On the existence of balanced metrics on six-manifolds of cohomogeneity one

We consider balanced metrics on complex manifolds with holomorphically trivial canonical bundle, most commonly known as balanced $\rm{SU}(n)$-structures. Such structures are of interest for both Hermitian geometry and string theory, since they provide the ideal setting for the Hull-Strominger system. In this paper, we provide a non-existence result for balanced non-K\"ahler $\rm{SU}(3)$-structures which are invariant under a cohomogeneity one action on simply connected six-manifolds.


Introduction
A U(n)-structure on a 2n-dimensional smooth manifold M is the data of a Riemannian metric g and a g-orthogonal almost complex structure J. The pair (g, J) is also known as an almost Hermitian structure on M . When J is integrable, i.e., (M, J) is a complex manifold, the pair (g, J) defines a Hermitian structure on M . In this case, the metric g is called balanced when dω n−1 = 0, ω := g (J·, ·) denoting the associated fundamental form, and we shall refer to (g, J) as a balanced U(n)-structure on M . Balanced metrics have been extensively studied in [4,10,11,12,13,23,25] (see also the references therein).
Balanced metrics are also interesting in the context of SU(n)-structures, especially in the six-dimensional case, thanks to their applications in physics. An SU(n)structure (g, J, Ψ) on a 2n-dimensional smooth manifold M , is a U(n)-structure (g, J) on M together with a (n, 0)-form of nonzero constant norm Ψ = ψ + + iψ − satisfying the normalization condition Ψ∧Ψ = (−1) n(n+1) 2 (2i) n ω n n! . An SU(n)-structure (g, J, Ψ) on M with underlying balanced U(n)-structure (g, J) for which dω = 0 and dΨ = 0 will be referred to as a balanced non-Kähler SU(n)-structure.
In 1986, Hull and Strominger [22,30], independently, introduced a system of pdes, now known as the Hull-Strominger system, to formalize certain properties of the inner space model used in string theory. Let M be a 2n-dimensional complex manifold equipped with a nowhere-vanishing holomorphic (n, 0)-form Ψ and let E be a holomorphic vector bundle on M endowed with the Chern connection. The Hull-Strominger system consists of a set of pdes involving a pair of Hermitian metrics (g, h) on (M, E). One of these equations dictates the metric g on M to be conformally balanced, more precisely d Ψ ω ω n−1 = 0, where Ψ ω is the norm of Ψ given explicitly by Ψ ∧ Ψ = (−1) n(n+1) 2 i n n! Ψ 2 ω ω n . When one assumes all structures to be invariant under the smooth action of a certain Lie group G, the aforementioned condition reduces to the balanced equation dω n−1 = 0, since the norm of Ψ is constant. Notice that in these cases (g, J, Ψ) is a balanced SU(n)-structure on M , up to a suitable uniform scaling of Ψ.
The issue of the existence and uniqueness of a general solution to the Hull-Strominger system is still an open problem. Nonetheless, solutions have been found under more restrictive hypotheses; for the non-Kählerian case, we refer the reader, for instance, to [6,7,14,15,16,18,24]. Other interesting solutions are given in [8], where a class of invariant solutions to the Hull-Strominger system on complex Lie groups was provided; these solutions extend to solutions on all compact complex parallelizable manifolds, by Wang's classification theorem [33]. Moreover, in [10], it was shown that a compact complex homogeneous space with invariant complex volume admitting a balanced metric is necessarily a complex parallelizable manifold. Then, the invariant solutions given in [8] exhaust the complex compact homogeneous case. If one allows the Lie group acting on the homogeneous space to be real, many other solutions to the Hull-Strominger system are known in the literature, see for instance [18,27,28,31]. Then, one may wonder what happens in the cohomogeneity one case. A cohomogeneity one manifold M is a connected smooth manifold with an action of a compact Lie group G having an orbit of codimension one. Currently, there are no known examples of balanced non-Kähler SU(n)-structures invariant under a cohomogeneity one action. In this paper, we investigate their existence. In particular, we focus on the simply connected 2n = 6-dimensional case. Recall that, when a cohomogeneity one manifold M has finite fundamental group, then M/G is homeomorphic to an interval I, see [5]. If we denote by π : M → M/G the canonical projection onto the orbit space, we shall call π −1 (t), for every t ∈ • I, principal orbits and the inverse images of the boundary points singular orbits. Denoting by M princ the union of all principal orbits, which is a dense open subset of M , and by K the isotropy group of a principal point, which is unique up to conjugation along M princ , the pair (G, K) completely determines the principal part M princ of the cohomogeneity one manifold, up to G-equivariant diffeomorphisms. Given a Lie group H, we denote its Lie algebra Lie(H) by the corresponding gothic letter h.
