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EXTRACTA

MATHEMATICAE

Volumen 33, Número 2, 2018

instituto de investigación de matemáticas de la

universidad de extremadura

EXTRACTA MATHEMATICAE

Vol. 36, Num. 1 (2021), 99 – 145

doi:10.17398/2605-5686.36.1.99

Available online January 28, 2021

Ancient solutions of the homogeneous Ricci
flow on flag manifolds

S. Anastassiou 1, I. Chrysikos 2

1 Center for Research and Applications of Nonlinear Systems (CRANS)
Department of Mathematics, University of Patras, Rion 26500, Greece

2 Faculty of Science, University of Hradec Králové
Rokitanskeho 62, Hradec Králové 50003, Czech Republic

sanastassiou@gmail.com , ioannis.chrysikos@uhk.cz

Received September 13, 2020 Presented by Carolyn S. Gordon
Accepted January 6, 2021

Abstract: For any flag manifold M = G/K of a compact simple Lie group G we describe non-

collapsing ancient invariant solutions of the homogeneous unnormalized Ricci flow. Such solutions
emerge from an invariant Einstein metric on M, and by [13] they must develop a Type I singularity

in their extinction finite time, and also to the past. To illustrate the situation we engage ourselves

with the global study of the dynamical system induced by the unnormalized Ricci flow on any flag
manifold M = G/K with second Betti number b2(M) = 1, for a generic initial invariant metric. We

describe the corresponding dynamical systems and present non-collapsed ancient solutions, whose α-

limit set consists of fixed points at infinity of M G. Based on the Poincaré compactification method,
we show that these fixed points correspond to invariant Einstein metrics and we study their stability

properties, illuminating thus the structure of the system’s phase space.

Key words: Ricci flow, homogeneous spaces, flag manifolds, ancient solutions, scalar curvature.

MSC (2020): Primary: 53C44, 53C25, 53C30, Secondary: 37C10.

Introduction

Given a Riemannian manifold (Mn,g), recall that the unnormalized Ricci
flow is the geometric flow defined by

∂g(t)

∂t
= −2 Ricg(t) , g(0) = g , (0.1)

where Ricg(t) denotes the Ricci tensor of the one-parameter family g(t). The
above system consists of non-linear second order partial differential equations
on the open convex cone M of Riemannian metrics on M. A smooth family{
g(t) : t ∈ [0,T) ⊂ R

}
∈ M defined for some 0 < T ≤ ∞, is said to be a

solution of the Ricci flow with initial metric g, if it satisfies the system (0.1)

ISSN: 0213-8743 (print), 2605-5686 (online)

c©The author(s) - Released under a Creative Commons Attribution License (CC BY-NC 3.0)

https://doi.org/10.17398/2605-5686.36.1.99
mailto:sanastassiou@gmail.com
mailto:ioannis.chrysikos@uhk.cz
https://publicaciones.unex.es/index.php/EM
https://creativecommons.org/licenses/by-nc/3.0/


100 s. anastassiou, i. chrysikos

for any x ∈ M and t ∈ [0,T). The Ricci flow was introduced in the celebrated
work of Hamilton [33] and nowadays is the essential tool in the proof of the
famous Poincaré conjecture and Thurston’s geometrization conjecture, due to
the seminal works [48, 49] of G. Perelman.

In general, and since a system of partial differential equations is involved,
it is hard to produce explicit examples of Ricci flow solutions. However, the
Ricci flow for an initial invariant metric reduces to a system of ODEs. More
precisely, homogeneity implies bounded curvature (see [21]), and thus the
isometries of the initial metric will be in fact the isometries of any involved
metric. Hence, when g is an invariant metric, any solution g(t) of (0.1) is
also invariant. As a result, in some cases it is possible to solve the system
explicitly and proceed to a study of their asymptotic properties, or even spec-
ify analytical properties related to different type of singularities and deduce
curvature estimates, see [16, 28, 7, 29, 18, 38, 11, 13, 1, 30], and the articles
quoted therein. Especially for the non-compact case, note that during the
last decade the Ricci flow for homogeneous, or cohomogeneity-one metrics,
together with the so-called bracket flow play a key role in the study of the
Alekseevsky conjecture, see [47, 41, 42, 43, 39, 40, 12, 14].

In this work we examine the Ricci flow, on compact homogeneous spaces
with simple spectrum of isotropy representation, in terms of Graev [31, 32], or
of monotypic isotropy representation, in terms of Buzano [18], or Pulemotov
and Rubinstein [52]. Nowadays, such spaces are of special interest due to their
rich applications in the theory of homogeneous Einstein metrics, prescribed
Ricci curvature, Ricci iteration, Ricci flow and other (see [4, 53, 10, 15, 31, 8,
9, 18, 32, 24, 11, 50, 52, 30]). Here, we focus on flag manifolds M = G/K of a
compact simple Lie group G and examine the dynamical system induced by the
vector field corresponding to the homogeneous Ricci flow equation. On such
cosets (even for G semisimple), the homogeneous Ricci flow cannot possess
fixed (stationary) points, since by the theorem of Alekseevsky and Kimel’fel’d
[3] invariant Ricci flat metrics must be flat and so they cannot exist. However,
as follows from the maximum principle, the homogeneous Ricci flow on any
flag manifold M = G/K must admit ancient invariant solutions, which in
fact by the work of Böhm [11] must have finite extinction time (note that
since any flag manifold is compact and simply connected and carries invariant
Einstein metrics, e.g., invariant Kähler-Einstein metrics always exist, the first
conclusion above occurs also by a result of Lafuente [38, Corollary 4.3]).

To be more specific, recall that a solution g(t) of the Ricci flow is called
ancient if it has as interval of definition the open set (−∞,T), for some T < ∞.



ancient solutions of the homogeneous ricci flow 101

Such solutions are important, because they arise as limits of blow ups of
singular solutions to the Ricci flow near finite time singularities, see [25]. For
our case it follows that for any flag manifold M = G/K one must be always
able to specify a G-invariant metric g, such that any (maximal) Ricci flow
solution g(t) with initial condition g(0) = g has an interval of definition of
the form (ta,T), with −∞ ≤ ta < 0 and T < ∞. Indeed, for any flag space
below we will provide explicit solutions of this type, which are ancient. They
arise by using an invariant Kähler-Einstein metric, which always exists, or any
other possible existent invariant Einstein metric g0, and they are defined on
open intervals of the form (−∞,T), where

T =
1

2λ
=
n

2
Scal(g0)

−1,

with λ =
Scal(g0)

n
> 0 being the corresponding Einstein constant (see Propo-

sition 2.4). All such solutions become extinct when t → T, in the sense that
g(t) → 0, i.e., they tend to 0. As ancient solutions, they have positive scalar
curvature Scal(g(t)) ([20]) and the asymptotic behaviour of Scal(g(t)), at least
for the trivial one, can be easily treated, see Proposition 2.4 which forms a
specification of [38, Theorem 1.1] on flag manifolds (see also Example 3.3 and
Example 3.4). Moreover, for any invariant Einstein metric on M = G/K with
b2(M) = 1, we show the existence of an unstable manifold and compute its
dimension. For the invariant Einstein metrics which are not Kähler, we obtain
a 2- or 3-dimensional unstable manifold, depending on the specific case, which
implies the existence of non–trivial ancient solutions emerging from the corre-
sponding Einstein metric. Actually, by [13], it also follows that these ancient
solutions develop a Type I singularity. This means (see [25, 18, 11, 44])

lim
t→−∞

(
|t| · supx∈M‖Rm(g(t))‖g(t)(x,t)

)
< ∞,

where Rm(g(t)) denotes the curvature tensor of (M = G/K,g(t)), or equiva-
lently that there is a constant 0 < Cg0 < ∞ such that

(T − t) · supM‖Rm(g(t))‖g(t) ≤ Cg0

for any t ∈ (−∞,T = 1
2λ

). Finally, by [13] we also deduce that the predicted
non-trivial ancient solutions are non-collapsed (and the same the trivial one,
see Corollary 2.8). In this point we should mention that not any compact
homogeneous space M = G/K of a compact (semi)simple Lie group G ad-
mits (unstable) Einstein metrics (see for example [55, 46] for non-existence



102 s. anastassiou, i. chrysikos

results). So, even assuming that the universal covering of Mn = G/K is not
diffeomorphic to Rn (which for the compact case is equivalent to say that Mn
is not a n-torus), the predicted solutions of Lafuente can be in general hard
to be specified.

