Bach Flow of Simply Connected Nilmanifolds

The success of the Ricci flow and other second order geometric flows has led to a desire to better understand higher order geometric flows. In particular, the study of fourth order flows is a natural progression from second order flows. Fourth order flows can be more difficult to since we no longer have access to maximum principles. An important class of fourth order flows are the gradient flows of quadratic curvature functionals.

In [BahuaudHelliwellSTEFSHOGF] Bahuad and Helliwell introduced a family of higher order flows called the ambient obstruction flow which involves the ambient obstruction tensor of Fefferman and Graham introduced in [FeffermanGrahamCI]. In dimension 4, the ambient obstruction tensor coincides with the Bach tensor and the corresponding flow is the Bach flow:

Here $\operatorname{B}(g)$ is the Bach tensor defined in (2) below, $s=s(g)$ is the scalar curvature and $\Delta$ is the Laplace-Beltrami operator. This flow corresponds to a conformally modified gradient flow of the $L^{2}$-norm of the Weyl curvature, and so is fourth order (see the introduction in [StreetsTLTBOFOCF]). While this might not be the first choice of a Bach flow due to the $\frac{1}{12}(\Delta s)g$ term, this is needed to counteract the invariance of the Bach tensor under the conformal group [LopezAOF, Section 1.2]. In any case, if a solution $g(t)$ to (1) has constant scalar curvature at each point in time, as is the case for homogeneous solutions, then $\Delta s\equiv 0$. Short time existence and uniqueness for the flow on compact manifolds was proved by Bahaud and Helliwell in [BahuaudHelliwellSTEFSHOGF, BahuaudHelliwellUFSHOGF].

The Bach flow was studied by Helliwell in [HelliwellBFOHP] on homogeneous manifolds of the form $M^{4}=S^{1}\times N^{3}$ where $N^{3}$ is a closed, homogeneous 3-manifold and $M$ is equipped with the product metric. By lifting to the universal cover, this reduces to the study of the Bach flow on 9 simply connected homogeneous spaces of the form $\widetilde{M}=\mathbb{R}\times\widetilde{N}$. In [GriffinGAOSOHM], Griffin uses the computations of the Bach tensor from [HelliwellBFOHP] and a theorem of Petersen and Wylie [PetersenWylieROHGSMARE, Theorem 3.6] to study gradient Bach solitons on the product manifolds from Helliwell’s paper.

The Bach tensor of a left-invariant metric is determined by its value at the identity, and therefore can be computed explicitly using metric coefficients and structure constants of the of the Lie algebra. In practice, however, this is difficult due to the fourth order nature of the tensor. For Riemannian products this formula greatly simplifies [HelliwellBFOHP, Section 2.1]. In order to make the problem more tractable in the non-product case we have chosen to study the flow on Lie group whose Lie algebra has a large number of structure constants which vanish.

In this article we study the Bach flow the simply connected nilpotent Lie group, $N^{4}$, whose Lie algebra is the unique four-dimensional indecomposable nilpotent Lie algebra. By indecomposable, we mean that the Lie algebra cannot be written as a direct product of two lower dimensional Lie algebras. Our main result is the following theorem.

A Bach soliton is a solution to (1) which is self similar. That is, the solution $g(t)$ to (1) beginning at a soliton $g$ can be written $g(t)=c(t)\varphi(t)^{*}g$ for some positive function $c$ and time dependent diffeomorphisms $\varphi(t)$. We say a soliton is expanding, steady or shrinking if the scaling constant is increasing, constant or decreasing respectively.

