Harmonic ﬂow of quaternion-Kähler structures

We formulate the gradient Dirichlet ﬂow of Sp(2)Sp(1) -structures on 8 -manifolds, as the ﬁrst systematic study of a geometric quaternion-Kähler (QK) ﬂow. Its critical condition of harmonicity is especially relevant in the QK setting, since torsion-free structures are often topologically obstructed. We show that the conformally parallel property implies harmonicity, extending a result of Grigorian in the G 2 case. We also draw several comparisons with Spin(7) -structures. Analysing the QK harmonic ﬂow, we prove an almost-monotonicity formula, which implies to long-time existence under small initial energy, via ε -regularity. We set up a theory of harmonic QK solitons, constructing a non-trivial steady example. We produce explicit long-time solutions: one, converging to a torsion-free limit on the hyperbolic plane; and another, converging to a limit which is harmonic but not torsion-free, on the manifold SU(3) . We also study compactness and the formation of singularities.


Introduction
In this paper we study the harmonic flow of H-structures introduced in [LSE19] for H = Sp(2)Sp(1) on 8manifolds.We refer to it simply as the quaternion-Kähler harmonic flow.The corresponding flows for H = G 2 , Spin(7), U(n) have been studied quite extensively in recent years cf.[DGK19, Gri19, DLSE21, HL21].To the best of our knowledge, this is the first study in the literature of a geometric flow of quaternion-Kähler structures.
Harmonic structures arise naturally as the critical points of the L 2 -energy of the intrinsic torsion of an Hstructure (wih H ⊂ SO(n)) and as such can be interpreted as the 'best' representative H-structure in a given isometric class.The harmonic flow is precisely the negative gradient flow of this energy functional.A wellknown result of Poon-Salamon in [PS91] asserts that there are only three compact (torsion-free) quaternion-Kähler 8-manifolds, and these are all symmetric spaces.Thus, harmonic structures can be viewed as the next most special objects on any other compact 8-manifolds admitting Sp(2)Sp(1)-structures.This article provides the first step towards a more general study of geometric flows of Sp(n)Sp(1)-structures on 4n-manifolds, and it can be read as a detailed instance of the abstract theory simultaneously formulated in [FLMSE22].One long-term prospect of Sp(n)Sp(1)-flows would be an analytic approach to the LeBrun-Salamon conjecture, which asserts that all compact quaternion-Kähler manifolds are symmetric spaces.Our exposition is organised as follows.
In Section 2 we review the basic properties of Sp(2)Sp(1)-structures, the geometry of which is determined by an algebraically special 4-form Ω.By comparing with Spin(7)-structures, determined by a different 4-form Φ, we derive several new identities.We emphasise the similarities and differences between the underlying structures.In Section 3 we derive the notion of harmonic Sp(2)Sp(1)-structures from the general framework of harmonic H-structures introduced in [LSE19].We illustrate its relevance in §3.2, by constructing explicit examples of harmonic Sp(2)Sp(1)-structures which are not torsion-free.
The analytic core of the paper is covered in Sections 4 and 5.We formulate the quaternion-Kähler harmonic flow and the corresponding notion of soliton, studying basic properties such as evolution of torsion and parabolic rescalings.By relying upon the analogy with the harmonic Spin(7)-flow [DLSE21], we establish in the quaternion-Kähler setting results such as a compactness theorem, almost-monotonicity formulae and ε-regularity, as well as a description of the singular set of the flow.One key difference of our approach is that we use the representation theory of Sp(2)Sp(1), rather than overly involved local computations, to simplify several proofs and thus illustrate the usefulness of a more unified approach to harmonic H-flows.Moreover, by adapting the work of He-Li in [HL21] in the context of harmonic U(n)-structures, we derive an improved monotonicity formula, in tandem with a similar development in [FLMSE22]; in fact, by further mobilising that paper's abstract theory for harmonic flows, we conclude long-time existence given small initial energy.
Finally, Section 6 illustrates different regimes of the harmonic flow with concrete examples.In §6.1 we study the flow on certain Lie groups and exhibit explicit solutions converging to harmonic Sp(2)Sp(1)-structures in infinite time.In particular, we exhibit a harmonic Sp(2)Sp(1)-structure on the manifold SU(3).In §6.2 we construct an example of a steady gradient soliton of the flow, which to our knowledge is the first explicit nontrivial soliton of a harmonic flow of geometric structures.
Since the stabiliser of Ω in GL(8, R) is isomorphic to Sp(2)Sp(1) ⊂ SO(8) (see also Proposition 2.7), it follows that Ω defines, up to homothety, both a metric g Ω and volume form vol Ω .In the above notation, these are pointwise given by g Ω = dx 2 1 + dx 2 2 + dx 2 3 + dx 2 4 + dx 2 5 + dx 2 6 + dx 2 7 + dx 2 8 , (2.5) The action of Sp(2)Sp(1) on R 8 corresponds to the usual left-action of Sp(2) on H 2 and right-action by Sp(1).This representation can be seen more concretely be way of Salamon's E-H formalism, as follows.The complexified (co)tangent bundle can be viewed as the Sp(2)Sp(1)-module where E (respectively H) is the associated vector bundle to the standard representation of Sp(n) (respectively Sp(1)) on C 2n (respectively C 2 ), see also [Sal82].In what follows we shall often ignore the fact that we are complexifying the tensor bundles of M , and use the same notation for the complexified and the underlying real vector bundles, as all these spaces admit a real structure owing to the quaternionic structure.

