$S^{1}$-invariant Laplacian flow

A symplectic $SU(3)$-structure on a $6$-manifold $P^{6}$ is given by a pair $(\omega,\Omega)$, where $\omega$ is a symplectic form and $\Omega$ is a complex $(3,0)$-form with real and imaginary parts given by $\Omega^{+}$ and $\Omega^{-}$ respectively. Our main goal in this article is to derive the evolution equations for such an $SU(3)$-structure on $P^{6}$, together with a Higgs fields $H:P^{6}\to\mathbb{R}^{+}$ and a connection $1$-form $\xi$ on an $S^{1}$ bundle $L^{7}\to P^{6}$, such that

is a solution to the Laplacian flow (see below for the definition). Put differently we want to understand how the $SU(3)$-structure on the quotient of an $S^{1}$-invariant solution to the Laplacian flow evolves. This investigation is further motivated from the results of Lotay-Wei in [18] that assert that if the initial $3$-form $\varphi_{0}$ on a compact manifold $L^{7}$ is $S^{1}$-invariant then so is the flow for as long as it exists. We study two simple cases and as a result we construct the first examples of inhomogeneous shrinking Laplacian solitons. It is known that shrinking solitons do not occur on compact manifolds and that the steady ones are actually torsion free [18]. Indeed the examples that we construct are non-compact. There are currently no known compact examples of expanding solitons. We prove that such expanders, if they exist at all, in fact do not admit any infinitesimal symmetry i.e. there are no Killing vector fields which preserve the associated $G_{2}$ $3$-form $\varphi$. This gives a partial explanation why such solitons have not (yet) been found.

A closed $G_{2}$-structure on a $7$-manifold $L^{7}$ is given by a closed non-degenerate $3$-form $\varphi$, which in turn determines a Riemannian metric $g_{\varphi}$ and volume form $\operatorname{vol}_{\varphi}$. In particular, $\varphi$ also defines a Hodge star operator $*_{\varphi}$. If $*_{\varphi}\varphi$ is also closed then the holonomy of $g_{\varphi}$ is a subgroup of $G_{2}$ and consequently the metric $g_{\varphi}$ is Ricci-flat. The Laplacian flow, defined as the initial value problem

where $\mathrm{\Delta}_{\varphi}:=dd^{*_{\varphi}}+d^{*_{\varphi}}d$, was introduced by Bryant in [5] as a way of potentially deforming $\varphi_{0}$ within its cohomology class to a torsion free one. The short time existence of the flow was proved by Bryant and Xu in [6], and long time existence, uniqueness and compactness results were proved by Lotay and Wei in [18].

In recent years there have been many works on the Laplacian flow cf. [10, 11, 12, 16, 17, 19] but the flow is nonetheless still very complicated to study in its full generality. A natural strategy to simplifying the equations is to impose symmetry. Aside from the homogeneous cases, three notable works in this direction were carried out by Fine and Yao in [10], where the authors, motivated by a question of Donaldson, study the flow on $L^{7}=M^{4}\times\mathbb{T}^{3}$ as a way of deforming a hypersymplectic triple on $M^{4}$ to a hyperKähler one, by Lotay and Lambert in [16], where the authors reinterpret the flow on $L^{7}=B^{3}\times\mathbb{T}^{4}$ as a spacelike mean curvature flow on $B^{3}\subset\mathbb{R}^{3,3}$, and by Fino and Raffero in [11], where the authors study the flow for warped $G_{2}$-structures on $L^{7}=P^{6}\times S^{1}$. In each case the presence of a free abelian group action allows for a reduction of the Laplacian flow to a lower dimensional space. It is exactly this common feature that motivates the work undertaken here.