We first give a local result for the existence of balanced non-Kähler SU(3)-structures by working on M princ .
Theorem A. Let M be a 6-dimensional simply connected cohomogeneity one manifold under the almost effective action of a connected Lie group G, and let K be the principal isotropy group. Then, the principal part M princ admits a G-invariant balanced non-Kähler SU(3)-structure (g, J, Ψ) if and only if M is compact and (g, k) = (su(2) ⊕ 2R, {0}).
We then prove that none of these local solutions can be extended to a global one. This leads us to state our main theorem: Theorem B. Let M be a six-dimensional simply connected cohomogeneity one manifold under the almost effective action of a connected Lie group G. Then, M admits no G-invariant balanced non-Kähler SU(3)-structures.
In [13], balanced metrics were constructed on the connected sum of k ≥ 2 copies of S 3 ×S 3 . However, it is not known whether S 3 ×S 3 admits balanced structures. In [23, Example 1.8], Michelsohn proved that S 3 × S 3 endowed with the Calabi-Eckmann complex structure does not admit any compatible balanced metric. By [2,Remark 1], in a manifold with six real dimensions, there is no non-Kähler Hermitian metric which is simultaneously balanced and strong Kähler-with-torsion (a.k.a SKT). In [11], Fino and Vezzoni conjectured that on non-Kähler compact complex manifolds it is never possible to find an SKT metric and also a balanced metric. In [17], an example of a SKT structure on S 3 × S 3 is provided. The key case that needs to be tackled in Theorem B is precisely S 3 × S 3 .
The paper is organized as follows. In Sect. 2, we review some basic facts about cohomogeneity one manifolds and SU(3)-structures which will be useful for our discussion. In Sect. 3, we present our problem, write a classification of the pairs (g, k) that can occur, and use the hypothesis of simply connectedness to reduce the list to only three possibilities. At the end of Sect. 3, we state Theorem A, which we prove in Sect. 4 via a case-by-case analysis. Finally, in Sect. 5, we prove Theorem B.
Acknowledgements. The first named author wants to thank Andrew Dancer and Jason Lotay for introducing the problem and their support and help with it. The second named author would like to thank Lucio Bedulli for introducing the problem and for useful conversations and comments and Anna Fino for her constant support, encouragement and patient guidance. The second named author would like to thank also Alberto Raffero and Fabio Podestà for helpful discussions and remarks.
Definition 2.1. A cohomogeneity one manifold is a connected differentiable manifold M with an action α : G × M → M of a compact Lie group G having an orbit of codimension one. We denote byα : G → Diff (M ) the Lie group homomorphism induced by the action.
From now on, let us assume that M is a simply connected cohomogeneity one manifold, and G is connected. By the compactness of G, the action α is proper and there exists a G-invariant Riemannian metric g on M ; this is equivalent to saying that G acts on the Riemannian manifold (M, g) by isometries. Moreover, we assume that the action α is almost effective, namely kerα is discrete. As usual, we denote by π : M → M/G the canonical projection and we equip M/G with the quotient topology relative to π. By a result of Bérard Bergery [5], the quotient space M/G is homeomorphic to a circle or an interval. As we are assuming that M is simply connected, we have that M/G is homeomorphic to an interval I. The inverse images of the interior points of the orbit space M/G are known as principal orbits, while the inverse images of the boundary points are called singular orbits. We denote by M princ the union of all principal orbits, which is an open dense subset of M , and by G p the isotropy group at p ∈ M .