Representation theory and intrinsic torsion
A description of the tensor bundles on quaternion-Kähler 8-manifolds in the E-H notation can be found in [Sal89,Swa91], but for our purposes we shall need the following more concrete description, found in [FS22].
We shall also need the decomposition of the space of 4-forms.First note that the Hodge star operator * splits the space of 4-forms into self-dual and an anti-self-dual components: These further decompose into irreducible Sp(2)Sp(1)-modules as follows: Recall also that there is a natural action of Λ 2 ∼ = so(8) ⊂ End(R 8 ) on the space of k-forms given by When k = 2, observe that this is just the usual Lie bracket operation in so(8).Since in our situation we have the quaternion-Kähler 4-form Ω, this gives rise to an 'infinitesimal action' operator ⋄ : Λ 2 → Λ 4 , defined on simple 2-forms by Proof.Since ⋄ is an Sp(2)Sp(1)-equivariant map, and Λ 2 0 E ⊗ S 2 H is the only Sp(2)Sp(1)-module contained in both Λ 2 and Λ 4 , by Schur's lemma it suffices to check that it is non-zero.
Remark 2.1.1.More generally, the diamond operator ⋄ can be naturally extended to the action of on Ω as above by (α ⊗ β) ⋄ Ω := α ∧ (β ♯ Ω).Then the same argument as in Lemma 2.1 shows that there is an isomorphism 15 of the Sp(2)Sp(1)-structure determined by Ω can now be defined as (2.12) In order to extract T from the above expression we need to invert the isomorphism ⋄ : Λ 2 15 → Λ 4+ 15 .To do so we first note that given an arbitrary 4-form κ one can define a triple contraction operator 3 : Λ 4 → Λ 2 , given on simple 4-forms, by In particular, by taking κ = Ω we get the following Sp(2)Sp(1)-equivariant map By inspecting the decomposition of 2and 4-forms into irreducible Sp(2)Sp(1)-modules, and by Schur's lemma, we know that either ι 3 is zero or it restricts to an isomorphism Λ 4+ 15 → Λ 2 15 .
Lemma 2.2.The operator ι 3 satisfies (2.14) Proof.Since the result is algebraic, it suffices to work at a point in M 8 .Given a simple 2-form α ∧ β, we want to compute (α ∧ β) ⋄ Ω.We now make two convenient assumptions, without loss of generality.Since Ω is invariant by Sp(2)Sp(1), and Sp(2) acts transitively on S 7 , we can set α = dx 1 while leaving Ω unchanged; furthermore, as the stabiliser of dx 1 in Sp(2) is isomorphic to Sp(1), we can assume that β = b•dx 2 +c•dx 3 +u•dx 4 +v •dx 5 , for some constants b, c, u, v. Hence our typical 2-form can be written as where π i j : Λ i → Λ i j denotes the projection map.A direct computation now shows that Using Lemma 2.2 we can now rewrite the intrinsic torsion tensor as (2.15) Finally we also record one key identity between the diamond operator and the triple contraction, for later use. (2.16) Proof.Again this is an algebraic relation, which can be assessed at a point.By choosing geodesic normal coordinates, we can assume that we are working on (R 8 , Ω).Since ⋄ and 3 are both linear operators, it suffices to consider the case when α and β are of the form v ⊗ w ∈ Λ 2 15 ∼ = Λ 2 0 E ⊗ S 2 H, for some unit vectors v, w.Furthermore we know that Sp(2)Sp(1) acts as SO(5 as required (compare with Remark 2.3.1, below).Now, as Sp(2)Sp(1)-modules, we have the following decomposition: Observe that there is no . Indeed, in the above example, The fact that the last term lies in Λ 2 10 is easily checked using (2.9), and this concludes the proof.
While in this paper we aim to deal with situations in which T does not vanish identically, it is worth recalling some properties of torsion-free quaternion-Kähler structures (in all dimensions).Quaternion-Kähler manifolds are always Einstein i.e.Ric(g) = λg cf.[Bes08,Sal82].If moreover λ = 0, then M is locally a hyperKähler manifold, so this case is usually excluded from the definition of quaternion-Kähler manifolds.If λ > 0, then M is compact, while if λ < 0 then M is non-compact.Poon and Salamon showed that the only compact quaternion-Kähler 8-manifolds are the symmetric spaces [PS91]: and G 2 SO(4) .
By contrast, LeBrun showed in [LeB91] that there are infinitely many examples in the non-compact case, see also [Bes08,Fow21] for other non-compact examples.Furthermore, by analysing the decomposition where K is irreducible Sp(2)Sp(1)-module defined by Swa91]).The intrinsic torsion T = 0 if, and only if, dΩ = 0 and the differential ideal ω 1 , ω 2 , ω 3 is algebraic.
While, for quaternion-Kähler structures in dimensions strictly greater than 8, being torsion-free is equivalent to the 4-form Ω being closed, there do exist quaternion-Kähler 4-forms in dimension 8 which are closed but not torsion-free [Sal01].
Remark 2.4.1.Since, in dimension 8, Ω is a self-dual 4-form, if dΩ = 0 then one often calls the induced quaternion-Kähler structure harmonic [CM15], which is is an altogether different meaning from our notion of harmonicity in the present context.Spin(7)-structures are another type of geometric structure arising on 8-manifolds by an algebraically special 4-form, under favourable topological conditions.Throughout this article we shall see that there is a rather close relation between harmonic Sp(2)Sp(1)and Spin(7)-structures, although their respective algebraic properties are quite different.Next we describe some common features of Sp(2)Sp(1)and Spin(7)-structures which, to the best of our knowledge, have not so far been described in the literature.

