v (t)=t ∂k k k=0 − 2 0 and ∂∂fk2 k=0 = ∂∂kz¯ k=0f1((cid:0)0). Now,(cid:1)k|eepingthe(cid:0)senotations: zk(t)(cid:1)=z¯+h(µt−K), so that|∂zk =| ∂z¯ +cos(√2t π). Thus, the vector field ∂k k=0 ∂k k=0 − 2 | | ∂z¯ (tf +e )+√2 cos(√2t)+1 f +sin(√2t)e 1 3 1 3 ∂k k=0 − | (cid:16) (cid:17) along γ is a Jacobi field. We deduce that 1 cos(√2t) f + 1 sin(√2t)e is − 1 √2 3 Jacobi itself and then the fourth column of the matrix. In the second case, we (cid:0) (cid:1) get ∂ u (t) = √2sin(√2t), while ∂z¯ = 1 and ∂zk = ∂z¯ ∂k k k=0 − ∂k k=0 ∂k k=0 ∂k k=0 − | | | | cos(√2t). Hence, the vector field (cid:0) (cid:1) t √2sin(√2t) f + 1 cos(√2t) e 1 3 − − (cid:16) (cid:17) (cid:16) (cid:17) along γ is Jacobi, so that √2sin(√2t)f +cos(√2t)e is Jacobi itself. 1 3 1.2.2 Linearized flow along a geodesic of type B Among geodesics of type B, only those directed by e will be considered. Let 3 then γ: t R γ + te Sol be such a geodesic, where γ Sol. The 0 3 0 ∈ 7→ ∈ ∈ orthogonalplane to γ (t) in TSol is generated by X and Y, therefore ′ ξ(γ(t),γ′(t)) = X,X˙,Y,Y˙ . | h i Let 1 1 g := X +X˙ , g :=X , g := Y +Y˙ , g :=Y . 1 2 3 4 2 −2 Proposition 1.2. Let γ: t R γ +te Sol be a geodesic of type B, where 0 3 ∈ 7→ ∈ γ Sol. The linearization of the geodesic flow of Sol along γ restricted to the 0 ∈ contact distribution ξ has the following matrix in the basis (g ,g ,g ,g ): 1 2 3 4 et 0 0 0 0 e t 0 0  −  , 0 0 e t 0 − 0 0 0 et     5 while the canonical symplectic form ♭ (dp dq) on the contact distribution ξ ∗ ∧ along γ is given by dg dg +dg dg . 1 2 3 4 ∧ ∧ Proof. The expression of the symplectic form follows from the formula ( ) ob- ∗ tained in the proof of Lemma 1.1, since along γ, x˙ = y˙ = 0. The geodesic γ is the intersection of the leaves of and containing it, which are totally ′ ′′ H H geodesic. Hence the direct sum decomposition ξ = ξ ξ , where ξ is the ′ ′′ ′ ⊕ contact distribution of S , and ξ is the contact distribution of S . The ∗ ′ ′′ ∗ ′′ H H geodesic flow restricted to ξ or ξ is the geodesic flow of the hyperbolic plane. ′ ′′ Thefieldse ande areKilling,providingthesecondandfourthcolumnsofthe 1 2 matrix. Wecanassumethatγ =0. GeodesicsoftypeB of passingthrough 0 ′ H 0 Sol at t=0 write ∈ sinh(t) γ (t)=a e ln(cosh(t) c sinh(t))e a 1 0 3 cosh(t) c sinh(t) − − 0 − with a2 +c2 = 1, see also [26, 5.2]. Therefore sinh(t)X is Jacobi. Likewise, 0 § geodesics of type B of passing through 0 Sol at t=0 write ′′ H ∈ sinh(t) γ (t)=b e +ln(cosh(t)+c sinh(t))e b 2 0 3 cosh(t)+c sinh(t) 0 with b2+c2 =1, so that sinh(t)Y is Jacobi. Hence the result. 0 1.3 Closed Sol-manifolds 1.3.1 Classification Recall the following: Lemma 1.2. Let L be the suspension of a diffeomorphism of the torus R2/Z2 definedbyalinearmapA Gl (Z). Assumethat(A I)isinvertibletoo. Then, 2 ∈ − the homology with integer coefficients of L satisfy the following isomorphisms H (L;Z)=Z ; H (L;Z)=Z Z2/(A I)(Z2) ; 0 ∼ 1 ∼ ⊕ − Z if det(A)>0(cid:0) (cid:1) Z if det(A)>0 H (L;Z)= and H (L;Z)= . 2 ∼(Z/2Z if det(A)<0 3 ∼(0 if det(A)<0 Note besides that in the situation of Lemma 1.2, if Λ is the fundamental group of L based at some point x L and Λ = Z2 is the fundamental group 0 ∈ 0 ∼ of the fiber ofL R/Z containingx , then the exactsequence 0 Λ Λ 0 0 → → → → Z 0splits. Thereforethederivedsubgroup[Λ,Λ]coincideswith(A I)(Λ ). 0 → − From Hurewicz’s isomorphism, we deduce the relation TorsH (L;Z)=Λ /(A I)(Λ ). 