Bach Flow on Homogeneous Products
The qualitative behavior of Bach flow is established on compact four-dimensional locally homogeneous product manifolds. This is achieved by lifting to the homogeneous universal cover and, in most cases, capitalizing on the resultant group structure. The resulting system of ordinary differential equa...
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| citation_txt | Bach Flow on Homogeneous Products. Dylan Helliwell. SIGMA 16 (2020), 027, 35 pages |
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| description | The qualitative behavior of Bach flow is established on compact four-dimensional locally homogeneous product manifolds. This is achieved by lifting to the homogeneous universal cover and, in most cases, capitalizing on the resultant group structure. The resulting system of ordinary differential equations is carefully analyzed on a case-by-case basis, with explicit solutions found in some cases. The limiting behavior of the metric and the curvature is determined in all cases. The behavior of quotients of ℝ×𝕊³ proves to be the most challenging and interesting.
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 16 (2020), 027, 35 pages
Bach Flow on Homogeneous Products
Dylan HELLIWELL
Department of Mathematics, Seattle University, 901 12th Ave, Seattle, WA 98122, USA
E-mail: helliwed@seattleu.edu
Received September 03, 2019, in final form March 29, 2020; Published online April 11, 2020
https://doi.org/10.3842/SIGMA.2020.027
Abstract. Qualitative behavior of Bach flow is established on compact four-dimensional
locally homogeneous product manifolds. This is achieved by lifting to the homogeneous uni-
versal cover and, in most cases, capitalizing on the resultant group structure. The resulting
system of ordinary differential equations is carefully analyzed on a case-by-case basis, with
explicit solutions found in some cases. Limiting behavior of the metric and the curvature
are determined in all cases. The behavior on quotients of R × S3 proves to be the most
challenging and interesting.
Key words: high-order geometric flows; Bach flow; locally homogeneous manifold; three-di-
mensional Lie group
2020 Mathematics Subject Classification: 53C44; 53C30; 34C40
1 Introduction
In four dimensions, Bach flow is a solution to
∂tg = B +
1
12
∆Sg, g(0) = h,
where B is the Bach tensor and S is scalar curvature for the metric g. This serves as a concrete
motivating example of a higher-order intrinsic curvature flow. Such flows, including flow by the
ambient obstruction tensor and flow by the gradient of the total curvature energy functional, have
been of interest recently. See for example [1, 2], and with related work found in [3, 10, 17, 20].
Analyzing Bach flow on an arbitrary locally homogeneous 4-manifold is a challenging en-
deavor, so our goal here is to understand Bach flow on a more restricted family that is more
tractable. Specifically, we study Bach flow on (M, g) where M = S1 × N , (N, g̃) is a closed
locally homogeneous three-dimensional Riemannian manifold, and g = gS1 + g̃ is the product
metric. By lifting to the universal cover M̂ of M , this analysis reduces to analysis of Bach flow
on one of nine simply connected homogeneous spaces.
The specific details for each of the nine cases can be found in Sections 5 and 6. As a summary,
we find:
• if N̂ = R3 or H3, Bach flow is static;
• if N̂ = Nil, ŜL(2,R), R× S2, or R×H2, Bach flow collapses to a flat surface;
• if N̂ = Solv, Bach flow collapses to a curve;
• if N̂ = E(2), Bach flow converges to a flat four-dimensional manifold;
• if N̂ = S3, Bach flow can collapse to a flat three-dimensional manifold, collapse to a flat
surface, or converge to a curved four-dimensional manifold, depending on the initial con-
ditions.
In this paper, Sn is the n-dimensional sphere, Hn is n-dimensional hyperbolic space, Nil is the
Heisenberg group consisting of 3 × 3 upper triangular matrices with 1’s on the diagonal, Solv
mailto:helliwed@seattleu.edu
https://doi.org/10.3842/SIGMA.2020.027
2 D. Helliwell
is the Poincaré group for 2-D Minkowski space R2 o O(1, 1), E(2) is the group of Euclidean
transformations of the plane, and ŜL(2,R) is the universal cover of SL(2,R).
The method here is similar to that of [11] and [12], where the qualitative behavior of volume-
normalized Ricci flow on locally homogeneous three- and four-dimensional manifolds was de-
termined. See [9] for an alternative approach to analyzing Ricci flow on homogeneous three-
dimensional manifolds, and see [14] for analysis of the quasi-convergence of locally homogeneous
manifolds under Ricci flow. Other geometric flows have also been analyzed on locally homo-
geneous spaces. See for example [13] for analysis of Cotton flow, [4] and [6] for analysis of
cross curvature flow, [5] for analysis of backward Ricci flow, and [8] for analysis of second-order
renormalization group flow.
Here, for most spaces, the analysis is a bit more challenging than for Ricci flow, since the
polynomials in the systems determined by Ricci flow are third order while those for Bach flow
are seventh order. These higher order expressions are more difficult to analyze for the purposes
of qualitative analysis.
Determining the behavior of Bach flow on model spaces has so far been limited to flow on
locally homogeneous 2×2 products by [7] and S1×Solv by [10]. Additionally, in [20], flow by the
gradient of the total curvature energy functional was analyzed on two specific four-dimensional
homogeneous spaces: S2 × H2 and R × S3. Bach flow is related to this flow, and comparing
and contrasting the qualitative behavior of these flows helps to understand this relationship. On
S2×H2 the equations determined by the two flows are essentially the same. On M̂ = R×S3, only
round metrics on S3 were considered in [20], and on compact quotients, the resulting product
metric was found to collapse to a three-dimensional space, with the S1 slice shrinking. Here, we
find that Bach flow is static in this case.
The general approach to understanding Bach flow on the spaces of interest is similar to that
found in [11]. The universal cover N̂ is either a Lie group or it is not. In the case where N̂
is a Lie group, the set of homogeneous metrics can be identified with the set of left-invariant
metrics on N̂ , which in turn are identified with the set of inner products on the tangent space at
the identity. Curvature can then be expressed in terms of the structure constants and the inner
product. The Lie groups of interest have the property that a basis can be found where the inner
product is diagonal and the structure constants can be written in a convenient form. As was
true for the Ricci tensor in [11], we find here that the Bach tensor in this setting is diagonal and
so Bach flow preserves this structure. The resulting system is analyzed, with explicit solutions
found in some cases, and limiting behavior is determined. If N̂ is not a Lie group, the analysis
proves to be somewhat simpler, owing to the fact that there are fewer homogeneous metrics on
these spaces. The resulting systems can all be solved explicitly.
This paper is organized as follows: In Section 2, the Bach tensor and Bach flow are discussed.
Additionally, formulas for the Bach tensor on products are provided. In Section 3, details
surrounding the locally homogeneous spaces and Lie groups of interest are provided, including
curvature formulas in terms of structure constants. In Section 4, useful results about ordinary
differential equations are provided. Section 5 is devoted to the derivation and analysis of Bach
flow on locally homogeneous 1 × 3 products and in Section 6, Bach flow is analyzed on locally
homogeneous 2 × 2 products. Finally, in Section 7, the qualitative results for Bach flow are
compared and contrasted with those of Ricci flow.
2 The Bach tensor and Bach flow
On a four-dimensional Riemannian manifold (M, g), the Bach tensor B is given by
Bjk = glqPjk;lq − glqPjl;kq + P ilWijkl,
Bach Flow on Homogeneous Products 3
where P is the Schouten tensor, which, for an n-dimensional manifold is defined as
P =
1
n− 2
(
Ric− S
2(n− 1)
g
)
and W is the Weyl tensor. Throughout this paper, curvature and index conventions follow those
found in [15]. The Bach tensor is a symmetric, trace free, divergence free tensor that is fourth
order in the metric and is conformally invariant: if ḡ = ρ2g, then B̄ = ρ−2B. It can be realized
as −1
4 grad(W) where W is the Weyl energy functional:
W =
∫
M
|W |2dµ
with |W |2 = gipgjqgkrglsWijklWpqrs.
In [1] and [2] short time existence and uniqueness are established for solutions to the geometric
flow
∂tg = B +
1
12
∆Sg, g(0) = h.
Here, and throughout, ∆ = gij∇i∇j . The positive multiple of ∆Sg is included to ensure that
the resulting flow is well posed. The fraction 1
12 could be replaced by any positive constant α.
In general, if α = 0, the analysis in [1] and [2] no longer applies and, as far as the author is
aware, it is not known if the case α = 0 is well posed.
On the other hand if, for a solution to the flow above, the scalar curvature S(t) is constant
on M , then the flow reduces to
∂tg = B, g(0) = h. (2.1)
In this paper, the flow is analyzed on locally homogeneous product manifolds and local homo-
geneity ensures that the scalar curvature is constant on M .
One useful consequence of the fact that the Bach tensor is trace free is that the volume form
is constant along the flow:
∂tdµg = 0.
If fixed coordinates are chosen, this is equivalent to saying det g(t) = deth is constant in
time.
2.1 Bach tensor on products
In general, the Bach flow equations lead to a complicated nonlinear system. Making use of the
product structure significantly simplifies the resulting equations.
Let
(
N (1), g̃(1)
)
and
(
N (2), g̃(2)
)
be Riemannian manifolds. Let M = N (1) × N (2). The
product metric g on M is
g = g(1) + g(2),
where g(i) = π∗i
(
g̃(i)
)
are the pullbacks of the component metrics by the natural projections.
Greek indices (α, β, γ, etc.) will be used for N (1), and lower case roman indices (i, j, k, etc.)
will be used for N (2). In the case where N (1) is one-dimensional, the subscript 0 will be used.
Abusing notation slightly, the tildes used above will be dropped. To clarify when dealing with
an object on N (1) or N (2) (as opposed to M) a parenthetical superscript will be used to indicate
the component.
4 D. Helliwell
For a general product,
Ricαβ = Ric
(1)
αβ , Ricjk = Ric
(2)
jk , Ricαk = 0
and
S = S(1) + S(2).
In particular, for 1× 3 products Ric00 = 0 and S = S(2).
The Bach tensor is somewhat more complicated. While the Bach tensor splits in the sense
that the components with mixed indices are zero, the components corresponding to one factor
depend on the curvature from the other factor.
The 1× 3 and 2× 2 cases are as follows: If dim
(
N (1)
)
= 1 and dim
(
N (2)
)
= 3 then
B00 =
(
− 1
12
(
∆(2)S(2)
)
− 1
4
[(
|Ric|(2)
)2 − 1
3
(
S(2)
)2])
g00, (2.2)
Bjk =
1
2
∆(2)Ric
(2)
jk −
1
12
∆(2)S(2)gjk −
1
6
S
(2)
;jk − 2 tr(2)
(
Ric(2) ⊗ Ric(2)
)
jk
+
7
6
S(2)Ric
(2)
jk +
3
4
(
|Ric|(2)
)2
gjk −
5
12
(
S(2)
)2
gjk, (2.3)
and
B0k = 0.
Here, tr(Ric⊗ Ric)jk = gilRicijRiclk. If dim
(
N (1)
)
= dim
(
N (2)
)
= 2 then
Bαβ = −1
6
S
(1)
;αβ +
1
6
[
∆(1)S(1) − 1
2
∆(2)S(2) +
1
4
((
S(1)
)2 − (S(2)
)2)]
gαβ (2.4)
and similarly
Bjk = −1
6
S
(2)
;jk +
1
6
[
∆(2)S(2) − 1
2
∆(1)S(1) +
1
4
((
S(2)
)2 − (S(1)
)2)]
gjk (2.5)
and
Bαk = 0.