In this article we consider the more general case when $L^{7}$ only admits a free $S^{1}$ action. In this case the Laplacian flow descends to a flow on a symplectic $SU(3)$-structure on $P^{6}$, also called almost Kähler, together with the data of a Higgs field and a connection $1$-form. This leads us to try and understand the resulting flow. In section 3 we prove a Gibbons-Hawking type construction for $S^{1}$-invariant closed $G_{2}$-structures (see Theorem 3.3) and we express the intrinsic torsion of the $G_{2}$-structure in terms of that of the $SU(3)$-structure. In other words we encode that data of an $S^{1}$-invariant closed $G_{2}$-structure only in terms of data on the quotient. In section 4 we then derive the evolution equations for such data when $\varphi$ is evolving by the Laplacian flow. We also prove that certain cohomology classes have to vanish on a compact manifold with an exact $G_{2}$-structure (see Theorem 4.15) and as a consequence we deduce that compact expanding solitons cannot arise from our construction. Even if $(L^{7},\varphi)$ is torsion free the $SU(3)$-structure on the quotient $P^{6}:=L^{7}/S^{1}$ is generally not torsion free, though it is always symplectic. For a flow of $SU(3)$-structures a natural question one can ask is if any class of $SU(3)$-structure is preserved. For instance in [11], the authors show that under certain constraints if $(\omega,\Omega)$ is initially symplectic half-flat then this is preserved by the flow. In section 5 we first consider the original example of the Laplacian flow studied by Bryant and show that the symplectic form on the quotient, for a given $S^{1}$ action, in fact remains constant under the flow, though the intrinsic torsion of the quotient $SU(3)$-structure is generic. Motivated by the work of Apostolov and Salamon in [1] where the authors construct (incomplete) examples of $G_{2}$ metrics which admit Kähler reduction, we next search for solutions to the Laplacian flow on these spaces. We show that the flow indeed preserves the Kähler property in this setting and we find explicit (complete) shrinking gradient solitons. These are the only currently known examples of inhomogeneous shrinkers.

Note that condition (2.2) is just a consequence of the fact that $\omega$ is of type $(1,1)$. Although an $SU(3)$-structure consists of the data $(g_{\omega},\omega,J,\Omega)$, one can in fact recover the whole $SU(3)$-structure only from the pair $(\omega,\Omega^{+})$, or $(\omega,\Omega^{-})$. This observation is due to Hitchin in [13] where he shows that $\Omega^{+}$, or $\Omega^{-}$, determines $J$. Abstractly this follows from the fact that the stabiliser of $\Omega^{\pm}$ in $GL^{+}(6,\mathbb{R})$ is congruent to $SL(3,\mathbb{C})\subset GL(3,\mathbb{C})$. The metric is then determined by (2.1) and $\Omega^{-}=J(\Omega^{+})=*_{\omega}\Omega^{+}$, where $*_{\omega}$ is the Hodge star operator determined by $g_{\omega}$ and the volume form

As $SU(3)$ modules the space of exterior differential forms splits as

where

and we get a corresponding decomposition for $\Lambda^{4}$ and $\Lambda^{5}$ via $*_{\omega}$. The $[\![\Lambda^{p,q}]\!]$ and $[\Lambda^{p,p}]$ notation refers to taking the corresponding real underlying vector space i.e.

cf. [7, 21]. As $SU(3)$ modules the spaces $\Lambda^{\bullet}_{6}$ are all isomorphic. In computations we will often need to interchange between these spaces and to do so we use the following lemma which follows from a simple calculation.

The intrinsic torsion of the $SU(3)$-structure $(\omega,\Omega^{+})$ is determined by

where $\sigma_{0},\pi_{0}$ are functions, $\nu_{1},\pi_{1}\in\Lambda^{1}_{6}$, $\pi_{2},\sigma_{2}\in\Lambda^{2}_{8}$ and $\nu_{3}\in\Lambda^{3}_{12}$, cf. [3]. Many well-known geometric structures can be recast using this formulation:

The reason for this nomenclature stems from the fact that the subgroup of $GL(7,\mathbb{R})$ which stabilises $\varphi_{0}$ is isomorphic to the Lie group $G_{2}$. Since $G_{2}$ is a subgroup of $SO(7)$ [4, 21] it follows that $\varphi$ determines (up to a choice of a constant factor) a Riemannian metric $g_{\varphi}$ and volume form $\operatorname{vol}_{\varphi}$ on $L^{7}$. Explicitly we define these by

where $U,V$ are vector fields on $L^{7}$. In particular, $\varphi$ defines a Hodge star operator $*_{\varphi}.$ The set of $3$-forms defining a $G_{2}$-structure is an open set in the space of sections of $\Lambda^{3}$ which we denote by $\Lambda^{3}_{+}(L^{7})$. In this article we will only consider the situation when $\varphi$ is closed. In this case we have

for a unique $2$-form $\tau\in\mathfrak{g}_{2}\cong\Lambda^{2}_{14}\hookrightarrow\Lambda^{2}\cong\mathfrak{so}(7)$ where