1 ∼ 0 − 0 Definition 1.1. AlinearmapA Gl (Z)iscalledhyperboliciffithastworeal 2 ∈ eigenvalues different from 1. ± 6 Lemma 1.3. Let L be the suspension of a diffeomorphism of the torus R2/Z2 defined by a hyperbolic linear map A Gl (Z). There exists a lattice Λ of 2 ∈ Isom(Sol) such that L is diffeomorphic to the quotient Λ Sol. Moreover, Λ is \ generated by a lattice Λ of K and an isometry 0 l: (x,y,z) Sol ε eλx,ε e λy,z+λ Sol 1 2 − ∈ 7−→ ∈ where λ R∗, ε1,ε2 1 . (cid:0) (cid:1) ∈ ∈{± } Proof. WeidentifyR2 withthederivedsubgroupK ofSolinawaythate ,e is 1 2 a basis of eigenvectorsof A associatedto the eigenvalues ε eλ and ε e λ where 1 2 − λ R ,ε ,ε 1 . ThesubgroupZ2 isthenidentifiedwithalatticeΛ K ∗ 1 2 0 ∈ ∈{± } ⊂ invariant by A. Let l be the product of the left multiplication by λe with the 3 isometry (x,y,z) Sol (ε x,ε y,z) Sol. 1 2 ∈ 7→ ∈ Denote by Λ the subgroup of Isom(Sol) generated by l and the left translations by elements of Λ , this is a lattice of Isom(Sol) which satisfies the split 0 exact sequence 0 Λ Λ Z 0, where the action of l by conjugation on 0 → → → → Λ0 coincideswiththeactionofA. ThequotientΛ SolisdiffeomorphictoL. \ LetLbethesuspensionofadiffeomorphismofthetorusR2/Z2 definedbya hyperbolic linear map A Gl2(Z). We provide L:= Λ Sol with the metric Sol ∈ \ givenbyLemma1.3. ThebasisB =S1 ofthe fibrationL B isthenendowed ∼ → with a metric induced by the one of L. The morphism P induces a morphism P : Isom(L) Isom(B) between their respective isometry groups. L → Note that the involution ρ2 induces an isometry of L which belongs to the kernel of P . Likewise, a translation (x,y,z) Sol (x+α,y+β,z) Sol L ∈ 7−→ ∈ induces an isometry of L if and only if (α,β) (A I) 1(Λ ). We denote by − 0 ∈ − F :=(A I) 1(Λ )/Λ this group of translations. − 0 0 − Lemma 1.4. Let L be the suspension of a hyperbolic diffeomorphism of the torus endowed with its metric Sol given by Lemma 1.3. Then, the kernel of the morphism P is generated by ρ2 and F while its image is finite. The latter is L reduced to isometries which preserve the orientation of B when L is nonori- entable. Proof. The group Isom(L) coincides with the quotient by Λ of the normalizer of Λ in Isom(Sol). An element of the kernel of P preserves all the leaves of . L F Itcannotinduce anyreflectiononthoseleavessincetheaxesofthesereflections wouldbedirectedbye ore ,butΛ doesnotcontainanynontrivialmultipleof 1 2 0 these elements. It follows that, up to multiplication by ρ2, it is a translationin thefibersandthen,anelementofF. TheimageofP isasubgroupofIsom(B) L which cannot be dense since the action on K by conjugation by an element of Sol close to K is a linear map close to the identity, which cannot preserve Λ . 0 Indeed, the fibers of L close to a given fiber are not isometric to it. Thus, the image of P is a finite subgroup of Isom(B). If k˜ is such an isometry which L reverses the orientation of B, it has a lift k of the form k(x,y,z)= η eθy+α,η e θx+β,θ z 1 2 − − with η1,η2 1 , θ,α,β R(cid:0). We get (cid:1) ∈{± } ∈ lklk−1(x,y,z)= ε1ε2(x α)+ε1eλα,ε1ε2(y β)+ε2e−λβ,z − − therefore such an isomet(cid:0)ry does not belong to Λ0 when the sign ε1ε(cid:1)2 of the determinant of A is negative. Hence the result. 7 Let L be a Sol variety given by Lemma 1.4 and k be a cyclic group of h i isometries of L acting without fixed point. If P (k) is an isometry of the base L B which preserves the orientation, then the quotient of L by k is also the h i suspension of a hyperbolic diffeomorphism of the torus. Should the opposite occur, P (k) is a reflection of B and we can assume that k is of order 2. The L quotient L/ k is no longer a bundle over B and is orientable. Indeed, L is h i necessarily orientable from Lemma 1.4, while over a fixed point of P (k), the L linearmapassociatedtokcannotbearotationbyanangleof π mod π,itmust 2 bethenareflectionintheassociatedfibers,thereforekpreservestheorientation of L. Definition 1.2. Following[24], wecallsapphirethe quotientofaSol-bundle L given by Lemma 1.4 by an involutive isometry acting without fixed point and inducing a reflection on the basis B. The second homologygroup with integer coefficients of a sapphire vanishes, its first homologygroupis torsion. We