Note now that for any flag manifold M = G/K of a compact simple Lie
group G, the symmetric space of invariant metrics M G is the phase space of
the homogeneous Ricci flow and it is flat, i.e., M G ∼= Rr+ for some r ≥ 1.
Therefore, the dynamical system of the homogeneous Ricci flow can be con-
verted to a qualitative equivalent dynamical system of homogeneous polyno-
mial equations and the well-known Poincaré compactification ([51]) strongly
applies. The main idea back of this method is to identify Rn with the northern
and southern hemispheres through central projections, and then extend X to
a vector field p(X) on Sn (see Subsection 2.2). Here, for any (non-symmetric)
flag manifold M = G/K of a compact simple Lie group G with b2(M) = 1,
we present the global study of the dynamical system induced by the vector
field corresponding to the unnormalized Ricci flow for an initial invariant met-
ric, which is generic. In particular, the main contribution of this work is the
description via the Poincaré compactification method, of the fixed points of
the homogeneous Ricci flow at the so-called infinity of M G (see Definition
2.11 and Remark 2.13). Based on this method we can study the stability
properties of such fixed points, which we prove that are in bijective corre-
spondence with the existent invariant Einstein metrics on M = G/K, and
moreover that coincide with the α-limit set of an invariant line, i.e., a solution
of the homogeneous Ricci flow which has as trace a line of M G. It turns out
that such solutions are ancient and non-collapsed, and develop Type I sin-
gularities. Note that through the compactification procedure of Poincaré, we
are able to distinguish the unique invariant Kähler-Einstein metric from the
other invariant Einstein metrics in terms of (un)stable manifolds, in particular
for every invariant Einstein metric we compute the dimension of the corre-
sponding (un)stable manifold. Moreover, for the case r = 2 we discuss the
ω-limit of any solution of the homogeneous Ricci flow, for details see Theorem
3.1. Since we are interested in the unnormalized Ricci flow on flag mani-
folds with b2(M) = 1, we should finally mention that this dynamical system
has been very recently examined by [30], for flag spaces with three isotropy
summands (however via a different method of the Poincaré compactification),
while a related study of certain examples of flag manifolds with two isotropy
is given in [29]. Note finally that for flag manifolds with r = 2, this work is
complementary to [18], in the sense that there were studied homogeneous an-



ancient solutions of the homogeneous ricci flow 103

cient solutions on compact homogeneous spaces with two isotropy summands.
However, the specific class of flag manifolds with r = 2 was excluded (see at
the end of the article [18]), and the filling of this small gap was a motivation
of the present work.

The structure of the paper is given as follows: In Section 1 we refresh
basics from the theory of homogeneous spaces, introduce the homogeneous
Ricci flow and recall some details from the structure and geometry of flag
manifolds. Next, in Section 2 we shortly present some basic results for ancient
solutions emerging from an invariant Einstein metric, and also the Poincaré
compactification adapted to our scopes. Finally, Section 3 is about the global
study of the homogeneous Ricci flow for all non-symmetric flag manifolds
M = G/K of a compact simple Lie group G with b2(M) = 1, where a proof
of our main theorem, i.e., Theorem 3.1, is presented.

1. Preliminaries

We begin by recalling preliminaries of the homogeneous Ricci flow. After
that we will refresh useful notions of the structure and geometry of generalized
flag manifolds.

1.1. Homogeneous Ricci flow. Recall that a homogeneous Rieman-
nian manifold is a homogeneous space M = G/K (see [36, 19, 5] for details
on homogeneous spaces) endowed with a G-invariant metric g, that is τ∗ag = g
for any a ∈ G, where τ : G×G/K → G/K denotes the transitive G-action.
Equivalently, is a Riemannian manifold (M,g) endowed with a transitive ac-
tion of its isometry group Iso(M,g). If M is connected, then each closed
subgroup G ⊆ Iso(M,g) which is transitive on M induces a presentation of
(M,g) as a homogeneous space, i.e., M = G/K, where K ⊂ G is the stabilizer
of some point o ∈ M. In this case, the transitive Lie group G can be also
assumed to be connected (since the connected component of the identity of
G is also transitive on M). Usually, to emphasize on the transitive group
G, we say that (M,g) is a G-homogeneous Riemannian manifold. However,
note that may exist many closed subgroups of Iso(M,g) acting transitively on
(M,g). Next we shall work with connected homogeneous manifolds.

As it is well-known, the geometric properties of a homogeneous space can
be examined by restricting our attention to a point. Set o = eK for the
identity coset of (Mn = G/K,g) and let ToG/K be the corresponding tangent
space. Since we assume the existence of a G-invariant metric g, K ⊂ G can be
identified with a closed subgroup of O(n) ≡ O(ToG/K) (or of SO(n) if G/K is



104 s. anastassiou, i. chrysikos

oriented), so K is compact and hence any homogeneous Riemannian manifold
(Mn = G/K,g) is a reductive homogeneous space. This means that there
is a complement m of the Lie algebra k = Lie(K) of the stabilizer K inside
the Lie algebra g = Lie(G) of G, which is AdG(K)-invariant, i.e., g = k ⊕ m
and AdG(K)m ⊂ m, where AdG ≡ Ad : G → Aut(g) denotes the adjoint
representation of G. Note that in general the reductive complement m may not
be unique, and for a general homogeneous space G/K a sufficient condition
for its existence is the compactness of K. On the other hand, once such a
decomposition has been fixed, there is always a natural identification of m
with the tangent space ToG/K = g/k, given by

X ∈ m ←→ X∗o =
d

dt

∣∣
t=0

τexp tX(o) ∈ ToG/K ,

where exp tX is the one-parameter subgroup of G generated by X. Under the
linear isomorphism m = ToG/K, the isotropy representation χ : K → Aut(m),
defined by χ(k) := (dτk)o for any k ∈ K, is equivalent with the representation
AdG |K : K × m → m. Hence, χ(k)X = AdG(k)X for any k ∈ K and X ∈ m.
In terms of Lie algebras we have χ∗(Y )X = [Y,X]m for any Y ∈ k and X ∈ m,
or in other words χ∗(Y ) = ad(Y )|m.

The homogeneous spaces M = G/K that we will examine below (with K
compact), are assumed to be almost effective, which means that the kernel
Ker(τ) (which is a normal subgroup both of G and K), is finite. Thus, the
isotropy representation χ is assumed to have a finite kernel, and then we
may identify k with the Lie algebra χ∗(k) = Lie(χ(K)) of the linear isotropy
group χ(K) ⊂ Aut(m). When only the identity element e ∈ G acts as the
identity transformation on M = G/K, then the G-action is called effective
and the isotropy representation χ is injective. If we assume for example that
G ⊆ Iso(M,g) is a closed subgroup, then the action of G to G/K is effective.
Note that an almost effective action of G on G/K gives rise to an effective
action of the group G′ = G/Ker(τ) (of the same dimension with G), so we
will not worry much for the effectiveness of M = G/K.