Using the formulation for the Bach tensor in the 1× 3 setting, we have the following:
Proposition 2.1. Let dim
(
N (1)
)
= 1 and dim
(
N (2)
)
= 3, and suppose S = S(2) is constant.
Then B = 0 if and only if g(2) is Einstein.
Proof. If g(2) is Einstein then Ric(2) = S(2)
3 g(2) and as a result, equations (2.2) and (2.3) both
simplify to zero.
In the other direction, if g(2) is not Einstein, then
(
|Ric|(2)
)2
>
(
S(2)
)2
3
.
Since ∆(2)S(2) = 0, then in particular, B00 < 0. �
An immediate consequence of this result and its proof is the following:
Proposition 2.2. Let M be a 1×3 product with product metric g solving equation (2.1). Suppose
that for all time, the scalar curvature is constant on M . Then g00 is static if and only if B = 0,
in which case, all components of the metric are static. Otherwise, g00 is strictly decreasing.
Bach Flow on Homogeneous Products 5
3 Locally homogeneous spaces and Lie groups
A Riemannian manifold (M, g) is locally homogeneous if for all points p, q in M , there exist
neighborhoods U and V about p and q respectively, and an isometry
ϕ : U → V
with ϕ(p) = q. If, for all pairs of points, the isometry can be chosen to be global, so that
ϕ : M → M , then (M, g) is homogeneous. If M is closed and locally homogeneous, then its
universal cover is homogeneous. A straightforward, but useful, result is that if a manifold is
locally homogeneous, its scalar curvature is constant. There are nine three-dimensional simply
connected homogeneous manifolds with compact quotients, six of which are Lie groups. The Lie
groups support a larger class of homogeneous metrics and require a more sophisticated analysis
than the three non-Lie groups. See [11] for more details surrounding these definitions and results.
3.1 Structure constants and curvature
Let G be a Lie group with Lie algebra g, and let {ei} be a left-invariant basis for g. The bracket
can be expressed in terms of structure constants Cij
k
[ei, ej ] = Cij
kek.
Given a left-invariant metric g, and working with a left-invariant frame, covariant derivatives,
and then curvature can be expressed in terms of structure constants. The Ricci and scalar
curvatures are
Ricjk = −1
2
(
C lj
p
+ Cpj
l
)
Clkp +
1
4
C lpjClpk +
1
2
C lpl(Cpjk + Cpkj) (3.1)
and
S = −1
4
C lkpClkp −
1
2
CpklClkp − C lplCkpk. (3.2)
Additionally, with a bit more calculation, the Laplacian of a left-invariant tensor can be expressed
in terms of structure constants. For this paper, the Laplacian of a left-invariant symmetric
(
2
0
)
-
tensor T is needed and we have the following:
(∆T )ij =
1
2
Tpq
(
Cki
p
Ckj
q + CkpiCk
q
j + Ci
pkCj
q
k − C
k
i
p
Ck
q
j − Ckj
p
Ck
q
i
− Cki
p
Cj
q
k − C
k
j
p
Ci
q
k + CkpiCj
q
k + CkpjCi
q
k
)
+
1
4
Tqj
((
Ckpi − Cki
p
+ Ci
pk
)(
Ck
q
p − Ckpq + Cp
q
k
)
+ 2Ckpk
(
Cp
q
i − Cpi
q
))
+
1
4
Tqi
((
Ckpj− Ckj
p
+ Cj
pk
)(
Ck
q
p− Ckpq+ Cp
q
k
)
+ 2Ckpk
(
Cp
q
j− Cpj
q
))
. (3.3)
3.2 Three-dimensional Lie groups
As seen in [19], the six three-dimensional simply connected Lie groups with compact quotients
are all unimodular and all have the property that for each group, there is a basis for the Lie
algebra such that the structure constants can be represented by
Cij
k = εijsE
ks,
where εijk is the Levi-Civita symbol which captures the parity of the permutation generating
“ijk” with ε123 = 1, and where E is a 3× 3 matrix specific to the group. See Fig. 1.
To simplify the later analysis, the Bach flow equations will be determined in a basis where
the structure constants have the form indicated here and where the initial metric is diagonal.
As shown in [18], such an initial set-up is always possible:
6 D. Helliwell
Bianchi
Group E
Type
I R3 0
II Nil diag(1, 0, 0)
VI0 Solv diag(−1, 1, 0)
VII0 E(2) diag(−1,−1, 0)
VIII ŜL(2,R) diag(−1, 1, 1)
IX S3 id
Figure 1. The six three-dimensional simply connected Lie groups with compact quotients. The 3 × 3
matrix E encapsulates the structure constants.
Theorem 3.1. Given a three-dimensional Lie algebra with structure constants of the form
Cij
k = εijlE
lk, (3.4)
and an inner product g, there is a basis where
• g is diagonal (the basis is orthogonal),
• the structure constants can still be written in the form (3.4),
• the matrix E is unchanged.
The proof of this theorem follows from the principal axis theorem and the fact that structure
constants can be rescaled by rescaling the basis. We call the basis guaranteed by Theorem 3.1
a diagonalizing basis.
In light of the structure afforded by Theorem 3.1, we note the following general facts about
the Ricci tensor and Bach tensor:
Proposition 3.2. Let N be a three-dimensional Lie group with structure constants of the form
of equation (3.4), and left-invariant metric g. Then, in a basis where E and g are diagonal, the
Ricci tensor is diagonal and, on S1 ×N , the Bach tensor is diagonal.
For the Ricci tensor, this was established in [11] for the specific matrices in Fig. 1. This proof
shows that the property is a consequence of the diagonal structure of E and g, and not specific
to particular matrices. The proof of the general result follows from careful accounting of the
indices in each term found in the formulas for the Ricci and Bach tensors, using the fact that in
three dimensions, the indices are restricted to just three values.
Proof. First observe that if E is diagonal then equation (3.4) shows that Cij
k can only be
nonzero if i, j, and k are all different. Moreover, if g is diagonal, then the same must be true
for any raising or lowering of any of the indices. Hence any structure constant with a repeated
index must be zero, and in any double sum involving a pair of structure constants, the two free
indices must be equal in order for the result to be nonzero. Based on these observations, every
term in equation (3.1) must be zero unless j = k so Ric must be diagonal.
The analysis for the Bach tensor is similar. Looking at equation (2.3), note first that the
second and third terms are zero since scalar curvature is constant, and the fifth, sixth, and
seventh terms are diagonal since Ric(2) and g are diagonal. So the only terms to check are the
first and the fourth. For the fourth term, we have
tr(Ric⊗ Ric)jk = gilRicijRiclk.
Bach Flow on Homogeneous Products 7
Since g is diagonal, the terms in this sum are only nonzero when i = l, and then, since Ric is
diagonal, we can only have a nonzero term when j = k.
Finally for the first term, we use equation (3.3) with T = Ric(2) to analyze
(
∆(2)Ric(2)
)
ij
.
Equation (3.3) has three large terms in it. For the first term, since T is diagonal, the only way
any of the sums of products can be nonzero is if p = q, but then each product becomes a double
sum and so must be zero unless i = j. For the second and third terms, one piece is zero because
of a structure constant with a repeated index. For the rest, the double sums again require the
third pair of indices to match in order to produce something nonzero, and since T is diagonal,
the only nonzero terms appear when i = j. �
To help with the analysis of curvature along the flow, we have the following:
Lemma 3.3. Let {e1, e2, e3} be an orthogonal basis for the tangent space of a point in a 3-
dimensional manifold. Then at that point, the sectional curvatures are given by
K(ei, ej) =
Ricii
gii
+
Ricjj
gjj
− S
2
.
Proof. On a 3-dimensional manifold, Riemann curvature can be expressed completely in terms
of Ricci and scalar curvature as
R = Ric ◦ g − S
4
g ◦ g,
where A ◦B is the Kulkarni–Nomizu product. Using an orthogonal basis, this reduces to
Rijji = Riciigjj + Ricjjgii −
S
2
giigjj .
Sectional curvature is given by
K(v, w) =
R(v, w,w, v)
|v|2|w|2 − 〈v, w〉2
,
so
K(ei, ej) =
R(ei, ej , ej , ei)
|ei|2|ej |2 − 〈ei, ej〉2
=
Riciigjj + Ricjjgii − S
2 giigjj
giigjj
=
Ricii
gii
+
Ricjj
gjj
− S
2
as desired. �
Lemma 3.3 is all that is needed in this paper since the four-dimensional manifolds considered
are 1 × 3 products, so the formula above can be used for the three-dimensional slice, and the
sectional curvatures involving the one-dimensional slice are zero.
4 Ordinary differential equations
The ordinary differential equations to which Bach flow reduces on homogeneous products are
analyzed using standard techniques which are recalled here. First, we appeal to existence and
uniqueness of solutions regularly and without mention. In some instances, the equations of
interest are separable and explicit solutions may be found. When such explicit solutions cannot
be found, the Escape Lemma, which states that if a maximal flow does not exist for all time
then it cannot lie in a compact set, may be used to help determine the qualitative behavior of
solutions. See [16] for details surrounding these results. In addition to these methods, we make
use of a couple more specialized results which follow.
The following technical lemma provides a technique for determining the images of the integral
curves for a system of two equations involving homogeneous functions.
8 D. Helliwell
Lemma 4.1. Let (x(t), y(t)) solve the following system
dx
dt
= p(x, y),
dy
dt
= q(x, y),
where p and q are both homogeneous of degree k. Suppose x 6= 0, p(x, y) 6= 0, and q(x,y)
p(x,y) 6=
y
x .
Then (x(t), y(t)) will lie in the curve
x = ηeΨ( y
x),
where
Ψ(v) =
∫
1
q(1,v)
p(1,v) − v
dv
and η is a constant depending on the initial conditions.
To prove this result, briefly, express dy
dx in terms of p and q and then compute the derivative
of v = y
x with respect to x, substitute and use separation of variables. The details are left to
the reader.
In general, an integral curve for a vector field can be bounded but fail to converge to a limit.
The next lemma shows that if a coordinate of a bounded solution does converge, then that
component of the vector field must go to zero. The proof is left to the reader.
Lemma 4.2. Let x(t) =
(
x1(t), . . . , xn(t)
)
be a bounded solution to
d
dt
x = V (x),
where V is a continuous vector field on a domain D. Suppose
lim
t→∞
xi(t) = L
and let {xk} = {x(tk)}, tk →∞ be a sequence of points on the curve that converges to x∞ ∈ D.
Then V i(x∞) = 0.
5 Bach flow on locally homogeneous 1 × 3 products
In this section, the main results of this paper are proved for 1 × 3 products that are not also
2×2 products. For each universal cover, explicit formulas for the Bach tensor are found and the
evolution of the metric under Bach flow is determined. In some cases, explicit solutions are found.
When explicit solutions are not found, qualitative behavior is determined. Limiting behaviors
of both the metric and its curvature are also found, and convergence, in the Gromov–Hausdorff
or pointed Gromov–Hausdorff topology, is described.
The general method is as follows: Given an initial metric, a diagonalizing basis is found so
that the metric is diagonal
h = diag(h00, h11, h22, h33).