Bianchi identity for the torsion
We now derive a 'Bianchi-type identity' for the torsion tensor T , which will be useful later on in the derivation of a monotonicity formula.The terminology comes the fact that this identity arises due to the diffeomorphisminvariance of the torsion tensor, just as the usual Bianchi identity arises from the invariance of Riemann curvature, cf.[Kar07].
Proposition 2.5.The torsion tensor T satisfies the following 'Bianchi-type identity' where we are viewing R(Y, X) as a 2-form.Moreover, Proof.From (2.15), we have where we used the fact that ∇ preserves g and hence 3 .Skewsymmetrising in X and Y , we get and the first part of the Proposition now follows from Lemma 2.2.For the second part, observe that and hence as a 2-form it lies entirely in S 2 H ⊕ S 2 E ⊂ Λ 2 ; likewise for (∇ X Ω) 3 (∇ Y Ω).This concludes the proof.
An important consequence of the Bianchi identity (2.18) is that the skew-symmetrisation of the covariant derivative of T is fully controlled by the 15-dimensional component of the curvature tensor (which depends only on the metric) and a quadratic term involving T .Remark 2.5.1.In [DLSE21] an analogous Bianchi-type identity is derived for the torsion of a Spin(7)-structure, say determined by Φ.The proof there is more computational in nature but the argument follows exactly as described above by replacing Ω by Φ and Ω 2 15 ∼ = (sp(2) ⊕ sp(1)) ⊥ by Ω 2 7 ∼ = spin(7) ⊥ ; so by analogy with (2.22), we have the Spin(7)-module decomposition More generally, suppose that we have the orthogonal reductive splitting so(n) = h ⊕ m and that H ∼ = stab(ξ) for some tensor ξ (in our case G = SO(8), H = Sp(2)Sp(1) and ξ = Ω), then we know that the torsion tensor T ∈ Ω 1 ⊗ m cf.[Sal89].The last term in (2.18) essentially corresponds to the Lie bracket of T (X) and T (Y ) and hence belongs to [m, m].So, if g = h ⊕ m corresponds to the Lie algebra decomposition of a symmetric space, then [m, m] ⊂ h.This is indeed the case in our situation and also eg. when G = SO(8) with H = Spin(7) [DLSE21], G = SO(2n) and H = U(n) [HL21].Thus, such a Bianchi identity must always hold in those contexts.This insight allows us to interpret proofs in these various contexts from a unified perspective, and thereby avoid unnecessarily complicated computations, as we shall illustrate below.
To establish the second claim, recall that the curvature operator in fact lies in the kernel of the skewsymmetrisation map defined by wedging the 2-forms in sp(2) ⊕ sp(1) cf.[Sal89]; this corresponds to the symmetry of the algebraic Bianchi identity.On the other hand, we have the irreducible decomposition where we again use the E-H formalism of (2.7).The traceless component of the Ricci tensor belongs to Comparing with the irreducible decomposition of Λ 4 , we see that the kernel of A always contains a copy of R (the curvature tensor of HP 2 ) and of S 4 E. Testing a few simple examples shows that the map A has a non-zero image in each irreducible component of Λ 4 ; for instance one can consider the wedge products of ω i ∈ sp(1) and dx 12 − dx 34 ∈ sp(2).Hence from Schur's Lemma it follows that A must be an isomorphism on all the remaining modules and this gives the result.
Remark 2.6.1.A similar argument was used in [Kar07, Corollary 4.12] to give a direct proof that G 2 -manifolds are indeed Ricci-flat (although the proof therein relies on a calculation in index notation for the G 2 -structure 3-form ϕ, the essence is the same).Our argument above shows that in fact, given a Bianchi-type identity, a similar proof can be used to show Ricci flatness for other special holonomy groups, cf.[FLMSE22].
2.3 Relations between Sp(2)Sp(1)-structures and Spin(7)-structures A Spin(7)-structure on an 8-manifold M is determined by an algebraically special 4-form Φ pointwise modelled on with ω i as defined above by (2.2)-(2.4).Since Spin( 7) is a subgroup of SO(8), it follows that Φ determines (up to homothety) both a metric g Φ and a volume form In the above pointwise coordinates, these coincide with the expressions (2.5) and (2.6), respectively.It is worth pointing out that a Spin(7)-structure endows each tangent space of M with the algebraic structure of the octonions O, while an Sp(2)Sp(1)-structure endows the tangent space with the algebraic structure of the quaternic plane H 2 .
As demonstrated by Karigiannis in [Kar05, Theorem 4.3.5], the metric g Φ can be explicitly extracted from Φ via the expression where X, e i ∈ T p N form a positively oriented basis of T p N , i.e.
In fact, [Kar05, Lemma 4.3.3]shows that the right-hand side of (2.24) is independent of the choice of extension of X to the basis {X, e i } of T p N : if one chooses a different extension {e ′ i } so that then the numerator of (2.24) changes by a factor of (det(P ) 2 det(P ) 7 ) 1/3 , but so does also the denominator and hence the quotient is indeed invariant.An inspection of the proof of the latter assertion reveals that the invariance of the right-hand side still holds if Φ is replaced by any 4-form Υ which is non-degenerate, i.e.Υ ∧ Υ > 0. In particular, this leads to the following analogous result: Proposition 2.7.The quaternion-Kähler metric g Ω is obtained from Ω via the expression where X ∈ T p N and {X, e i } ∈ T p N is any extension of X to a postively oriented basis of T p N .
Proof.We know from [Kar05, Lemma 4.3.3]that the right-hand side of (2.25) is independent of the extension {e i }, and it is Sp(2)Sp(1)-invariant, so it suffices to check that (2.25) holds for a preferred extension.Identifying Ω at p with the standard Ω 0 on R 8 , we let X be an arbitrary vector, e i := I i (X), for i = 1, 2, 3, and we choose {e 4 , e 5 , e 6 , e 7 } to be orthogonal to {X, e 1 , e 2 , e 3 }.The result will now follow from the next Lemma.
Lemma 2.8.Given V, W ∈ Γ(T N ), consider the orthogonal decomposition W = aV + W 1 + W 2 , where W 1 denotes the projection of W onto I 1 V, I 2 V, I 3 V and W 2 denotes the projection of W onto the orthogonal complement of the quaternionic span of V .Then we have In particular, given another vector field U = bV + U 1 + U 2 , by polarising the above we get (2.27) Proof.Since the result is algebraic, we may work on R 8 without loss of generality.Furthermore, Sp(2) acts transitively on the unit sphere, so we can assume V = ∂ x 1 , and hence Since the stabiliser in Sp(2) of a unit vector is isomorphic to Sp(1), we can also set W 2 = ∂ x 5 .The result now follows from a straightforward computation.
Remark 2.8.1.Given a unit vector V ∈ T p N , the subspace I 1 V, I 2 V, I 3 V can now be defined as the span of those unit vectors W for which the right-hand side of (2.26) is equal to 18vol Ω .This gives a concrete way of defining the 2-sphere of almost complex structures, i.e. the twistor space, starting from Ω only.
3 Harmonic Sp(2)Sp(1)-structures In this section we describe how the notion of harmonic Sp(2)Sp(1)-structures arises from the general framework of harmonic H-structures introduced in [LSE19].

Harmonic homogeneous Sp(2)Sp(1)-sections
We begin by describing Sp(2)Sp(1)-structures as sections of a homogeneous fibre bundle.We shall be brief here and refer the reader to [LSE19] for more details.First we fix an oriented Riemannian 8-manifold (M, g) and denote by p : F → M its orthonormal frame bundle, with fibre G = SO(8).Since Sp(2)Sp(1) ⊂ SO(8), cf.Proposition 2.7, the quotient q : F → N := F/Sp(2)Sp(1) defines a principal H = Sp(2)Sp(1)-bundle, which in turn is a smooth fibre bundle π : N → M with the homogeneous space SO(8)/Sp(2)Sp(1) as typical fibre.It follows that Sp(2)Sp(1)-structures compatible with the metric g are determined by sections of π.
Let Λ 2 15 → N be the vector bundle associated to q with fibre Λ 2 15 , whose points are the Sp(2)Sp(1)equivalence classes defined by the infinitesimal action of w ∈ Λ 2 15 on z ∈ F i.e.
where w * z is a fundamental left-invariant vector field.
This defines a vector bundle isomorphism and, since the 15 ) of the Levi-Civita connection is Sp(2)Sp(1)-equivariant and q-horizontal, it projects to a homogeneous connection form f ∈ Ω 1 (N, Λ 2 15 ) defined by: Recall that a vector v ∈ T N describes an incidence condition at a Sp(2)Sp(1)-class of frames.On πvertical vectors, the connection form f coincides with the canonical isomorphism (3.1), while π-horizontal vectors lie in the kernel i.e.
Assuming that M is compact, endowing the fibres of N with the metric induced by the bi-invariant metric on SO(8), and considering the metric induced by g on H, we define the energy functional where d V denotes the projection of dσ on V. From the aforementioned discussion, we have that Note also that, since the horizontal space H is endowed with the metric g, the total energy of σ is equal to (3.5) and a constant multiple of the volume of M cf.[LSE19, Lemma 3].In what follows, we shall denote by ∇ the associated Levi-Civita connection to the latter metric on N .The results in [LSE19] show that (3.5) corresponds to the L 2 -norm on the intrinsic torsion associated to Ω σ : For the reader's convenience we summarise a few key relevant results in [LSE19]: Proposition 3.1.The covariant derivative of the universal section Ξ is given by Harmonic Sp(2)Sp(1)-structures, defined as the critical points of (3.5), satisfy the Euler-Lagrange equation: where X ∈ T M and ∇ ω denotes the induced connection on π * Λ 2 .
The harmonic section flow, defined as the negative gradient flow of (3.5), starting from an Sp(2)Sp(1)structure defined by σ 0 , is given by Appealing to Proposition 3.1 and using the map I, we can then reinterpret the above flow more concretely in terms of a geometric flow for Ω t (see §4 below).This is the same procedure that leads to the harmonic flows of G 2 -, Spin(7)and U(n)-structures in [DGK19, Gri19, DLSE21, HL21].