The Bach soliton $g_{\infty}$ is a non-trivial algebraic Bach soliton (see [LauretRSHN, LauretGFATSOHS] for literature on algebraic Ricci solitons). That is, if $B:\mathfrak{n}_{4}\to\mathfrak{n}_{4}$ is the endomorphism defined by $\operatorname{B}(g_{\infty})_{e}=g(B\cdot,\cdot)$ then,

where $D\in\operatorname{Der}(\mathfrak{n}_{4})$ is a non-zero derivation. To prove Theorem 1 we use the bracket flow technique introduced by Lauret to study the Ricci flow in homogeneous manifolds [LauretRFOHM] (Lauret extends the bracket flow technique to more general geometric structures in [LauretGFATSOHS]). The bracket Bach flow is an ODE in the variety of Nilpotent Lie brackets $\mathcal{N}^{4}\subset\Lambda^{2}(\mathbb{R}^{4})^{*}\otimes\mathbb{R}^{4}$ which is equivalent to the Bach flow. Algebraic solitons correspond to solutions of the bracket flow which evolve only by scaling. These are easier to detect than general solitons since we have removed the action of the diffeomorphism group. Our analysis follows Lauret’s analysis of the Ricci flow on simply-connected nilmanifolds in [LauretTRFFSCN].

We also remark that the soliton $g_{\infty}$ is not of gradient type due to a Theorem of Petersen and Wylie [PetersenWylieROHGSMARE, Theorem 3.6]. To our knowledge, this is the first example of a non-gradient Bach soliton.

The paper is structured as follows. In §2 we introduce the Bach tensor and its basic properties. In §3 review some group actions on the variety of nilpotent Lie algebras and recall the classification of four-dimensional Lie algebras. In §4 we review the structure of four-dimensional simply connected nilmanifolds. In particular, we show that up to isometry, left-invariant metrics on $N^{4}$ depend on 3 real parameters. By suitably gauging the Bach flow (Proposition 10), we obtain an ODE in three variables, $a,b,c$, which is equivalent to the bracket Bach flow the simply connected Lie group $N^{4}$. By studying the long time behaviour of the quantities $a,b,c$ we are able to understand the long time behaviour of the Bach flow. Self similar solutions of the Bach flow are studied in §6. In terms of the variables $a,b,c$ the soliton condition becomes a set of algebraic equations. Finally, we considered the normalised flow in §7.

Let $M^{4}$ be a four dimensional Riemannian manifold. The Bach tensor is the tensor given in local coordinates by

Here, $W$ is the Weyl tensor, $R^{kl}=g^{ka}g^{lb}R_{ab}=g^{ka}g^{lb}(\operatorname{Rc})_{ab}$ are the components of the Ricci tensor with both indices raised.

The Bach tensor is conformally invariant ($\operatorname{B}(\rho g)=\rho^{-1}\operatorname{B}(g)$ for any positive function $\rho$) and arises in the study of manifolds which are conformally Einstein [KuhnelRademacherCTOPRM, Section 6]. In particular, Theorem 6.6 in [KuhnelRademacherCTOPRM] says $\operatorname{B}(g)\equiv 0$ is a necessary condition for $(M^{4},g)$ to be conformal to an Einstein manifold. The converse holds if $g$ is conformal to a metric with harmonic Weyl tensor [KuhnelRademacherCTOPRM, Corollary 6.8]. The Bach tensor is trace and divergence free [GriffinGAOSOHM].

On a compact manifold the Bach tensor is $-1/4$ times the gradient of the Riemannian functional

[BesseEM, 4.76] (note that the definition given here is only agrees with the definition in [BesseEM] up to a factor of $-4$).

