Closed SL(3,C) -structures on nilmanifolds
CCLOSED SL (3 , C ) -STRUCTURES ON NILMANIFOLDS ANNA FINO AND FRANCESCA SALVATORE
Abstract.
A closed SL(3 , C )-structure on an oriented 6-manifold is given bya closed definite 3-form ρ . In this paper we study two special types of closedSL(3 , C )-structures. First we consider closed SL(3 , C )-structures ρ which are meanconvex, i.e. such that d ( J ρ ρ ) is a semi-positive (2 , J ρ denotes theinduced almost complex structure. This notion was introduced by Donaldson inrelation to G -manifolds with boundary and as a generalization of nearly-K¨ahlerstructures. In particular, we classify nilmanifolds which carry an invariant meanconvex closed SL(3 , C )-structure. A classification of nilmanifolds admitting invari-ant mean convex half-flat SU(3)-structures is also given and the behaviour withrespect to the Hitchin flow equations is studied. Then we examine closed SL(3 , C )-structures which are tamed by a symplectic form Ω, i.e. such that Ω( X, J ρ X ) > X . In particular, we show that if a solvmanifoldadmits an invariant tamed closed SL(3 , C )-structure, then it has also an invariantsymplectic half-flat SU(3)-structure. introduction An SL(3 , C )-structure on an oriented manifold of real dimension 6 is defined by adefinite real 3-form ρ , i.e. by a stable 3-form ρ inducing an almost complex structure J ρ (see [27, 38]). We shall say that the SL(3 , C )-structure ρ is closed if dρ = 0. Asremarked in [13], closed SL(3 , C )-structures obey an h -principle, since any hypersur-face in R acquires a closed SL(3 , C )–structure.A special case of closed SL(3 , C )-structure is given by a closed SU(3) -structure ,i.e. by the data of an almost Hermitian structure (
J, g, ω ) and a (3 , i ∧ Ψ = 23 ω , d (Re(Ψ)) = 0 . Indeed the 3-form ρ = Re(Ψ) defines a closed SL(3 , C )-structure such that J ρ = J .As shown in [13], a closed SL(3 , C )-structure always determines a real 3-formˆ ρ := J ρ ρ such that d ˆ ρ is of type (2 ,
2) with respect to J ρ . Moreover ˆ ρ is the imaginarypart of a complex (3 , , C )-structure is meanconvex if the (2 , d ˆ ρ is semi-positive. Note that J ρ is integrable if and onlyif d ( J ρ ρ ) = 0 . A special class of mean convex closed SL(3 , C )-structures is given by nearly-K¨ahler structures. Indeed, a nearly-K¨ahler structure can be defined as anSU(3)-structure ( ω, Ψ) satisfying the following conditions: dω = − ν Re(Ψ) , d (Im(Ψ)) = ν ω , where ν ∈ R − { } and therefore, up to a change of sign of Re(Ψ), we can suppose ν >
0. The nearly-K¨ahler condition forces the induced Riemannian metric g tobe Einstein and, up to now, very few examples of manifolds admitting complete Mathematics Subject Classification.
Key words and phrases.
SL(3 , C )-structures, mean convex, half-flat SU(3)-structures, symplecticforms. a r X i v : . [ m a t h . DG ] O c t ANNA FINO AND FRANCESCA SALVATORE nearly-K¨ahler structures are known [5, 21, 24, 25, 36, 37]. More in general, an SU(3)-structure ( ω, Ψ) such that d (Re(Ψ)) = 0 and d ( ω ∧ ω ) = 0 is called half-flat , see forinstance [2, 4, 6, 8, 11, 18, 22, 28, 29] for general results on this types of structures. Inparticular, every oriented hypersurface of a Riemannian 7-manifold with holonomyin G is naturally endowed with a half-flat SU(3)-structure and, conversely, using theHitchin flow equations, a 6-manifold with a real analytic half-flat SU(3)-structurecan be realized as a hypersurface of a 7-manifold with holonomy in G [4, 28].Nilmanifolds, i.e. compact quotients Γ \ G of connected, simply connected, nilpo-tent Lie groups G by a lattice Γ, provide a large class of compact 6-manifolds admit-ting invariant closed SL(3 , C )-structures [6, 7, 8, 10, 19], where by invariant we meaninduced by a left-invariant one on the nilpotent Lie group G . Note that nilmanifoldscannot admit invariant nearly K¨ahler structures, since by [35] the Ricci tensor ofa left-invariant metric on a non-abelian nilpotent Lie group always has a strictlynegative direction and a strictly positive direction.Since a nilmanifold is parallelizable, its Stiefel-Whitney numbers and Pontryaginnumbers are all zero, hence by well-known theorems of Thom and Wall, it boundsorientably, i.e. it is diffeomorphic to the boundary of a compact connected manifold N . So it would be a natural question to see if, given a 6-dimensional nilmanifoldendowed with an invariant mean convex closed SL(3 , C )-structure ρ , there existson N a closed G -structure with boundary value an “enhancement” of ρ (see [13,Section 3.1] for more details).In this paper we classify 6-dimensional nilpotent Lie algebras admitting meanconvex closed SL(3 , C )-structures (Theorem 4.1). According to [23, 32] there are 34isomorphism classes of 6-dimensional real nilpotent Lie algebras g i , i = 1 , . . . , , listed in Table 1. We show that, if M = Γ \ G is a nilmanifold such that the Liealgebra g of G is isomorphic to any of six Lie algebras g i , i = 1 , , , , ,
34, then M does not admit any invariant mean convex closed SL(3 , C )-structures. If g is notisomorphic to any of those Lie algebras, M admits an invariant mean convex closedSU(3)-structure. Using the classification of half-flat nilpotent Lie algebras (see [8]),we prove that 16 of the 24 isomorphism classes admit a mean convex half-flat SU(3)-structure (Theorem 5.2). An explicit mean convex closed (half-flat) SU(3)-structurefor every Lie algebra is given in Table 2. Moreover, in Section 6 we show that themean convex condition is preserved by the Hitchin flow equations in some specialcases. More generally, since in our examples the property is preserved for smalltimes, it would be interesting to determine if this is always the case.Given a closed SL(3 , C )-structure ρ on a 6-manifold, another natural condition tostudy is the existence of a symplectic form Ω taming J ρ , i.e. such that Ω( X, J ρ X ) > X . This is equivalent to the positivity in the standardsense of the (1 , , of Ω. We shall say that a closed SL(3 , C )-structure ρ is tamed if there exists a symplectic form Ω such that Ω , > , C )-structure on a compact 6-manifold cannotbe tamed by any symplectic form. If we remove the assumption of mean convexity,examples of tamed closed SL(3 , C )-structures are given by symplectic half-flat struc-tures ( ω, Ψ), i.e., by half-flat SU(3)-structures ( ω, Ψ) with dω = 0. In this case ρ = Re(Ψ) is tamed by the symplectic form ω , since ω is of type (1 ,
1) with respectto J ρ . In [10], nilmanifolds admitting invariant symplectic half-flat structures wereclassified. Later, this classification was generalized to solvmanifolds, i.e. to compactquotients Γ \ G of connected, simply connected, solvable Lie groups G by latticesΓ (for more details, see [17]). In this paper we prove that, if a solvmanifold Γ \ G admits an invariant tamed closed SL(3 , C )-structure, then Γ \ G also has an invariant LOSED SL(3 , C )-STRUCTURES ON NILMANIFOLDS 3 symplectic half-flat structure (Theorem 7.1). Explicit examples of closed SL(3 , C )-structures tamed by a symplectic form Ω such that d Ω , (cid:54) = 0 are provided. Theseexamples provide new examples of closed G -structures on the product M × S , where M = Γ \ G is a 6-dimensional solvmanifold endowed with an invariant tamed closedSL(3 , C )-structure. It would be interesting to see if there exist compact manifoldswhich have tamed closed SL(3 , C )-structures but do not admit any symplectic half-flat structures.The paper is organized as follows. In Section 2 we review the general theory ofsemi-positive ( p, p )-forms focusing on the case p = 2. In Section 3 we study theintrinsic torsion of closed SU(3)-structures in relation to the mean convex condition.Section 4 contains the classification of nilmanifolds admitting an invariant meanconvex closed SL(3 , C )-structure. In Section 5 we focus on mean convex half-flatSU(3)-structures and, in Section 6, we study their behaviour under the Hitchin flowequations. Finally, in Section 7 we classify solvmanifolds admitting invariant tamedclosed SL(3 , C )-structures (Theorem 7.1). Acknowledgements.