Examples of harmonic quaternion-Kähler structures
In this section we construct several explicit examples of strictly harmonic QK structures i.e. harmonic QK structures which are not torsion-free.We also refer the reader to §6.1.2below for an example on the Lie group SU(3).The main result of this section is that conformally parallel Sp(2)Sp(1)-structures are harmonic.In fact our proof applies to G 2 -and Spin(7)-structures as well, thereby extending the result of Grigorian in the G 2 case, cf.[Gri19, Theorem 4.3].
It is well-known that if (M, g) a complete Einstein manifold, aside from the round sphere, then g is the unique Einstein metric in its conformal class (up to homothetic rescaling), cf.[KR95].In particular, if g is a quaternion-Kähler metric, then f 2 g cannot be an Einstein metric unless f is constant.So any harmonic structure in the class [f 2 g], if any exists, cannot be torsion-free.While it is natural to expect that the conformally rescaled metric will converge back under the Ricci flow to the Einstein metric (which is a Ricci soliton), the latter behaviour cannot happen in our case, since the harmonic flow preserves the metric.So this naturally motivates searching for harmonic structures in such conformal classes.

Example on a flat torus
Consider the flat torus T 8 with the quaternion-Kähler structure Ω 0 defined by expression (2.1).We define a conformally flat quaternion-Kähler structure by and denote the associated Sp(2)Sp(1) coframing by e i := f (x 1 )dx i and its dual by e i := f (x 1 ) −1 ∂ x i .In terms of decomposition (2.17), we know that the torsion T takes values in Ω 1 8 , and it is essentially determined by the 1-form df .Indeed a direct calculation shows that the torsion tensor is explicitly given by T (e 1 ) = 0, and from this one finds that ∇ e i (T (e i )) = 0, for i = 1, . . .8.
Another simple computation shows that ∇ ∇e i e i Ω = 0.
Combining the above, we have Thus, we have just shown that Ω, as defined by (3.12), determines a harmonic section for the conformally flat metric f (x 1 ) 2 g T 8 .Observe that T = 0 if and only if f (x 1 ) is constant, as expected.

Example on the hyperbolic quaternionic plane
Let us now consider the hyperbolic quaternionic plane HH 2 .Topologically HH 2 is diffeomorphic to R 8 , but as a Riemannian manifold it is the symmetric space Sp(2,1) Sp(2)Sp(1) i.e. the non-compact dual of HP 2 = Sp(3) Sp(2)Sp(1) .In particular, it has holonomy group equal to Sp(2)Sp(1).As a cohomogeneity one manifold, under the action of the quaternion Heisenberg group, whose Lie algebra is given by (0, 0, 0, 0, 12 + 23, 13 − 24, 14 + 23), we can express the quaternion-Kähler metric as where the 1-forms α i are defined by and s ∈ (−∞, 1), cf.[Fow21,GLPS02].As in the previous example, we consider conformal metrics given by f (s) 2 g HH 2 .We define an Sp(2)Sp(1)-coframing by setting One again verifies that ∇ ∇e i e i Ω = 0, and that the torsion tensor is explicitly given by As above we find that ∇ e i (T (e i )) = 0, for each i, and hence div(T ) = 0.
Thus, the above conformally parallel quaternion-Kähler structures indeed define harmonic sections.
The above examples seem to suggest that conformally parallel Sp(2)Sp(1)-structures might always be harmonic; in fact this is known to be true in the G 2 case, cf.[Gri19, Theorem 4.3].We shall now show that this holds for a larger class of H-structures, including Sp(2)Sp(1) and Spin(7).
Proposition 3.2.Let (M, g, ξ) be a Riemannian manifold with holonomy group contained in H ⊂ SO(n), where ξ is a parallel k-form which determines the holonomy reduction.Consider the conformal data given by g = e 2f g and ξ = e kf ξ on M .The intrinsic torsion T , defined by where π m denotes the orthogonal projection in Λ 2 ∼ = h ⊕ m cf.§3.1 and [LSE19, Part I].
Proof.Let ∇ (respectively ∇) denote the Levi-Civita connection of g (respectively g).For any p-form α, we have Applying the above to α = ξ and using the fact that ∇ξ = 0, we have that where we decorate with˜quantities defined with respect to g, and we also used that (X ♭ ∧df )⋄α = ( X♭ ∧df ) ⋄ α.
It now follows that the intrinsic torsion of ξ is given by (3.17) It is worth pointing out that the projection map π m only depends on the conformal class of ξ, so there is no ambiguity here.Moreover, such a formula for the intrinsic torsion was indeed to be expected, because T vanishes if and only if f is constant, so T has to correspond to some pairing between g and df .The reader can also observe this in the explicit examples given in § §3.2.1, 3.2.2.Since g has holonomy contained in H, it follows that ∇ and π m commute.Hence applying (3.15) again, with α = π m ( X♭ ∧ df ), gives Using (3.15) yet again, with α = X♭ , and substituting in the above yields the result.
Proof.Taking ξ = Ω in Proposition 3.2, we have , where E i denotes a local orthonormal framing with respect to g. Working at a point and identifying Ω with Ω 0 and E i with ∂ x i , it suffices to check directly that the last two terms in (3.18) vanish as well, when summing over the E i .Since Sp(2)Sp(1) acts transitively on the unit sphere, we can also identify ∇f with c∂ x 1 at a point, to further ease computation.In those terms it is straightforward to check that div g( T ) = 0.
The argument in the above proof can also be applied to the G 2 and Spin(7) cases.This reduces the problem of computing the divergence of torsion for a conformally parallel structure to verifying that that the last two terms in (3.18) vanish when summing over i, which is essentially just a pointwise computation.The result in the G 2 case is already known, cf.[Gri19, Theorem 4.3]), and our argument extends easily to the Spin(7) case: Corollary 3.4.A locally conformally parallel Spin(7)-structure on an 8-manifold is harmonic.
Proof.Repeat the proof of Theorem 3.3 with Ω replaced by the Spin(7)-structure 4-form Φ, which is pointwise modelled on (2.23).
4 Quaternion-Kähler harmonic flow: basic properties In this section we derive the harmonic flow equation for quaternion-Kähler (QK) structures and define the corresponding notion of soliton.
We can express the harmonic flow (3.10) for H = Sp(2)Sp(1) in terms of an evolution equation for the defining 4-form Ω(t).First, in terms of the isomorphism (3.1) between the vertical component of T N and the bundle m, we have where we used Proposition 3.1 and expression (2.15) for the intrinsic torsion.Extending the connection form f to M 8 × R t , and performing the same computation as above, we get where we inverted the diamond operator ⋄ in (3.7) using the operator 3 .
In view of the results of the previous section, the harmonic flow of a Sp(2)Sp(1)-structure starting at Ω 0 becomes: We shall also refer to the above flow as the QK harmonic flow.As an instance of the general theory of harmonic H-flows, we already know that the flow admits a unique short-time solution, given smooth initial data.Moreover, if the flow exists for a maximal time T max , then In this section we will develop the technical results necessary to study the behaviour of the flow as t → T max , and investigate under what circumstances T max can be extended to infinity.