Recall that the space of G-invariant symmetric covariant 2-tensors on a
(almost) effective homogeneous space M = G/K with a reductive decompo-
sition g = k⊕m, is naturally isomorphic with the space of symmetric bilinear
forms on m, which are invariant under the isotropy action of K on m. As a
consequence, the space M G of G-invariant Riemannian metrics on M = G/K
coincides with the space of inner products 〈 , 〉 on m satisfying

〈X,Y 〉 = 〈χ(k)X,χ(k)Y 〉 = 〈AdG(k)X, AdG(k)Y 〉 ,



ancient solutions of the homogeneous ricci flow 105

for any k ∈ K and X,Y ∈ m. The correspondence is given by 〈X,Y 〉 =
g(X,Y )o. Moreover, when K is compact and m = h

⊥ with respect to B =
−Bg, where Bg is the Killing form of g, then one can extend the above cor-
respondence between elements g = 〈 , 〉 ∈ M G and AdG(K)-invariant B-
selfadjoint positive-definite endomorphisms L : m → m of m, i.e., 〈X,Y 〉 =
B(LX,Y ), for any X,Y ∈ m. To simplify the text, whenever is possible
next we shall relax the notation AdG(K) to Ad(K) and scalar products on m
as above, will be just referred to as Ad(K)-invariant scalar products. Note
that the Ad(K)-invariance of 〈 , 〉 implies its ad(k)-invariance, which means
that the endomorphism ad(Z)|m : m → m is skew-symmetric with respect to
〈 , 〉, for any Z ∈ k. When K is connected, the inclusions Ad(k)m ⊂ m and
[k,m] ⊂ m are equivalent and hence one can pass from the Ad(K)-invariance
to ad(k)-invariance and conversely. From now on will denote by P(m)Ad(K)

the space of all Ad(K)-invariant inner products on m.

Given a Riemannian manifold (Mn,g), a solution of the Ricci flow is a fam-
ily of Riemannian metrics {gt}∈ M satisfying the system (0.1). If the initial
metric g = g(0) ∈ M is a G-invariant metric with respect to some closed
subgroup G ⊆ Iso(M,g), i.e., (M = G/K,g) is a homogeneous Riemannian
manifold and so g ∈ M G, then the solution {g(t)} is called homogeneous, i.e.,
{g(t)}∈ M G. Indeed, the isometries of g are isometries for any other evolved
metric, and by [37] it is known that the isometry group is preserved under the
Ricci flow. Thus, after considering a reductive decomposition g = k ⊕ m of
(M = G/K,g), the homogeneity of g allows us to reduce the Ricci flow to a
system of ODEs for a curve of Ad(K)-invariant inner products on P(m)Ad(K),
where m ∼= ToG/K is a reductive complement. In particular, due to the iden-
tification M G ∼= P Ad(K)(m) we may write g(t) = 〈 , 〉t and then (0.1) takes
the form

d

dt
〈 , 〉t = −2 Ric〈 , 〉t , 〈 , 〉0 ≡〈 , 〉 = g ,

where Ric〈 , 〉t denotes the Ad(K)-invariant bilinear form on m, corresponding
to the Ricci tensor of g(t). Note that since g0 = g(0) is an invariant metric,
the solution g(t) of (0.1) must be unique among complete, bounded curvature
metrics (see [21]).

Remark 1.1. When one is interested in more general homogeneous spaces
M = G/K and a reductive decomposition may not exist, the above setting
can be appropriately transferred to g/k ∼= ToG/K. However, the “reductive
setting” serves well the goals of this paper and it is sufficient for our subsequent
computations and description.



106 s. anastassiou, i. chrysikos

1.2. Flag manifolds. Let G be a compact semisimple Lie group with
Lie algebra Lie(G) = g. A flag manifold1 is an adjoint orbit of G, i.e.,

M = Ad(G)w = {Ad(G)w : g ∈ G}⊂ g

for some left-invariant vector field w ∈ g. Let K = {g ∈ G : Ad(g)w = w}⊂
G be the isotropy subgroup of w and let k = Lie(K) be the corresponding Lie
algebra. Since G acts on M transitively, M is diffeomorphic to the (compact)
homogeneous space G/K, that is Ad(G)w = G/K. In particular, k = {X ∈ g :
[X,w] = 0} = ker ad(w), where ad : g → End(g) is the adjoint representation
of g. Moreover, the set Sw = {exp(tw) : t ∈ R} is a torus in G and the
isotropy subgroup K is identified with the centralizer in G of Sw, i.e., K =
C(Sw). Hence rankG = rankK and K is connected. Thus, equivalently a
flag manifold is a homogeneous space of the form G/K, where K = C(S) =
{g ∈ G : ghg−1 = h for all h ∈ S} is the centralizer of a torus S in G.
When K = C(T) = T is the centralizer of a maximal torus T in G, the
G/T is called a full flag manifold. Flag manifolds admit a finite number of
invariant complex structures, in particular flag spaces G/K of a compact,
simply connected, simple Lie group G exhaust all compact, simply connected,
de Rham irreducible homogeneous Kähler manifolds (see for example [4, 8,
23, 2] for further details).

So, for any flag manifold M = G/K we may work with G simply connected
(if for instance a flag manifold of SO(n) is given, one can always pass to its
universal covering by using the double covering Spin(n)). For our scopes, it is
also sufficient to focus on the de Rham irreducible case, which is equivalent to
say that G is simple, see [36]. Hence, in the following we can always assume
that M = G/K satisfies these conditions, and as before we shall denote by
Bg the Killing form of the Lie algebra g. The Ad(G)-invariant inner product
B := −Bg induces a bi-invariant metric on G by left translations, and we
may fix, once and for all, a B-orthogonal Ad(K)-invariant decomposition g =
k⊕m. G-invariant Riemannian metrics on G/K will be identified with Ad(K)-
invariant inner products 〈 , 〉 on the reductive complement m = ToG/K. Note
that the restriction B

∣∣
m

induces the so-called Killing metric gB ∈ M G, which
is the unique invariant metric for which the natural projection π : (G,B) −→
(G/K,gB) is a Riemannian submersion.

The second Betti number of any flag manifold M = G/K is encoded in
the corresponding painted Dynkin diagram. To recall the procedure, let gC =
hC ⊕

∑
α∈R g

C
α be the usual root space decomposition of the complexification

1Also called complex flag manifold, or generalized flag manifold.



ancient solutions of the homogeneous ricci flow 107

gC of g, with respect to a Cartan subalgebra hC of gC, where R ⊂
(
hC
)∗

is the

root system of gC. Via the Killing form of gC we identify
(
hC
)∗

with hC. Let
Π = {α1, . . . ,α`} (dim hC = `) be a fundamental system of R and choose a
subset ΠK of Π. We denote by RK =

{
β ∈ R : β =

∑
αi∈ΠK kiαi

}
the closed

subsystem spanned by ΠK. Then, the Lie subalgebra k
C = hC ⊕

∑
β∈RK g

C
β

is a reductive subalgebra of gC, i.e., it admits a decomposition of the form
kC = Z

(
kC
)
⊕ kCss, where Z(kC) is its center and kCss =

[
kC,kC

]
the semisimple

part of kC. In particular, RK is the root system of k
C
ss, and thus ΠK can be

considered as the associated fundamental system. Let K be the connected
Lie subgroup of G generated by k = kC ∩ g. Then the homogeneous manifold
M = G/K is a flag manifold, and any flag manifold is defined in this way, i.e.,
by the choise of a triple

(
gC, Π, ΠK

)
, see also [4, 8, 23, 2].

Set ΠM = Π\ΠK and RM = R\RK, such that Π = ΠK t ΠM , and
R = RK t RM , respectively. Roots in RM are called complementary roots.
Let Γ = Γ(Π) be the Dynkin diagram of the fundamental system Π.

Definition 1.2. Let M = G/K be a flag manifold. By painting black
the nodes of Γ corresponding to ΠM , we obtain the painted Dynkin diagram
of G/K (PDD in short). In this diagram the subsystem ΠK is determined as
the subdiagram of white roots.