Its Ricci and scalar curvatures are calculated using equations (3.1) and (3.2), and then using
equations (2.2), (2.3), and (3.3) the Bach tensors are calculated. As indicated by Proposition 3.2,
the Bach tensor is also diagonal, so the fact that the metric is diagonal is preserved along the
flow. The solution will be denoted
g = diag(g00, g11, g22, g33).
Bach Flow on Homogeneous Products 9
One quantity that makes a regular appearance is
β =
1
6(det g)2
=
1
6(deth)2
.
This quantity depends on the initial metric, but is constant along the flow. As a consequence,
once an initial metric is chosen, β can be treated as a constant for the whole system.
After explicitly determining the Bach tensor in each case, a general structure emerges. Speci-
fically, the nonzero components of the Bach tensor have the form
Bii = αiβpi(g11, g22, g33)(g00)2gii,
where αi is a constant and pi is a homogeneous fourth degree polynomial. This structure makes
a great deal of qualitative analysis possible when explicit solutions are not found. The details of
the analysis vary from space to space, although there are similarities when the spaces themselves
have similar structure constant matrices E.
One general fact is that g00 is decreasing, as indicated by Proposition 2.2. This fact will
not be explicitly included in the specific theorems for each space. Another general fact is that
the flow is defined (at least) on the interval [0,∞). In cases where the flow remains bounded,
this follows from the escape lemma. In cases where the flow does not remain bounded, this is
discovered after the analysis of each flow is completed and follows from the work in [17], which
shows that the maximal time is finite only if there is curvature blow-up, and the fact that in all
of our cases, curvature remains bounded.
5.1 R3
For this manifold the matrix E used to determine the structure constants in Theorem 3.1 is the
zero matrix, so regardless of the initial metric, the structure constants are all zero. Hence the
Ricci tensor, scalar curvature are zero, and on R×R3 the Bach tensor is zero and so the metric
is static under Bach flow.
5.2 Nil
For this manifold the matrix used to determine the structure constants in Theorem 3.1 is
E = diag(1, 0, 0).
For any metric g, using a diagonalizing basis, the Ricci tensor is diagonal with
Ric11 =
(g11)2
2g22g33
, Ric22 = − g11
2g33
, Ric33 = − g11
2g22
and scalar curvature is
S = −g00(g11)2
2 det g
.
The Bach tensor on R×Nil is diagonal with
B00 = −β(g00)3(g11)4, B11 = −5β(g00)2(g11)5,
B22 = 3β(g00)2(g11)4g22, B33 = 3β(g00)2(g11)4g33.
With the Bach tensor in hand, we have the following theorem:
10 D. Helliwell
Theorem 5.1. On M̂ = R × Nil the solutions to equation (2.1) in a diagonalizing basis for h
are given by
g00(t) =
(
γt+ (h00)−22
)− 1
22 , g11(t) = α
(
γt+ (h00)−22
)− 5
22 ,
g22(t) = h22(h00)3
(
γt+ (h00)−22
) 3
22 , g33(t) = h33(h00)3
(
γt+ (h00)−22
) 3
22 ,
where
α =
h11
(h00)5
and γ = 22α4β =
11
3(deth)2
(
h11
(h00)5
)4
.
Proof. Note that the first and second equations are coupled and the third and fourth equations
depend on the first and second solutions, but are otherwise uncoupled. Because everything is
multiplicative, we can solve explicitly for g00, and g11, and then g22 and g33.
Starting with the following ansantz
g11 = α(g00)k
and then comparing the resulting differential equations produces
g00(t) =
(
γt+ (h00)−22
)− 1
22 and g11(t) = α
(
γt+ (h00)−22
)− 5
22 ,
where
γ = 22α4β.
Then we can solve for g22 and g33:
g22(t) = h22(h00)3
(
γt+ (h00)−22
) 3
22 and g33(t) = h33(h00)3
(
γt+ (h00)−22
) 3
22 . �
With these solutions in hand, we find two dimensions collapse in the limit as t → ∞.
The “g00” direction collapses more slowly than the first dimension in Nil. Meanwhile, the
other two dimensions grow at the same rate, preserving their aspect ratio. These solutions are
immortal, but not ancient.
All components of the Ricci tensor converge to zero in the limit, and using Lemma 3.3 we
have the following:
Theorem 5.2. Let M be a compact quotient of R×Nil and let p ∈M . Let g solve equation (2.1)
where h is locally homogeneous. Then (M, g, p) collapses to a flat surface in the pointed Gromov–
Hausdorff topology.
5.3 Solv
For this manifold the matrix used to determine the structure constants in Theorem 3.1 is
E = diag(−1, 1, 0).
For any metric g, using a diagonalizing basis, the Ricci tensor is diagonal with
Ric11 =
(g11)2 − (g22)2
2g22g33
, Ric22 =
(g22)2 − (g11)2
2g11g33
, Ric33 = −(g11 + g22)2
2g11g22
and scalar curvature is
S = −(g11 + g22)2
2g11g22g33
.
Bach Flow on Homogeneous Products 11
The Bach tensor is diagonal with
B00 = −βpI(g11, g22)(g00)3, B11 = −βpII(g11, g22)(g00)2g11,
B22 = −βpII(g22, g11)(g00)2g22, B33 = 3βpI(g11, g22)(g00)2g33,
where
pI(x, y) = x4 + x3y + xy3 + y4, pII(x, y) = 5x4 + 3x3y − xy3 − 3y4.
With the Bach tensor in hand, we have the following theorem:
Theorem 5.3. On M̂ = R× Solv every solution to equation (2.1) in a diagonalizing basis has
the following properties:
• g00, g11, g22 → 0;
• g33 →∞ monotonically;
• g11
g22
→ 1.
If h11 = h22, then
g00 = µ
1
2
(
24µβt+ (h11)−6
)− 1
6 , g11 = g22 =
(
24µβt+ (h11)−6
)− 1
6 ,
g33 = (h11)3h33
(
24µβt+ (h11)−6
) 1
2 .
Otherwise, if (without loss of generality) h11 < h22, then
• g11 < g22 for the entire flow;
• g22 is decreasing;
• g11
g22
is increasing;
• g11 and g22 are related by
(g11g22)25 = η(g22 − g11)4
(
2(g22)2 + g11g22 + 2(g11)2
)3
,
where
η =
(h11h22)25
(h22 − h11)4
(
2(h22)2 + h11h22 + 2(h11)2
)3 .
It turns out that in addition to appearing in the Bach tensor above, the two polynomials pI
and pII also make an appearance in the next section so we establish some facts about them for
use here and later.
Lemma 5.4. The polynomial pI(x, y) is symmetric, homogeneous of degree 4, positive when x
or y is nonzero, and can be factored as
pI(x, y) = (x+ y)2
(
x2 − xy + y2
)
.
The polynomial pII(x, y) is homogeneous of degree 4 and can be factored as
pII(x, y) = (x+ y)
(
5x3 − 2x2y + 2xy2 − 3y3
)
.
The cubic factor has exactly one real factor (αx−y) where α is about 1.23. If x > y, pII(x, y) > 0.
12 D. Helliwell
The proof of this lemma is left to the reader. While α, as the root of a cubic, can be found
exactly, this exact form is not important for the analysis here. With these facts about pI and pII
established, we proceed with the proof of Theorem 5.3.
Proof of Theorem 5.3. Since B00 and B33 are so similar, we can compute
d
dtg33
d
dtg00
=
B33
B00
=
−3g33
g00
,
which implies
g33 = γ(g00)−3, (5.1)
where γ = (h00)3h33. Since, by Proposition 2.2, g00 is decreasing, this shows g33 must be
increasing. Since det g is constant, using equation (5.1) we have
(g00)2 = µg11g22, (5.2)
where
µ =
γ
deth
.
Incorporating these identities into the formulas for B11 and B22 we have
d
dt
g11 = −µβpII(g11, g22)(g11)2g22 (5.3)
and
d
dt
g22 = −µβpII(g22, g11)(g22)2g11.
Because of the symmetry in these equations, we may, without loss of generality, restrict our
attention to the region defined by 0 ≤ g11 ≤ g22.
If h11 = h22 then, focusing on g11 we get
d
dt
g11 = −4µβg7
11,
which is separable. Solving, we get
g11 = g22 =
(
24µβt+ (h11)−6
)− 1
6 .
Next, we solve for g00 and g33 using equations (5.1) and (5.2). We have
g00 = µ
1
2
(
24µβt+ (h11)−6
)− 1
6 and g33 = (h11)3h33
(
24µβt+ (h11)−6
) 1
2 .
In this special case, we see that under Bach flow, any compact quotient of R× Solv collapses to
a curve in the limit.
If h11 < h22 then g11 < g22 by existence and uniqueness, since we have a solution that
preserves the equality g11 = g22. With this inequality preserved, from the properties of pII,
looking at equation (5.3) we find that g22 is decreasing. Also
d
dt
(
g11
g22
)
=
(
d
dtg11
)
g22 − g11
(
d
dtg22
)
(g22)2
= 4µβ(g11 + g22)(g22 − g11)
(
2(g11)2 + g11g22 + 2(g22)2
)
(g11)2 > 0
Bach Flow on Homogeneous Products 13
so we find that g11
g22
is increasing. This fraction is bounded above by 1, so it must converge. We
will see below that it converges to 1.
The fact that this fraction is increasing also implies that g11 cannot converge to zero unless g22
does as well. This, combined with Lemma 4.2 implies that g22 and hence g11 must converge to 0
since the only points in the domain of interest where d
dtg22 is zero are along the g22 axis.
Next, letting v = g11
g22
we have
dg11
dg22
=
pII(v, 1)v2
pII(1, v)v
= v
(
5v3 − 2v2 + 2v − 3
)(
5− 2v + 2v2 − 3v3
) .
Therefore by Lemma 4.1, the solution curves for our original differential equation satisfy the
equation
g22 =
η̃
(
1− g11
g22
) 1
10
(
2 + g11
g22
+ 2
(g11
g22
)2) 3
40
(g11
g22
) 5
8
,
where η̃ is a constant determined by the initial conditions. Multiplying both sides by
(g11
g22
) 5
8 ,
we have
(
g11
g22
) 5
8
g22 = η̃
(
1− g11
g22
) 1
10
(
2 +
g11
g22
+ 2
(
g11
g22
)2
) 3
40
. (5.4)
This can be rewritten as
(g11g22)25 = η(g22 − g11)4
(
2(g22)2 + g11g22 + 2(g11)2
)3
,
where η = η̃40. This is true in particular at t = 0, so
η =
(h11h22)25
(h22 − h11)4
(
2(h22)2 + h11h22 + 2(h11)2
)3 .
Taking the limit as t→∞, the left side of equation (5.4) must be zero, and therefore so must the
right. Since the second factor is positive, it follows that lim
t→∞
(
1− g11
g22
) 1
10 = 0 and so lim
t→∞
g11
g22
= 1.
From the analysis above, we know that in general, g11 and g22 go to zero as t goes to infinity.
From equations (5.2) and (5.1), we then know that g00 also goes to zero and g33 grows to infinity.
Therefore, under Bach flow, any compact quotient of R× Solv collapses to a curve in the limit.
These facts, combined with the fact that g11
g22
goes to 1 imply that a general solution approaches
the specific solution found above in the limit. �
We now have the following:
Theorem 5.5. Let M be a compact quotient of R × Solv and let p ∈ M . Let g solve equa-
tion (2.1) where h is locally homogeneous. Then (M, g, p) collapses to a line in the pointed
Gromov–Hausdorff topology.