call Sol-manifold any manifold obtained as a quotient of Sol by a discrete subgroup of isometries acting without fixed point. Recall the Theorem 1.1. The closed Sol-manifolds are the sapphires and the suspensions of diffeomorphisms of the torus R2/Z2 defined by hyperbolic linear maps. Proof. By definition, sapphires are closed Sol-manifolds while suspensions of hyperbolic diffeomorphisms of the torus are Sol by Lemma 1.3. Conversely, let Λ Isom(Sol) be a cocompact discrete subgroup acting ⊂ wihout fixed point on Sol. Let Λ be the kernel of the restriction of P to Λ. 0 An element of Λ writes gh where g is a translation of vector αe +βe Sol, 0 1 2 ∈ α,β R and h id,r ,r , since Λ acts without fixed point and preserves X Y 0 ∈ ∈ { } all the leaves of . The subgroup of translations of Λ is of index at most 2 in 0 F Λ and is necessarily of rank 2, see for example [25, Theorem 4.17]. 0 Let id = l Λ be such that P(l) preserves the orientation of B. The 6 ∈ quotient of Sol by the subgroup generated by l and the translations of Λ is 0 a torus bundle over the circle with hyperbolic monodromy. The result then follows from Lemma 1.4. 1.3.2 Closed geodesics Let L = Λ Sol be a closed Sol-manifold given by Theorem 1.1. The lattice Λ \ satisfies the exact sequence 0 → Λ0 −→ Λ −P→L Λ/Λ0 → 0, where Λ0 ⊂ K. We denotebyp: L Btheassociatedmap,whereB =R/P (Λ)ishomeomorphic −→ L toanintervalifLisasapphireandtothecircleotherwise. Anyperiodicgeodesic γ: R L has a lift which is a geodesic γ: R Sol. We will say that γ is of −→ −→ type A, B, or C if γ is of type A, B, or C in the sense of [26]. Closed geodesics of type A of L are in particular quotients of geodesics of type A of Sol directed e by elements of Λ0.eThese geodesics are contained in the fibers of p and then belong only to a dense countable subset of such fibers. Lemma 1.5. Let L be a closed Sol-manifold. Then, any closed geodesic of type C of L is homotopic to a closed geodesic of type A of L. Furthermore, closed geodesics of type B of L are quotients of geodesics of type B of Sol directed by e , that is intersection of hyperbolic leaves of and in Sol. 3 ′ ′′ H H 8 Proof. Let γ: R L be a periodic geodesic of type C and let γ: t R −→ ∈ −→ (x˜(t),y˜(t),z˜(t)) Sol be a lift of γ. There exists l Λ such that for every 0 0 ∈ ∈ t R, γ(t+T) = l0 γ(t), where T is the minimal period of γ. Ien particular, ∈ · the coordinatezõf γ is T-periodic,by [26, 4.4]. This forcesT to be a multiple of 4K, where K and µ are the quantities i§ntroduced in [26]. We deduce from µ e e the equation of geodeesics of type C obtained in [26, 4.4] the relation § γ(t+T) γ(t)=2 Lµ ab T ( X Y) . − | | ± ± (cid:16) p (cid:17) Hence, writing Te=n 4µK ,en∈N∗, we deduce that (cid:16) (cid:17) 8nLK ab ( X Y)=l , 0 | | ± ± (cid:16) p (cid:17) so that the closedgeodesic γ is homotopicto the closedgeodesicof type A of L defined by l . The latter’s length is a multiple of the quantity 8LK ab, with 0 | | the notationsof[26]. Likewise,letγ: R L be a periodicgeodesicoftype B, −→ p of minimal period T, and let γ: t R (γh(t),γv(t)) Sol/Λ = K/Λ ⋊R ∈ −→ ∈ 0 0 (cid:16) (cid:17) bealiftofγ toetheinfinitecyecliccoeveringofL. Thereexistsl ∈Λ/Λ0 ofinfinite order such that for all t R, γ(t+T) = l γ(t). The action of l on the torus ∈ · K/Λ is defined by a hyperbolic linear map A. We deduce that for all t R, 0 ∈ (A I)(γ (t))=0. Hence, γ isnecessarilyconstantandequaltoafixedpoint h he e − of A. e e Remark 1.1. TheproofofLemma1.5providesanestimateofthelengthofclosed geodesicsoftype A homotopicto closedgeodesicsoftype C. Thisestimate will be crucial in the proof of Proposition 1.3. Likewise, if L is the suspension of a diffeomorphism of the torus defined by a hyperbolic linear map A Gl (Z), 2 ∈ we deduce that closed geodesics of type B of L are in