Evolution of the intrinsic torsion
Let us derive the evolution of the torsion tensor T under the harmonic flow (4.2).
Proposition 4.1.Under the harmonic flow (4.2) the intrinsic torsion T evolves by for X ∈ Γ(T M ).Moreover, Proof.From (2.15), we have where for the first equality we used the fact that g is unchanged along the flow and hence so are ∇ and 3 , and for the second equality we use (4.2).The first part of the proposition now follows from where we again use the fact that 3 only depends on g and hence is invariant under ∇.For the second part we use the same argument as in the proof of Proposition 2.5 i.e.
The evolutions of Dirichlet energy and density now follow immediately: Corollary 4.2.The norm square of T evolves by Note that (4.7) was to be expected, since the harmonic flow is just the negative gradient flow of the energy functional.Furthermore, however much the L 2 -norm of T decreases under the flow, it can still concentrate over certain points on M , thereby resulting in singularities.In order to analyse such behaviour, we need a monotonicity formula, which will derive in §5.2.Next we show that the harmonic flow admits a parabolic rescaling.

Parabolic rescaling
In the study of geometric flows one often encounters finite-time singularities.These singularities are in many cases modelled on soliton solutions to the flow, and hence classifying those becomes an important problem.To find these solitons as one approaches a singularity, one performs a parabolic scaling i.e. a rescaling of geodesic distance by x → cx while time scales by t → c 2 t, for some constant c.Provided that we have a compactness theorem, this allows one to take a suitable limit of the flow and thus to extract information about the singularity; this procedure is well-known for eg.for the Ricci and mean curvature flows.To perform the analogous scaling in our context, we first consider the behaviour of the intrinsic torsion under a homothetic rescaling, in accordance with the homogeneity degree of the QK 4-form.
Proof.Observe that the homothetically rescaled metric is given by gΩ = c 2 g Ω , while the Levi-Civita connection remains unchanged i.e. ∇ = ∇.Thus, we compute where ⋄ denotes the associated operator to gΩ .It is worth emphasising that by definition the operator ⋄ acting on 2-forms depends on the metric g.Corollary 4.4.If Ω t is a solution to (4.2) defined for t ∈ [0, T max ) then under the parabolic rescaling (Ω t , t) → ( Ωt := c 4 Ω, t := c 2 t), Ωt is again a solution to (4.2) but now defined for t ∈ [0, c 2 T max ).
Proof.We compute directly d Ωt d t = c 2 div g (T ) ⋄ Ω = div g( T ) ⋄ Ω, using (4.2) for the first equality and Lemma 4.3 for the second one.
Next we introduce the notion of solitons for the harmonic QK flow (4.2).

Harmonic Sp(2)Sp(1) solitons
The simplest solutions to a geometric flow are those that evolve by scaling symmetry of the flow equation; these are called solitons and arise naturally when analysing singularities of the flow (see Theorem 5.12 below).We now describe what harmonic solitons look like in our context.Definition 4.5.A solution {Ω(t)} of the harmonic QK flow (4.2) is said to be self-similar if there exist a function ρ(t), with ρ(0) = 1, and a family of diffeomorphisms {f (t) : M → M }, with f (0) = Id, such that (4.8) We shall now justify the name self-similar solution.Denoting by W (t) ⊂ X (M ) the infinitesimal generator of f (t) ⊂ Diff(M ), the stationary vector field of a self-similar solution is defined by (4.9) From (4.8) we immediately deduce that the metric evolves by On the other hand, since the harmonic flow is isometric, i.e. its time-derivative g ′ (t) vanishes, In particular, this shows that ρ(t) completely determines f (t) (up to isometry).Furthermore, specialising [DLSE21, Lemma 2.9] to the case H = Sp(2)Sp(1), we know that the torsion tensor T (t) of Ω(t) satisfies The above can be also shown quite easily using (4.2) and (4.8).We should emphasise that the projection map π 2 15 : Λ 2 → Λ 2 15 is also time-dependent, since it is determined by Ω(t).The above motivates the following definition.
Definition 4.6.A harmonic Sp(2)Sp(1)-soliton on a Riemannian manifold (M 8 , g) is given by a triple (Ω, X, c), where Ω induces the metric g, X is a vector field and c is a constant such that (4.12) According to whether c < 0, c = 0 or c > 0, the corresponding soliton is said to be shrinking, steady or expanding, respectively.
We can now show that solitons indeed give rise to self-solution solutions of (4.2).
Note that a soliton does not determine a unique self-similar solution, in fact for any function h, depending only on t, such that h(0) = 0 and h ′ (0) = − c 2 , we can set For instance, setting h(t) = − c 2 t we get eternal skrinkers and expanders as well.It is a classical result that the only complete Riemannian manifold with a non-Killing homothetic vector field is Euclidean space, cf.[Tas65].We immediately deduce that: Corollary 4.8.Shrinking and expanding solitons of the QK harmonic flow (3.10) are always isometric to Euclidean R 8 .
Note that there are plenty of non-parallel structures Ω on R 8 inducing the Euclidean metric, so it natural to ask whether there exists any non-torsion-free shrinking or expanding soliton.We shall answer in the affirmative with an explicit example of a steady soliton in §6.2, by means of the following simple idea.If X = ∇f is some gradient vector field, then dX ♭ = 0 and hence gradient harmonic solitons satisfy div T = T (∇f ). (4.13) 5 Quaternion-Kähler harmonic flow: long-time existence and singularities By exploiting the similarities with the harmonic flow of Spin(7)-structures, we readily obtain a compactness theorem for the harmonic quaternion-Kähler flow.We also prove an almost-monotonicity formula, by building upon the recent work in [HL21] in the context of almost-Hermitian structures.Our monotonicity formula also applies to the Spin(7) case and hence leads to a stronger convergence result than in [DLSE21].In fact our proof of the monotonicity formula extends to a much more general class of H-structures, as established independently in [FLMSE22].In the last part we describe the singular set of the flow.