The Bach flow has been studied on homogeneous $4$-manifolds whose universal cover has a product structure (that is, $(\widetilde{M}^{4},g)=(N_{1}\times N_{2},g_{1}\oplus g_{2})$ where $(N_{i},g_{i})$ are simply connected homogeneous manifolds of lower dimension which admit compact quotients) by Helliwell in [HelliwellBFOHP]. When $\dim N_{2}=3$ we must have that $N_{2}$ is a unimodular Lie group and that $N_{1}=\mathbb{R}$. In this case, Helliwell uses Milnor frames to diagonalise the Ricci and Bach tensor and then studies the resulting ODE’s. In particular, this includes the study of the Bach flow on $(\mathbb{R}^{4},\bar{g})$ and $(\mathbb{R}\times H^{3},\bar{g}\oplus g)$ where $\bar{g}$ denotes the Euclidean metric on $\mathbb{R}^{n}$ and $g$ is a left invariant metric on the three-dimensional Heisenberg group $H^{3}$. For $\dim N_{1}=\dim N_{2}=2$ the only possibilities are $M(c_{1})^{2}\times M(c_{2})$ where $M(c)^{2}$ denotes the simply connected space form with constant curvature $c\in\mathbb{R}$. In this case the flow remains a product of two space forms and the flow is equivalent to a system of ODE’s for the curvature of the factors.

Any bilinear skew-symmetric map $\mu:\mathbb{R}^{4}\times\mathbb{R}^{4}\to\mathbb{R}^{4}$ which satisfies the Jacobi identity defines a Lie algebra structure structure on $\mathbb{R}^{4}$. There is a natural ‘change of basis’ action of $\operatorname{GL}_{4}(\mathbb{R})$ on $\Lambda^{2}(\mathbb{R}^{4})^{*}\otimes\mathbb{R}^{4}$, the vector space of bilinear skew-symmetric maps, given by

Two Lie brackets $\mu_{1}$ and $\mu_{2}$ define isomorphic Lie algebras if and only if there is a $h\in\operatorname{GL}_{4}(\mathbb{R})$ such that $h\cdot\mu_{1}=\mu_{2}$.

Moreover, any inner product $\left\langle\cdot,\cdot\right\rangle$ on $\mathbb{R}^{4}$ induces an inner product on $\Lambda^{2}(\mathbb{R}^{4})^{*}\otimes\mathbb{R}^{4}$ [LauretTRFFSCN]:

where $\{e_{i}\}_{i=1}^{4}$ is any orthonormal basis.

If the map $(\operatorname{ad}_{\mu}x)$, defined by $(\operatorname{ad}_{\mu}x)y:=\mu(x,y)$ for all $y\in\mathbb{R}^{4}$, is a nilpotent endomorphism for all $x\in\mathbb{R}^{4}$, we say that $\mu$ is nilpotent. A bracket $\mu$ which satisfies the Jacobi identity and is nilpotent defines a nilpotent Lie algebra [VaradarajanLGLAATR, Theorem 3.5.4]. In this way, the set

parametrises structure constants of $4$-dimensional, nilpotent Lie algebras.

For any element $\mu\in\mathcal{N}_{4}$, $(\mathbb{R}^{4},\mu)$ is isomorphic to one of the three following Lie algebras [deGraafCOSLA, Section 5.5]:

1. (1)

   $\mathbb{R}^{4}$: with the abelien Lie bracket $\mu(x,y)=0$ for all $x,y\in\mathbb{R}^{4}$.

2. (2)

   $\mathbb{R}\oplus\mathfrak{h}_{3}$: This is the product of the 3 dimensional Heisenberg algebra with $\mathbb{R}$. The non-trivial bracket relations are $\mu(e_{1},e_{2})=-\mu(e_{2},e_{1})=e_{3}$.

3. (3)

   $\mathfrak{n}_{4}$: This is not a product Lie algebra. The non-trivial bracket relations are

   $$\mu(e_{1},e_{2})=-\mu(e_{2},e_{1})=e_{3},\quad\mu(e_{1},e_{3})=-\mu(e_{3},e_{1})=e_{4}.$$

Note that 1. and 2. are decomposable Lie algebras.

In this paper, we are concerned with simply connected Lie groups, $N^{4}$, such that $\operatorname{Lie}(N)\simeq\mathfrak{n}_{4}$. Any such simply connected Lie group can be identified with $\mathbb{R}^{4}$ under the operation

See [VaradarajanLGLAATR, Section 3].