Proof. From the previous theorem, we know that three dimensions collapse, while one expands.
Moreover, working in a diagonalizing basis, we find that Ric11 and Ric22 converge to zero while
Ric33 converges to −2. Therefore, by Lemma 3.3, all the sectional curvatures go to zero in the
limit. �
14 D. Helliwell
5.4 E(2)
For this manifold the matrix used to determine the structure constants in Theorem 3.1 is
E = diag(−1,−1, 0).
For any metric g, using a diagonalizing basis, the Ricci tensor is diagonal with
Ric11 =
(g11)2 − (g22)2
2g22g33
, Ric22 =
(g22)2 − (g11)2
2g11g33
, Ric33 = −(g11 − g22)2
2g11g22
and scalar curvature is
S = −(g11 − g22)2
2g11g22g33
.
The Bach tensor is diagonal with
B00 = −βpI(−g11, g22)(g00)3, B11 = −βpII(−g11, g22)(g00)2g11,
B22 = −βpII(g22,−g11)(g00)2g22, B33 = 3βpI(−g11, g22)(g00)2g33,
where pI and pII were defined is Section 5.3.
Theorem 5.6. On M̂ = R× E(2) every solution to equation (2.1) in a diagonalizing basis has
the following properties:
• g11 and g22 are related by
(g11g22)25 = η(g22 + g11)4
(
2(g22)2 − g11g22 + 2(g11)2
)3
,
where
η =
(h11h22)25
(h22 + h11)4
(
2(h22)2 − h11h22 + 2(h11)2
)3 ;
• the flow exists for all time and as t→∞,
g11, g22 → (432η)
1
40 , g00 →
(
(h00)3h33
deth
) 1
2
(432η)
1
40 ,
g33 →
(
(deth)3
(h00)3h33
) 1
2
(432η)−
3
40 ;
• g33 is increasing.
If h11 = h22, then the solution is static. Otherwise, if (without loss of generality) h11 < h22,
then
• g11 < g22 for the entire flow,
• g11 is increasing,
• g22 is decreasing.
Bach Flow on Homogeneous Products 15
Proof. Since the only difference between this system and that of R× Solv is the minus sign on
one of the variables in pI and pII, most of the initial analysis of the previous section carries over
and we have
g33 = γ(g00)−3. (5.5)
with γ = (h00)3h33. Because of this inverse relationship, since g00 is decreasing, we find that g33
must be increasing. We also have
(g00)2 = µg11g22, (5.6)
where
µ =
γ
deth
.
Incorporating these identities into the formulas for B11 and B22 we have
d
dt
g11 = −µβpII(−g11, g22)(g11)2g22,
d
dt
g22 = −µβpII(g22,−g11)(g22)2g11.
Because of the symmetry in these equations, we may, without loss of generality, restrict our
attention to the region defined by 0 ≤ g11 ≤ g22. If h11 = h22, then B = 0 and we have a set of
stationary solutions corresponding to the flat metrics on E(2). If h11 < h22 then g11 < g22 for all
time and from the poperties of pII, we find that g11 is increasing and g22 is decreasing. Therefore,
both must converge and by Lemma 4.2 this can only happen at a point where g11 = g22.
As with Solv, we can say a bit more about the curves traced out by the solutions using
Lemma 4.1. Except for two minus signs, the analysis here is almost identical to that for Solv
and we find
(g11g22)25 = η(g22 + g11)4
(
2(g22)2 − g11g22 + 2(g11)2
)3
, (5.7)
where
η =
(h11h22)25
(h22 + h11)4
(
2(h22)2 − h11h22 + 2(h11)2
)3 .
Let gii(∞) be the limit of gii as t → ∞. Then we know that g11(∞) = g22(∞) and using
equation (5.7) we find(
g11(∞)
)50
= η
(
2g11(∞)
)4(
3(g11(∞))2
)3
and so
g11(∞) = g22(∞) = (432η)
1
40 .
Then by equation (5.6)
g00(∞) = µ
1
2 (432η)
1
40 =
(
(h00)3h33
deth
) 1
2
(432η)
1
40
and by equation (5.5)
g33(∞) = γ
[
µ
1
2 (432η)
1
40
]−3
=
(
(deth)3
(h00)3h33
) 1
2
(432η)−
3
40 . �
Finally, we have the following
Theorem 5.7. Let M be a compact quotient of R × E(2). Let g solve equation (2.1) where
h is locally homogeneous. Then (M, g) converges to a flat four-dimensional manifold in the
Gromov–Hausdorff topology.
Proof. None of the components of the metric converge to zero, so there is no collapse. Since
g11 − g22 → 0 as t → ∞, looking at the Ricci curvature, we find that the manifold becomes
Ricci-flat in the limit. By Lemma 3.3, so do the sectional curvatures. �
16 D. Helliwell
5.5 ŜL(2,R)
For this manifold the matrix used to determine the structure constants in Theorem 3.1 is
E = diag(−1, 1, 1).
For any metric g, using a diagonalizing basis, the Ricci tensor is diagonal with
Ric11 =
(g11)2 − (g22 − g33)2
2g22g33
, Ric22 =
(g22)2 − (g11 + g33)2
2g11g33
,
Ric33 =
(g33)2 − (g11 + g22)2
2g11g22
and scalar curvature is
S = −(g11)2 + (g22)2 + (g33)2 + 2(g11g22 + g11g33 − g22g33)
2g11g22g33
.
The Bach tensor is diagonal with
B00 = −βqI(−g11, g22, g33)(g00)3, B11 = −βqII(−g11, g22, g33)(g00)2g11,
B22 = −βqII(g22,−g11, g33)(g00)2g22, B33 = −βqII(g33,−g11, g22)(g00)2g33,
where
qI(x, y, z) = x4 − x3(y + z) + x2yz + x
(
−y3 + y2z + yz2 − z3
)
+ y4 − y3z − yz3 + z4
and
qII(x, y, z) = 5x4 − 3x3(y + z) + x2yz + x
(
y3 − y2z − yz2 + z3
)
− 3y4 + 3y3z + 3yz3 − 3z4.
The sign choices made in the formulas for the Bach tensor here come from the fact that qI and qII
are also used in the next section for S3, where no minus signs are needed in the expressions for
the Bach tensor.
With the Bach tensor in hand, we have the following theorem:
Theorem 5.8. On M̂ = R× ŜL(2,R), every solution to equation (2.1) in a diagonalizing basis
has the following properties:
• g00, g11 → 0;
• g22, g33 →∞;
• g33 − g22 → 0.
Before proving this, we establish some supporting lemmas. The two polynomials qI and qII
also make an appearance in the next section so we provide some facts about them here. For qI,
we have the following lemma, the proof of which is left to the reader.
Lemma 5.9. The polynomial qI has the following properties:
• it is symmetric;
• qI(−x,−y,−z) = qI(x, y, z);
• it is always nonnegative;
• it is equal to zero if and only if x = y = z or two variables are equal and the third is zero.
Bach Flow on Homogeneous Products 17
Note that qII is symmetric in the last two variables. Because of this, and the fact that the
flow equations for g22 and g33 are essentially the same, we say that without loss of generality,
g22 ≤ g33.
The qualitative behavior of the flow is determined through a number of estimates which arise
from monotonicity of various quantities. To keep things clear, these monotonicity results are
presented in the following lemmas.
Lemma 5.10. Suppose h22 < h33. Then g33
g22
decreases along the flow.
Proof. Writing out the quotient rule and plugging in the differential equations for g22 and g33,
we have
d
dt
g33
g22
= −β
(
qII(g33,−g11, g22)− qII(g22,−g11, g33)
)
(g00)2 g33
g22
.
Writing out and simplifying the factor involving the qII’s, we get
qII(g33,−g11, g22)− qII(g22,−g11, g33)
= 2(g33 − g22)
[
4(g33)3 + 2(g33)2g22 + 2g33(g22)2 + 4(g22)3
+ g11
(
3(g33)2 + 2g33g22 + 3(g22)2
)
+ (g11)3
]
and this is positive since g22 < g33. �
Lemma 5.11. Suppose h22 ≤ h33. Then g00g22 increases along the flow.
Proof. Writing out the product rule and plugging in the differential equations for g00 and g22,
we have
d
dt
(g00g22) = −β
(
qI(−g11, g22, g33) + qII(g22,−g11, g33)
)
(g00)3g22.
Writing out and simplifying the factor involving the qI and qII, we get
qI(−g11, g22, g33) + qII(g22,−g11, g33)
= −2
[
(g33 − g22)
[
(g33)3 + (g33)2g22 + g33(g22)2 + 3(g22)3
+ g11
(
(g33)2 + g33g22 + 2(g22)2
)]
+ (g11)3(g11 + g33)
]
.
This is negative when g22 ≤ g33 so d
dt(g00g22) is positive. �
Since g00 decreases along the flow, an immediate consequence of this lemma is that if h22 ≤ h33
then g22 by itself increases along the flow. In fact, a consequence of the proof is that d
dtg22 is
positive.
Lemma 5.12. Suppose h22 ≤ h33. Then (g11)
3
5 g22 increases along the flow.
Proof. Writing out the product rule and plugging in the differential equations for g11 and g22,
we have
d
dt
[
(g11)
3
5 g22
]
= −β
5
(
3qII(−g11, g22, g33) + 5qII(g22,−g11, g33)
)
(g00)2(g11)
3
5 g22.
Writing out and simplifying the factor involving qII we get
3qII(−g11, g22, g33) + 5qII(g22,−g11, g33)
= −2
[
(g33 − g22)
[
12(g33)3 + 5(g33)2g22 + 5g33(g22)2 + 8(g22)3
+ g11(9(g33)2 + 5g33g22 + 6(g22)2)
]
+ g2
11[g33g22 + g11(3g33 − 2g22)]
]
.
This is always negative since g22 ≤ g33, so d
dt
[
(g11)
3
5 g22
]
is positive. �
18 D. Helliwell
Lemma 5.13. Suppose g22 and g33 diverge and g11 converges to zero. Then the quantity g33−g22
g11
converges to zero along the flow.
Proof. First note that if g22 = g33 then the result is true immediately. If, without loss of
generality, g22 < g33, we proceed in two steps. The first step is similar to the lemmas above.
We have
d
dt
(g33 − g22) = −βs(g11, g22, g33)(g00)2(g33 − g22),
where
s(x, y, z) = −3x4 − x3(y + z)− x2yz + x
(
3y3 + 5y2z + 5yz2 + 3z3
)
+ 5y4 + 5y3z + 4y2z2 + 5yz3 + 5z4.
With this we have
d
dt
g33 − g22
(g11)2
= −β
[
s(g11, g22, g33)− 2qII(−g11, g22, g33)
]
(g00)2 g33 − g22
(g11)2
.
Looking at the polynomial in brackets, we have
s(g11, g22, g33)− 2qII(−g11, g22, g33)
= g33
[
(g33)2 + (g22)2
]
(g33 − g22) + 7g33
[
(g33)3 − (g11)3
]
+ 7g22
[
(g22)3 − (g11)3
]
+ 3g22g33
[
g22g33 − (g11)2
]
+ g11
[
5(g22)3 + 3(g22)2g33 + g22(g33)2 + 5(g33)3
]
+ 3(g33)4 + 4(g22)4 − 13(g11)4.