Compactness
If a solution to the harmonic flow (4.2) has a finite-time singularity, then we obtain a new sequence of solutions by performing parabolic rescalings.In order to analyse the singularity, we need to be able to take a limit of such a sequence, following the standard method used for instance for the Ricci flow and mean curvature flow.We begin by specifying the notion of limit in our context: Definition 5.1.Let (M 8 i , Ω i , g i ) be a sequence of complete Riemannian manifolds, with Sp(2)Sp(1)-structures determined by Ω i and marked points p i ∈ M 8 i .Then we call (M 8 , Ω, p) a limit of the sequence, and write if there exists a sequence of compact sets {U i } exhausting M 8 with p i ∈ int(U i ), and a sequence of diffeomorphisms {F i : such that, on every compact set K ⊂ M 8 and for each ε > 0, there exixts i 0 (depending on ε) such that where g 0 denotes a fixed reference metric on M 8 and ∇ is its corresponding Levi-Civita connection.
We can now state the compactness theorem for the QK harmonic flow.
Theorem 5.2.Let M i be a sequence of compact 8-manifolds with marked points p i ∈ M i , and let {Ω i (t)} denote a sequence of solutions to the QK harmonic flow (4.2) on M i defined for t ∈ (a, b).Suppose the following assumptions hold: where inj denotes the injectivity radius and C k are uniform constants independent of i.Then there exist a manifold M 8 , with a marked point p ∈ M , and a solution Ω(t) to (4.2) on M , defined for t ∈ (a, b), arising as the subsequential limit Proof.Since the proof is analogous to the G 2 and Spin(7) cases, we shall only highlight the key parts of the argument, referring the reader to [DGK19, Theorem 3.13] and [DLSE21, Theorem 4.19] for further detail.In order to obtain the limit space (M, g, p) as a complete pointed Riemannian manifold, we resort to the Cheeger-Gromov compactness theorem, cf.[Ham95,Theorem 2.3].This relies on hypotheses (5.2) and (5.3): condition (5.2) ensures that collapsing/degeneration-type phenomena does not occur, and (5.3) ensures that curvature does not concentrate along the limiting process.More precisely, Cheeger-Gromov compactness gives rise to an exhausting family of compact nested sets U i ⊂ M 8 and diffeomorphisms Note that here we are also using the fact that the metric is unchanged under the flow, i.e. g i (0) = g i (t).
Next we need to obtain the limit 4-form Ω(t), which, by contrast to the metric, does vary with time.In view of the Shi-type estimates for the general harmonic flow of H-structures [DLSE21, Proposition 2.16], Assumption (5.1) guarantees uniform bounds on all derivatives of torsion, for all time t ∈ (a, b).Appealing to the Arzelá-Ascoli theorem, we can extract a 4-form Ω(t) on M as the limit of F * i Ω i (t).Now, it is not a priori clear that the limit Ω(t) also defines an Sp(2)Sp(1)-structure on M .To deduce to latter, we take the limit of (2.25) and use the fact that the Riemannian metric g arises as the Cheeger-Gromov limit of (g i ).This concludes the proof.

