Compact Solvmanifolds with a Closed G 2-Structure

We consider a parametrized family of compact G2-calibrated solvmanifolds, and construct associative (so volume-minimizing submanifolds) 3-tori with respect to the closed G2-structure. We also study the Laplacian flow of this closed G2 form on the solvable Lie group underlying to each of these solvmanifolds, and show long time existence of the solution.


Introduction
A G 2 -structure on a seven-dimensional manifold M is defined by a positive 3-form ϕ (the G 2 form) on M, which induces a Riemannian metric g ϕ and a volume form dV ϕ on M such that ( ) for any vector fields X, Y on M. If the 3-form ϕ is covariantly constant with respect to the Levi-Civita connection of the metric g ϕ or, equivalently, the 3-form ϕ is closed and coclosed [1], then the holonomy group of g ϕ is a subgroup of the exceptional Lie group G 2 , and the metric g ϕ is Ricci-flat.
When this happens, the G 2 -structure is said to be torsion-free [2].The first compact examples of Riemannian manifolds with holonomy G 2 were constructed first by Joyce [3], and then by Kovalev [4].Recently, other examples of compact manifolds with holonomy G 2 were obtained in [5] [6].
There are many different G 2 -structures attending to the behavior of the exterior derivative of the G 2 form [1] [7].In the following, we will focus our attention on G 2 -structures where the 3-form ϕ is closed.In this case, the G 2 -structure is said to be closed (or calibrated).The first example of a compact G 2 -calibrated manifold, which does not admit any torsion-free G 2 -structure, was obtained in [8].This example is a compact nilmanifold, that is a compact quotient of a simply connected nilpotent Lie group by a lattice, endowed with an invariant calibrated G 2 -structure.In [9], Conti and the first author classified the 7-dimensional compact nilmanifolds admitting a left invariant closed G 2 -structure.More examples were given in [10] [11] [12] [13].
Calibrated geometry was introduced by Harvey and Lawson in [14] and it concerns to a special type of minimal submanifolds of a Riemannian manifold, which are defined by a closed form (the calibration) on the manifold.Such submanifoldds are called calibrated submanifolds (see Section 5 for details).
In addition to compact Kähler manifolds and compact 7-manifolds with a torsion-free G 2 -structure, 7-manifolds with a closed G 2 -structure are also calibrated manifolds.In fact, if M is a 7-manifold with a closed G 2 -structure ϕ , then ϕ is a calibration [14].The 3-dimensional orientable submanifolds Y M ⊂ calibrated by the G 2 form ϕ , that is, those 3-dimensional submanifolds Y M ⊂ such that ϕ restricted to Y is a volume form for Y, are called associative 3-folds of ( ) In this paper, we consider a parametrized family of 7-dimensional compact solvmanifolds By [16] [17], a closed G 2 -structure on a compact manifold cannot induce an Einstein metric, unless the induced metric has holonomy contained in G 2 .It is still an open problem to see if the same property holds on noncompact manifolds.For the homogeneous case, a negative answer has been recently given in [18].Indeed, in [18] it is proved that if a solvable Lie algebra has a closed G 2 -structure then the induced inner product is Einstein if and only if it is flat.
Natural generalizations of Einstein metrics are given by Ricci solitons, which have been introduced by Hamilton in [19] The other motivation of this paper comes from the Laplacian flow on 7-manifolds admitting closed G 2 -structures.Let M be a 7-dimensional manifold with a closed G 2 -structure ϕ .The Laplacian flow on M starting from ϕ is given by where ( )  [16] as a tool to find torsion-free G 2 -structures on compact manifolds.Short-time existence and uniqueness of the solution, in the case of compact manifolds, were proved in [20].Properties of this flow were proved in [21] [22] [23].
The first noncompact examples with long-time existence of the solution were obtained on seven-dimensional nilpotent Lie groups in [24], but in those examples the Riemannian curvature tends to 0 as t goes to infinity.Further solutions on solvable Lie groups were described in [25] [26] [27] [28].Moreover, a cohomogeneity one solution converging to a torsion-free G 2 -structure on the 7-torus was worked out in [29].
In Section 6, we consider the solvable (non-nilpotent) Lie group

( )
H k , and we prove that it is defined on a time interval of the form ( ) , T ∞ , where 0 T < is a real number.(This solution was previously given in [25] from a family of symplectic half-flat structures on a 6-dimensional ideal of the Lie algebra ( ) is independent of the time t, and so the solution ( ) does not converge to a torsion-free G 2 -structure as t goes to infinity.