Remark 1.3. Conversely, given a PDD, one may determine the associated
flag manifold M = G/K as follows: The group G is defined as the unique
simply connected Lie group generated by the unique real form g of the com-
plex simple Lie algebra gC (up to inner automorphisms of gC), which is re-
constructed by the underlying Dynkin diagram. Moreover, the connected
Lie subgroup K ⊂ G is defined by using the encoded by the PDD splitting
Π = ΠK tΠM . The semisimple part of K is obtained from the (not necessar-
ily connected) subdiagram of white simple roots, while each black root, i.e.,
each root in ΠM , gives rise to a U(1)-summand. Thus, the PDD determines
the isotropy group K and the space M = G/K completely. By using certain
rules to determine whether different PDDs define isomorphic flag manifolds
(see [4]), one can obtain all flag manifolds G/K of a compact simple Lie group
G (see for example the tables in [2]).

Proposition 1.4. ([17, 4]) The second Betti number of a flag manifold
M = G/K equals to the totality of black nodes in the corresponding PDD,
i.e., the cardinality of the set ΠM .



108 s. anastassiou, i. chrysikos

Note that for any flag manifold M = G/K, the B-orthogonal reductive
complement m decomposes into a direct sum of Ad(K)-inequivalent and irre-
ducible submodules, which we call isotropy summands, see [4, 8, 2] and the
references therein. This means that when m ∼= ToM is viewed as a K-module,
then there is always a B-orthogonal Ad(K)-invariant decomposition

m = m1 ⊕···⊕mr , (1.1)

for some r ≥ 1, such that:

• K acts (via the isotropy representation) irreducibly on any m1, . . . ,mr;

• mi � mj are inequivalent as Ad(K)-representations for any i 6= j.

In fact, a decomposition as in (1.1) satisfying the given conditions must be
unique, up to a permutation of the isotropy summands, see for example [52].
When r = 1, M = G/K is an isotropy irreducible compact Hermitian sym-
metric space (HSS in short), and all these cosets can be viewed as flag man-
ifolds with b2(M) = 1. Note that from the class of full flag manifolds only
CP1 ∼= S2 = SU(2)/U(1) is an irreducible HSS, and hence a flag manifold
with b2(M) = 1. In this text we are mainly interested in non-symmetric flag
manifolds M = G/K with b2(M) = 1, and in this case r is bounded by the
inequalities 2 ≤ r ≤ 6 (see below).

Lemma 1.5. ([4, 8]) Let M = G/K be a flag manifold of a compact
simple Lie group. Then, the isotropy representation of M is monotypic and
decomposes as in (1.1), for some r ≥ 1. Moreover, any G-invariant metric on
M = G/K is given by

g = 〈 , 〉 =
r∑
i=1

xi ·B|mi , (1.2)

where xi ∈ R+ are positive real numbers for any i = 1, . . . ,r. Thus, M G
coincides with the open convex cone Rr+ = {(x1, . . . ,xr) ∈ R

r : xi > 0 for
any i = 1 . . .r}.

Invariant metrics as in (1.2) are called diagonal (see [55] for details). By
Schur’s Lemma, the Ricci tensor Ricg of such a diagonal invariant metric g,
needs to preserve the splitting (1.1) and consequently, Ricg is also diagonal,
i.e., Ricg(mi,mj) = 0, whenever i 6= j. As before, Ricg is determined by
a symmetric Ad(K)-invariant bilinear form on m, although not necessarily



ancient solutions of the homogeneous ricci flow 109

positive definite, and hence it has the expression

Ricg ≡ Ric〈 , 〉 =
r∑
i=1

yi ·B
∣∣
mi

=
r∑
i=1

(xi · rici) ·B
∣∣
mi
,

for some yi = xi · rici ∈ R, where 〈 , 〉 ∈ P Ad(K)(m) is the Ad(K)-invariant
inner product corresponding to g ∈ M G and rici are the so-called Ricci com-
ponents. There is a simple description of rici, and hence of Ricg (and also of
the scalar curvature Scalg = tr Ricg), in terms of the metric parameters xi, the
dimensions di = dimR mi and the so-called structure constants of G/K with
respect to the decomposition (1.1),

ckij ≡
[
k

i j

]
=
∑
α,β,γ

B
(
[Xα,Yβ],Zγ

)2
, i,j,k ∈{1, . . . ,r},

where {Xα}, {Yβ}, {Zγ} are B-orthonormal bases of mi, mj, mk, respectively.
These non-negative quantities were introduced in [55] and they have a long
tradition in the theory of (compact) homogeneous Einstein spaces, see for
example [46, 53, 10, 15, 24]. Following [32], next we shall refer to rational
polynomials depending on some real variables x1,x

−1
1 , . . . ,xm,x

−1
m for some

positive integer m, by the term Laurent polynomials. If such a rational polyno-
mial is homogeneous, then it will be called a homogeneous Laurent polynomial.
Now, as a conclusion of most general results presented by [55, 46], one obtains
the following

Proposition 1.6. ([55, 46]) Let M = G/K be a flag manifold of a com-
pact simple Lie group G. Then,

(i) The components rick of the Ricci tensor Ricg corresponding to
g =

∑r
i=1 xi · B|mi ∈ M

G are homogeneous Laurent polynomials in
x1,x

−1
1 , . . . ,xr,x

−1
r of degree −1, given by

rick =
1

2xk
+

1

4dk

r∑
i,j=1

xk
xixj

[
k

i j

]
−

1

2dk

r∑
i,j=1

xj
xkxi

[
j

k i

]
, (k = 1, . . . ,r) .

(ii) The scalar curvature Scalg =
∑r

i=1 di·rici corresponding to g =
∑r

i=1 xi·
B|mi ∈ M

G is a homogeneous Laurent polynomial in x1,x
−1
1 , . . . ,xr,x

−1
r

of degree −1, given by

Scalg =
1

2

r∑
i=1

di
xi
−

1

4

r∑
i,j,m=1

[
m

i j

]
xm
xixj

.

Note that Scalg : M → R is a constant function.



110 s. anastassiou, i. chrysikos

2. The homogeneous Ricci flow on flag manifolds

In this section we fix a flag manifold M = G/K of a compact, simply
connected, simple Lie group G, whose isotropy representation m decomposes
as in (1.1). Since M G endowed with the L2-metric coincides with the open
convex cone Rr+, for the study of the Ricci flow, as an initial invariant metric
we fix the general invariant metric g =

∑r
i=1 xi·B|mi; This often will be simply

denoted by g = (x1, . . . ,xr) ∈ Rr+ ∼= M
G. Then, the Ricci flow equation (0.1)

with initial condition g(0) = g descents to the following system of ODEs (in
component form):{

ẋk = −2xk · rick(x1, . . . ,xr) , : 1 ≤ k ≤ r
}
. (2.1)

Since g ∈ M G, and any invariant metric evolves under the Ricci flow again
to a G-invariant metric, every solution of the homogeneous Ricci flow needs
to be of the form

g(t) =

r∑
i=1

xi(t) ·B|mi, g(0) = g = 〈 , 〉 ,

where the smooth functions xi(t) are positive on the same maximal interval
t ∈ (ta, tb) for which g(t) is defined. Usually, solutions have a maximal interval
of definition (ta, tb), where 0 ∈ (ta, tb), −∞ ≤ ta < 0 and tb ≥ +∞. Next we
are mainly interested in ancient solutions.

2.1. Invariant ancient solutions. Recall that (see for example
[18, 38]).

Definition 2.1. A solution g(t) of (2.1) which is defined on an interval
of the form (−∞, tb) with tb < +∞, is called ancient.

Note that ancient solutions typically arise as singularity models of the
Ricci flow and it is well-known that all ancient solutions have non-negative
scalar curvature, see for instance [20, Corollary 2.5]. We first proceed with
the following

Lemma 2.2. The homogeneous Ricci flow (2.1) on a flag manifold M =
G/K, does not possess fixed points in M G ∼= Rr+. In other words, considering
the associated flow to (2.1), i.e., the map

ψt : M
G −→ M G , g 7−→ g(t) ,



ancient solutions of the homogeneous ricci flow 111

where t ∈ (−�,�) for some � ∈ (0,∞), there is no g ∈ M G ∼= Rr+ such that
ψt(g) = g for all t.