Since g22 and g33 go to infinity, and g11 goes to zero, this must eventually become and stay
positive and so the fraction g33−g22
(g11)2
must eventually decrease. Since 1
g11
diverges, this impiles
that g33−g22
g11
must converge to zero. �
We are now in a position to prove Theorem 5.8. The proof requires considering a few different
possibilities and ruling out any option other than what is described in the theorem.
Proof of Theorem 5.8. Without loss of generality, we may restrict our attention to flows that
satisfy g22 ≤ g33. Now, first suppose g00 converges to a value greater than zero, with the goal of
ruling this possibility out. Consider two possibilities. Suppose first that g22 remains bounded
above. By Lemma 5.10, g33 must remain bounded as well. From this, since det g is constant, we
know that g11 remains bounded above and also below by some positive number.
By Lemma 5.11, g22 is increasing, since g00 is decreasing, so since g22 is bounded, it must
converge. By Lemma 4.2, there must be a point where d
dtg22 = 0. This contradicts the fact,
from the proof of Lemma 5.11, that d
dtg22 is positive in the given domain.
This implies that g22 goes to infinity, and so must g33. Since we are still working with the
possibility that g00 does not go to zero, we may conclude that g11 converges to zero, again since
det g is constant.
Now consider the product g00(g11)
6
5 g22g33. Note that this is equal to (det g)(g11)
1
5 which
must go to zero since det g is constant and g11 goes to zero. On the other hand
g00(g11)
6
5 g22g33 = g00
[
(g11)
3
5 g22
][
(g11)
3
5 g33
]
≥ g00
[
(g11)
3
5 g22
]2
.
By Lemma 5.12, the squared factor is increasing. But this implies that g00 must go to zero,
a contradiction.
Bach Flow on Homogeneous Products 19
So we may conclude that g00 converges to zero. Knowing this, since g00g22 is increasing by
Lemma 5.11, g22 and hence g33 must both diverge to ∞. But then, again by Lemma 5.11,
g00g22g33 diverges and so g11 must go to zero since g00g11g22g33 = det g is constant.
Finally, we have now established the hypotheses for Lemma 5.13 so we may conclude that
g33 − g22 → 0. �
With the limiting behavior of the metric established, the next step is to determine the cur-
vature.
Proposition 5.14. On M̂ = R× ŜL(2,R) for every solution to equation (2.1) in a diagonalizing
basis, Ric11
g11
converges to 0, and Ric22 and Ric33 both converge to −1. The scalar curvature
converges to 0 as well.
Proof. We have
Ric11
g11
=
(g11)2 − (g22 − g33)2
2g11g22g33
.
By Theorem 5.8, the numerator goes to zero and, since the determinant is constant, the denom-
inator goes to infinity.
For Ric22 we rewrite
Ric22 =
(g22)2 − (g11 + g33)2
2g11g33
=
g22 − g33
g11
1
2
(
g22
g33
+ 1
)
− g11
2g33
− 1.
By Lemma 5.13, the first factor in the first term goes to zero and the rest of the term is bounded.
The middle term also goes to zero. The computation for Ric33 is similar.
For scalar curvature, we rewrite to get
S = − g11
2g22g33
− g22 − g33
g11
g22 − g33
g22g33
− 1
g33
− 1
g22
and, by Theorem 5.8 and Lemma 5.13, all these terms go to zero. �
Finally, we have the following:
Theorem 5.15. Let M be a compact quotient of R × ŜL(2,R) and let p ∈ M . Let g solve
equation (2.1) where h is locally homogeneous. Then (M, g, p) converges to a flat surface in the
pointed Gromov–Hausdorff topology.
Proof. From the previous proposition, and by Lemma 3.3, the sectional curvatures all go to
zero along the flow. �
5.6 S3
For this manifold the matrix used to determine the structure constants in Theorem 3.1 is
E = id.
For any metric g, using a diagonalizing basis, the Ricci tensor is diagonal with
Ric11 =
(g11)2 − (g22 − g33)2
2g22g33
, Ric22 =
(g22)2 − (g11 − g33)2
2g11g33
,
Ric33 =
(g33)2 − (g11 − g22)2
2g11g22
20 D. Helliwell
and scalar curvature is
S = −(g11)2 + (g22)2 + (g33)2 − 2(g11g22 + g11g33 + g22g33)
2g11g22g33
.
The Bach tensor is diagonal with
B00 = −βqI(g11, g22, g33)(g00)3, B11 = −βqII(g11, g22, g33)(g00)2g11,
B22 = −βqII(g22, g33, g11)(g00)2g22, B33 = −βqII(g33, g11, g22)(g00)2g33,
where qI and qII were defined in Section 5.5.
On this space, there are a variety of possibilities for Bach flow, depending on the initial con-
ditions. To accommodate this richer structure, we break the results into a number of theorems.
Because of the symmetry in the equations, we may suppose, without loss of generality, that
h11 ≤ h22 ≤ h33. We first analyze the cases where at least two of the initial conditions are
equal. These results will begin to illustrate the complexity of the situation and begin to provide
some context for the remaining cases. Ultimately, all possibilities are analyzed, culminating in
Theorem 5.28.
Theorem 5.16. On M̂ = R × S3, let g solve equation (2.1) in a diagonalizing basis with
h11 = h22 = h33. Then g is static.
Note that in this case,
(
N (2), g(2)
)
is a round sphere.
Theorem 5.17. On M̂ = R × S3, let g solve equation (2.1) in a diagonalizing basis with
h11 = h22 < h33 or h11 < h22 = h33 < 4h11. Then
• g00 →
(
κ
3
) 3
8 (deth)−
1
2 ;
• g11, g22, g33 →
(
3
κ
) 1
8 (deth)
1
2 ;
• the components of g00 and g22 are related by(
4 deth− g00(g22)3
)
(g00)3g22 = κ,
where
κ =
(
4 deth− h00(h22)3
)
(h00)3h22;
• if h11 = h22, then g11 and g22 are increasing, and g33 is decreasing;
• if h22 = h33, then g11 is increasing.
Theorem 5.18. On M̂ = R × S3, let g solve equation (2.1) in a diagonalizing basis with
4h11 = h22 = h33. Then
g00(t) = 4(deth)
(
1
26
t+ (h33)2
)− 3
2
,
4g11(t) = g22(t) = g33(t) =
(
1
26
t+ (h33)2
) 1
2
.
Note that in this case, g00 → 0 and
(
N (2), g(2)
)
is self-similar as it expands.
Theorem 5.19. On M̂ = R × S3, let g solve equation (2.1) in a diagonalizing basis with
4h11 < h22 = h33. Then
Bach Flow on Homogeneous Products 21
• g00, g11 → 0;
• g22, g33 →∞;
• the components of g00 and g22 are related by(
g00(g22)3 − 4 deth
)
(g00)3g22 = κ,
where
κ =
(
h00(h22)3 − 4 deth
)
(h00)3h22;
• g22 and g33 are increasing.
Before proving these theorems, we introduce some new structure to help with the analysis.
To capitalize on the fact that det g is constant along the flow, and to exploit the symmetry
among the equations for g11, g22, and g33, we introduce three new variables:
a = (g00)
1
3 g11, b = (g00)
1
3 g22, c = (g00)
1
3 g33
so that abc = det g, and we rewrite our system using these. We have
d
dt
g00 = −βqI(a, b, c)(g00)
5
3 ,
d
dt
a = −2β
3
r(a, b, c)(g00)
2
3a,
d
dt
b = −2β
3
r(b, a, c)(g00)
2
3 b,
d
dt
c = −2β
3
r(c, a, b)(g00)
2
3 c,
(5.8)
where
r(x, y, z) = 8x4 − 5x3(y + z) + 2x2yz + x
(
y3 − y2z − yz2 + z3
)
− 4y4 + 4y3z + 4yz3 − 4z4.
For this new system, the solution curves lie in the surface {abc = deth}. Moreover, because of
the symmetry in the equations, we may restrict our attention to solutions that satisfy a ≤ b ≤ c.
From the determinant constraint, we know a = deth
bc so this inequality becomes
√
deth
c ≤ b ≤ c.
Thus, the flow is analyzed on the domain
D =
{
(g00, b, c) : g00 ≥ 0,
√
deth
c
≤ b ≤ c
}
.
In the following, while a can be eliminated, we find that it is useful to use in the analysis. As
such, a should always be thought of as a function of b and c. Let P0 be the point in D where
g00 = 0 and a = b = c, let L0 be the ray where g00 ≥ 0 and a = b = c, let ∂D0 be the set of
points in D where g00 = 0, let ∂Da=b be the set of points in D where a = b, let ∂Db=c be the
set of points in D where b = c. Note that ∂D = ∂D0 ∪ ∂Da=b ∪ ∂Db=c. Let P1 be the point
in D where g00 = 0 and 4a = b = c, and let L1 be the ray where g00 ≥ 0 and 4a = b = c. See
Figs. 2 and 3.
With this notation in place, before proving the theorems above, we note that ∂D0 corresponds
to degenerate metrics and points in ∂D0 are not really achievable from the perspective of the
original system. However, once an initial metric h is chosen, determining β, system (5.8) is well
22 D. Helliwell
b
c
P0
P1
∂D0 ∩ ∂Da=b
��*
MS
�
MU = ∂D0 ∩ ∂Db=c
AK
Figure 2. Essential features of ∂D0. While the points here correspond to degenerate metrics, studying
the behavior of Systems (5.8) and (5.10) here helps to clarify the behavior of solutions in D.
b
c
g00
P0
P1
L0
-
L1
-
∂D0
QQk
∂Db=c
QQk
∂Da=b
�
��
DS
�
��
Figure 3. Essential features of D. The regions DL0 and D∞, not labelled, are determined by the
surfaces shown, with DL0
between ∂Da=b and DS . Together, Theorems 5.16–5.22 describe the behavior
of solutions in each surface or region.
defined on ∂D0, and it is useful to explore the behavior here because it informs the behavior
on the interior. All solutions starting here are static and as a consequence, it is conceivable
that nondegenerate solutions converge to these points. We will find that with the exception
of solutions converging to P1, this is not the case. In the following proofs, unless otherwise
indicated, we restrict our attention to initial conditions with g00 > 0.
Bach Flow on Homogeneous Products 23
We now have the following:
Proof of Theorem 5.16. This case corresponds to L0. Here, r(a, b, c) = qI(a, b, c) = 0, and
we have static solutions. �
Proof of Theorem 5.17. First, we consider the case where h11 = h22, which corresponds
to ∂Da=b. Here, c = deth
b2
and, with this, the system reduces to two variables:
d
dt
g00 = −βqI(b, b, c)(g00)
5
3 ,
d
dt
b = −2β
3
r(b, b, c)(g00)
2
3 b.
Both qI and r simplify substantially:
qI(b, b, c) = c2(b− c)2, r(b, b, c) = −c2(b− c)(b− 4c)
and from this, we can see that b is increasing, since b < c. Hence a is also increasing. This
implies that g11 and g22 are also increasing. Furthermore, since abc is constant, c is decreasing.
By Lemma 4.2 the flow must converge to a point in either L0 or ∂D0 ∩ ∂Da=b. This second
possibility will be ruled out below.
Next, since r(b, b, c) is not zero, we have
dg00
db
=
d
dtg00
d
dtb
= −3
2
b− c
b− 4c
g00
b
.