The almost-monotonicity formula
This section is strongly based on the celebrated methods developed by Hamilton in [Ham93], which we invite the unfamiliar reader to consult.Let (M 8 , g) be a complete Riemannian manifold.For p ∈ M 8 , we denote by u (p,t 0 ) a positive fundamental solution of the backward heat equation, starting from the delta function at p, at time t 0 , i.e.
∂ ∂t and we set In what follows we shall simply write u = u (p,t 0 ) .Suppose now that we have a solution to the harmonic QK flow (4.2) on (M 8 , g), defined for t ∈ [0, t 0 ).Then, following [GH96], we define the functional which is invariant under parabolic rescaling -unlike the energy functional E. Our first goal is to prove that Θ satisfies an almost-monotonicity formula.We begin by proving the following key lemma.
Lemma 5.3.Under the harmonic QK flow (4.2), the functional Θ evolves by where {E i } denotes a local orthonormal framing.
Proof.Using the definition of u, a direct calculation shows that Let us consider the last summand in the above expression.Integrating by parts we have where we used the Bianchi type identity of Proposition 2.5 and the fact that T (E i ) ∈ Ω 2 15 for the penultimate equality.Another integration by parts, together with Corollary 4.2, shows that (5.6) Combining the above, we have Equipped with the above lemma, we now apply the same argument as in [DLSE21, Theorem 5.2] to obtain the following monotonicity result: Theorem 5.4 (Weak almost-monotonicity formula).Let {Ω(t)} be a solution of the harmonic QK flow (4.2) on (M, g), and let 0 < τ 1 < τ 2 < t 0 .The following assertions hold: 1.If M is compact, then there exist constants K 1 , K 2 > 0, depending only on the geometry of (M, g), such that 2. If M = R 8 with its Euclidean structure, then Θ(Ω(τ 2 )) ≤ Θ(Ω(τ 1 )). (5.8) Proof.The proof follows Hamilton's original argument, which also appears in detail in [DGK19, DLSE21], so we shall only outline the key steps.It is worth emphasising that, although we consider here the structure group H = Sp(2)Sp(1), whereas [DGK19] considers H = G 2 and [DLSE21] considers H = Spin(7), the argument is essentially the same with T ∈ Γ(Λ 1 ⊗ h ⊥ ).In other words, the proof is independent of the structure group H ⊂ SO(n), so long as a Bianchi-type identity holds.We illustrate this below by avoiding multi-index computations specific to some choice of H.
Integrating by parts the last term of (5.5), (5.9) and using again (2.18) gives an integral involving u, T , R and ∇R only.First note that the curvature terms only depend on g and hence are bounded.Since M u vol = 1 and E(Ω(t)) is decreasing, it follows that (5.9) is bounded by where C is a constant determined by the geometry of (M 8 , g).For the second term of (5.5), using again that E(Ω(t)) is decreasing, a standard argument shows that we can bound it by Combining the above, we have Let ξ(t) be a solution of the ODE Then we can rewrite (5.10) as and the first claim now follows.The second claim is immediate from the explicit expression of the backwards heat kernel, see (5.13) below.
The key application for the above monotonicity formula is the following ε-regularity theorem, which we shall use to study singularities of the flow in §5.3.We follow the approach employed by Grayson-Hamilton in the context of harmonic map heat flow [GH96].
Suppose {Ω(t)} t∈[0,t 0 ) is a solution to the harmonic QK flow (4.2), inducing g and satisfying E(Ω(0)) Proof.In view of the weak almost-monotonicity formula in Theorem 5.4, the proof of Theorem 5.5 is now completely analogous to those in [DGK19, Theorem 5.7] and [DLSE21, Theorem 5.5], so we shall only detail its key moments.Suppose by contradiction that, for any sequences ε i , ρi → 0, there exist ρ i ∈ (0, ρi ] such that, given any r i ∈ (0, ρ i ) and C i → ∞, there exist counterexamples {Ω i (t)} t∈[0,t i ) such that but for some , where (x i , ti ) denotes the point where the maximum is attained, we can consider the parabolic rescaled flow as in Corollary 4.4, with c = Q i .Using (5.12) and the definition of Q i , we find that Now compactness, from Theorem 5.2, implies that the limit of the rescaled flow (M, Ω i (t), xi ) is the ancient solution (R 8 , Ω ∞ (t), 0) and satisfies |T (Ω ∞ )(0, 0)| = 1.On the other hand, taking the limit of Θ in the monotonicity formula of Theorem 5.4 shows that |T (Ω ∞ )(0, 0)| = 0, which gives the desired contradiction.
Although the almost-monotonicity formula of Theorem 5.4 is sufficient for analysing singularities of the flow, we shall need a more refined monotonicity formula to obtain long time existence given small initial energy.To this end we modify the functional Θ as follows.
We may assume, without loss of generality, that (M 8 , g) has injectivity radius at least 1, and introduce geodesic normal coordinates x i in a unit ball around any given point p ∈ M via the exponential map Let φ be a test function on R 8 , with compact support in B(0, 1) and constant on B(0, 1/2), and let G denote the usual Euclidean backward heat kernel on R 8 : where We should emphasise that | • | here denotes the norm with respect to g (not the Euclidean metric) and also that the integrand is only supported on B(p, 1) ∼ = B(0, 1).In contrast to the functional Θ defined by (5.4), observe that now we are using the Euclidean heat kernel G, rather than u, and we are only working locally in geodesic unit balls where the function φ is supported.As with Θ(t), we shall compute the evolution of Z(t).
A subtle point here is that, in geodesic normal coordinates at p, the metric g is approximately Euclidean, and we already saw in Theorem 5.4 that Θ(t) is indeed monotone on Euclidean R 8 , so the trick is to exploit this approximation in B(p, 1) using the functional Z(t).This was done for the harmonic map heat flow (of maps) by Chen and Struwe [CS89], and it was recently adapted to the harmonic flow of almost-Hermitian structures by He and Li [HL21], albeit with some subtleties, see Remark 5.6.1 below.
Theorem 5.6.For any N > 1 and t 1 , t 2 such that the following monotonicity formula holds: where C is a constant depending only on (M, g), and Proof.First we compute Gφ 2 vol, using Corollary 4.2 for the second term, and the last term comes from differentiating G. Integrating by parts, we have denotes an endomorphism.Since vol = |g|dx, in local coordinates we compute where we now integrated by parts on R n .Combining all of the above, we have so far Since we are working in B(p, 1) with geodesic normal coordinates we know that in a neighbourhood of x = 0, and hence, using we get the bound Note that here we again used the fact that G can only concentrate at x = 0, but |∇φ| and the Γ k ij vanish at x = 0. We now have the estimate where we used that is uniformly bounded, for 0 < t < t 0 .Analogously, we get As argued above, for t 0 − t > 1/N we have G < CN n/2 , so On the other hand, if t 0 − t < 1/N , then Young's inequality gives Gathering all terms, Remark 5.6.1.A similar approach can be found in [HL21, Theorem 3.1], in the case of almost Hermitian structures, with H = U(n/2).We must highlight however two important differences, stemming from what we believe to be a minor overlook of some features in the original proof by Chen-Struwe [CS89] for the harmonic heat flow of maps, as opposed to tensors, eventually leading us to a different function f in the monotonicity formula.First, the authors integrate by parts in (5.15) with respect to the Euclidean metric, rather than g, and this results in an additional term involving derivatives of g (confusingly denoted by ∇g ij therein).This is indeed the procedure adopted in [CS89, Lemma 4.2], where it is not problematic because there are no covariant derivatives of tensors involved, only partial derivatives of maps.
Second, our integration by parts in (5.17) gives rise to a term involving g ik x i x j , which is equal to |x| 2 only to zeroth order in the unit geodesic ball, since g ij = δ ij + O(|x| 2 ).This yields a term of order |x| 4 in (5.18), which does not appear in [HL21].It is the bound on this term that finally requires a different choice of f .Remark 5.6.2.We now highlight the key features of the proof that generalise immediately to other structure groups H ⊂ SO(n).First, we need the squared norm of which is used in (5.14).Second, we need the Bianchi identity to establish that which is used in (5.16).Aside from these two ingredients, the rest of the calculations is completely independent of the structure group H. From the results in [DGK19, LSE19] and [DLSE21] we immediately deduce that Theorem 5.6 also applies to the cases of H = G 2 and H = Spin(7).
Next we define the functional and a similar argument as in [HL21, Theorem 3.2] now yields: Theorem 5.7.For any N > 1 and R 1 , R 2 such that the following monotonicity formula holds: where C 0 , C > 0 are constants depending only on (M, g) and The same argument applies to the term involving (div T ⋄ Ω) 3 (T ⋄ Ω) = −(T ⋄ Ω) 3 (div T ⋄ Ω).So combining the above we have On a compact manifold, the kernel of the rough Laplacian consists of parallel 2-forms, so indeed it is orthogonal to div T .
Note that Lemma 5.9 does differ depending on the H-structure.For instance, in the Spin(7)-case we have We refer the reader to [DGK19, Lemma 5.11] for the G 2 case.