First, we will suppose M is compact. It follows that M/G is homeomorphic to the closed interval I = [−1, 1]. Denote by O 1 and O 2 the two singular orbits π −1 (−1) and π −1 (1), respectively, and fix q 1 ∈ O 1 . By compactness of the G-orbits, there exists a minimizing geodesic γ q 1 : [−1, 1] → M from q 1 to O 2 which is orthogonal to every principal orbit. We call a normal geodesic a geodesic orthogonal to every principal orbit. Let γ : [−1, 1] → M be a normal geodesic between π −1 (−1) and π −1 (1); up to rescaling, we can always suppose that the orbit space M/G is such that π • γ = Id [−1,1] . Then, by Kleiner's Lemma, there exists a subgroup K of G such that G γ(t) = K for all t ∈ (−1, 1) and K is subgroup of G γ(−1) and G γ (1) .
For M non-compact, M/G is homeomorphic either to an open interval or to an interval with a closed end. In the former case, M is a product manifold M ∼ = I×G/K. In the latter case, there exists exactly one singular orbit, and M/G ∼ = I where I = [0, L) and L is either a positive number or +∞. Analogously to the compact case, there exists a normal geodesic γ : [0, L) → M such that γ(0) ∈ π −1 (0) and we can suppose π • γ = Id [0,L) . In addition, there exists a subgroup K of G such that G γ(t) = K for all t ∈ (0, L) and if H := G γ(0) , K is a subgroup of H.
So we have that: • for every p 1 , p 2 ∈ M princ , G · p 1 and G · p 2 are diffeomorphic. Therefore, up to conjugation along the orbits, when M is compact we have three possible isotropy groups H 1 := G γ(−1) , H 2 := G γ(1) and K := G γ(t) , t ∈ (−1, 1). When M is non-compact and has one singular orbit, instead, we have two possible isotropy groups H := G γ(0) and K := G γ(t) , t ∈ (0, L). From all of the above, we have that M princ ∼ = • I × G/K, and so, by fixing a suitable global coordinate system, we can decompose the Ginvariant metric g as g γ(t) = dt 2 + g t , where dt 2 is the (0, 2)-tensor corresponding to the vector field ξ := γ (t) evaluated at the point γ (t), and g t is a G-invariant metric on the homogeneous orbit G · γ (t) through the point γ (t) ∈ M . Now, we will assume M is compact. By the density of M princ in M and the Tube Theorem, M is homotopically equivalent to where the geodesic balls S γ(±1) := exp (B ε ± (0)), B ε ± (0) ⊂ T γ(±1) (G · γ (±1)) ⊥ , are normal slices to the singular orbits in γ (±1). Here, G × H i S γ(±1) is the associated fiber bundle to the principal bundle G → G/H i with type fiber S γ(±1) . By Bochner's linearization theorem, M is also homotopically equivalent to 3) The isotropy groups H i act on B ε ± (0) via the slice representation and, since the boundary of the tubular neighborhood Tub(O i ) := G × H i B ε ± (0), i = 1, 2, is identified with the principal orbit G/K and the G-action on Tub(O i ) is identified with the H i -action on B ε ± (0), then H i acts transitively on the sphere S l i := ∂B ε ± , l i > 0 still having isotropy K. The normal spheres S l i are thus the homogeneous spaces H i /K, i = 1, 2. The H i -action on S l i , i = 1, 2, may be ineffective, but it is sufficient to quotient H i by the ineffective kernel to obtain an effective action: transitive effective actions of compact Lie groups on spheres were classified by Borel and are summarized in Table 1. Table 1. Transitive effective actions of compact Lie groups on spheres The collection of G with its isotropy groups G ⊃ H 1 , H 2 ⊃ K is called the group diagram of the cohomogeneity one manifold M . Viceversa, let G ⊃ H 1 , H 2 ⊃ K be compact groups with H i /K = S l i , i = 1, 2. By the classification of transitive actions on spheres one has that the H i -action on S l i is linear and hence it can be extended to an action on B ε ± bounded by S l i , i = 1, 2. Therefore, (2.3) defines a cohomogeneity one manifold M . Analogously, if M is a non-compact cohomogeneity one manifold with one singular orbit, we define the group diagram of M to be the collection of G and the isotropy groups G ⊃ H ⊃ K, where the homogeneous space H/K will be a sphere. The converse is also true: the group diagram defines a non-compact cohomogeneity one manifold M . In these cases, M is homotopically equivalent to Let M i be cohomogeneity one manifolds with respect to the action of Lie groups G i , i = 1, 2. We say that the action of G 1 on M 1 is equivalent to the action of G 2 on M 2 if there exists a Lie group isomorphism ϕ : G 1 → G 2 and an equivariant diffeomorphism f : M 1 → M 2 with respect to the isomorphism ϕ. We shall study cohomogeneity one manifolds up to this type of equivalence.