Closed G 2 -Structures
In this section we collect some basic facts and definitions concerning G 2 forms on smooth manifolds (see [1]  , , e e  of the (local) 1-forms on M. Such a 3-form ϕ was introduced by Bonan in [35], and it induces a Riemannian metric g ϕ and a volume form dV ϕ on M satisfying (1).We say that the manifold M has a closed (or calibrated) G 2 -structure if there is a G 2 -structure ϕ on M such that ϕ is closed, that is 0 dϕ = , and so ϕ defines a calibration [14].
Now, let G be a 7-dimensional simply connected nilpotent Lie group with Lie algebra g .Then, a G 2 -structure on G is left invariant if and only if the corresponding 3-form ϕ is left invariant.Thus, a left invariant G 2 -structure on G corresponds to an element ϕ of ( )

*
Λ g that can be written as (2), that is, with respect to some orthonormal coframe { } , , e e  of the dual space * g .
We say that a G 2 -structure on g is calibrated if ϕ is closed, i.e.

0, dϕ =
where d denotes the Chevalley-Eilenberg differential on , which is called a compact solvmanifold; and if g has a calibrated G 2 -structure, the G 2 -structure on \ G Γ is also calibrated.

Formal Manifolds
First, we need some definitions and results about minimal models.Let ( ) , A d be a differential algebra, that is, A is a graded commutative algebra over the real numbers, with a differential d which is a derivation, that is, deg a is the degree of a.
A differential algebra ( ) A d is said to be minimal if it satisfies the following two conditions: 1) A is free as an algebra, that is, A is the free algebra V over a graded vector 2) there exists a collection of generators { } , a I τ τ ∈ , for some well-ordered index set I, such that ( ) < and each da τ is expressed in terms of preceding a µ ( µ τ < ).This implies that da τ does not have a linear part, that is, it lives in Morphisms between differential algebras are required to be degree-preserving algebra maps which commute with the differentials.Given a differential algebra ( ) We will say that ( ) , d  is minimal and there exists a morphism of differential graded algebras on cohomology.Halperin [36] proved that any connected differential algebra ( ) A d has a minimal model unique up to isomorphism.
A minimal model ( ) that induces the identity on cohomology.The formality of a minimal model can be distinguished as follows.
Theorem 3.1 [37] A minimal model ( ) and the space V decomposes as a direct sum V C N = ⊕ with ( ) 0 d C = , d is injective on N and such that every closed element in the ideal ( ) If M is a simply connected manifold, the dual of the real homotopy vector space  ( ) Many examples of formal manifolds are known: spheres, projective spaces, compact Lie groups, symmetric spaces, flag manifolds, and all compact Kähler manifolds [37].
We will also use the following property Lemma 3.3 Let 1 M and 2 M be differentiable manifolds.Then, the product manifold is formal if and only if 1 M and 2 M are formal.
In [39], the condition of formal manifold is weaken to s-formal manifold as follows.

Definition 3.4 Let ( )
, d  be a minimal model of a differentiable manifold M. We say that ( ) such that for each i s ≤ , the space i V of generators of degree i decomposes as a direct sum , where the spaces i C and i N satisfy the three following conditions: 1) ( ) 3) any closed element in the ideal The relation between the formality and the s-formality for a manifold is given in the following theorem.Theorem 3.5 Let M be a connected and orientable compact differentiable manifold of dimension 2n or ( )