This is separable and we get
g00 =
[
κ
[(
4 deth− b3
)
b
]−1] 3
8 , (5.9)
where
κ = (h00)
8
3
(
4 deth− (b(0))3
)
b(0) =
(
4 deth− h00(h22)3
)
(h00)3h22.
Substituting for b in equation (5.9) and rearranging gives us the desired relationship between g00
and g22.
The relationship given by equation (5.9) shows us two things. First since
g33 = (g00)−
1
3 c = deth(g00)−
1
3 b−2
we can substitute and then differentiate with respect to b to find that g33 is decreasing.
Second, in the limit, we find that g00 stays positive, so these solutions stay nondegenerate.
In the limit, a, b, and c converge to (deth)
1
3 so
g00 → 3−
3
8κ(deth)−
1
2 , g11, g22, g33 → 3
1
8κ−
1
3 (deth)
1
2 .
Next, we consider the case where h22 = h33, which, accounting for the allowable values for h11,
corresponds to those points in ∂Db=c that lie between L0 and L1. Here, a = deth
b2
and, as above,
the system reduces to two variables:
d
dt
g00 = −βqI(a, b, b)(g00)
5
3 ,
d
dt
b = −2β
3
r(b, a, b)(g00)
2
3 b.
Again, both qI and r simplify substantially:
qI(a, b, b) = a2(b− a)2, r(b, a, b) = −a2(b− a)(b− 4a).
24 D. Helliwell
Algebraically, this system is identical to the previous case, so the analysis is quite similar, and the
resulting relationship between g00 and b is determined by the same equation (5.9). This implies
the same relationship for g00 and g22. Important differences arises when analyzing the qualitative
behavior however. First, note that here, d
dtb < 0 so b and c are decreasing. This implies that a
is increasing, and so g11 must be increasing as well. We cannot conclude that g22 and g33 are
decreasing however, and it turns out that if the initial conditions are close enough to L1 then
in fact g22 and g33 will increase for a while before eventually decreasing. The transition occurs
when g22 = g33 = 3g11, which is found by analyzing the equation for d
dtg22 directly. Finally,
while the qualitative behavior differs somewhat from the previous case, the limiting behavior is
the same. �
Proof of Theorem 5.18. This case corresponds to L1. Here, r(a, b, b) = r(b, a, b) = 0 which
implies that d
dta = d
dtb = d
dtc = 0, and g22 = g33 = 4g11 for all time. Then, using the fact that
det g = g00g11g22g33 = 1
4g00(g33)3 and focusing on the equation for g33, we have
d
dt
g33 = −βqII
(
g33,
1
4
g33, g33
)
(g00)2g33 =
1
27
(g33)−1.
This is separable and we have
g33 =
(
1
26
t+ h2
33
) 1
2
.
Once this is known, the other three components are also known. We have
g22 = g33, g11 =
1
4
g33 and g00 = 4(deth)
(
1
26
t+ h2
33
)− 3
2
. �
Proof of Theorem 5.19. This case corresponds to those points in ∂Db=c that do not lie be-
tween L0 and L1. Algebraically, the system is the same as for the second case in Theorem 5.17.
In this case, since b > 4a, b and c are increasing, and a is decreasing, so g22 and g33 must be
increasing as well.
Since b is increasing, if it were bounded, it would have to converge to a point where r(b, a, b)=0
or where g00 = 0, by Lemma 4.2. We will find below that because of the algebraic relationship
between g00 and b, g00 is positive as long as b <∞ so the only possibility is r(b, a, b) = 0. Since
there are no points where this occurs other than b = a and b = 4a, we find that b and c, and
hence g22 and g33 must diverge in the limit. The fact that g11 → 0 follows from Lemma 5.24
which appears later and is used for solutions starting at other points in D as well.
For the algebraic relationship between g00 and g22, the system is the same as for the second
case in Theorem 5.17, and the analysis is essentially the same. Again, the fact that b > 4a alters
the formula for the trace of the solution so that instead of equation (5.9), we have
g00 =
[
κ
[(
b3 − 4 deth
)
b
]−1] 3
8 ,
where
κ = (h00)
8
3 ((b(0))3 − 4 deth)b(0) = (h00(h22)3 − 4 deth)(h00)3h22.
With this small change made, substituting for b and rearranging produces the result. �
Our next goal is to determine the qualitative behavior of solutions with initial conditions
that do not lie on the boundary. In light of the results above, we introduce a bit more notation
and structure before stating the theorems. First, observe that there are no equilibria aside from
Bach Flow on Homogeneous Products 25
those found on the boundary above. To see this, note that to have d
dtb = d
dtc = 0, we must have
r(b, a, c) = r(c, a, b) = 0 and so in particular,
r(c, a, b)− r(b, a, c) = 0.
Writing the left side out explicitly, we have
r(c, a, b)− r(b, a, c) = 3(c− b)
(
4c3 + 2c2b+ 2cb2 + 4b3 − 3c2a− 2abc− 3b2a− a3
)
.
Under the condition that a ≤ b ≤ c, we find that the large factor on the right is always positive
so the only way we can have an equilibrium point is if b = c.
As mentioned earlier, all the points in ∂D0 are equilibria making it difficult to determine
qualitative behavior near ∂D0. To resolve this, we adjust the system again. Specifically, we
rescale the system by multiplying the right hand sides by the nonzero factor β−1(g00)−
2
3 to
produce the new system
d
dt
g00 = −qI(a, b, c)g00,
d
dt
b = −2
3
r(b, a, c)b,
d
dt
c = −2
3
r(c, a, b)c.
(5.10)
The solutions to this system will just be reparameterizations of solutions to system (5.8). More-
over, this system extends to a (mostly) nonzero system on ∂D0 and, since d
dtg00 is still zero here,
solutions on this part of the boundary stay in this part of the boundary.
Restricting attention to ∂D0, note that, consistent with the observations above, there are
two equilibrium points P0 and P1. Disregarding the equation for g00, the linearization at P0 is
d
dt
(
b
c
)
= 6(deth)
4
3
(
−1 0
0 −1
)(
b
c
)
and we have a stable equilibrium. The linearization at P1 is
d
dt
(
b
c
)
= 2−
7
3 3(deth)
4
3
(
−106 107
107 −106
)(
b
c
)
resulting in a saddle.
The unstable manifold MU for the saddle is the line {b = c}. The stable manifold MS is a
curve that approaches P1 perpendicularly to MU . See Fig. 2.
Analyzing the ratio c
b , we have
d
dt
c
b
= −2
3
[
r(c, a, b)− r(b, a, c)
]c
b
.
As shown above, r(c, a, b)−r(b, a, c) is always positive when a < b < c so the fraction c
b decreases
as t increases. From this, we find that the solutions approach the boundary b = c.
The set ∂D0\MS comprises two components. By Lemma 4.2, and the fact that c
b is decreasing,
solutions starting in the component that includes P0 converge to P0 as t → ∞ while solutions
starting in the other component converge to a = 0, b = c =∞.
Motivated by these observations, let DS be the set of points in D where g00 ≥ 0 and
(0, b, c) ∈ MS , and note that D\DS comprises two components. Let DL0 be the component
that includes L0, and let D∞ be the component that avoids L0. See Fig. 3.
We now have
26 D. Helliwell
Theorem 5.20. On M̂ = R×S3, let g solve equation (2.1). In a diagonalizing basis, suppose h
corresponds to a point in DS. Then
• g00 → 0;
• g11, g22, g33 →∞;
• g22
g11
→ 4, and g22
g33
→ 1.
Theorem 5.21. On M̂ = R × S3, let g solve equation (2.1). In a diagonalizing basis, suppose
h corresponds to a point in D∞. Then
• g00, g11 → 0;
• g22, g33 →∞;
• g33 − g22 → 0.
Theorem 5.22. On M̂ = R×S3, let g solve equation (2.1). In a diagonalizing basis, suppose h
corresponds to a point in DL0. Then
• g00 does not converge to zero;
• g11, g22, and g33 converge to the same value.
We prove these by analyzing the behavior of system (5.10) and we note the following general
structure for its solutions. Let (0, b(t), c(t)) be a solution in ∂D0 and consider the solution with
initial condition (h00, b(0), c(0)). Since the equations for b and c do not depend on g00, b(t)
and c(t) still solve this system. Then
d
dt
g00 = −qI(a(t), b(t), c(t))g00,
which is separable and we have
g00 = h00e
Q(t),
where
Q(t) =
∫ t
0
−qI(a(τ), b(τ), c(τ))dτ. (5.11)
Note that, since Q does not depend on g00, the ratio of two solutions with initial conditions that
differ only in h00 will be constant.
We can now prove Theorems 5.20, 5.21, and 5.22. While Theorem 5.20 is straightforward, it
turns out that Theorems 5.21 and 5.22 are fairly subtle.
Proof of Theorem 5.20. Since h corresponds to a point in DS , we know that (0, b(0), c(0)) ∈
MS . Hence 4a(t), b(t), and c(t) all converge to the same value. The theorem will then be proved
once it is established that g00 → 0. Since the solution is bounded for t ≥ 0, the interval on which
it is defined includes [0,∞). Moreover, since the solution is converging to L1, qI(a(t), b(t), c(t))
is bounded below by a positive constant, so Q(t) → −∞ as t → ∞. Therefore g00 → 0, as
desired. �
For Theorem 5.21, we first establish a couple lemmas.
Lemma 5.23. Suppose a ≤ b ≤ c and b and c diverge (so a converges to zero). Then for all
m ∈ R, c−b
am → 0.
Bach Flow on Homogeneous Products 27
Proof. Observe first that if c−b
am′ is bounded for a particular exponent m′, then the result is true
for all m < m′. Therefore, it is enough to show that c−b
am is bounded for all m ≥ 1.
The equation solved by c− b is
d
dt
(c− b) = −2
3
u(a, b, c)(c− b),
where
u(x, y, z) = −4x4 + x3(y + z)− x2yz − x
(
5y3 + 7y2z + 7yz2 + 5z3
)
+ 8y4 + 7y3z + 6y2z2 + 7yz3 + 8z4,
and so
d
dt
c− b
am
= −2
3
[u(a, b, c)−mr(a, b, c)]c− b
am
.
Writing out u(a, b, c)−mr(a, b, c), we find
u(a, b, c)−mr(a, b, c) = 3b3 + 6b2c2 + 3c4 + b2(5b+ 7c)(b− a) + c2(7b+ 5c)(c− a)
+m(c− b)2
[
3b2 + 3bc+ 4c2 + (b− a)(b+ c)
]
+ (1 + 5m)a3(b+ c)
− a(1 + 2m)
(
abc+ 4a3
)
.
As a → 0 and b and c diverge to ∞, the first and second lines are positive and diverge, and
the third and fourth lines are positive. Only the last line is negative, but it converges to zero
(since abc = deth is constant). From this, we find that c−b
am is eventually decreasing, and hence
bounded above. �
Lemma 5.24. Suppose a ≤ b ≤ c and b and c diverge (so a converges to zero). Then g11 → 0.
Proof. We already know that b = (g00)
1
3 g22 diverges. We show here that (g00)
2
3 g22 also even-
tually increases. We have
d
dt
[
(g00)
2
3 g22
]
= −β
3
s(g11, g22, g33)(g00)
8
3 g22,
where
s(x, y, z) = 2qI(x, y, z) + 3qII(y, x, z)
= −
[
(z − y)
(
7z3 + 6yz2 + 6y2z + 17y3 − x
(
7z2 + 6yz + 11y2
))
+ x2
(
yz − xy − 7xz + 7x2
)]
.