Proof. Obviously, fixed points of the homogeneous Ricci flow need to cor-
respond to invariant Ricci-flat metrics, and conversely. Since M = G/K is
compact, according to Alekseevsky and Kimel’fel’d [3] such a metric must be
necessarily flat. But then M must be a torus, a contradiction.

However, the system (2.1) can admit more general solutions, different than
stationary points. Indeed, any flag manifold Mn = G/K is compact and
simply connected, and hence its universal covering is not diffeomorphic with
an Euclidean space. Hence, by [38, Corollary 4.3] it follows that

Proposition 2.3. ([38]) For any flag manifold M = G/K of a compact,
simply connected, simple Lie group G, there exists a G-invariant metric g
such that any Ricci flow solution g(t) with initial condition g(0) = g, has an
interval of definition of the form (ta, tb), with −∞≤ ta < 0 and tb < ∞.

In the following section, for all flag manifolds M = G/K with b2(M) =
1 we will construct solutions of (2.1), specify the maximal interval of their
definition (ta, tb) and study their asymptotic behaviour at the infinity of the
corresponding phase space M G (in terms of the Poincaré compactification, see
Definition 2.11). These solutions, both trivial and non-trivial, emerge from
invariant Einstein metrics, in particular the trivial are shrinking type solutions
of (2.1), in terms of [22, p. 98] for instance. Moreover, they are ancient and
hence the scalar curvature along such solutions is positive, while they extinct
in finite time. In fact, for any flag manifold M = G/K one can state the
following basic result.

Proposition 2.4. Let M = G/K be a flag manifold of a compact, con-
nected, simply connected, simple Lie group G, with M G ∼= Rr+, for some
r ≥ 1. Let g0 be any G-invariant Einstein metric on M = G/K with Einstein
constant λ and consider the 1-parameter family g(t) = (1 − 2λt)g0. Then,

(i) g(t) is an ancient solution of (2.1) defined on the open interval
(
−∞, 1

2λ

)
with g(t) → 0 as t → 1

2λ
(from below). Hence, its scalar curvature

Scal(g(t)) is a monotonically increasing function with the same interval
of definition, and satisfies

lim
t→−∞

Scal(g(t)) = 0 , lim
t→ 1

2λ

Scal(g(t)) = +∞ .

In particular, Scal(g(t)) > 0 for any t ∈
(
−∞, 1

2λ

)
.



112 s. anastassiou, i. chrysikos

(ii) Similarly, the Ricci components ricti ≡ rici(g(t)) corresponding to the
solution g(t) = (1 − 2λt)g0, satisfy the following asymptotic properties:

lim
t→−∞

ricti = 0 , lim
t→ 1

2λ

ricti = +∞ .

In particular, ricti > 0, for any t ∈
(
−∞, 1

2λ

)
.

Proof. (i) Let g0 be a G-invariant Einstein metric on M = G/K
(e.g., one can fix as g0 an invariant Kähler-Einstein metric, which always
exists). Set c(t) = (1 − 2λt) and note that c(t) = (1 − 2λt) > 0 for any
t ∈

(
−∞, 1

2λ

)
. Then, since g(t) = c(t)g0 is a 1-parameter family of invariant

metrics, we have

g(t) =

r∑
i=1

(1 − 2λt)x0i ·B|mi =
r∑
i=1

c(t)x0i ·B|mi =
r∑
i=1

xi(t) ·B|mi ,

where xi(t) := c(t)x
0
i , for any 1 ≤ i ≤ r with xi(0) = x

0
i , where without loss

of generality we assume that g0 = (x
0
1, . . . ,x

0
r) for some x

0
i ∈ R

r
+ and some

N 3 r ≥ 1. As it is well-known from the general theory of Ricci flow, and it
is trivial to see, g(t) is a solution of (2.1) which has as interval of definition
the open set

(
−∞, 1

2λ

)
. Hence it is an ancient solution, since 0 < 1

2λ
< +∞

(recall that λ > 0 is the Einstein constant of an invariant Einstein metric
on a compact homogenous space), and moreover limt→−∞g(t) = +∞ and
limt→ 1

2λ
g(t) = 0. On the other hand, by Proposition 1.6, the scalar curvature

Scalg0 ≡ Scal(g0) is a homogeneous Laurent polynomial of degree −1. Hence,

Scal(c(t)g0) = c(t)
−1 Scal(g0) =

1

(1 − 2λt)
Scal(g0) .

Consequently Scal(g(t)) = 1
(1−2λt) Scal(g0) > 0, for any t ∈

(
−∞, 1

2λ

)
, since

Scal(g0) > 0. Obviously,

Scal′(g(t)) =
2λScal(g0)

(1 − 2λt)2
> 0 ,

for any
(
−∞, 1

2λ

)
and when t tends to 1

2λ
from below, we see that Scal(g(t)) →

+∞. Since Scal(g(t)), as a smooth function of t, is defined only for t ∈(
−∞, 1

2λ

)
, we conclude. Moreover, for t →−∞ we get Scal(g(t)) → 0.



ancient solutions of the homogeneous ricci flow 113

(ii) Similarly, by Proposition 1.6, the components rici ≡ rici(g0) = ric0i
(1 ≤ i ≤ r) of the Ricci tensor of g0 are homogeneous Laurent polynomials of
degree −1. Hence,

rici(c(t)g0) =
1

c(t)
rici(g0) =

1

c(t)
ric0i ,

i.e., rici(g(t)) = c(t)
−1 ric0i . The conclusion now easily follows, since g0 is

Einstein and so ric0i = λ, which is independent of t, for any 1 ≤ i ≤ r.

Remark 2.5. (i) The conclusions for the asymptotic behaviour of scalar
curvature for the solutions g(t) verify a more general statement for the limit
behaviour of the scalar curvature of homogeneous ancient solutions, obtained
by Lafuente [38, Theorem 1.1 (i)] in terms of the so-called bracket flow 2. Later,
for the convenience of the reader, in Section 3 we will illustrate Proposition
2.4 by certain examples (see Example 3.3 and Example 3.4).

(ii) Recall by [22, p. 545] that

Rict ≡ Ric(g(t)) = Ric(g(0)) = λg0 =
λ

(1 − 2λt)
g(t) .

Thus, the trivial solutions g(t) given in Proposition 2.4, being homothetic
to invariant Einstein metrics, they also satisfy the Einstein equation (and
therefore the Ricci flow equation), and hence

lim
t→−∞

Rict = λg0 = lim
t→( 1

2λ
)−

Rict .

The Ad(K)-invariant g(t)-self-adjoint operator rg(t) ≡ rt : m → m (Ricci endo-
morphism) corresponding to the Ricci tensor Rict is defined by Rict(X,Y ) =
gt(rt(X),Y ) for any X,Y ∈ m. It satisfies the relation rt = λc(t)At = λA0 = r0,
for any t ∈ (−∞, 1

2λ
), where A0 =

∑r
i=1 x

0
i ·Idmi is the positive definite Ad(K)-

invariant g0-self-adjoint operator corresponding to the diagonal metric g0. The
endomorphism of the 1-parameter family g(t) = c(t)g0, given by At = c(t)A0,
is also positive definite for any t ∈ (−∞, 1

2λ
), and the same satisfies rt.

(iii) Obviously, Proposition 2.4 and the above conclusions can be extended
to any homogeneous space G/K of a compact semisimple Lie group G modulo
a compact subgroup K ⊂ G, with a monotypic isotropy representation admit-
ting an invariant Einstein metric g0. A simple example is given below. On the

2 The solutions g(t) described above have as maximal interval of definition the open set
(−∞, 1

2λi
), so the second part of [38, Thmeorem 1.1] does not apply in our situation.