The authors are supported by the Project PRIN 2017 “Realand complex manifolds: Topology, Geometry and Holomorphic Dynamics” and byG.N.S.A.G.A. of I.N.d.A.M. The authors would like to thank Simon Chiossi andAlberto Raffero for useful discussions and comments.2.
Preliminaries on semi-positive differential forms
In this section we review the definition and main results regarding semi-positive( p, p )-forms on complex vector spaces. We are interested in the case where thecomplex vector space is the tangent space to an almost complex manifold M but,in this section, we emphasize considerations involving only linear algebra. For moredetails we refer for instance to [12, 26].Let V be a complex vector space of complex dimension n and ( z , . . . , z n ) becoordinates on V . Note that V can be considered also as a real vector space ofdimension 2 n endowed by the complex structure J given by the multiplication by i .We denote by (cid:16) ∂∂z , . . . , ∂∂z n (cid:17) the corresponding basis of V and by ( dz , . . . , dz n ) itsdual basis of V ∗ .Consider the exterior algebraΛ V ∗ ⊗ C = (cid:77) Λ p,q V ∗ , where Λ p,q V ∗ is a shorthand for Λ p V ∗ ⊗ Λ q V ∗ . V has a canonical orientation, given by the ( n, n )-form τ ( z ) := 12 n idz ∧ dz ∧ . . . ∧ idz n ∧ dz n = dx ∧ dy ∧ dx n . . . ∧ dy n , (2.1)where z j = x j + iy j . In particular, an almost complex manifold always has a canonicalorientation.We shall say that a ( p, p )-form γ is real if γ = γ . One can introduce a naturalnotion of positivity for real ( p, p )-forms. Definition 2.1.
A real ( p, p )-form γ ∈ Λ p,p V ∗ is said to be semi-positive if, for all α j of Λ , V ∗ , 1 ≤ j ≤ n − p , γ ∧ i α ∧ α ∧ . . . ∧ i α n − p ∧ α n − p = λτ ( z ) , λ ≥ . We shall focus on the case n = 3 and using the results in [12] we shall provideequivalent definitions for semi-positive real forms of type (1 ,
1) and (2 , ANNA FINO AND FRANCESCA SALVATORE
Proposition 2.2.
Let α = i (cid:80) j,k a jk dz j ∧ dz k be a real (1 , -form on V . Then thefollowing are equivalent:(i) α is semi-positive,(ii) the Hermitian matrix of coefficients ( a jk ) is positive semi-definite,(iii) there exist coordinates ( w , . . . w n ) on V such that α = i n (cid:88) k =1 ˜ a kk dw k ∧ dw k , with ˜ a kk ≥ , ∀ k = 1 , . . . n. Proposition 2.3. If α , α are semi-positive real (1 , -forms, then α ∧ α is semi-positive. Definition 2.4.
A real (1 , α = i (cid:80) j,k a jk dz j ∧ dz k is positive if the matrixof coefficients ( a jk ) is positive definite.Now, for n = 3, we want to characterize the concept of semi-positivity for real(2 , γ be a real (2 , V . We can write γ = − (cid:88) i Let γ (cid:54) = 0 be a real (2 , -form on V . Then the following areequivalent:(i) γ is semi-positive,(ii) γ ∧ α > for every positive real (1 , -form α , i.e. γ ∧ α = λτ ( z ) where λ > ,(iii) the associated (1 , -form β is positive semi-definite. In particular, we can give the following Definition 2.6. A real (2 , γ on V is positive if the associated (1 , β is positive.As shown in [26, Theorem 1.2], a real (2 , γ is always diagonalizable, i.e.there exist coordinates ( w , w , w ) of V such that γ = − (cid:88) i 0, for every i < k . Inparticular, the diagonal matrix ( β mn ) associated to γ in these coordinates is positivesemi-definite. Moreover, γ is positive if and only if γ iikk > 0, for every i < k .3. Mean convexity and intrinsic torsion of SU (3) -structures In this section we study the mean convex property in the context of closed SU(3)-structures and provide necessary and sufficient conditions in terms of the intrinsictorsion of the SU(3)-structure.An SL(3 , C )-structure on a 6-manifold M is a reduction to SL(3 , C ) of the framebundle of M which is given by a definite real 3-form ρ , i.e. by a stable 3-form inducingan almost complex structure J ρ . We recall that a 3-form ρ on a real 6-dimensionalspace V is stable if its orbit under the action of GL( V ) is open. If we fix a volumeform ν ∈ Λ V ∗ and denote by A : Λ V ∗ → V ⊗ Λ V ∗ the canonical isomorphism induced by the wedge product ∧ : V ∗ ⊗ Λ V ∗ → Λ V ∗ , we can consider the map K ρ : V → V ⊗ Λ V ∗ , v (cid:55)→ A (( i v ρ ) ∧ ρ ) . A 3-form ρ on V is stable if and only if λ ( ρ ) = Tr( K ρ ) (cid:54) = 0 (see [27, 38] for furtherdetails). When λ ( ρ ) < 0, the 3-form ρ induces an almost complex structure J ρ := − (cid:112) − λ ( ρ ) K ρ and we shall say that ρ is definite . A simple computation shows that J ρ does notchange if ρ is rescaled by a non-zero real constant, i.e., J ρ = J sρ for every s ∈ R −{ } .Moreover, defining ˆ ρ := J ρ ρ , we have that ρ + i ˆ ρ is a complex (3 , J ρ .We shall say that an SL(3 , C )-structure ρ is closed if dρ = 0. According to [13], d ˆ ρ is a real (2 , Definition 3.1. Let ρ be a closed SL(3 , C )-structure on M . We shall say that ρ is mean convex (resp. strictly mean convex) if d ˆ ρ , pointwise, is a non-zero semi-positive(resp. positive) (2 , , C )-structure ρ on a 6-manifold M , if there exists a non-degeneratepositive (1 , ω on M such that ρ ∧ ˆ ρ = ω , then the pair ( ω, Ψ), whereΨ = ρ + iJ ρ ˆ ρ , defines an SU(3)-structure and the associated almost J ρ -Hermitianmetric g is given by g ( · , · ) := ω ( · , J ρ · ). Since Ψ is completely determined by its realpart ρ , we shall denote an SU(3)-structure simply by the pair ( ω, ρ ).In this case, at any point p ∈ M , one can always find a coframe (cid:0) f , . . . , f (cid:1) ,called adapted basis for the SU(3)-structure ( ω, ρ ), such that ω = f + f + f , ρ = f − f − f − f . (3.1)Here f ij ··· k stands for the wedge product f i ∧ f j ∧ · · · ∧ f k .We shall say that the SU(3)-structure ( ω, ρ ) is closed if dρ = 0 and in a similarway we can introduce the following Definition 3.2. A closed SU(3)-structure ( ω, ρ ) on a 6-manifold M is (strictly)mean convex if the SL(3 , C )-structure ρ is (strictly) mean convex. ANNA FINO AND FRANCESCA SALVATORE The intrinsic torsion of the SU(3)-structure ( ω, ρ ) can be identified with the pair( ∇ ω, ∇ Ψ), where ∇ is the Levi-Civita connection of g , and it is a section of the vectorbundle T ∗ M ⊗ su (3) ⊥ , where su (3) ⊥ ⊂ so (6) is the orthogonal complement of su (3)with respect to the Killing Cartan form B of so (6). Moreover, by [6, Theorem 1.1]the intrinsic torsion of ( ω, ρ ) is completely determined by dω , dρ and d ˆ ρ . Indeed,there exist unique differential forms ν , π ∈ C ∞ ( M ), ν , π ∈ Λ ( M ), ν , π ∈ [Λ , M ] , ν ∈ (cid:74) Λ , M (cid:75) such that dω = − ν ρ + 32 π ˆ ρ + ν ∧ ω + ν ,dρ = π ω + π ∧ ρ − π ∧ ω,d ˆ ρ = ν ω − ν ∧ ω + J π ∧ ρ, (3.2)where [Λ , M ] := { α ∈ [Λ , M ] | α ∧ ω = 0 } is the space of primitive real (1 , (cid:74) Λ , M (cid:75) := { η ∈ (cid:74) Λ , M (cid:75) | η ∧ ω = 0 } is the space of primitive real(2 , 1) + (1 , ν i , π j are called torsion forms of the SU(3)-structureand they completely determine its intrinsic torsion, which vanishes if and only if allthe torsion forms vanish identically.If ρ is closed, as a consequence of (3.2), we have d ˆ ρ = θ ∧ ω, where θ is the(1 , θ := ν ω − ν .We recall that, given a real (1 , α , the trace Tr( α ) of α is given by 3 α ∧ ω =Tr( α ) ω . Then, in terms of ν and the (1 , θ , we can prove the following Proposition 3.3. Let ( ω, ρ ) be a closed SU(3) -structure on M . Then(i) if ( ω, ρ ) is mean convex, then the torsion form ν is strictly positive andthe (1 , -form θ is not negative (semi-)definite. Moreover, its trace Tr( θ ) isstrictly positive,(ii) if θ is semi-positive, then the SU(3) -structure is mean convex.Proof. Let us assume that ( ω, ρ ) is a mean convex closed SU(3)-structure on M . By(3.2) we have d ˆ ρ = θ ∧ ω . Now, Proposition 2.5 implies d ˆ ρ ∧ α > , α . Then (i) follows by choosing α = ω ; indeed d ˆ ρ ∧ ω = ν ω , since ν ∈ [Λ , M ]. In particular Tr( θ ) = 3 ν > 0. (ii) follows from Proposition 2.3. (cid:3) A closed SU(3)-structure ( ω, ρ ) is called half-flat if dω = 0 and we shall refer toit simply as a half-flat structure. Half-flat structures are strictly related to torsionfree G -structures. We recall that a G -structure on a 7-manifold N is characterizedby the existence of a 3-form ϕ inducing a Riemannian metric g ϕ and a volume form dV ϕ given by g ϕ ( X, Y ) dV ϕ = 16 ι X ϕ ∧ ι Y ϕ ∧ ϕ, X, Y ∈ Γ( T M ) . By [16], the G -structure ϕ is torsion free , i.e. ϕ is parallel with respect to theLevi-Civita connection of g ϕ , if and only if ϕ is closed and co-closed, or equivalentlyif the holonomy group Hol( g ϕ ) is contained in G . A torsion free G -structure ϕ on N induces on each oriented hypersurface ι : M (cid:44) → N a natural half-flat structure( ω, ρ ) given by ρ = ι ∗ ϕ, ω = 2 ι ∗ ( ∗ ϕ ϕ ) . Conversely, in [28], the so-called Hitchin flow equations (cid:40) ∂∂t ρ ( t ) = dω ( t ) , ∂∂t ω ( t ) ∧ ω ( t ) = − d ˆ ρ ( t ) , (3.3) LOSED SL(3 , C )-STRUCTURES ON NILMANIFOLDS 7 have been introduced, proving that every compact real analytic half-flat manifold( M, ω, ρ ) can be embedded isometrically as a hypersurface in a 7-manifold N with atorsion free G -structure. Moreover, the intrinsic torsion of the half-flat structure canbe identified with the second fundamental form B ∈ Γ( S T ∗ M ) of M with respectto a fixed unit normal vector field ξ . As in [13], with respect to J ρ , we can write B = B , + B C , where B , is the real part of a Hermitian form and B C is the real partof a complex quadratic form. If we denote by β , = B , ( J ρ · , · ) the corresponding(1 , M , we have β , ∧ ω = d ˆ ρ , from which it follows that, if ( ω, ρ ) is meanconvex, then the mean curvature µ given explicitly by µρ ∧ ˆ ρ = d ˆ ρ ∧ ω is positivewith respect to the normal direction (for more details see [13, Prop. 1]). Moreover,since the wedge product with ω defines an injective map on 2-forms, comparing thiswith (3.2) yields θ = 2 β , . Then, by Proposition 3.3, if B , defines a positivesemi-definite Hermitian product, then the half-flat structure ( ω, ρ ) is mean convex.Special types of half-flat structures ( ω, ρ ) are called coupled , when dω = − ν ρ ,and double , when d ˆ ρ = ν ω . Notice that, by Proposition 3.3, double structures ( ω, ρ ) are trivially mean convexas long as ν > 0. However, it is straightforward to check that, if ( ω, ρ ) is a doublestructure such that ν < 0, then ( ω, − ρ ) is mean convex.In [7, Theorem 4.11], a classification of 6-dimensional nilpotent Lie algebras en-dowed with a double structure was given. Other examples of double structures on S × S were found in [31, 41].For a general Lie algebra we can show the following Proposition 3.4. If a Lie algebra g has a closed strictly mean convex SL(3 , C ) -structure, then g admits a double structure.Proof. Let ρ be a closed strictly mean convex SL(3 , C )-structure on g and denoteˆ ρ = J ρ ρ as usual. Then d ˆ ρ is a positive (2 , , α such that d ˆ ρ = α . Moreover, since α is positive with respectto J ρ , α is a positive multiple of the volume form ρ ∧ ˆ ρ . Since J ρ does not changefor a non-zero rescaling of ρ , this implies that there exists b (cid:54) = 0 such that ( bρ, α ) isa double structure on g . (cid:3) As a consequence, the classification of nilpotent Lie algebras admitting closedstrictly mean convex SL(3 , C )-structures reduces to Theorem 4.11 in [7]. Therefore,in the next two sections we weaken the condition asking for the existence of closed(non-strictly) mean convex SL(3 , C )-structures.4. Mean convex closed SL(3 , C ) -structures on nilmanifolds We recall that a nilmanifold M = Γ \ G is a compact quotient of a connected,simply connected, nilpotent Lie group G by a lattice Γ. We shall say that an SL(3 , C )-structure ρ (resp. SU(3)-structure ( ω, ρ )) is invariant if it is induced by a left-invariant one on the nilpotent Lie group G . Therefore, the study of these types ofstructure is equivalent to the study of SL(3 , C )-structures (resp. SU(3)-structures)on the Lie algebra g of G and we can work at the level of nilpotent Lie algebras.Six-dimensional nilpotent Lie algebras have been classified in [23, 32]. Up toisomorphism, they are 34, including the abelian algebra (see Table 1 for the list).Using this classification we can prove Theorem 4.1. Let M = Γ \ G be a -dimensional nilmanifold. If the Lie algebra g of G is isomorphic to any of the six Lie algebras ANNA FINO AND FRANCESCA SALVATORE g = (0 , , e , e , e + e , e − e ) , g = (0 , , e , e , e , e − e ) , g = (0 , , e , e , e + e , e + e ) , g = (0 , , , e , e − e , e + e ) , g = (0 , , , e , e , e + e ) , g = (0 , , , , , ,then M does not have any invariant mean convex closed SL(3 , C ) -structures. More-over, if the Lie algebra g of G is not isomorphic to any of the Lie algebras in theprevious list, M admits an invariant mean convex closed SU(3) -structure. An explicitmean convex closed SU(3) -structure for every Lie algebra g i , i / ∈ { , , , , , } ,is given in Table 2.Proof. Let g be the Lie algebra of G . Every invariant SL(3 , C )-structure on M is determined by an SL(3 , C )-structure on g and vice versa. First note that thepossibility that g is abelian is precluded by Definition 3.1. Then, in order to provethe first part of the theorem, we first show the