Moreover, if a cohomogeneity one manifold M has group diagram G ⊃ H 1 , H 2 ⊃ K or G ⊃ H ⊃ K, one can show that any of the following operations results in a Gequivariantly diffeomorphic manifold: (1) switching H 1 and H 2 , (2) conjugating each group in the diagram by the same element of G, is the data of a Riemannian metric g, a g-orthogonal almost complex structure J, and a (3, 0)-form of nonzero constant norm Ψ = ψ + + iψ − satisfying the normalization condition ψ + ∧ ψ − = 2 3 ω 3 . Following a result obtained in [29] and later reformulated in [19,Section 2], one can show that giving an SU(3)-structure is equivalent to giving a pair of differential forms (ω, ψ + ) ∈ Λ 2 (M )×Λ 3 (M ) satisfying suitable conditions. Here Λ k (M ) denotes the space of differential forms of degree k on M . To see this, let us briefly recall the concept of stability in the context of vector spaces.
Remark 2.4. From the formulas in [3], we have that if (g, J, Ψ) is a balanced non-Kähler SU(3)-structure on a six-dimensional differentiable manifold M , Scal(g) < 0, Scal(g) being the scalar curvature associated with the metric g.

Balanced SU(3)-structures on six-dimensional cohomogeneity one manifolds
Let (g, J, Ψ) be an SU(3)-structure on a simply connected cohomogeneity one manifold M of complex dimension 3 for the almost effective action of a compact connected Lie group G. We are thus requiring G to preserve the SU(3)-structure on M . For the convenience of the reader, recall that • G preserves the metric g if and only ifα (h) is an isometry for each h ∈ G, • G preserves the almost complex structure J if and only if J commutes with the differential dα (h) for each h ∈ G, • G preserves the 3-form Ψ if and only ifα (h) * Ψ = Ψ, for each h ∈ G. This in particular implies that the principal isotropy K acts on T p M preserving (g p , J p , Ψ p ) for any p ∈ M , which means that K is a subgroup of SU (3). Now, since the J-invariant K-action fixes the subspace ξ| p of T p M , then it fixes Jξ| p as well. Let us write T p M as which is a contradiction since the K-action is closed along the G-orbits. Therefore, for each h ∈ K, its action on T p M is described by a 6 × 6 block matrix (2) and hence K can be identified with a subgroup of SU (2). Therefore, k := Lie (K) is {0}, R, or su (2). As observed in [26], all the possible candidate pairs (g, k), with g compact, which may admit an SU(3)-structure in cohomogeneity one are: In particular, M is simply connected if and only if the image of π 1 (S l ) generates π 1 (G/K) under the natural inclusions.
We know that π 1 (S l ) is either {0} or Z. Now, we observe that for cases (a.1) and (c.1), π 1 (G/K) = Z 2 , for cases (a.2), (b.3) and (c.2), π 1 (G/K) = Z 5 , and for case (b.2), π 1 (G/K) is either Z 2 or Z 3 . If M is non-compact and has no singular orbits, π 1 (M ) = π 1 (G/K). Hence, when M is non-compact, we can discard the pairs (a.1), (a. Remark 3.2. In case (b.1), we shall need to divide the discussion depending on the embeddings of k = R in g = su(2)⊕su(2) which, up to isomorphism, are all generated by an element of the form  with fixed p, q ∈ N. Up to uniform rescalings, which do not change the immersion of k, we can assume either (p, q) = (1, 0) or p, q to be coprime if neither is zero. Notice that when (p, q) = (1, 1) or (p, q) = (1, 0), k induces a decomposition of g in Ad(K)-modules, some of which are equivalent. In the former case, we shall say that k is diagonally embedded in g, while in the latter k is said to be trivially embedded in one of the two su(2)-factors of g. When instead p, q are different and nonzero, the Ad(K)-modules are pairwise inequivalent.