114 s. anastassiou, i. chrysikos

other side, there are examples of effective compact homogeneous spaces, with
non-monotypic isotropy representation, i.e., mi ∼= mj for some 1 ≤ i 6= j ≤ r,
for which the invariant (Einstein) metrics are still diagonal. To take a taste,
consider the Stiefel manifold V2(R`+1) = G/K = SO(` + 1)/SO(` − 1) and
assume for simplicity that ` 6= 3. This is a compact homogeneous space,
admitting a U(1)-fibration over the Grassmannian

Gr+2 (R
`+1) = SO(` + 1)/SO(`− 1) ×SO(2) .

Let so(`+1) = so(`−1)⊕m be a B-orthogonal reductive decomposition. Then,
it is not hard to see that m = m0 ⊕ m1 ⊕ m2, where m0 is 1-dimensional and
m1 ∼= m2 are two irreducible submodules of dimension `−1, both isomorphic to
the standard representation of SO(`− 1). Hence, the isotropy representation
of V2(R`+1) is not monotypic. However, the invariant metrics on V2(R`+1)
can be shown that are still diagonal. This is based on the action of the
generalized Weyl group (gauge group) NG(K)/K on the space P(m)

Ad(K)

of Ad(K)-invariant inner products on m (see for example [45, 6]). For the
specific case of V2(R`+1), the group NG(K)/K is isomorphic to a circle and this
action was used in [34, p. 121] to eliminate the off-diagonal components of the
invariant metrics (note that for ` 6= 3, V2(R`+1) admits a unique SO(2` + 1)-
invariant Einstein metric). Hence, the results discussed above can also be
extended in that more general case.

Example 2.6. Let M = G/K be an isotropy irreducible homogeneous
space of a compact simple Lie group G. Consider a B-orthogonal reductive
decomposition g = k ⊕ m. Then, M G = R+ and the Killing form gB = B|m
is the unique invariant Einstein metric (up to a scalar). Hence, g(t) = (1 −
2λBt)gB is a homogeneous ancient solution of the corresponding homogeneous
Ricci flow, defined on the open interval (−∞, 1

2λB
), where λB > 0 is the

Einstein constant of gB. This applies in particular to any symmetric flag
manifold M = G/K of a compact simple Lie group G (i.e., a compact isotropy
irreducible HSS where r = 1).

Definition 2.7. A homogeneous ancient solution of (2.1) is called non-
collapsed, if the corresponding curvature normalized metrics have a uniform
lower injectivity radius bound.

Non-collapsed homogeneous ancient solutions of the Ricci flow on compact
homogeneous space have been recently studied in [13]. In this work, among
other results, the authors proved that:



ancient solutions of the homogeneous ricci flow 115

• (α) If G,K are connected and does not exist some intermediate group
K ⊂ L ⊂ G such that L/K is a torus, i.e., if G/K is not a homogeneous
torus bundle G/K → G/L over G/L, then any ancient solution on G/K
is non-collapsed ([13, Remark 5.3]).

• (β) Non-trivial homogeneous ancient solutions of Ricci flow must develop
a Type I singularity close to their extinction time and also to the past
(see [13, Corollary 2, page 2, and pages 24 – 25]).

Hence we deduce that

Corollary 2.8. Let M= G/K be a flag manifold as in Proposition 2.4.
Then any non-trivial ancient solution of the homogeneous Ricci flow, if it
exists, emanates from an invariant Einstein metric, is non-collapsed and de-
velops a Type-I singularity close to the extinsion time (and also to the past).
In particular, the trivial ancient solutions of the form g(t) = (1−2λt)g0, where
g0 ∈MG is an invariant Einstein metric on M with Einstein constant λ, are
non-collapsed and as t → 1

2λ
the volume of M= G/K with respect to g(t)

tends to 0, V olg(t)(G/K) → 0, i.e., M= G/K shrinks to a point in finite time.

Proof. For the record, let us assume that there exists some intermediate
closed subgroup K ⊂ L ⊂ G such that L/K is torus Ts for some s ≥ 1. Then
obviously, it must be rankL > rankK. But rankK = rankG and then inclusion
L ⊂ G gives a contradiction. Hence, M = G/K cannot be homogeneous torus
bundle. Thus, according to (α) any possible homogeneous ancient solution
of (2.1) on M = G/K must be non-collapsed. The other claims and the
assertions for Type I behaviour, follow now by (β).

Remark 2.9. Recall that on a compact homogeneous space M = G/K the
total scalar curvature functional

S(g) =

∫
M

Scal(g)dVg ,

restricted on the set M G1 of G-invariant metrics of volume 1, coincides with the
smooth function M G1 3 g 7−→ Scal(g) ≡ Scalg, since the scalar curvature Scalg
of g is a constant function on M. In this case, G-invariant Einstein metrics
of volume 1 on G/K are precisely the critical points of the restriction S|M G1 .
A G-invariant Einstein metric is called unstable if is not a local maximum of
S|M G1 . By [13, Lemmma 5.4] it also known that for any unstable homoge-
neous Einstein metric on a compact homogeneous space G/K, there exists a



116 s. anastassiou, i. chrysikos

non-collapsed invariant ancient solution emanating from it. For general flag
manifolds, up to our knowledge, is an open question if all possible existent
invariant Einstein metrics are unstable or not. However, by [9, Thmeorem
1.2] it is known that on a flag manifold with r = 2, i.e., with two isotropy
summands, there exist two non-isometric invariant Einstein metrics which are
both local minima of S|M G1 , and hence unstable in the above sense. Thus, for
r = 2 the existence of non-collapsed ancient solutions of system (2.1) can be
also obtained by a direct combination of the results in [9, 13].

2.2. The Poincaré compactification procedure. To study the
asymptotic behaviour of homogeneous Ricci flow solutions, even of more gen-
eral abstract solutions than these given in Proposition 2.4, one can successfully
use the compactification method of Poincaré which we refresh below, adapted
to our setting (see also [28, 7, 29]). Indeed, the Poincaré compactification
procedure is used to study the behaviour of a polynomial system of ordinary
differential equations in a neighbourhood of infinity, see [27, 54] for an explicit
description. We describe it here, in a form suitable for our purposes.

Let (M = G/K,g) be a flag manifold with M G ∼= Rr+ for some r ≥
1. By using the expressions of Proposition 1.6 and after multiplying the
right-hand side of the equations in (2.1), with a suitable positive factor, we
obtain a qualitative equivalent dynamical system consisting of homogeneous
polynomial equations of positive degree. Let us denote this system obtained
from the homogeneous Ricci flow via this procedure, by{

ẋk = RFk(x1, . . . ,xr) : k = 1, . . . ,r
}
. (2.2)

So, for any 1 ≤ k ≤ r, RFk(x1, . . . ,xr) are homogeneous polynomials, whose
maximum degree is defined to be the degree d of the system (2.2). Let us
proceed with the following definition.

Definition 2.10. The vector field associated to the system (2.2) will be
denoted by

X(x1, . . . ,xr) :=
(
RF1(x1, . . . ,xr), . . . ,RFr(x1, . . . ,xr)

)
and referred to as the homogeneous vector field associated to the homogeneous
Ricci flow on (M = G/K,g).

Consider the subset of the unit sphere Sr ⊂ Rr+1, containing all points
having non-negative coordinates, i.e.,

Sr≥ =
{
y ∈ Rr+1 : ‖y‖ = 1, yi ≥ 0, i = 1, . . . ,r + 1

}
⊂ Sr .



ancient solutions of the homogeneous ricci flow 117

It is also convenient to identify Rr≥ with the subset of R
r+1 defined by

T :=
{

(y1, . . . ,yr,yr+1) ∈ Rr+1 : yr+1 = 1, yi ≥ 0, ∀ i = 1, . . . ,r
} ∼= Rr≥ .