Analyzing s(g11, g22, g33), the large factor inside the first term is positive if g11 ≤ g22 ≤ g33. For
the second term, note that g22
g11
= b
a which diverges for the solutions under consideration. This
implies that eventually, g22 becomes, and stays, larger than 8g11. This, combined with the fact
that g22 ≤ g33 implies
g22g33 − g11g22 − 7g11g33 ≥ 0.
From this, we find that s(g11, g22, g33) eventually becomes, and stays, negative and so (g00)
2
3 g22
eventually increases.
Using this fact, rewrite det g as follows:
det g = g00g11g22g33 = g11
[
(g00)
2
3 g22
]
c.
Since det g is constant, (g00)
2
3 g22 is increasing, and c→∞, it must be the case that g11 → 0. �
28 D. Helliwell
We are now ready to prove Theorem 5.21
Proof of Theorem 5.21. Since h corresponds to a point in D∞, we know that b(t) and c(t)
diverge, and so a(t) → 0. Since g00 is decreasing, it must be the case that g22 and g33 diverge.
By Lemma 5.24, g11 → 0, so the theorem will then be proved once it is established that g00 → 0
as well.
For metrics in the given domain, r(b, a, c) < 0 so for any flow in this setting, b is strictly
increasing. Using this, we make a substitution to rewrite equation (5.11) to get
−Q(T ) =
∫ T
0
qI(a(t), b(t), c(t))dt =
∫ b(T )
b(0)
qI(a, b, c)
−2
3r(b, a, c)b
db,
where we recognize that a and c are now functions of b. We now estimate qI and r along the
flow. For qI, we have the following:
qI(a, b, c) = a2(b− a)(c− a) + (b− c)2
[
a(b+ c) + b2 + bc+ c2
]
≥ a2(b− a)(c− a).
For r, we have
−r(b, a, c) = a2(b− 4a)(c− a) + (c− b)
[
−a
(
5b2 + 3bc+ 4c2
)
+ 8b3 + 3b2c+ 3bc2 + 4c3
]
= a2(b− 4a)(c− a) +
(c− b)
a5
a5r̄(a, b, c)
= a2
[
(b− 4a)(c− a) +
(c− b)
a5
a3r̄(a, b, c)
]
,
where r̄ is a cubic polynomial. Since abc is constant, ab and ac go to zero along the flow, and
this implies that a3r̄(a, b, c)→ 0. By Lemma 5.23, (c−b)
a5
→ 0 as well. Hence, (c−b)
a5
a3r̄(a, b, c) is
bounded by a positive constant K along the flow (for t ≥ 0) and so
−r(b, a, c) ≤ a2([b− 4a)(c− a) +K] ≤ La2(b− 4a)(c− a) ≤ La2b(c− a),
where the second inequality follows from the fact that (b − 4a)(c − a) is bounded below by
a positive constant along the flow.
Combining the estimate for qI and for r, we have∫ ∞
b(0)
qI(a, b, c)
−2
3r(b, a, c)b
db ≥ 3
2
∫ ∞
b(0)
a2(b− a)(c− a)
La2b(c− a)b
db =
3L
2
(∫ ∞
b(0)
1
b
db−
∫ ∞
b(0)
a
b2
db
)
.
The first integral diverges while the second integral stays finite so Q(t) → −∞ along the flow,
and g00 → 0. �
Before proving Theorem 5.22, we establish some estimates for qI and r near P0. In the
following, keep in mind that since a = deth
bc , its value changes when comparing the functions in
question at different points.
Lemma 5.25. There is a neighborhood U of P0 such that for all points (0, b, c) in U ∩ ∂D0,
r(c, a, b) ≥ r(c, a, c) ≥ 0.
Proof. As a first step, we show that r(c, a, b) > 0 for points in the given domain near a = b = c.
In fact, to help with the argument later, we show that for each c the function is minimized at
b = c. First, note that when b = a =
√
deth
c ,
r(c, a, a) = 2c2(4c− a)(c− a)
Bach Flow on Homogeneous Products 29
and when b = c, so that a = deth
c2
,
r(c, a, c) = a2(c− a)(4a− c).
These are both positive as long as a < c < 4a. Next we compute the derivative with respect
to b. Since a = deg h
bc , we have ∂ba = −deth
c b−2 = −ab−1 so
∂br(c, a, b) = −5c3 − abc− 3a3b−1c− 16b3 − 8a3
+ 5ab−1c3 + 3b2c+ 8ab2 + 16a4b−1 + 2a2c
and the second derivative
∂2
b r(c, a, b) = 12a3b−2c− 48b2 + 24a3b−1 − 10ab−2c3 + 6bc+ 8ab− 80a4b−2 − 2a2b−1c.
This is negative at P0, so must be negative in a neighborhood of this point. This implies that
as b varies, r(c, a, b) is minimized at one of the endpoints above.
To determine which endpoint is the minimum, comparing the two expressions algebraically
proves difficult. To more easily compare, let c = v(deth)
1
3 (and note that v = 1 corresponds to
the point P0). Then define
fa(v) = r(c, a, a) = (deth)
4
3
(
8v4 − 10v
5
2 + 2v
)
.
Then
(fa)
′(v) = (deth)
4
3
(
32v3 − 25v
3
2 + 2
)
and so (fa)
′(1) = 9(deth)
4
3 . Also
(fa)
′′(v) = (deth)
4
3
(
96v2 − 75
2
v
1
2
)
and so (fa)
′′(1) = 117
2 (deth)
4
3 .
On the other hand define
fc(v) = r(c, c, a) = (deth)
4
3
(
−4v−8 + 5v−5 − v−2
)
.
Then
(fc)
′(v) = (deth)
4
3
(
32v−9 − 25v−5 + 2v−3
)
and so (fc)
′(1) = 9(deth)
4
3 , which matches (fa)
′(1). Also
(fc)
′′(v) = (deth)
4
3
(
−288v−10 + 125v−6 − 6v−4
)
so (fc)
′′(1) = −139(deth)
4
3 . This shows that fa and fc agree to first order, but that near P0, fa
eventually grows faster and we can conclude that for each c close to P0, r(c, a, b) is minimized
when b = c. �
Lemma 5.26. There is a neighborhood U of P0 such that for all points (0, b, c) in U ∩ ∂D0,
0 ≤ qI(a, b, c) ≤ qI(a, c, c).
30 D. Helliwell
Proof. We have
qI(a, a, c) = c2(c− a)2 and qI(a, c, c) = a2(c− a)2.
For points between these two, we compute the partial derivative with respect to b. As in the
previous lemma, a = deg h
bc so ∂ba = −ab−1 and we have
∂bqI(a, b, c) = −4a4b−1 + 2a3 + 3a3b−1c− a2c− 2ab2 + abc+ ab−1c3 + 4b3 − 3b2c− c3.
The second derivative is
∂2
b qI(a, b, c) = 20a4b−2 − 6a3b−1 − 12a3b−2c+ 2a2b−1c− 2ab− 2ab−2c3 + 12b2 − 6bc.
At P0, this is positive so qI is concave up near P0 and we may conclude that it is maximized at
one endpoint. To determine which endpoint is larger, let c = v(deth)
1
3 and define
fa(v) =
(
qI(a, a, c)
) 1
2 = c(c− a) = (deth)
2
3
(
v2 − v
1
2
)
and
fc(v) =
(
qI(a, c, c)
) 1
2 = a(c− a) = (deth)
2
3
(
v−1 − v−4
)
.
Then fa(1) = fc(1). Computing derivatives, we have
(fa)
′(v) = (deth)
2
3
(
2v − 1
2
v−
1
2
)
and
(fc)
′(v) = (deth)
2
3
(
−v−2 + 4v−5
)
.
Hence (fa)
′(1) = 3
2 while (fc)
′(1) = 3 and we may conclude that near P0, fc grows faster, so qI
is maximized when b = c. �
We are now ready to prove Theorem 5.22.
Proof of Theorem 5.22. Since h corresponds to a point in DL0 , we know that a(t), b(t),
and c(t) converge to the same value. The theorem will then be proved once it is established
that g00 does not go to zero. Note that, since the solution is bounded, the interval on which it
is defined includes [0,∞).
By Lemma 5.25, r(c, b, a) is positive near a = b = c, so d
dtc is negative and c is strictly
decreasing. From this, we can reparameterize the integral above and then use Lemmas 5.25
and 5.26 to get
−Q(T ) =
∫ T
0
qI(a(t), b(t), c(t))dt =
∫ c(T )
c(0)
qI(a, b, c)
−2
3r(c, a, b)c
dc =
3
2
∫ c(0)
c(T )
qI(a, b, c)
r(c, a, b)c
dc
≤ 3
2
∫ c(0)
c(T )
qI(a, c, c)
r(c, a, c)c
dc =
3
2
∫ c(0)
c(T )
a2(c− a)2
a2(c− a)(4a− c)c
dc =
3
2
∫ c(0)
c(T )
(c− a)
(4a− c)c
dc.
The last integrand on the second line is bounded and the interval of integration stays bounded,
so the integral stays finite as T → ∞. (We are allowed to cancel the factor a2(c − a) because
the a being used is the same for the numerator and the denominator, since the estimates are
both taken on the same side of the boundary.) �
With the limiting behavior of the metric established, the next step is to determine curvature.
Bach Flow on Homogeneous Products 31
Proposition 5.27. On M̂ = R× S3, let g be a solution to equation (2.1) with initial metric h.
Then in a diagonalizing basis,
• if h corresponds to a point in DS, then Ric11 converges to 1
32 , Ric22 and Ric33 both converge
to 7
8 , and S converges to 0;
• if h corresponds to a point in D∞, then Ric11
g11
converges to zero, Ric22 and Ric33 both
converge to 1, and S converges to 0;
• if h corresponds to a point in DL0, then Ric11, Ric22, and Ric33 all converge to 1
2 , and S
converges to a positive value.
Proof. Since the Ricci tensor is invariant under uniform rescaling of the metric, we can use the
components of the metric directly, or we can use a, b, and c to determine the Ricci curvature.
We have three cases.
If h corresponds to a point in DS , then we can simply plug in the fact that, in the limit,
4a = b = c to get the values indicated for the Ricci tensor. For scalar curvature, we have
S = −(g11)2 + (g22)2 + (g33)2 − 2(g11g22 + g11g33 + g22g33)
2g11g22g33
= −(g00)
1
3
a2 + b2 + c2 − 2(ab+ ac+ bc)
2 deth
→ 0
since g00 → 0.
If h corresponds to a point in D∞, we have
Ric33 =
c2 − (b− a)2
2ab
=
(c− b)(c+ b)
2ab
+ 1− a
2b
.
The first term goes to zero since c−b
a → 0 by Lemma 5.23, and c+b
b stays bounded since c
b → 1.
The last term also goes to zero so Ric33 → 1. The analysis for Ric22 is similar.
For Ric11
g11
we have
Ric11
g11
=
a2 − (c− b)2
2g11bc
=
g
1
3
[
a2 − (c− b)2
]
2abc
.
Here, the numerator goes to zero while the denominator stays constant.