( )
G k be the simply connected and solvable Lie group of dimension 5 consisting of matrices of the form where i x ∈  , for 1 5 i ≤ ≤ , and k is a real number such that k k e e − + is an integer number different from 2. Then a global system of coordinates { } ,1 5 x a x = , and a standard calculation shows that a basis for the right invariant 1-forms on ( ) We notice that the Lie group ( ) G k may be described as a semidirect product ( ) , where  acts on 4  via the linear transformation ( )  given by the matrix , , a a = a  and similarly for x .Therefore ( ) , where  is a connected abelian subgroup, and 4  is the nilpotent commutator subgroup.
Now we show that there exists a discrete subgroup ( ) it suffices to find some real number 0 t such that the matrix defining ( ) to an element A of the special linear group ( ) SL 4,  with distinct real eigenvalues λ and 1 λ − .Indeed, we could then find a lattice 0 Γ in which is invariant under ( ) , and take ( ) ( ) . To this end, we choose the matrix  given by with double eigenvalues

−
. Taking , we have that the matrices ( ) and A are conjugate.In fact, put Then a direct calculation shows that ( ) is the transpose of the vector ( ) , , , m m m m , where 1 2 3 4 , , , m m m m ∈  , the lattice 0 Γ in 4  defined by ( ) , , , , is invariant under the subgroup  .Thus ( ) ( ) is a 5-dimensional compact solvable manifold.
Alternatively, ( ) S k may be viewed as the total space of a T 4 -bundle over the circle , , , , , , , , , and S is the quotient ( )
Next, we consider the 7-dimensional compact manifold ( ) ( ) where T 2 is the 2-torus 2 G k and their projections on ( ) S k by the same symbols.Then, if we denote by 6 7  , e e the (right invariant) closed 1-forms on the 2-torus T 2 whose cohomology classes generate the De Rham cohomology group ( ) 1 2 , H T  , we have that the 1-forms i e ( ) M k are such that and such that at each point of ( ) , , , , , , e e e e e e e is a basis for the 1-forms on ( ) Here 15  e stands for 1 5  e e ∧ , and so on.Then, the real cohomology groups of ( ) Thus, the Betti numbers of ( ) S k is 2-formal and so formal.Therefore, ( ) ( ) Proof.To prove that ( ) S k is 2-formal, we see that its minimal model must be a differential graded algebra ( ) , where  is the free algebra of the form ( ) ( ) , where the generator 1 a has degree 1, the generators 2 2 2 2 , , , a b c e have degree 2, and the differential d is given by , inducing an isomorphism on cohomology, is defined by Hence, ( ) S k is 2-formal, and so formal by Theorem 3.5.Now, Lemma 3.3 implies that ( ) ( ) We define the 3-form ϕ on ( )

( )
H k and the structure equations of ( ) H k are given by (11).So, the closed G 2 form k ϕ defined in ( 14) is a right invariant closed G 2 form on ( ) Let N be a simply connected solvable Lie group of dimension n, and denote by n its Lie algebra.Recall that a right invariant metric g on N is called a Ricci solsoliton metric (or simply solsoliton metric) if its Ricci endomorphism ( ) Ric g differs from a derivation D of n by a scalar multiple of the identity map n I , i.e. if there exists a real number λ such that ( ) Not all solvable Lie groups admit solsoliton metrics, but if a solsoliton exists, then it is unique up to automorphism and scaling [41].

Associative 3-Folds in M 7 (k)
In this section, we show associative 3-folds of the compact G 2 -calibrated solvmanifold ( ) M k defined in (10) with the closed G 2 form k ϕ given by ( 14).First, we need some definitions and results about calibrations (see [14] [15] for details).

Let ( )
, M g be a Riemannian manifold.An oriented tangent V on M is a subspace V of some tangent space p T M to M, with dimV k = and equipped with an orientation.If V is an oriented tangent k-plane on M, then g is a Euclidean metric on V. So, combining | V g with the orientation on V gives a natural volume form Let θ a closed k-form on a Riemannian manifold ( ) , M g .We say that θ is a calibration on M if for any p M ∈ and every oriented k-dimensional subspace V of the tangent space p T M we have , for some 1 λ ≤ (see [14] and [15] 3.7).Thus, if Y is an oriented submanifold of M with dimension k then, for any p Y ∈ , the tangent space p T Y is an oriented tangent k-plane on M. We say that Y is a calibrated submanifold if ( ) All calibrated submanifolds are minimal submanifolds.Even more, every compact calibrated submanifold is volume-minimizing in its homology class ([15] Proposition~3.7.2). [14] proved that any closed G 2 form ϕ on a 7-manifold M is a calibration on M. = is an embedded associative 3-fold.Furthermore, if N is compact then so is P.