Consider now the central projection f : Rr≥ ∼= T → S
r
≥, assigning to every

p ∈ T the point T(p) ∈ Sr≥, defined as follows: T(p) is the intersection of the
straight line joining the initial point p with the origin of Rr+1. The explicit
form of f is given by

f(y1, . . . ,yr, 1) =
1

‖(y1, . . . ,yr, 1)‖
(y1, . . . ,yr, 1) .

Through this projection, Rr≥ can be identified with the subset of S
r
≥ with

yi+1 > 0. Moreover, the equator of S
r
≥, that is

Sr−1≥ =
{
y ∈ Sr≥ : yr+1 = 0

}
,

is identified as the infinity of Rr≥. To be more explicit, let us state this as a
definition.

Definition 2.11. A point at infinity of M G ∼= Rr≥ is understood to be
point of the equator Sr−1≥ .

Note that Sr≥ is diffeomorphic with the standard r–dimensional simplex

∆r =
{

(y1, . . . ,yr+1) ∈ Rr+1 :
∑

iyi = 1, ∀ yi ≥ 0, i = 1, . . . ,r + 1
}

and thus with the so-called non-negative part of the real projective space
RPr≥ =

(
Rr+1≥ \{0}

)
/R+ (see [26, Chapter 4]). Via push-forward, the central

projection carries the vector field X onto Sr≥. The vector field obtained by
this procedure, i.e., the vector field

p(X)(y) := yd−1r+1f∗X(y)

is called the Poincaré compactification of X, and this is an analytic vector
field defined on all Sr≥. Actually, in order to perform computations with p(X)
one needs its expressions in a chart. Thus, consider the chart

U =
{
y ∈ Sr : y1 > 0

}
of Sr, and project it to the plane

{
(y1, . . . ,yr+1) ∈ Rr+1 : y1 = 1)

} ∼= Rr≥.
Then, via the corresponding central projection, assign to every point of U the



118 s. anastassiou, i. chrysikos

point of intersection of the straight line joining the origin with the original
point. This second central projection denoted by F : U → Rr≥, has obviously
the form F(y1,y2, . . . ,yr+1) =

(
1,

y2
y1
, . . . ,

yr+1
y1

)
. We can now compute the

local expression of p(X) in the chart U; As above, let us denote by xi the
coordinates on Rr≥. Then, we obtain

Proposition 2.12. The local expression of the Poincaré compactification
p(X) of the homogeneous vector field X associated to the Ricci flow on (M =
G/K,g), reads as

ẋi = x
d
r

(
−xiRF1 + RFi+1

)
for any i = 1, . . . ,r − 1 ,

ẋr = x
d
r

(
−xrRF1

)
,

(2.3)

where 3

RFi(x1, . . . ,xr) := RFi
( 1
xr
,
x2
xr
, . . . ,

xr−1
xr

)
.

Remark 2.13. In the expressions given by (2.3), the factor xdr is canceled
by the dominators of the rational polynomials RFi(x1, . . . ,xr). After doing
so and by locating the fixed points of the resulting dynamical system, under
the condition xr = 0, we obtain the so-called fixed points at the infinity of
M G ∼= Rr+ of the homogeneous Ricci flow on (M = G/K,g).

Example 2.14. For r = 2 the expression of p(X) in the local chart U is
given by

ẋ1 = x
d
2

(
−x1RF1 + RF2

)
, ẋ2 = x

d
2

(
−x2RF1

)
,

where RFi(x1,x2) = RFi
(

1
x2
, x1
x2

)
for any i = 1, 2. We may omit the term

1
‖(x1,x2,1)‖

, by using a time reparametrization. Fixed points of the homoge-

neous Ricci flow on (M = G/K,g) at infinity of M G = R2+, can be studied
by setting x2 = 0. For r = 3 the expression of p(X) in the local chart U reads
by

ẋ1 = x
d
3

(
−x1RF1 + RF2

)
, ẋ2 = x

d
3

(
−x2RF1 + RF3

)
, ẋ3 = x

d
3

(
−x3RF1

)
,

where RFi = RFi
(

1
x3
, x1
x3
, x2
x3

)
, for any i = 1, 2, 3. In an analogous way, fixed

points at infinity can be studied by setting x3 = 0, while similarly are treated
cases with r > 3.

3 Here, we have omitted the term 1‖(x1,...,xr,1)‖
, since it does not affect the qualitative

behaviour of the system.



ancient solutions of the homogeneous ricci flow 119

3. Global study of HRF on flag spaces
M = G/K with b2(M) = 1

We turn now our attention to the global study of the classical Ricci flow
equation for an initial invariant metric on (non symmetric) flag manifolds
M = G/K with b2(M) = 1. Again we can work with G simple. Let us begin
first with a few details about this specific class of flag spaces.

3.1. Flag manifolds M = G/K with b2(M) = 1. According to
Proposition 1.4, flag manifolds M = G/K of a compact simple Lie group G
with b2(M) = 1 are defined by painting black in the Dynkin diagram of G only
a simple root, i.e., ΠM = {aio} for some aio ∈ Π. The number of the isotropy
summands of a flag space M = G/K with b2(M) = 1 can be read from the
PDD, at least when we encode with it the so-called Dynkin marks of the simple
roots; These are the positive integers coefficients appearing in the expression
of the highest root of G as a linear combination of simple roots. For flag
manifolds with b2(M) = 1, i.e., ΠM = Π/ΠK = {aio : 1 ≤ io ≤ ` = rankG}
we have the relation r = Dynk(αio) (see [23, 24]), where r is the integer ap-
pearing in (1.1). In other words, a flag manifold M = G/K with b2(M) = 1
and r isotropy summands is obtained by painting black a simple root αio with
Dynkin mark r, and conversely. Since for a compact simple Lie group G the
maximal Dynkin mark equals to 6 (and occurs for G = E8 only), we result
with the bound 1 ≤ r ≤ 6.

So, from now on assume that M = G/K is a flag manifold as above
with b2(M) = 1 and 2 ≤ r ≤ 6. The classification of such flag manifolds
can be found in [24]; There, in combination with earlier results of Kimura,
Arvanitoyeorgos and the second author [35, 8, 9] is proved that any such
space admits a finite number of non-isometric (non-Kähler) invariant Einstein
metrics, a result which supports the still open finiteness conjecture of Böhm-
Wang-Ziller ([15]). Note that M = G/K admits a unique invariant complex
structure, and thus a unique invariant Kähler-Einstein metric, with explicit
form (see [17, 24])

gKE =

r∑
i

i ·B|mi = B|m1 + 2B|m2 + . . . + rB|mr . (3.1)

The isotropy summands satisfy the relations

[k,mi] ⊂ mi , [mi,mi] ⊂ k + m2i , [mi,mj] ⊂ mi+j + m|i−j| (i 6= j).



120 s. anastassiou, i. chrysikos

Hence, the non-zero structure constants are listed as follows:

r non-zero structure constants ckij

2 c211 ([9])

3 c211, c
3
12 ([35, 7])

4 c211, c
3
12, c

4
13, c

4
22 ([8])

5 c211, c
3
12, c

4
13, c

5
14, c

4
22, c

5
23 ([24])

6 c211, c
3
12, c

4
13, c

5
14, c

6
15, c

4
22, c

5
23, c

6
24, c

6
33 ([24])

For r = 2, 3, the use of the Kähler-Einstein metric is sufficient for an explicit
computation of ckij, and this yields a general expression of them in terms of
di = dim mi (see [35, 9, 7]). For 4 ≤ r ≤ 6 more advanced techniques are
necessary for the computation of ckij, which depend on the specific coset (see
[8, 24]).