From now on, for each p ∈ M princ , let m p =: m be an Ad(K)-invariant complement of k in g. For each p ∈ M princ , we have that T p M = ξ| p ⊕ m| p , where for every X ∈ g, we denote by X the action field It is known that since M princ ∼ = Recall that if the m i 's are pairwise inequivalent, then they are orthogonal with respect the metric g t , for every t (see (2.1)). Otherwise, the expression of the metric strongly depends on the specific equivalence of the modules. In all cases, we recover the whole SU(3)-structure from a pair of G-invariant stable forms (ω, ψ + ) of degree two and three, respectively.
To fix the notations, in what follows, we shall denote by • B the opposite of the Killing-Cartan form on g, • {ẽ i } i=1,2,3 the standard basis for su (2) given bỹ Moreover, we recall some basic facts about G-actions which will be useful for our discussion: • Since g · γ p = γ g·p for the uniqueness of the normal geodesic γ starting from the point g · p, we have that ΦX 1 • Φ ξ t (p) = Φ ξ t • ΦX 1 (p), where Φ v t denotes the flow of the vector field v evaluated at time t. This is equivalent to [ξ,X] = 0, for each X ∈ g; • A k-form α on M princ of the form

Proof of Theorem A
From all the above discussion and the previous lemmas, the only possible pairs allowing M princ to support a balanced SU(3)-structure are (a.1) with M compact, (c.3), and (b.1). We investigate these three cases separately.

Case (c.3). g = su(3), k = su(2).
Consider the B-orthogonal basis of g given by Then, k = f 1 , f 2 , f 3 . Let a := f 8 and n := f 4 , f 5 , f 6 , f 7 , hence, m = a ⊕ n. Since the Ad(K)-invariant irreducible modules in the decomposition of g are pairwise inequivalent, the metric g on M princ is diagonal and, in particular, it is of the form for some positive h, f ∈ C ∞ ( • I). Moreover, with respect to the frame {e i } i=1,...,6 of M princ , the structure equations are given by Fix the volume form Ω = e 1...6 . One can easily show that a pair of generic G-invariant forms (ω, ψ + ) on M princ of degree two and three is given, respectively, by In particular, from p 4 = 0, it follows with λ = −4(p 2 1 (p 2 3 +p 2 6 )+(p 2 p 6 −p 3 p 5 ) 2 ). We suppose that ψ + is stable with λ < 0. Then, q 4 = 0 if and only if p 1 = 0. Since p 1 has to be equal to zero, it can be shown that the compatibility condition ψ + ∧ ω = 0 is equivalent to the following system of equations: Moreover, the positive-definiteness of g implies h 1 > 0. Then, the normalization condition ψ + ∧ ψ − = 2 3 ω 3 is equivalent to The balanced condition dω 2 = 0 is satisfied if and only if Since k = {0}, we can write T p M ∼ = e 1 | p ⊕ĝ p , for each p ∈ M princ . Moreover, every k-form α on M princ of the form However, using the results from [32], we can check that this example cannot be extended to the singular orbits to give a smooth metric on the whole manifold. This concludes the proof of Theorem A.