For scalar curvature, we have
S = −(g11)2 + (g22)2 + (g33)2 − 2(g11g22 + g11g33 + g22g33)
2g11g22g33
= −(g00)
1
3
a2 + b2 + c2 − 2(ab+ ac+ bc)
2 det g
= −(g00)
1
3
(c− b)2 + a2 − 2a(c+ b)
2 det g
=
−(g00)
1
3
2 det g
[
(c− b)2 + a2
]
+ (g00)
1
3
(
1
b
+
1
c
)
.
In this form, we can see that both terms go to zero.
If h corresponds to a point in DL0 , then in the limit, a = b = c, and we get the desired values
for the Ricci tensor. For scalar curvature,
S = −(g11)2 + (g22)2 + (g33)2 − 2(g11g22 + g11g33 + g22g33)
2g11g22g33
= −(g00)
1
3
a2 + b2 + c2 − 2(ab+ ac+ bc)
2 deth
→ 3
2
a2(g00)
1
3
deth
=
3
2g11
,
which is positive. �
32 D. Helliwell
Finally, we have the following:
Theorem 5.28. Let M be a quotient of R × S3 and let p ∈ M . Let g solve equation (2.1)
where h is locally homogeneous. Then
• if h corresponds to a point in DS, then (M, g, p) collapses to a flat three-dimensional
manifold in the pointed Gromov–Hausdorff topology;
• if h corresponds to a point in D∞, then (M, g, p) collapses to a flat surface in the pointed
Gromov–Hausdorff topology;
• if h corresponds to a point in DL0, then (M, g) converges to a quotient of the product of
a circle and the round sphere in the Gromov–Hausdorff topology.
Proof. These results follow from the previous proposition and Lemma 3.3. �
5.7 H3
This space is not a Lie group so the techniques used above do not apply. In fact, the analysis here
is much simpler. There is a one parameter family of homogeneous metrics for H3 and they are
all constant scalar multiples of the standard hyperbolic metric and hence Einstein. Therefore,
by Proposition 2.1, R×H3 is static under Bach flow.
5.8 R × S2 and R × H2
While these spaces can be thought of as three-dimensional factors for various 1 × 3 products,
they are more naturally viewed in terms of 2×2 products which are discussed in the next section.
6 Bach flow on locally homogeneous 2 × 2 products
Bach flow on products of homogeneous surfaces was explored in [7]. While not new, we reproduce
the analysis here for completeness because of the fact that three families of 1× 3 products can
also be viewed as 2× 2 manifolds, namely quotients of R× N̂ where N̂ is R×S2, R×H2, or R3.
Working as before on the universal cover, it seems at first that there are essentially six different
cases to consider: R2 ×R2, R2 × S2, R2 ×H2, S2 × S2, S2 ×H2, H2 ×H2. However, it turns out
that the Bach tensor does not distinguish between the spherical and hyperbolic slices and the
analysis reduces to three cases, one of which is trivial (since the Bach tensor vanishes). To see
this, because of the constancy of the scalar curvatures, equations (2.4) and (2.5) reduce to
Bαβ =
1
24
((
S(1)
)2 − (S(2)
)2)
gαβ and Bjk =
1
24
((
S(2)
)2 − (S(1)
)2)
gjk.
Since the scalar curvatures of the slices are squared, there is no way to distinguish between
a positively curved space and a negatively curved space.
For any of these spaces we can write a homogeneous product metric as
g = f1g
(1) + f2g
(2),
where g(i) is the standard metric for the ith slice and fi > 0.
The constancy of the volume form along the flow implies that
γ = f1(t)f2(t) = f1(0)f2(0)
is a constant determined by the initial metric.
Bach Flow on Homogeneous Products 33
6.1 R2 × R2, R2 × S2, and R2 × H2
These spaces can also be thought of as 1× 3 products and the analysis here completes the 1× 3
product cases. The space R2 × R2 is flat, so the metric is static under Bach flow. For the
remaining two spaces, we have the following:
Theorem 6.1. Let M be a compact quotient of R2 × S2 or R2 ×H2 and let p ∈M . Let g solve
equation (2.1) where h is locally homogeneous. Then (M, g, p) converges to a flat surface in the
pointed Gromov–Hausdorff topology.
Proof. Using the structure introduced above, the curvature of the first slice is zero and that of
the second slice is
S(2) = ±2f−1
2 ,
(positive for S2 and negative for H2). Bach flow then reduces to
d
dt
f1 = −1
6
f−2
2 f1,
d
dt
f2 =
1
6
f−1
2 .
The second equation is separable. Once its solution is found, it can be plugged into the first
and we can solve the resulting separable equation for f1. We end up with
f1(t) = γ
(
1
3
t+ f2(0)2
)− 1
2
, f2(t) =
(
1
3
t+ f2(0)2
) 1
2
.
We find that the solutions are immortal but not ancient. The flat slice shrinks while the
curved slice blows up and its scalar curvature goes to zero. �
6.2 S2 × S2, S2 × H2, and H2 × H2
These spaces are not 1× 3 products. They are included here for completeness.
Theorem 6.2. Let M be a compact quotient of S2 × S2, S2 × H2, or H2 × H2. Let g solve
equation (2.1) where h is locally homogeneous. Then (M, g) converges to a Bach-flat four-
dimensional manifold in the Gromov–Hausdorff topology. The difference in the magnitude of
the curvature of each slice converges to zero, and the scalar curvature converges to
• 4γ−
1
2 on S2 × S2,
• 0 on S2 ×H2,
• −4γ−
1
2 on H2 ×H2.
Proof. For these spaces the scalar curvatures are
S(i) = ±2f−1
i ,
positive for spheres and negative for hyperbolic spaces, and the Bach tensor can be written
Bαβ =
1
6
(
f−2
1 − f−2
2
)
f1
(
g(1)
)
αβ
, Bjk =
1
6
(
f−2
2 − f−2
1
)
f2
(
g(2)
)
jk
.
Note that if f1(0) = f2(0), then the solution is constant. Otherwise, suppose f1(0) < f2(0). The
constant γ allows us to reduce the system (2.1) to a single equation.
d
dt
f1 =
1
6
(
f−1
1 − γ−2f3
1
)
.
34 D. Helliwell
This is separable and we can solve to get
f1 =
√
γ tanh
(
t
3γ
+ µ
)
, f2 =
√
γ coth
(
t
3γ
+ µ
)
,
where
µ =
1
2
ln
(
f1(0) + f2(0)
f2(0)− f1(0)
)
.
If f1(0) > f2(0) the solutions are swapped. From this we see that, again, solutions are immortal,
but not ancient. Here, the absolute values of the curvatures of the slices converge to the same
value, and the manifold converges to a Bach-flat four-dimensional manifold. The limiting scalar
curvatures can be calculated directly. �
7 Comparison with Ricci flow
We finish with a comparison of the qualitative behavior between Ricci flow, as determined in [11]
and [12], and Bach flow. There are a number of ways this might be done. First, there is the
choice of whether to include the one-dimensional component for Ricci flow. Also, there is the
choice of whether or not to use volume-normalized Ricci flow or unmodified Ricci flow. For
a product metric, the Ricci tensor splits and on a one-dimensional manifold, the Ricci tensor is
zero. This implies that unmodified Ricci flow on S1 ×N leaves the one-dimensional component
fixed and the behavior on the three-dimensional slice is the same as for Ricci flow on just N . For
volume-normalized Ricci flow, the behavior on S1 ×N will be somewhat different from that of
volume-normalized Ricci flow on just N . First, there is a dimensional constant in the modifying
term, and second, volume normalized flow on S1 × N does not preserve the volume of N . In
the end, the differences in all these flows are somewhat cosmetic. Rescaling space and time
in appropriate ways allows the solution to one of these problems to be modified so as to solve
another. For a bit more detail, see the discussions in [12] including the analysis for those cases
that relate to the results in [11].
At first, it might seem most natural to compare Bach flow on S1 ×N to volume-normalized
Ricci flow on S1 × N since both flows are acting on the same space, and both flows preserve
volume. However for volume-normalized Ricci flow the behavior of the one-dimensional slice
depends on the scalar curvature of N , while by Proposition 2.2, under Bach flow the one-
dimensional slice never expands. On the other hand, for volume-normalized Ricci flow on N the
static solutions are Einstein. Similarly, the static solutions gS1 + g̃ for Bach flow are those for
which g̃ is Einstein on N by Proposition 2.1. As such, the qualitative behavior can more easily
be compared. We find that on most spaces, the qualitative behavior is the same, but there are
two notable differences. If N = S3, volume-normalized Ricci flow always converges to the round
sphere, while for Bach flow, the eventual qualitative behavior depends on the initial metric. If
N = S1 × S2, volume-normalized Ricci flow experiences curvature blow-up in finite time, while
Bach flow collapses to a flat surface as t→∞.
Acknowledgements
The author would like to thank Eric Bahuaud for the many valuable discussions while developing
this paper, and the referees for their in-depth, candid feedback, and constructive suggestions for
improvement.
Bach Flow on Homogeneous Products 35
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1 Introduction
2 The Bach tensor and Bach flow
2.1 Bach tensor on products
3 Locally homogeneous spaces and Lie groups
3.1 Structure constants and curvature
3.2 Three-dimensional Lie groups
4 Ordinary differential equations
5 Bach flow on locally homogeneous 1 3 products
5.1 R3
5.2 Nil
5.3 Solv
5.4 E(2)
5.5 SL"0362SL(2,R)
5.6 S3
5.7 H3
5.8 R S2 and R H2
6 Bach flow on locally homogeneous 2 2 products
6.1 R2 R2, R2 S2, and R2 H2
6.2 S2 S2, S2 H2, and H2 H2
7 Comparison with Ricci flow
References
|
| id | nasplib_isofts_kiev_ua-123456789-210583 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-17T12:04:16Z |
| publishDate | 2020 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Helliwell, Dylan 2025-12-12T10:29:36Z 2020 Bach Flow on Homogeneous Products. Dylan Helliwell. SIGMA 16 (2020), 027, 35 pages 1815-0659 2020 Mathematics Subject Classification: 53C44; 53C30; 34C40 arXiv:1803.07733 https://nasplib.isofts.kiev.ua/handle/123456789/210583 https://doi.org/10.3842/SIGMA.2020.027 The qualitative behavior of Bach flow is established on compact four-dimensional locally homogeneous product manifolds. This is achieved by lifting to the homogeneous universal cover and, in most cases, capitalizing on the resultant group structure. The resulting system of ordinary differential equations is carefully analyzed on a case-by-case basis, with explicit solutions found in some cases. The limiting behavior of the metric and the curvature is determined in all cases. The behavior of quotients of ℝ×𝕊³ proves to be the most challenging and interesting. The author would like to thank Eric Bahuaud for the many valuable discussions while developing this paper, and the referees for their in-depth, candid feedback and constructive suggestions for improvement. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Bach Flow on Homogeneous Products Article published earlier |
| spellingShingle | Bach Flow on Homogeneous Products Helliwell, Dylan |
| title | Bach Flow on Homogeneous Products |
| title_full | Bach Flow on Homogeneous Products |
| title_fullStr | Bach Flow on Homogeneous Products |
| title_full_unstemmed | Bach Flow on Homogeneous Products |
| title_short | Bach Flow on Homogeneous Products |
| title_sort | bach flow on homogeneous products |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/210583 |
| work_keys_str_mv | AT helliwelldylan bachflowonhomogeneousproducts |