Harvey and Lawson in
Remark 5.2 Note that Proposition 10.8.1 in [15] is stated for the G 2 -structures that are closed and coclosed, but the coclosed condition is not used in the proof.Proposition 5.3 There exist nine disjoint copies of 3-tori in ( ) : , , , , , , , , , , , , , that is σ is the product of the involutions ( ) ( ) with the identity map of 2  , where 1 σ is defined by ( ) ( ) : , , , , , , , , .
, and so 1 σ descends to the 5-dimensional compact manifold ( ) ( ) ( ) . Hence, σ defines also an involution of ( ) M k .From now on, we denote by : M k induced by the involution σ of ( ) H k defined in (15).Then, taking into account (5), we have that the induced action on the 1-forms i e is given by * * , Therefore, the G 2 form k ϕ on ( ) M k defined in ( 14) is preserved by the involution σ of ( )
Let P be the fixed locus of σ .Then, P consists of all the 3-dimensional spaces P a given as follows: , , , a a a a = a with ( ) ( ) ( ) ( ) ( ) Consequently, P is a disjoint union of 9 copies of a 3-torus T 3 .
Since the G 2 form k ϕ on ( )

7
M k defined in ( 14) is preserved by the involution σ of ( )

7
M k , each of the 9 torus P a in ( ) : , k M k ϕ by Proposition 5.1.

The Laplacian Flow
The purpose of this section is to prove that the Laplacian flow of k ϕ on the ( ) t ϕ .This flow was introduced by Bryant in [16] to study seven-dimensional manifolds admitting calibrated G 2 -structures.Notice that the stationary points of the flow Equation in (17) are harmonic G 2 -structures, which coincide with torsion-free G 2 -structures on compact manifolds.Short-time existence and uniqueness of the solution of ( 17) when M is compact were proved in [20].In the following theorem, we determine a global solution of the Laplacian flow of the closed G 2 form k ϕ given by ( 14) on the Lie group where  From now on, we write ( ) ( ) ( ) ( ) ( ) The equations (23) with the initial conditions ( )

( ) 7 M 7 M
k with an invariant closed G 2 -structure k ϕ , which is not coclosed, where k is a real number such that k = .Moreover, we construct associative calibrated (so volume-minimizing) 3-tori in ( ) k with respect to the closed G 2 form k ϕ (Proposition 5.3).

7 M
k , and we show that the Laplacian flow of k ϕ on ( ) H k exists for all time.In fact, in Theorem 6.2, we explicitly determine the solution ( ) k t ϕ for the flow of k ϕ on

2 H
the simply connected solvable (non-nilpotent) Lie group ( ) ( ) and let k ϕ be the right invariant closed G 2 form on

7 M k from the fixed locus of a G 2 -involution of the compact manifold ( ) 7 M k applying the following. Proposition 5 . 1 (
The 3-dimensional orientable submanifolds Y M ⊂ calibrated by the G 2 form ϕ , i.e. those submanifolds Y M p Y ∈ and for some unique orientation of Y, are called associative 3-folds.Next, we shall produce examples of associative 3-folds in ( ) [15] [Proposition 10.8.1])Let N be a 7-manifold with a closed G 2 form φ , and let : N N σ → be an involution of N satisfying * σ φ φ = and such that σ is not the identity map.Then the fixed point set

Theorem 6 . 1
Assume that M is compact.Then, the Laplacian flow (17) has a unique solution defined for a short time [ ) ε depending on 0 ϕ .
Ricci soliton metric.For the metric determined by the invariant closed G 2 form k for some λ ∈  and some derivation D of the corresponding Lie algebra, where I is the identity map.A natural question is thus to see if a closed G 2 -structure on a noncompact manifold induces a (non-Einstein) )