Proof of Theorem B
We will finally prove our main theorem. By [21], a six-dimensional compact simply connected cohomogeneity one manifold M whose corresponding principal part is given by the pair (g, k) = (su(2) ⊕ 2R, {0}) at the Lie algebra level is G-equivariantly diffeomorphic to the product of two threedimensional spheres, i.e., M ∼ = S 3 × S 3 . If we denote by H i for i = 1, 2, the singular isotropy groups for the G-action on M and by h i = Lie(H i ), for i = 1, 2, their Lie algebras, we have that both h 1 and h 2 are isomorphic to R so that both the singular orbits of M are four-dimensional compact submanifolds of M . Let b i be the i-th Betti number of M , then, up to G-equivariant diffeomorphisms, we may assume b 4 = 0. By Michelsohn's obstruction [23,Corollary 1.7], if M admitted any 4-dimensional compact complex submanifold S, then M would not admit a balanced metric. Therefore, we can make a few considerations by focusing on one tubular neighborhood of singular orbit G ⊃ H ⊃ K at a time. In particular, we divide the discussion depending on the immersion of h ⊂ g. Let S be the singular orbit given by the group diagram G ⊃ H ⊃ K. We notice that if S is J-invariant, a complex structure on M would give rise to a complex structure on S, so we can discard all these cases by Michelsohn's obstruction. In particular, we have that T q M = T q S ⊕ V where V = T q S ⊥ is the slice at q ∈ S; since S is four-dimensional, V is always a 2-plane. We recall that the H-action on T q S is given by the adjoint representation while the H-action on V is given via slice representation, and since V is bi-dimensional, this action is just a rotation on V of a certain speed, say a. Let us start by considering the case when h is contained in the center of g, ξ(g). In this case, the H-action on T q S is trivial. Therefore, T q S ⊕ V are inequivalent modules of the H-action on T q M and, since J commutes with the H-action, J preserves T q S for each q ∈ S, i.e., S is an almost complex manifold and we may apply Michelsohn's obstruction to discard this case. Therefore, we may suppose that h has a non-trivial component in the su(2)-factor of g. In particular, since rank(su(2)) = 1 and the adjoint action ignores components in the center of g, we may assume, without loss of generality and using the notation from Sect. 4.3, h = f 1 . Moreover, if we denote by m the tangent space to S via the usual identification, the decomposition of m in irreducible H-modules is given by where H acts on l 0 trivially and on l 1 via rotation of speed d. Therefore, when the integer a is different from d, the modules l 0 , l 1 and V are inequivalent for the H-action and again, since J commutes with the H-action, it cannot exchange two different modules. In particular, J(T q S) ⊆ T q S and we may apply Michelsohn's obstruction as before. For the remaining case a = d, we have that the two modules l 1 and V are equivalent, hence, J(l 1 ⊕ V ) ⊆ l 1 ⊕ V but not necessarily J(l 1 ) ⊆ l 1 . In particular, when this case occurs, the orbit S is not J-invariant, and we do not have obstructions to the existence of balanced metrics. Therefore, from now on, we assume this is the case.
Let ∂/∂x be a vector field such that (ξ| q , ∂/∂x| q ) is an orthonormal basis for the slice V and T * q M = e 1 | q , dx| q , e 3 | q , e 4 | q , e 5 | q , e 6 | q . Let ϕ : h → End(T q M ) be the haction on T q M . Then in order to have l 1 and V h-equivalent, ϕ(f 1 ) * acts on 1-forms given in the previous basis as where p j ∈ C ∞ ((−1, 1)) for any j = 1, . . . , 20.
Remark 5.1. We also note that in case (a.1) and when h = R, we can remove the hypothesis of simply connectedness from the non-compact case and still get a non existence result. Let M be a six-dimensional non-compact cohomogeneity one manifold under the almost effective action of a connected Lie group G and let K, H be the principal and singular isotropy groups, respectively, with (g, h, k) = (su(2) ⊕ 2R, R, {0}). Then, M admits no G-invariant balanced non-Kähler SU(3)structures.
From Theorem B, we get the following Corollary.

Corollary 5.2.
There is no non-Kähler balanced SU(3)-structure on S 3 × S 3 which is invariant under a cohomogeneity one action.
Remark 5.3. In the non-simply-connected case, by Theorem A, we can discard cases (b.1) and (c.3), as these do not admit local solutions to conditions (1)- (7). Moreover, as observed in [26,Section 3], one can also rule out cases (b.3) and (c.2), as the Gaction would not be almost effective, as well as case (c.1) since it would give rise to a three-dimensional J-invariant subspace, a contradiction.
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