Let $\left\langle\cdot,\cdot\right\rangle$ be the standard inner product on $\mathbb{R}^{n}$. Then any other inner product on $\mathbb{R}^{n}$ can be written as

for some $h\in\operatorname{GL}_{n}(\mathbb{R})$. If $(\cdot,\cdot)$ is an inner product on $\mathbb{R}^{n}$, we denote by $g_{\mu,(\cdot,\cdot)}$ the left-invariant metric on $(\mathbb{R}^{n},\cdot_{\mu})$ which agrees with $(\cdot,\cdot)$ on $T_{0}\mathbb{R}^{n}$ [LauretTRFFSCN]. When $(\cdot,\cdot)$ is the standard inner product on $\mathbb{R}^{n}$, we will write $g_{\mu}$ instead of $g_{\mu,(\cdot,\cdot)}$. We then have the following proposition.

Note that by the previous proposition, up to isometry we may assume that a left invariant metric agrees with the standard inner product on $T_{0}\mathbb{R}^{n}$.

Suppose now that $\mu\in\mathcal{N}_{4}$ is a Lie bracket such that $(\mathbb{R}^{4},\mu)\simeq\mathfrak{n}_{4}$ and $\left\langle\cdot,\cdot\right\rangle$ is the standard inner product on $\mathbb{R}^{4}$. Proposition 2 below gives a convenient basis of $\mathbb{R}^{4}$ for us to work in.

If $\mu\in\mathcal{N}_{4}$ has the structure coefficients (6) with respect to the standard basis of $\mathbb{R}^{4}$ then we will write $\mu=\mu_{a,b,c}$.

Let us define

It follows from the above discussion that if $(\mathbb{R}^{n},g_{\mu})$ is a simply connected Nilpotent Lie group with left-invariant metric $g_{\mu}$, then up to isometry we may assume that $\mu\in\mathcal{O}$. Clearly we can identify $\mathcal{O}$ with the set

We will use this in §5 in order to reduce our study of the bracket flow, to be introduced later on. Of course, there is no guarantee that a solution to the bracket flow (9) will remain in $\mathcal{O}$. However, we will show in §5 that we can gauge our flow so that the solution does remain within $\mathcal{O}$. In particular, our problem reduces to the study of an ODE in an open subset of $\mathbb{R}^{3}$.

If $\mu=\mu_{a,b,c}\in\mathcal{O}$, then the norm of $\mu$ induced from the inner product in (4) reduces to

It will be useful to have an explicit description of an arbitrary derivation $D\in\operatorname{Der}(\mu)$ of a bracket $\mu\in\mathcal{O}$. We can obtain this by differentiating the description of an automorphism of $\mathfrak{n}_{4}$ given in [VanThuongMO4DULG].

Suppose now that $(M^{4},g)=(\mathbb{R}^{4},g_{\mu})$ is a Nilpotent Lie group with left-inavriant metric and that we have a $(\mathbb{R}^{4},\cdot_{\mu})$-invariant solution $(g(t))_{t\in(a,b)}$ to the Bach flow with $g(0)=g_{\mu}$ (a solution $(g(t))_{t\in(a,b)}$ is $(\mathbb{R}^{4},\cdot_{\mu})$-invariant if it is for all $t\in(a,b)$). Since the scaler curvature is constant, we find that $\operatorname{B}(g(t))(0)$ satisfies the ODE

Here, we have written $\operatorname{B}(\left\langle\cdot,\cdot\right\rangle_{t})=\operatorname{B}(g(t))(0)$. Conversely, given a solution $(\cdot,\cdot)_{t}$ to the ODE (8) we obtain a $(\mathbb{R}^{4},\cdot_{\mu})$-invariant solution $g(t)$ to the Bach flow by defining $g(t)=g_{\mu,(\cdot,\cdot)_{t}}$ for all $t$. By the usual existence and uniqueness of ODE’s, we are guaranteed a unique $(\mathbb{R}^{4},\cdot_{\mu})$-invariant solution with a maximal interval of existence [LauretGFATSOHS]. The need for this reasoning is that uniqueness of the Bach flow is an open problem on a general manifold.