Singularities of the flow
In this section we investigate the formation of singularities along the harmonic QK flow.Let us consider a solution {Ω(t)} to (4.2) defined for t ∈ [0, T max ), and define the singular set S of the flow by where ε and ρ are as in Theorem 5.5.This wording is justified by the next result, in the same vein as [GH96, Theorem 4.3].
Proof.We adapt a similar scheme of proof as in [GH96, Theorem 4.3], also found in [DLSE21,Theorem D].First let's assume that H 6 (S) is finite.Since S is a closed set of finite 6-dimensional measure, Theorem 4.2 of [GH96] asserts that there exists S ′ ⊂ S such that Now, the solution to the backwards heat equation is given by and it satisfies u S ′ (x, t) ≤ C T max − t . (5.23) From the definition of S in (5.22), we have that Θ (y,Tmax) (Ω(T max − ρ 2 ))dH 6 (y), so using definition 5.4 of Θ and (5.23) we have To conclude the proof note that if instead H 6 (S) was infinite then one could choose subset S ′ ⊂ S with arbitrarily large 6-dimensional Hausdorff measure, but repeating the above argument would give a contraction.So we must have that indeed H 6 (S) < ∞ and this yields the result.
Note that, for each x ∈ S, one can find a sequence (x i , t i ) → (x, T max ) such that We shall now show that type-I singularities of the harmonic QK flow are in fact modelled on shrinking solitons.Based on the analogy with the harmonic map flow, motivated from the results in [GH96] we define type-I singularities as follows: Definition 5.11.A solution {Ω(t)} t∈[0,Tmax) to the harmonic QK flow (4.2) is said to encounter a type-I singularity at T max if where C > 0 is a constant.If a singularity does not satisfy the above bound, then it is said to be a type-II singularity.
Proof.From Corollary 4.4, we see that {Ω i (t)}) is well-defined for t ∈ [−λ −2 i T max , 0).The fact that the limit is an ancient solution on R 8 now follows from the compactness in Theorem 5.2.We conclude that the limit is a shrinking soliton by Theorem 5.4, see also [GH96, Theorem 5.1].
It was recently shown that finite-time singularities do occur for the harmonic flow of almost Hermitian structures [HL21].Their construction can be easily adapted to our setting, see also [FLMSE22,Theorem 2.16].However, it is worth mentioning that those examples are based on a non-constructive argument, and as such the concrete nature of the singularity is unknown.Next we shall construct several explicit solutions to the harmonic flow illustrating long-time existence and convergence to both torsion free and (non-trivial) harmonic Sp(2)Sp(1)structures.

Explicit solutions of the harmonic flow
In this section we construct explicit solutions to the harmonic QK flow (4.2).In particular, we exhibit convergence to harmonic QK structures in infinite time; examples of a similar flavour for the harmonic G 2 flow on S 7 were found in [LMSES22].We also construct a steady harmonic Sp(2)Sp(1)-soliton (see Definition 4.6), which to the best of our knowledge is in fact the first nontrivial concrete example of a harmonic soliton for any H-structure.

Eternal solutions
We find two eternal solutions to the harmonic QK flow (4.2).The first example we describe is on R 8 , endowed with the quaternionic hyperbolic metric as in sub-section 3.2.2.We modify the torsion-free QK 4-form in a suitable way to a non torsion-free one then show that the harmonic flow indeed converges back to the torsionfree solution in infinite time.The second example we describe is on SU(3), which we endow with a left-invariant metric.From the results of Poon and Salamon in [PS91], we know that there are no torsion-free QK structures on SU(3), so it is especially interesting to understand harmonic structures in this situation as the next 'best' possible QK structures.We show that the flow in this case converges to a left-invariant harmonic QK structure and that moreover it induces a hypercomplex structure on SU(3) cf.[Joy92, Section 3, Example 1].The latter is a new example of a strictly harmonic QK structure.
We begin by proving the following elementary result: Proposition 6.1.Suppose that Ω 0 is invariant under an isometry f : M → M and that Ω(t) is the solution to the harmonic flow (4.2).Then Ω(t) is also invariant under f .
Proof.Given Ω(t), we can define another solution where we used the facts that ⋄ and ∇ only depend on the metric g, and that f is an isometry.Since f * Ω(0) = Ω(0), by uniqueness of the flow cf.[LSE19, Theorem 1] it follows that f * Ω(t) = Ω(t).
As a consequence, if Ω 0 is G-invariant then so is Ω(t).In particular, for invariant Ω 0 on a homogeneous space, the harmonic flow reduces to an ODE system in t, so this provides a natural set up to study long-time behaviour and (possible) finite-time singularities.Our examples shall exhibit the former behaviour.

Convergence to a torsion-free solution on HH 2
Having shown in Section 3.2 that there indeed exist non-torsion-free QK structures with divergence-free intrinsic torsion, let us describe an explicit eternal solution to the flow converging (in infinite time) to a torsion-free solution.
As in §3.2.2, we shall again consider the hyperbolic quaternionic plane HH 2 , but now viewed as a solvable Lie group with the coframe {E i } satisfying the following structure equations: The equivalence with the cohomogeneity one description of HH 2 in §3.2.2 can easily be seen by setting T (E 7 ) = T (E 8 ) = 0.
From this one finds that div T = −192a(E 12 + E 34 − E 56 − E 87 ).(6.1) Using the definition of the infinitesimal diamond action (2.11), we can compute the harmonic QK flow (4.2) and it turns out that the flow preserves the above ansatz (which is in fact what motivated this choice in the first place).
Due to the symmetry of the problem, the latter reduces to the single ODE: One easily solves the latter to find the eternal solution and b(t) = tanh (768t), ∀t ∈ R. is well-defined and indeed converges to 0 as t → ∞.The above example shows that, even if the harmonic QK flow is not rigorously speaking the gradient flow of the Dirichlet energy functional on this non-compact manifold, it still exhibits some of its informally expected properties.Next we exhibit a compact example.
6.1.2Convergence to a harmonic solution on SU(3) which is not torsion-free We shall now take M 8 = SU(3) and construct an SU (3)-invariant solution to (4.2).We first begin by expressing the Maurer-Cartan form of SU(3) explicitly as where θ i denote a left-invariant coframing.We define a left invariant metric by g = θ 2 1 + θ 2 2 + θ 2 3 + θ 2 4 + θ 2 5 + θ 2 6 + θ 2 7 + θ 2 where f ∈ [0, 2π) is an arbitrary constant.With ω i as above we can now define a QK 4-form Ω on SU(3) by (2.1), which is indeed compatible with g.Note that f can in fact be taken to be any function on SU(3) and the above will still hold, but since we are only interested in SU(3)-invariant structures we shall restrict to the situation when f is constant.Moreover, we also note the following special case: Proposition 6.2.When f = 0, the quaternionic structure defined by the 4-form Ω is in fact a hypercomplex structure.
In what follows we denote by {E i } the dual frame to {θ i }.One can view SU(3) as an SU(2)-bundle over S 5 , where the S 3 fibres are generated by the left invariant vector fields E 2 , E 3 , E 4 .Furthermore, the Hopf fibration exhibits S 5 as an U(1)-bundle over CP 2 , where the S 1 fibres correspond to the orbit of the vector field E 1 .This illustrates the diagonal embedding of U(2) = U(1)SU(2) in SU(3): CP 2 In view of the above, we can interpret the U(1) rotation defined by f as lying in the SU(2) fibre.It is worth pointing out that the metric g is in fact SU(3) × U(2)-invariant, where the U(2) corresponds to the right action generated by E 1 , E 2 , E 3 , E 4 .Note that g is not the bi-invariant metric of SU(3), the bi-invariant Einstein metric g E is instead given by g E = 3θ 2 1 + θ 2 2 + θ 2 3 + θ 2 4 + θ 2 5 + θ 2 6 + θ 2 7 + θ 2 8 .
Proof.It suffices to verify that L E i Ω = 0 for i = 1, .., 4 when f = 0 but if f = 0 then the latter only holds for i = 1, 2.