non existence result for the five Liealgebras g , g , g , g and g . For any of these Lie algebras, let us consider a genericclosed 3-form ρ = (cid:88) i 0. Then ρ induces an almostcomplex structure J ρ and we may ask if the induced (2 , d ˆ ρ is semi-positive.Notice that the 1-forms ζ k = e k − iJ ρ e k , for k = 1 , . . . , 6, generate the space Λ , g ∗ i of (1 , J ρ on g i , i = 1 , , , , . Here we are using theconvention J ρ α ( v ) = α ( J ρ v ) for any α ∈ g ∗ , v ∈ g . So, for any closed definite 3-form ρ , we extract a basis ( ξ , ξ , ξ ) for Λ , g ∗ i , where ξ j = ζ k j for some k j ∈ { , . . . , } and j = 1 , , 3. Then, ( ξ , ξ , ξ , ξ , ξ , ξ ) is a complex basis for g ∗ i ⊗ C and we canwrite d ˆ ρ in this new basis as d ˆ ρ = − (cid:88) i 0, thesystem (4.2) in the variables p ijk has no solutions.Let us see this explicitly for g i , i = 1 , 2. By a direct computation, for the genericclosed 3-form ρ on g we have λ ( ρ ) = (cid:2) ( p + 2 p ) p + p p + p (cid:3) +4 p p ( p − p p + p p )and, for the generic closed 3-form ρ on g , we get λ ( ρ ) = (cid:0) p + p p + 2 p p (cid:1) + 4 p p ( − p p + p p + p p ) . Notice that, if at least one between p and p is equal to zero, then λ ( ρ ) ≥ 0. So letus assume that both p , p are non-zero. Then ( e , J ρ e , e , J ρ e , e , J ρ e ) definesa basis of g ∗ i , for i = 1 , 2, hence ( ξ = e − iJ ρ e , ξ = e − iJ ρ e , ξ = e − iJ ρ e ) is abasis of (1 , g i , i = 1 , 2. By a direct computation, it can be shown that inthese cases the matrix coefficient β vanishes and so β β − | β | = −| β | ≤ β = 0 implies λ ( ρ ) = 0 which is a contradiction.By a very similar discussion, we may discard cases g , g and g as well. In orderto prove the second part of the theorem, we construct an explicit mean convex closedSU(3)-structure ( ω, ρ ) on the remaining nilpotent Lie algebras (see Table 2). (cid:3) Mean convex half-flat structures on nilmanifolds In [8], a classification up to isomorphism of 6-dimensional real nilpotent Lie alge-bras admitting half-flat structures was given. The non-abelian ones are twenty threeand they are listed in Table 1. So, in order to classify nilpotent Lie algebras admittinga mean convex half-flat structure, we restrict our attention to this list. An explicit ex-ample of mean convex half-flat structure on g i , i = 6 , , , , , , , , , , , , , , , 33, is already given in Table 2. Therefore, we only need to provenon-existence of mean convex half-flat structures on the remaining Lie algebras g i , i = 4 , , , , , , 27. By Theorem 4.1, we may immediately exclude the Lie al-gebras g i , i = 4 , , 12, since mean convex half-flat structures are in particular meanconvex closed SL(3 , C )-structures.For the remaining Lie algebras g i , i = 11 , , , 27, whose first Betti number is3 or 4, we first collect some necessary conditions to the existence of mean convexclosed SU(3)-structures ( ω, ρ ) in terms of a filtration of J ρ -invariant subspaces U i of g ∗ , and then, by working in an SU(3)-adapted basis, we exhibit further obstructions.Let us start by defining the filtration { U i } as in [7]. Let ( ω, ρ ) be an SU(3)-structure on a 6-dimensional nilpotent Lie algebra g and let ( g, J ρ ) be the inducedalmost Hermitian structure on g . By nilpotency there exists a basis (cid:0) α , . . . , α (cid:1) of g ∗ such that, if we denote V j := (cid:10) α , . . . , α j (cid:11) , then dV j ⊂ Λ V j − and, by construction,0 ⊂ V ⊂ . . . ⊂ V ⊂ V = g ∗ . We notice that the basis ( e i ) whose correspondingstructure equations are given in Table 1 satisfies the previous conditions and V i =ker d when b ( g ) = i . In the following, we consider V i = (cid:10) e , . . . , e i (cid:11) . As in [7], let U j := V j ∩ J ρ V j be the maximal J ρ -invariant subspace of V j for each j . Then, since J ρ is an automorphism of the vector space g , a simple dimensional computation showsthat dim R U , dim R U ∈ { , } , dim R U ∈ { , } and dim R U = 4. Note that thefiltration { U i } depends on V i and the almost complex structure J ρ .We can prove the following Lemma 5.1. Let ρ be a mean convex closed SL(3 , C ) -structure on a nilpotent Liealgebra g . If g is isomorphic to g = (0 , , , e , e , e + e + e ) or g = (0 , , , e , e , e + e ) , then U = U . If g is isomorphic to g = (0 , , , e , e , e + e ) or g = (0 , , , , e , e + e ) , then dim R U = 2 , or equivalently (cid:10) e , e (cid:11) is J ρ -invariant. Moreover, on g , up toisomorphism, we also have dim R U = 4 .Proof. On each Lie algebra g i , i = 11 , , , 27, we consider the generic closed 3-form ρ = (cid:88) i Theorem 5.2. A nilmanifold M = Γ \ G has an invariant mean convex half-flatstructure if and only if the Lie algebra g of G is isomorphic to any of the Lie algebras g i , i = 6 , , , , , , , , , , , , , , , , as listed in Table 1. LOSED SL(3 , C )-STRUCTURES ON NILMANIFOLDS 11 Proof. Starting from the classification of half-flat nilpotent Lie algebras given in [8],we divide the discussion depending on the first Betti number b of g .When b ( g ) = 2, the claim follows directly by Theorem 4.1. In particular wehave seen that g cannot admit mean convex closed SL(3 , C )-structures and, for theremaining Lie algebras g , g and g from Table 1, we provide an explicit example inTable 2 on the respective Lie algebras. We note that these examples on g , g and g are double.Analogously, when b ( g ) = 3, an explicit example of mean convex half-flat struc-ture on g i , i = 10 , , , , , 24, is given in Table 2. By Theorem 4.1, we mayexclude the existence of mean convex half-flat structures on g and g . For theremaining Lie algebras g i , i = 11 , , 21, let ( ω, ρ ) be a mean convex half-flat struc-ture on g i . Then, by Lemma 5.1, with respect to the fixed nilpotent filtration V i = (cid:10) e , . . . , e i (cid:11) , we may assume dim R U = 2. Using this and the information on U we collected in Lemma 5.1, we shall show that on the three Lie algebras there ex-ists an adapted basis ( f i ) with dual basis ( f i ) such that df = df = 0 and f ∈ ξ ( g i ),where by ξ ( g i ) we denote the center of g i .To see this, let us consider the case of g , first. Then we may assume dim R U = 4.This occurs if and only if V = J ρ V . In particular, we may choose a g -orthonormalbasis (cid:0) f , f (cid:1) of U such that J ρ f = − f , take f , f ∈ U ⊥ ∩ U of unit normsuch that J ρ f = − f , and complete it to a basis for g ∗ by choosing f ∈ U ⊥ ∩ V and f ∈ U ⊥ ∩ J ρ V of unit norm such that J ρ f = − f . Then, by construction, (cid:0) f , . . . , f (cid:1) is an adapted basis for