Let us fix once and for all initial metric $g_{0}=g_{\mu_{0}}$ which is invariant under the Nilpotent Lie group $(\mathbb{R}^{4},\cdot_{\mu_{0}})$. Note that we have we have tacitly assumed that the initial inner product on $T_{0}\mathbb{R}^{n}$ is the standard inner product, but this is not an issue since up to isometry this is always the case (see §3).

By the above discussion, there is a unique curve of left-invariant metrics of inner products $\left\langle\cdot,\cdot\right\rangle_{t}$ on $\mathbb{R}^{4}$ satisfying (8) which corresponds to the unique $(\mathbb{R}^{4},\cdot_{\mu_{0}})$-invariant solution of the Bach flow beginning at $g_{\mu_{0}}$. It follows that for each $t$ there is a $h(t)\in\operatorname{GL}_{4}(\mathbb{R})$ such that

One can show that the one parameter family of matrices $h(t)$ can be chosen to be a smooth curve (see [LauretGFATSOHS, Section 4.1] or Proposition 3 below). The corresponding curve of brackets is given by $\mu(t)=h(t)\cdot\mu_{0}$. The bracket flow, introduced by Lauret to study Ricci flow on homogeneous manifolds in [LauretRFOHM], is motivated by considering what equation the curve $\mu(t)\in\operatorname{GL}_{4}(\mathbb{R})\cdot\mu_{0}$ satisfies. For more examples of the bracket flow technique see [LauretTRFFSCN, BohmLafuenteIHRF, StanfieldPHCF].

Note that $\pi$ is the derivative of the $\operatorname{GL}_{n}(\mathbb{R})$ representation on $\Lambda^{2}(\mathbb{R}^{n})^{*}\otimes\mathbb{R}^{n}$ defined in (3). Moreover, the solution $\mu(t)$ of (9) remains in the orbit $\operatorname{GL}_{4}(\mathbb{R})\cdot\mu_{0}$ since $\frac{1}{2}\pi(B_{\mu})\in T_{\mu}(\operatorname{GL}_{4}(\mathbb{R})\cdot\mu)$ (see [LauretRFOHM, Lemma 3.2]).

Beginning at the inital metric $g_{\mu_{0}}$ we now have two families of Riemannian manifolds

where $g(t)$ is the unique $(\mathbb{R}^{n},\cdot_{\mu_{0}})$-invariant solution of the Bach flow and $\mu(t)$ is the solution of the bracket flow (9). The next proposition shows that these are equivalent in a precise way.

For a proof of Proposition 3, one should consult [LauretTRFFSCN, Theorem 5.1] (note that the proof in [LauretTRFFSCN] is for the Ricci flow however only symmetry of the Ricci tensor is used). In particular, Proposition 3 shows that the solutions of (1) and (9) have the same maximal interval of existence and the same curvature (see the Remark after Theorem 3.3 in [LauretRFOHM]).

Recall from §3 that if $\lambda\in\operatorname{O}(4)\cdot\mu$ then the metrics $g_{\mu}$ and $g_{\lambda}$ are isometric. This is due to the fact that if $h\in\operatorname{GL}_{4}(\mathbb{R})$ gives rise to the inner product $(\cdot,\cdot)=\left\langle h\cdot,h\cdot\right\rangle$ and $k\in\operatorname{O}(4)$ then $hk\in\operatorname{GL}_{4}(\mathbb{R})$ gives rise to the same inner product. It will be useful to exploit this $\operatorname{O}(4)$ equivariance when studying the bracket flow (9). Böhm and Lafuente describe this as a refinement of Uhlenbeck’s trick of moving frames (see Sections 2 and 3 in [BohmLafuenteIHRF]).