correspondence with the periodic points of A: R2/Z2 R2/Z2. −→ Proposition 1.3. Let L be a closed three-dimensional manifold given by The- orem 1.1 and let Π be a finite subset of homotopy classes of L. There exists a Sol-metric on L such that no element of Π gets realized by a closed geodesic of type C of L. Furthermore, this metric can be chosen such that closed geodesics of type A of L homotopic to elements of Π are of Morse-Bott index 1. Proof. From Theorem 1.1, the Sol-manifold L is diffeomorphic to the quotient of Sol by a lattice Λ Isom(Sol) satisfying the exact sequence 0 Λ 0 ⊂ → −→ Λ −P→L Λ/Λ0 → 0 where Λ0 ⊂ K is a lattice. The fundamental group of L is therefore isomorphic to Λ, and from Lemma 1.5, only classes in Π Λ can be 0 ∩ realized by closed geodesics of type A or C. Up to multiplication of the lattice Λ by a constant 0 < ε 1, we can assume that all the elements of Π Λ 0 0 ≪ ∩ have length bounded from above by 4 π. Such a Sol-metric fits. Indeed, from − Lemma 1.5 and Remark 1.1, every closed geodesic of type C of L is homotopic to a closed geodesic of type A of length a multiple of 8LK ab, adopting the | | notations of [26]. Now, taking again the notations of [26], we get p 8 K 8LK ab = E (1 k2) where 0 k 1. | | √2√1+k2 − 2 − ≤ ≤ (cid:18) (cid:19) p 9 π 2 Moreover, E = 1 k2sin2θdθ 1 and − ≥ Z0 p π 2 1 k2 π K 1 k2 = − dθ . p − Z0 s1−k2sin2θ ≤ 2 Wegettheestimate8LK ab 4 πwhichpreventsthegeodesicoftypeC to | |≥ − behomotopictoanelementofΠ. Likewise,thelengthofclosedgeodesicsoftype p A homotopic to elements of Π are less than 4 π < 2π. From Proposition1.1, − √2 the Conley-Zehnder index of these geodesics in the trivialisation (h ,...,h ) of 1 4 ξ is1. Indeed,theConley-Zehnderindexoftherotationblockis1bydefinition, 1 t while the (Bott-)Conley-Zehnder index of the unipotent block U = 0 1 (cid:20) (cid:21) vanishes, see the thesis of F. Bourgeois. Indeed, this block is solution of the 1 0 0 1 differential equation U˙ =S U with U(0)=I, S = and = . J 0 0 J 1 0 (cid:20) (cid:21) (cid:20)− (cid:21) By definition, the (Bott-)Conley-Zehnder index of this block is the Conley- Zehnder index of the solution of the differential equation V˙ = (S δI) V − J with V(0)=I and 0<δ 1, which is hyperbolic. The result follows from [27, ≪ Theorem 3.1], [5, Proposition 1.7.3] which identifies this Conley-Zehnder index to the Morse-Bott index. The Sol-metrics given by Proposition 1.3 are metrics for which the area of the fibers of the map p: L B is not too large compared to the length of B. −→ In fact, without changing the length of B, it is possible to expand or contract the fibers of p as much as we want, keeping the Sol feature of the metric. This observation was crucial in the proof of Proposition 1.3 and will be very useful in Section 2. 2 Sol Lagrangian submanifolds in uniruled symplectic manifolds 2.1 Statement of the results Definition 2.1. We say that a closed symplectic manifold (X,ω) is uniruled iff it has a non vanishing genus 0 Gromov-Witten invariant of the form [pt] ;[pt],ωk X , whereA H (X;Z), k 2,and[pt] representsthe Poincaré h k iA ∈ 2 ≥ k dualofthepointclassinthemodulispace ofgenus0stablecurveswith 0,k+1 M k+1 marked points. This Definition 2.1 differs from [10, Definition 4.5] where ωk is replaced by anyfinitesetofdifferentialformsonX. Nevertheless,from[16,Theorem4.2.10], complex projective uniruled manifolds are all symplectically uniruled in the sense of Definition 2.1. The advantage for us to restrict ourselves to Defini- tion 2.1 is that for every Lagrangian submanifold L of X, the form ω has a Poincarédual representative disjoint from L. Our goal is to prove the following results. Theorem2.1. Let(X,ω)beacloseduniruledsymplecticmanifoldofdimension six. For any Lagrangian submanifold L of X homeomorphic to the suspension 10