the SU(3)-structure ( ω, ρ ). In particular, since V = (cid:10) f , f , f , f , f (cid:11) , the inclusion dV j ⊂ Λ ( V j − ) implies f ∈ ξ ( g ). There-fore, since f , f ∈ V = ker d , we have df = df = 0.Now we consider g and g . By Lemma 5.1, we can assume dim R U = 2 forboth Lie algebras. As shown in [7], since U , V ⊂ V , we have dim R ( U ∩ V ) ≥ (cid:0) f , f (cid:1) to be a unitary basis of U with f ∈ V . Then, since U ⊂ V = ker d , we may suppose df = df = 0. Analogously, since dim R ( V ∩ J ρ V ) ≥ U ∩ V = V ∩ J ρ V ∩ V = V ∩ J ρ V , then dim R ( U ∩ V ) ≥ 3, from whichdim R ( U ∩ V ∩ U ⊥ ) ≥ (cid:0) f , f (cid:1) to be a unitary basisof U ⊥ ∩ U with f ∈ V . Finally, since dim R ( U ⊥ ∩ V ) ≥ 1, we may take a unitarybasis (cid:0) f , f (cid:1) of U ⊥ with f ∈ V . By construction, (cid:0) f , f , . . . , f (cid:1) is an adaptedbasis for ( ω, ρ ). In particular, since U ⊂ V , we also have V = (cid:10) f , f , f , f , f (cid:11) ,which implies f ∈ ξ ( g i ). for i = 11 , 14. This proves our claim.Now, we shall show that the three Lie algebras g i , i = 11 , , 21, do not admitany mean convex half-flat structures. By contradiction, let us suppose there exists anilpotent Lie algebra g endowed with a mean convex half-flat structure ( ω, ρ ) whichis isomorphic to g , g or g . By the previous discussion, without loss of generality,we may assume that there exists an adapted basis ( f i ), i.e. satisfying ω = f + f + f , ρ = f − f − f − f , ˆ ρ = f + f + f − f , and such that df = df = 0, f ∈ ξ ( g ). In particular, g has structure equations df = df = 0 , df k = − (cid:88) i 6, and that( ω, ρ ) is half-flat, we can show by a direct computation that, if c (cid:54) = 0, then the Jacobi identities d f i = 0, i = 3 , . . . , 6, are equivalent to the conditions c = c = c = c = c = c = c = c = c = 0 , which imply b ( g ) ≥ 4, so we can exclude this case. Then we must have c = 0.Let us assume c (cid:54) = 0. Again a straightforward computation shows that d f = 0implies c = c = c = 0 , c = − c , c = − c − c . Now let us look at the mean convex condition. Since we are working in the adaptedbasis ( f i ), using (4.1) we obtain that the matrix ( β mn ) associated to d ˆ ρ , with respectto the basis ( ξ = f + if , ξ = f + if , ξ = f + if ), is given by c − i ( c + c )0 c + i ( c + c ) − c − c + c − c . Therefore d ˆ ρ is semipositive if and only if c = 0, c = − c and − c − c + c − c > 0. In particular, c = 0 and c = − c imply that the Jacobi identitieshold if and only if c = c = 0. However, this also implies df = df = 0 so that b ( g ) ≥ c = c = 0 and, asa consequence, df = − c f − ( c + c ) f − c f − c f − c f − c f ,df = − c f − c f − c f − c f − ( c + c ) f − c f ,df = − ( c + c + c ) f − c f + ( c + c ) f − c f − c f + ( c + c + c ) f ,df = − c f − c f − c f − c f − c f − ( c − c ) f . (5.1)In particular, f is a non-exact 2-form belonging to Λ (ker d ) such that f ∧ d g ∗ = 0.On the other hand, a simple computation shows that for any Lie algebra g i , for i = 11 , , 21, a 2-form α ∈ Λ (ker d ) such that α ∧ d g ∗ i = 0 is necessarily exact, sowe get a contradiction. This concludes the non-existence part of the proof in thecase b = 3.Now we consider the remaining case b ( g ) ≥ 4. An explicit example of meanconvex half-flat structure on g i , i = 25 , , , , , , 33, is given in Table 2.Then, we only need to prove the non-existence of mean convex half-flat structureson g .Let ( ω, ρ ) be a mean convex half-flat structure on g . We claim that on g there exists an adapted basis ( f i ) such that df = df = df = 0 and f ∈ ξ ( g ).By Lemma 5.1, we can assume U = U with dim R U = 2. We recall that U has dimension 2 or 4. Let us suppose dim R U = 4, first. We note that in thiscase the existence of an adapted basis ( f i ) for ( ω, ρ ) such that f ∈ ξ ( g ) and V = U = (cid:10) f , f , f , f (cid:11) follows from the previous discussion on g , where weonly used dim R U = 2 and dim R U = 4. In particular, since V = ker d on g , inthis case we also have df = df = df = df = 0. When dim R U = 2 instead, since U = U = U , the discussion is the same as for g and g , where we only used U = U to find an adapted basis such that df = df = 0 and f lying in the center.In particular, since by construction f , f , f ∈ V , on g we also have df = 0, since V = ker d . This proves our claim on g . Now, using this claim we shall show that g does not admit any mean convex half-flat structures. Like in the previous cases,by contradiction, let us suppose there exists a nilpotent Lie algebra g isomorphic to g admitting a mean convex half-flat structure ( ω, ρ ). Then we may assume that LOSED SL(3 , C )-STRUCTURES ON NILMANIFOLDS 13 there exists on g an adapted basis ( f i ) for ( ω, ρ ) such that df = df = df = 0 and V = (cid:10) f , f , f , f , f (cid:11) , so that f ∈ ξ ( g ). Then df k = − (cid:88) i 6, if and only if c = c = c = c = c = 0 , from which itfollows that b ( g ) = 4 holds if and only if c = − c c , c = c c − c c c . Then g must have structure equations df = df = df = 0 ,df = c f − c f − c f ,df = − c c c f + ( c ) c f + c f ,df = − c f − c c − c c c f − c f − c f . (5.3) Note that, by (5.3), g has the same central and derived series as g and, if c = 0, g is almost abelian, so it cannot be isomorphic to g . Thus we can suppose c (cid:54) = 0.By [8], a 6-dimensional 3-step nilpotent Lie algebra having b = 4 and admitting ahalf-flat structure must be isomorphic to either g or g . In addition, b ( g ) = 6,while b ( g ) = 7. We shall show that we cannot have b ( g ) = 7 and so we shall get acontradiction. To this aim we need to compute the space Z of closed 2-forms. By adirect computation using (5.3) and c (cid:54) = 0, it follows that dim Z = dim Λ V +2 = 8.Therefore, in order to get b ( g ) = 7, we have to require that the space B of exact2-forms is one-dimensional. This is equivalent to asking that the linear map d | (cid:104) f ,f ,f (cid:105) : (cid:10) f , f , f (cid:11) → Λ g ∗ , has rank equal to 1. Let us denote by E the matrix associated to d | (cid:104) f ,f ,f (cid:105) in theinduced basis ( f ij ) of Λ g ∗ . Eliminating all the zero rows, one has E = c − c c c − c − c ( c ) c − c c − c c c − c c − c − c . Then E has rank 1 if and only if E is not the zero matrix and all the 2 × E vanish. Notice that the minor c c is different from zero, since we have alreadyexcluded both cases c = 0 and c = 0. Then g cannot be isomorphic to g andwe obtain a contradiction. This concludes the case b ≥ (cid:3) Remark . By Theorem 5.2, we notice that, on a 6-dimensional nilpotent