In particular, the solutions $\mu,\bar{\mu}$ to the bracket flow (9) and the gauged bracket flow (10) have the same maximal interval of existence and the same curvature.

We now want to study the bracket flow for $(\mathbb{R}^{4},g_{\mu})$ when $(\mathbb{R}^{4},\mu)\simeq\mathfrak{n}_{4}$.

We have noted in §3 that the solution $\mu(t)$ to the bracket flow (9) beginning at $\mu_{0}\in\mathcal{O}$ may not remain in $\mathcal{O}$. However, we have also seen in §3 that for any $\mu\in\mathcal{N}_{4}$, the orbit $\operatorname{O}(4)\cdot\mu$ intersects $\mathcal{O}$. That is to say, for each $t$ we can find a $k(t)\in\operatorname{O}(4)$ such that $k(t)\cdot\mu(t)\in\mathcal{O}$. Since $\mu$ and $k\cdot\mu$ determine isometric Riemannian manifolds for $k\in\operatorname{O}(4)$, we may use $\operatorname{O}(4)$ to gauge our flow, readjusting at each point in time to ensure that the solution remains in $\mathcal{O}$. We formalise this in Proposition 6 below by appealing to Proposition 10 in §5.

Therefore, the bracket flow (9) is equivalent to the following ODE for $\mu=\mu_{a,b,c}\in\mathcal{O}$:

With these in hand we are in a position to study the long time behaviour of the flow.

Lemma 2 gives the following important corollaries.

Corollary 2 shows that the bracket flow beginning at an arbitrary bracket $\mu_{0}\in\mathcal{O}$ converges to the Euclidean bracket as $t\to\infty$.

In this section we study solitons of the Bach flow. A Bach soliton is a Riemannian manifold $(M^{4},g)$ such that

for a constant $\lambda\in\mathbb{R}$, and a complete vector field, $X\in\mathfrak{X}(M)$ ($\mathcal{L}_{X}$ is the Lie derivative in the direction $X$). The soliton is called expanding, steady or shrinking if $\lambda>0$, $\lambda=0$ or $\lambda<0$ respectively. If the vector field arises as the gradient of a potential function $u\in C^{\infty}(M)$, then (17) becomes

In this case, we say the soliton is a gradient soliton. For homogeneous solitons, the Laplacian term in (17) vanishes since the scalar curvature is constant. Homogeneous gradient Bach solitons were studied by Griffin in [GriffinGAOSOHM]. Griffin applies Theorem 3.6 from [PetersenWylieROHGSMARE] to reduce the study to Riemannian products of manifolds of dimension less than $4$. Proposition 7 below is a simple corollary of [PetersenWylieROHGSMARE, Theorem 3.6] which we will use to deduce that any non-product solitons we find cannot be gradient.

Bach solitons correspond to solutions of the Bach flow which evolve by scaling and pullback by diffeomorphisms [LauretGFATSOHS, Theorem 4.10].

If $(M^{4},g)=(\mathbb{R}^{n},g_{\mu})$ is a simply connected Nilpotent Lie group, then an analogous condition to the metric evolving self similarly under the Bach flow is that the solution $\mu(t)$ to the bracket flow evolves only by scaling (see the discussion before Theorem 6 in [LauretGFATSOHS]). If this is the case, the bracket $\mu=\mu(0)$ satisfies

A simply connected Nilpotent Lie group $(\mathbb{R}^{n},g_{\mu})$ whose Bach endomorphism satisfies (19) is called an algebraic Bach soliton. Algebraic solitons were introduced by Lauret to study Ricci Nilsolitons (Ricci solitons on Nilpotent Lie groups) in [LauretRSHN]. In [LauretGFATSOHS] Lauret generalises the the bracket flow technique and algebraic solitons to a large class of geometric structures. An important observation made by Lauret is that an algebraic soliton is indeed a soliton in the sense of (17). For convenience, we have summarised this for the Bach tensor in Proposition 8 below.