The $2$-form $\tau$ is called the intrinsic torsion of the closed $G_{2}$-structure. On $(L^{7},\varphi)$ there is also a natural $G_{2}$ equivariant map given by

The kernel of $j$ is

and as $G_{2}$ modules we have

where $\Lambda^{3}_{27}$ is identified with the space of traceless symmetric $2$-tensors via $j$ while $j(\varphi)=6g_{\varphi}$. If $\varphi$ is both closed and coclosed i.e. $\tau=0$, then the holonomy of $g_{\varphi}$ is contained in $G_{2}$ and $\operatorname{Ric}(g_{\varphi})=0$. We refer the reader to the classical references [4, 21] for proofs of the aforementioned facts. The problem of constructing examples with holonomy equal to $G_{2}$ is very hard. The Laplacian flow was introduced as a means to tackle this problem.

The Laplacian flow (1.1)-(1.3) preserves the closed condition i.e. $d\varphi(t)=0$ for all $t$ and thus (1.1) is equivalent to

In the compact setting, Hitchin gives the following interpretation of the flow. Consider the functional $\Psi:[\varphi_{0}]^{+}\to\mathbb{R}^{+}$ defined by

where $[\varphi_{0}]^{+}:=\{\varphi_{0}+d\beta\in\Lambda^{3}_{+}(L)\ |\ \beta\in\Lambda^{2}(L)\}$ denotes an open set in $[\varphi_{0}]\in H^{3}(L,\mathbb{R}).$ Then computing the Euler-Langrange equation Hitchin finds that the critical points of $\Psi$ satisfy $d*_{\rho}\rho=0$. The Laplacian flow is the gradient flow of $\Psi$ with respect to an $L^{2}$ norm induced by $g_{\rho}$ on $[\varphi_{0}]^{+}.$ The Hessian of $\Psi$ at a critical point is non-degenerate transverse to the action of the diffeomorphism group and in fact is negative definite. Thus $\Psi$ can be interpreted as a Morse-Bott functional and the torsion free $G_{2}$-structures correspond to the local maxima. In the non-compact setting this interpretation is not valid but the Laplacian flow is still well-defined and critical points are still torsion free $G_{2}$-structures cf. [18].

In [5], Bryant computes the evolution equations for the following geometric quantities under the Laplacian flow

We see immediately that up to lower order terms (2.18) coincides with the Ricci flow of $g_{\varphi}$. In [5, (4.30)] Bryant also derives an expression for the Ricci tensor in terms of the torsion form only and thus, one can express (2.18) only in terms of the torsion form as

The simplest solutions to (1.1) are those that evolve by the symmetry of the flow. If $\varphi_{0}$ satisfies

for a vector field $V$ and constant $\lambda$ then

where $F_{t}$ is the diffeomorphism group generated by $U(t)=(1+\frac{2}{3}\lambda t)^{-\frac{2}{3}}V$, is a solution to flow and $\varphi_{0}$ is called a Laplacian soliton [18]. Depending on whether $\lambda$ is positive, zero or negative the soliton is called expanding, steady or shrinking respectively. If $V$ is a gradient vector field then we call them gradient solitons. We refer the reader to [18] for foundational theory for the Laplacian flow.

Consider the Laplacian flow starting from an $S^{1}$-invariant closed $G_{2}$-structure. Then by the existence and uniqueness of the flow, at least in the compact case, it follows that this symmetry persists i.e the solution to the flow can be expressed as (3.2) for as long as it exists cf. [18, Corollary $6.7$]. Note also that since $\mathcal{L}_{Y}\frac{\partial}{\partial t}(\xi)=\frac{\partial}{\partial t}(\mathcal{L}_{Y}\xi)=0$ and $Y\raise 1.0pt\hbox{\large$\lrcorner$}\>\frac{\partial}{\partial t}(\xi)=0$ it follows that $\frac{\partial}{\partial t}(\xi)$ is a basic $1$-form i.e. it corresponds to a $1$-form on $P^{6}$. Thus, in the $S^{1}$-invariant setting, from (3.20) we see that (2.16) becomes equivalent to the following evolution equations on $(\omega,\Omega,H,\xi)$:

We omit writing the dependence on $t$ to ease the notation.