Liealgebra g with b ( g ) = 2, whenever a mean convex half-flat SU(3)-structure exists,a double example can also be found. This is not true for different values of the firstBetti number.Under the hypothesis of exactness, we can prove the following Theorem 5.4. Let g be a -dimensional nilpotent Lie algebra admitting an exactmean convex SL(3 , C ) -structure. Then g is isomorphic to g or g . Moreover, upto a change of sign, every exact definite -form ρ on g and g is mean convex,and g is the only nilpotent Lie algebra admitting mean convex coupled structures,up to isomorphism.Proof. Among the 6-dimensional nilpotent Lie algebras admitting half-flat struc-tures, as shown in the proof of [19, Theorem 4.1], the only Lie algebras that canadmit exact SL(3 , C )-structures are isomorphic to g , g or g . Therefore, by The-orem 4.1, g is the only nilpotent Lie algebra among them which can admit a meanconvex structure. In particular, a coupled mean convex structure on g is given inTable 2. This example was first found in [19], up to a change of sign of the definite3-form. For the remaining nilpotent Lie algebras g i , for i = 3 , , , , , , , , which can admit mean convex SL(3 , C )-structures by Theorem 4.1, we prove that g is the only one that admits exact definite 3-forms. To see this, let ( e j ) be thebasis of g ∗ i as listed in Table 1. Then the generic exact 3-form ρ on g i is given by dη ,where η = (cid:88) i 0. Finally, on g , λ ( ρ ) = − p . Then, if p (cid:54) = 0, ρ = dη is a definite 3-form on g . Moreover, ( e − iJ ρ e , e − iJ ρ e , e − iJ ρ e )is a basis for Λ , g ∗ and, with respect to this basis, the matrix ( β mn ) associatedto the (2 , d ˆ ρ is diag(0 , , − p ). Then, when p < ρ is mean convex,otherwise − ρ is. By a direct computation one can check that the same conclusionshold also for g . In particular, the generic exact 3-form ρ = dη , with η as in (5.4), isdefinite as long as p (cid:54) = 0. Moreover, ( e − iJ ρ e , e − iJ ρ e , e − iJ ρ e ) is a basis ofΛ , g ∗ , for every exact definite ρ and, with respect to this basis, the matrix ( β mn )associated to the (2 , d ˆ ρ is diag(0 , , − p ). (cid:3) Hitchin flow equations In this section we study the mean convex property in relation to the Hitchin flowequations (3.3). We recall that the solution ( ω ( t ) , ρ ( t )) of (3.3) starting from a half-flat structure remains half-flat as long as it exists. However, the same does nothappen in general for special classes of half-flat structures. Then, a natural questionis whether the Hitchin flow equations preserve the mean convexity of the initial data( ω (0) , ρ (0)). A first example of solution preserving the mean convex condition of theinitial data, up to change of sign of ρ (0), was found in [20, Proposition 5.4]. In thiscase the initial structure is coupled.More generally, when the Hitchin flow solution ( ω ( t ) , ρ ( t )) preserves the coupledcondition of the initial data, then ρ ( t ) = f ( t ) ρ (0), where f : I → R is a non-zerosmooth function with f (0) = 1 (for more details see [20, Proposition 5.2]). Then, acoupled solution preserves the mean convexity of the initial data as long as it exists.Some further remarks can be made in other special cases. If ( ω ( t ) , ρ ( t )) is asolution of (3.3) starting from a strictly mean convex half-flat structure ( ω, ρ ), bycontinuity the solution remains mean convex, at least for small times. This occurs,for instance, for double structures. In particular cases, the mean convex property ofthe double initial data is preserved for all times: Proposition 6.1. Let M be a connected -manifold endowed with a double structure ( ω, ρ ) . If ( ω ( t ) , ρ ( t )) is a double solution of (3.3) defined on some I ⊆ R , ∈ I ,i.e. d ˆ ρ ( t ) = ν ( t ) ω ( t ) for each t ∈ I for some smooth nowhere vanishing function ν : I → R , then there exists a nowhere vanishing smooth function f : I → R suchthat ω ( t ) = f ( t ) ω (0) . Conversely, if ( ω ( t ) , ρ ( t )) is a solution of (3.3) with ω ( t ) = f ( t ) ω (0) , then it is a double solution.Proof. Let ( ω ( t ) , ρ ( t )) be a solution with ω ( t ) = f ( t ) ω (0). From (3.3) one gets d ˆ ρ ( t ) = − ∂∂t (cid:0) ω ( t ) (cid:1) = − ∂∂t (cid:0) f ( t ) ω (0) ∧ ω (0) (cid:1) = − f ( t ) ˙ f ( t ) ω (0) . Then ω ( t ) = f ( t ) ω (0) is a double solution with ν ( t ) = − ddt ln f ( t ). Conversely, if d ˆ ρ ( t ) = ν ( t ) ω ( t ) , then ∂∂t ω ( t ) ∧ ω ( t ) = − d ˆ ρ ( t ) = − ν ( t ) ω ( t ) . Since the wedge product with ω ( t ) is injective on 2-forms, this is equivalent to ∂∂t ω ( t ) = − ν ( t ) ω ( t ), whose unique solution is ω ( t ) = f ( t ) ω (0), with f ( t ) = e − (cid:82) t ν ( s ) ds . (cid:3) We now provide an explicit example of double solution to (3.3) and show that adouble solution with double initial data may not exist. Example 6.2. Consider the double SU(3)-structure ( ω, ρ ) given in Table 2 on g .The solution of the Hitchin flow equations with initial data ( ω, ρ ) is double and it isexplicitly given by ω ( t ) = (cid:18) − t (cid:19) ω,ρ ( t ) = − (cid:18) − t (cid:19) e + e + e + e . In particular d ˆ ρ ( t ) = ν ( t ) ω ( t ) with ν ( t ) = (2 − t ) − > t in the maximalinterval of definition I = ( −∞ , ). Consider now the double SU(3)-structure ( ω, ρ )given in Table 2 on g . The solution of the Hitchin flow equation with initial data( ω, ρ ) is given by ω ( t ) = f ( t ) (cid:0) e − e (cid:1) − f ( t ) e ,ρ ( t ) = h ( t ) e + ( h ( t ) − e − e − e + e − e + h ( t ) e , where f ( t ) , f ( t ) , h ( t ) , h ( t ) satisfy the following autonomous ode system: ˙ f = f f (2 h − , ˙ f = − f f (2 f + f (2 h − , ˙ h = − f , ˙ h = − f , with initial conditions f (0) = f (0) = h (0) = 1, h (0) = 0, which, by knowntheorems, admits a unique solution with given initial data. In particular, this solutionis not a double solution. A direct computation shows that the eigenvalues λ i ( t ) ofthe matrix ( β mn ( t )) associated to d ˆ ρ ( t ) are λ = λ = (cid:113) − h + h + h , λ = (1 − h ) (cid:113) − h + h + h . In particular the mean convex property is preserved for small times as expected.To our knowledge, the question of whether the Hitchin flow preserves the meanconvexity of the initial data when the (2 , M be a compact real analytic 6-dimensional manifold endowed with a half-flat mean convex SU(3)-structure ( ω, ρ ).Since the unique solution of (3.3) starting from ( ω, ρ ) is a one-parameter family ofhalf-flat structures ( ω ( t ) , ρ ( t )), we can write d ˆ ρ ( t ) = ( ν ( t ) ω ( t ) − ν ( t )) ∧ ω ( t ) , where ν ( t ) ∈ C ∞ ( M ) and ν ( t ) ∈ Λ , M is a primitive (1 , J ρ ( t ) for each t ∈ I , where I is the maximal interval of definition of the flow. Then d ˆ ρ ( t ) ∧ ω ( t ) = ν ( t ) ω ( t ) and, since ν (0) > ν ( t ) > ν ( t ) > 0, the volume form ω ( t ) is pointwise decreasing: ∂∂t ( ω ( t ) ) = ∂∂t ( ω ( t ) ) ∧ ω ( t ) + ∂∂t ω ( t ) ∧ ω ( t ) = − d ˆ ρ ( t ) ∧ ω ( t ) = − ν ( t ) ω ( t ) . Moreover, ω ( t ) is a positive (2 , J ρ ( t ) for all t ∈ I and, fromthe second equation in (3.3), we know that − ∂ t ( ω ( t )) remains a (2 , J ρ ( t ) for each t ∈ I such that − ∂ t ( ω ( t )) (cid:12)(cid:12) t =0 = 2 d ˆ ρ (0) is semi-positive. LOSED SL(3 , C )-STRUCTURES ON NILMANIFOLDS 17 Then the Hitchin flow solution preserves the mean convexity of the initial data ifand only if − ∂ t ( ω ( t )) = 2 d ˆ ρ ( t ) remains semi-positive. The essential difficulty in thisproblem lies in the fact that the link between the positivity of ω ( t ) and the meanconvexity of the initial data is not sufficient to ensure the mean convexity of thesolution since also the almost complex structure evolves in a non-linear way underthe equation ∂ t ( ρ ( t )) = dω ( t ). Let us look at the behaviour of (3.3) on a specificexample. Example 6.3. Consider the mean convex half-flat structure ( ω, ρ ) given in Table 2on g and consider the family of solutions to the second equation in (3.3), startingfrom ( ω, ρ ): ω ( t ) = − a ( t ) e + 1 a ( t ) e + a ( t ) e ,ρ ( t ) = e + b ( t ) e − e − e + b ( t )( e − e ) , where a ( t ) , a ( t ) , b ( t ) , b ( t ) satisfy the following ode system: (cid:40) ˙ a = − a a (cid:0) a b + 1 (cid:1) , ˙ a = a (cid:0) a b − (cid:1) , (6.1)subject to the normalization condition (cid:112) b − b = a , with initial data a (0) = a (0) = b (0) = 1, b (0) = 0. This system defines a family of solutions to ∂ t ( ω ( t ) ) = − d ˆ ρ ( t ) depending on b ( t ). Then, if b ( t ) = a ( t ) − 1, for instance, d ˆ ρ ( t ) is not semi-positive, at least for small times t > 0. Anyway, the unique solution to (3.3) startingfrom ( ω, ρ ), given by (6.1) together with (cid:40) ˙ b = − a , ˙ b = a , preserves the mean convexity of the initial data.By a direct computation, one can show that the mean convexity of the initial datais preserved by (3.3), for small times, also in all the other examples of half-flat meanconvex structures given in Table 2.7. Tamed closed SL (3 , C ) -structures A closed SL(3 , C )-structure ρ is called tamed if there exists a symplectic formΩ taming J ρ , i.e. if ω := Ω , is positive. As already observed in [13], compact6-manifolds cannot admit tamed mean convex SL(3 , C )-structures.Notice that, if we denote as usual ˆ ρ = J ρ ρ , when the normalization condition ρ ∧ ˆ ρ = ω is satisfied and dω = 0, then the pair ( ω, ρ ) defines a symplectic half-flatstructure.In this section we study the existence of invariant tamed closed SL(3 , C )-structureson solvmanifolds. Since the structures are invariant we can work as in the previoussections at the level of solvable unimodular Lie algebras. Theorem 7.1. Let Γ \ G be a -dimensional solvmanifold, not a torus. Then Γ \ G admits an invariant tamed closed SL(3 , C ) -structure if and only if the Lie algebra g of G has symplectic half-flat structures.If g is nilpotent, then it is isomorphic to g or g as listed in Table 1.If g is solvable, then it is isomorphic to one among g , , g , − , , g , − , − , , e (1 , ⊕ e (1 , , A − ,β, − β , , A α, − α, , ⊕ R , as listed in Table 3. Moreover, all the eight Lie algebras admit closed SL(3 , C ) -structures tamed by asymplectic form Ω such that d Ω , (cid:54) = 0 .Proof. First we prove the theorem in the nilpotent case. 6-dimensional symplec-tic nilpotent Lie algebras were classified in [23] (see also [40]) and their struc-ture equations are listed in Table 1. For any such Lie algebra we consider a pair( ρ, Ω) ∈ Λ g ∗ i × Λ g ∗ i explicitly given by ρ = (cid:88) i 0, wemay then apply this result on each g i by considering the almost complex structure J ρ induced by ρ . We notice that, for any g i listed in Table 1, e ∈ ξ ( g i ). A directcomputation on each g i for i = 3 , , , , , , , , , , , , , , 30, shows that J ρ e ∈ [ g i , g i ], for any J ρ induced by a closed 3-form ρ . On g i , for i = 23 , , 33, thesame obstruction holds since an explicit computation shows that the map π ◦ J ρ : ξ ( g i ) → g i , has non-trivial kernel, where π denotes the projection onto g i / [ g i , g i ]. This meansthat, for each ρ , one can find a non-zero element in the center of g i whose imageunder J ρ lies entirely in [ g i , g i ]. For all the other cases, let Ω = Ω , + Ω , + Ω , bethe decomposition of Ω in types with respect to J ρ , and denote by ω the (1 , , := (Ω + J ρ Ω). Then, in order to have a closed SL(3 , C )-structure tamed by Ωwe have to require that ω is positive, i.e., that the symmetric 2-tensor g := ω ( · , J ρ · ) ispositive definite. Denote by g ij := g ( e i , e j ) the coefficients of g with respect the dualbasis ( e , . . . , e ) of g . Then, a direct computation on g i , for i = 11 , , , , g always vanishes, so we may discard these cases as well. We maythen restrict our attention to the remaining Lie algebras g and g . Since, asshown in [10, Theorem 2.4], these are the only 6-dimensional non-abelian nilpotentLie algebras carrying a symplectic half-flat structure. Explicit examples of closedSL(3 , C )-structures tamed by a symplectic form Ω such that d Ω , (cid:54) = 0 are given by ρ = − e − e − e − e − e − e − e , Ω = e + 12 e − e + e + e + e , on g , and by ρ = e + 2 e + e + e + e + e , Ω = e − e − e + e , on g . This proves the first part of the theorem.Using the classification results in [30, Th. 2] for 6-dimensional symplectic uni-modular (non-nilpotent) solvable Lie algebras, for each Lie algebra one can computethe metric coefficients g ij of g with respect to the basis ( e , . . . , e ) for g as listedin Table 3. It turns out that, if g is one among g , − , , g , , , g − , , , , g , − , , , g , , g , , , g , , A − , ⊕ R , A − , ,γ , , A , ⊕ R , A − , ⊕ R , A , ,γ , ⊕ R , A , ⊕ R , A − , , ⊕ R , A , ⊕ R or A , ⊕ R , each closed definite 3-form ρ induces a J ρ such that g = 0.In a similar way, if g is g − , or g − , − , , then g = 0, while when g is n ± , , e (2) ⊕ R or e (1 , ⊕ R , g = 0. Finally, when g = e (1 , ⊕ h , then g = 0. In some othercases g cannot ever be positive definite since, for each closed ρ inducing an almost LOSED SL(3 , C )-STRUCTURES ON NILMANIFOLDS 19 complex structure J ρ , g rr = − g kk for some r (cid:54) = k . In particular, when g = g , , ,then g = − g , when g = e (2) ⊕ e (2), then g = − g , and when g is e (2) ⊕ e (1 , e (2) ⊕ h , then g = − g . As shown in [17, Prop. 3.1, 4.1 and 4.3], for theremaining Lie algebras g , , g , − , , g , − , − , , e (1 , ⊕ e (1 , A − ,β, − β , , A α, − α, , ⊕ R aslisted in Table 3, a symplectic half-flat structure always exists. Moreover, on theseLie algebras, an explicit example of closed SL(3 , C )-structure tamed by a symplecticform Ω such that d Ω , (cid:54) = 0 is given Table 3. (cid:3) Remark . (1) By [17, Remarks 3.2 and 4.4], the solvable Lie groups cor-responding to each solvable Lie algebra admitting closed tamed SL(3 , C )-structures admit compact quotients by lattices (for further details see [3, 15,42, 43]).(2) As shown in [13], given an SL(3 , C )-structure ρ tamed by a 2-form Ω on areal 6-dimensional vector space V , the 3-form ϕ = ρ + Ω ∧ dt, defines a G - structure on V ⊕ R . 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LOSED SL(3 , C )-STRUCTURES ON NILMANIFOLDS 21 Appendix Table 1 contains the isomorphism classes of 6-dimensional real nilpotent Lie al-gebras g i , i = 1 , . . . , , including their first Betti numbers and an indication ofwhether they admit half-flat structures and symplectic forms. In Table 2 we givean explicit example of mean convex closed SU(3)-structure, indicating which onesare half-flat. Table 3 contains all 6-dimensional symplectic solvable (non-nilpotent)unimodular Lie algebras, specifying which admit tamed closed SL(3 , C )-structures.An explicit example of a closed tamed SL(3 , C )-structure is also included. Table 1. g Structure constants b ( g ) Half-flat Symplectic g (0 , , e , e , e + e , e − e ) 2 – – g (0 , , e , e , e , e − e ) 2 – – g (0 , , e , e , e , e ) 2 – (cid:51) g (0 , , e , e , e + e , e + e ) 2 (cid:51) (cid:51) g (0 , , e , e , e , e + e ) 2 – (cid:51) g (0 , , e , e , e , e ) 2 (cid:51) (cid:51) g (0 , , e , e , e , e − e ) 2 (cid:51) (cid:51) g (0 , , e , e , e , e + e ) 2 (cid:51) (cid:51) g (0 , , , e , e − e , e + e ) 3 (cid:51) (cid:51) g (0 , , , e , e , e + e ) 3 (cid:51) (cid:51) g (0 , , , e , e , e + e + e ) 3 (cid:51) (cid:51) g (0 , , , e , e , e + e ) 3 (cid:51) (cid:51) g (0 , , , e , e , e ) 3 (cid:51) (cid:51) g (0 , , , e , e , e + e ) 3 (cid:51) – g (0 , , , e , e , e + e ) 3 (cid:51) – g (0 , , , e , e , e − e ) 3 (cid:51) – g (0 , , , e , e , e ) 3 – – g (0 , , , e , e − e , e + e ) 3 – (cid:51) g (0 , , , e , e , e − e ) 3 – (cid:51) g (0 , , , e , e + e , e ) 3 – (cid:51) g (0 , , , e , e , e + e ) 3 (cid:51) (cid:51) g (0 , , , e , e , e ) 3 (cid:51) (cid:51) g (0 , , , e , e , e ) 3 – (cid:51) g (0 , , , e , e , e ) 3 (cid:51) (cid:51) g (0 , , , , e , e + e ) 4 (cid:51) – g (0 , , , , e , e ) 4 – (cid:51) g (0 , , , , e , e + e ) 4 (cid:51) (cid:51) g (0 , , , , e − e , e + e ) 4 (cid:51) (cid:51) g (0 , , , , e , e + e ) 4 (cid:51) (cid:51) g (0 , , , , e , e ) 4 (cid:51) (cid:51) g (0 , , , , e , e ) 4 (cid:51) (cid:51) g (0 , , , , , e + e ) 5 (cid:51) – g (0 , , , , , e ) 5 (cid:51) (cid:51) g (0 , , , , , 0) 6 (cid:51) (cid:51) Table 2. Explicit examples of mean convex closed SU(3)-structures g Mean convex closed SU(3)-structures Half-flat mean convex example g ω = − e − e − e ρ = − e + e − e − e − e + e – g ω = − e − e − e ρ = e − e − e + 2 e – g ω = e − e − e ρ = e − e − e − e − e − e (cid:51) g ω = − e + e − e ρ = − e + e − e + e − e + e (cid:51) g ω = e − e − e ρ = e − e − e − e − e − e (cid:51) g ω = − e + e − e ρ = e − e + e − e − e + e (cid:51) g ω = e + e + e − e + e + e + e + e + e ρ = 2 e + e − e + e + e + e − e + e − e – g ω = e + e + e ρ = − e + e + e + e − e − e (cid:51) g ω = e − e + e ρ = − e − e + e + e – g ω = e + e − e ρ = e + e − e + e − e + e (cid:51) g ω = e + e − e ρ = 2 e − √ e − e + √ e (cid:51) g ω = e + e + e ρ = − e + 2 e + e + e – g ω = e − e − e ρ = e − √ e + √ e + e + e – g ω = − e + e − e ρ = e + e − e + e – g ω = − e − e + e ρ = − e − e + e − e + e + e – g ω = − e − e + e ρ = − e + e + e + e – g ω = e + e + e ρ = e − e − e − e (cid:51) g ω = e + e + e ρ = 2 e + e + e − e – g ω = − e + e − e ρ = − e + e + e + e (cid:51) g ω = − e + e + e ρ = e + e − e − e (cid:51) g ω = e + e − e + e ρ = − e + e + e − e + e – g ω = − √ e − e + e ρ = e + e + e + e − e – g ω = − e − e + e ρ = − e + e + e + e (cid:51) g ω = e + e − e ρ = e − e + e − e (cid:51) g ω = e − e + e ρ = e − e + e + e + e + e (cid:51) g ω = − e − e + e ρ = − e + e − e − e (cid:51) g ω = −√ e − e − e ρ = − e + e − e + 2 e (cid:51) g ω = − e − e − e ρ = − e + e − e + e (cid:51) LOSED SL(3 , C )-STRUCTURES ON NILMANIFOLDS 23 Table 3. g Structure constants Tamed closed SL(3 , C )-structure g , − , ( e , e , , e , − e , 0) – g , , ( e , e , , e , − e , 0) – g − , , , ( − e + e , − e , e , e , , 0) – g , − , , ( − e + e , e , − e , e , , 0) – g − , ( e , e , − e , e + e , e − e , 0) – g − , − , ( e , − e , e , e + e , − e , 0) – g , ( e , , e , e , − e , 0) – g , , ( e , , e , − e , e , 0) – g , ( e , − e , e , e − e , e + e , ρ = − e − e + e − e Ω = − e + e − e + e g , − , ( e + e , − e + e , e , − e , , ρ = e − e + e + e Ω = e + e + e + e g , , ( − e + e , e + e , − e , e , , 0) – g , ( − e + e , e , e + e + e , e , − e , 0) – g , − , − , ( − e + e , − e − e , e − e , e + e , , ρ = e + e + e − e + e Ω = e + e + e n ± , ( − e , − e − e , − e + e ∓ e , e , − e , 0) – e (2) ⊕ e (2) (0 , − e , e , , − e , e ) – e (1 , ⊕ e (1 , 1) (0 , − e , − e , , − e , − e ) ρ = − e − e + e − e − e + e + e Ω = − e + e − e e (2) ⊕ R (0 , − e , e , , , 0) – e (1 , ⊕ R (0 , − e , − e , , , 0) – e (2) ⊕ e (1 , 1) (0 , − e , e , , − e , − e ) – e (2) ⊕ h (0 , − e , e , , , e ) – e (1 , ⊕ h (0 , − e , − e , , , e ) – A − ,β, − β , ( e , − e , βe , − βe , , ρ = − e − e − e − e Ω = − e + e + e + e ( β = − A − , ⊕ R ( e , , e , − e , , 0) – A − , ,γ , ( e , − e , γe , − γe , , 0) – A , ⊕ R ( e , , e , − e , , 0) – A − , ⊕ R ( e + e , e , − e + e , − e , , 0) – A , ,γ , ⊕ R ( e , − e , γe , − γe , , 0) – A α, − α, , ⊕ R ( αe + e , − e + αe , − αe + e , − e − αe , , ρ = e + e + e + e − e Ω = − e + e − e A , ⊕ R ( e + e , − e + e , e , − e , , 0) – A − , , ⊕ R ( − e + e , e , − e , e , , 0) – A , ⊕ R ( e + e , e − e , e , , , 0) – A , ⊕ R (2 e + e , e + e , − e + e , , , 0) – (A. Fino) Dipartimento di Matematica “G. Peano”, Universit`a degli Studi di Torino,Via Carlo Alberto 10, 10123 Torino, Italy Email address : [email protected] (F. Salvatore) Dipartimento di Matematica “G. Peano”, Universit`a degli Studi diTorino, Via Carlo Alberto 10, 10123 Torino, Italy Email address ::