Case r = 2 : All flag manifolds M = G/K with G simple and r = 2 have
b2(M) = 1, see [9] for details. We recall that:

c211 =
d1d2

d1 + 4d2
,

ric1 =
1

2x1
− c211

x2
2d1x

2
1

, ric2 =
1

2x2
+
c211
4d2

(x2
x21
−

2

x2

)
,

Scalg =
2∑
i=1

di · rici =
1

2

(d1
x1

+
d2
x2

)
−
c211
4

(x2
x21

+
2

x2

)
. (3.2)

Case r = 3 : Let M = G/K be a flag manifold with b2(M) = 1 and
r = 3. Such flag spaces have been classified by [35], see also [7, 30]. They all
correspond to exceptional compact simple Lie groups, but the correspondence
is not a bijection. One should mention that not all flag manifolds with r = 3
are exhausted by this type of homogeneous spaces; This means that still one
can construct flag spaces with r = 3 and b2(M) = 2. We recall that

c211 =
d1d2 + 2d1d3 −d2d3
d1 + 4d2 + 9d3

,

c312 =
d3(d1 + d2)

d1 + 4d2 + 9d3
,

ric1 =
1

2x1
−
c211x2
2d1x

2
1

+
c312
2d1

( x1
x2x3

−
x2
x1x3

−
x3
x1x2

)
,



ancient solutions of the homogeneous ricci flow 121

ric2 =
1

2x2
+
c211
4d2

(x2
x21
−

2

x2

)
+
c312
2d2

( x2
x1x3

−
x1
x2x3

−
x3
x1x2

)
,

ric3 =
1

2x3
+
c312
2d3

( x3
x1x2

−
x1
x2x3

−
x2
x1x3

)
,

Scalg =
1

2

(d1
x1

+
d2
x2

+
d3
x3

)
−
c211
4

(x2
x21

+
2

x2

)
(3.3)

−
c312
2

( x1
x2x3

+
x2
x1x3

+
x3
x1x2

)
.

Case r = 4 : Let M = G/K be a flag manifold with G simple, r = 4 and
b2(M) = 1. There exist four such homogeneous spaces and all of the them
correspond to an exceptional Lie group. To give the reader a small taste of
PDDs, we present these flag spaces below, together with the corresponding
PDD and Dynkin marks. As for the case r = 3, one should be aware that still
exist flag manifold with r = 4, but b2(M) = 2.

M = G/K with b2(M) = 1 and r = 4 PDD

F4 /SU(3) ×SU(2) ×U(1)
cα1
2

cα2
3

>sα3
4

cα4
2

E7 /SU(4) ×SU(3) ×SU(2) ×U(1)
cα1
1

cα2
2

cα3
3

sα4
4 cα72

cα5
3

cα6
2

E8 /SO(10) ×SU(3) ×U(1)
cα1
2

cα2
3

sα3
4

cα4
5 cα83

cα5
6

cα6
4

cα7
2

E8 /SU(7) ×SU(2) ×U(1)
cα1
2

cα2
3

cα3
4

cα4
5 cα83

cα5
6

sα6
4

cα7
2

We also recall that

ric1 =
1

2x1
−
c211
2d1

x2
x21

+
c312
2d1

( x1
x2x3

−
x2
x1x3

−
x3
x1x2

)
+
c413
2d1

( x1
x3x4

−
x3
x1x4

−
x4
x1x3

)
,

ric2 =
1

2x2
−
c422
2d2

x4
x22

+
c211
4d2

(x2
x21
−

2

x2

)
+
c312
2d2

( x2
x1x3

−
x1
x2x3

−
x3
x1x2

)
,



122 s. anastassiou, i. chrysikos

ric3 =
1

2x3
+
c312
2d3

( x3
x1x2

−
x2
x1x3

−
x1
x2x3

)
+
c413
2d3

( x3
x1x4

−
x1
x3x4

−
x4
x1x3

)
,

ric4 =
1

2x4
+
c422
4d4

(x4
x22
−

2

x4

)
+
c413
2d4

( x4
x1x3

−
x1
x3x4

−
x3
x1x4

)
,

Scalg =
1

2

4∑
i=1

di
xi
−
c312
2

( x1
x2x3

+
x2
x1x3

+
x3
x1x2

)
−
c413
2

( x1
x3x4

+
x3
x1x4

+
x4
x1x3

)
−
c211
4

(x2
x21

+
2

x2

)
−
c422
4

(x4
x22

+
2

x4

)
.

Let us finally present the values of ckij and the corresponding dimensions:

M = G/K c422 c
2
11 c

3
12 c

4
13

F4 /SU(3) ×SU(2) ×U(1) 2 2 1 2/3

E7 /SU(4) ×SU(3) ×SU(2) ×U(1) 2 8 4 4/3

E8 /SO(10) ×SU(3) ×U(1) 2 16 8 8/5

E8 /SU(7) ×SU(2) ×U(1) 14/3 14 7 14/5

d1 d2 d3 d4

F4 /SU(3) ×SU(2) ×U(1) 12 18 4 6

E7 /SU(4) ×SU(3) ×SU(2) ×U(1) 48 36 16 6

E8 /SO(10) ×SU(3) ×U(1) 96 60 32 6

E8 /SU(7) ×SU(2) ×U(1) 84 70 28 14

Case r = 5 : According to [24], there is only one flag manifold M = G/K
with G simple, b2(M) = 1 and r = 5; This is the coset space

M = G/K = E8 /U(1) ×SU(4) ×SU(5)

and is determined by painting black the simple root α4 of E8, i.e., ΠM = {α4},
with Dynk(α4) = 5 (and hence r = 5). The Ricci components rici are given
by

ric1 =
1

2x1
−
c211
2 d1

x2
x12

+
c312
2 d1

( x1
x2x3

−
x2
x1x3

−
x3
x1x2

)
+
c413
2 d1

( x1
x3x4

−
x3
x1x4

−
x4
x1x3

)
+
c514
2 d1

( x1
x4x5

−
x4
x1x5

−
x5
x1x4

)
,



ancient solutions of the homogeneous ricci flow 123

ric2 =
1

2x2
+
c211
4 d2

( x2
x12
−

2

x2

)
−
c422
2 d2

x4
x22

+
c312
2 d2

( x2
x1x3

−
x1
x2x3

−
x3
x2x1

)
+
c523
2 d2

( x2
x3x5

−
x3
x2x5

−
x5
x2x3

)
,

ric3 =
1

2x3
+
c312
2 d3

( x3
x1x2

−
x2
x3x1

−
x1
x3x2

)
+
c413
2 d3

( x3
x1x4

−
x1
x3x4

−
x4
x1x3

)
+
c523
2 d3

( x3
x2x5

−
x2
x3x5

−
x5
x3x2

)
,

ric4 =
1

2x4
+
c422
4 d4

( x4
x22
−

2

x4

)
+
c413
2 d4

( x4
x1x3

−
x1
x3x4

−
x3
x4x1

)
+
c514
2 d4

( x4
x1x5

−
x1
x4x5

−
x5
x1x4

)
,

ric5 =
1

2x5
+
c523
2 d5

( x5
x2x3

−
x2
x3x5

−
x3
x2x5

)
+
c514
2 d5

( x5
x1x4

−
x1
x4x5

−
x4
x1x5

)
.

The non-zero structure constants have been computed in [24, Proposition 6]
and it is useful to recall them:

c211 = 12 , c
3
12 = 8 , c

4
13 = 4 , c

5
14 = 4/3 , c

4
22 = 4 , c

5
23 = 2 .

Moreover, d1 = 80, d2 = 60, d3 = 40, d4 = 20 and d5 = 8.

Case r = 6 : By [24] it is known that there is also only one flag manifold
M = G/K with G simple, b2(M) = 1 and r = 6. This is isometric to the
homogeneous space

M = G/K = E8 /U(1) ×SU(2) ×SU(3) ×SU(5),

which is determined by painting black the simple root α5 of E8, i.e., ΠM =
{α5}, with Dynk(α5) = 6. We know that d1 = 60, d2 = 60, d3 = 40, d4 = 30,
d5 = 12 and d6 = 10. Also, the values of the non-zero structure constants
have the form (see [24, Proposition 12])

c211 = 8 , c
3
12 = 6 , c

4
13 = 4 , c

5
14 = 2 , c

6
15 = 1 ,

c422 = 6 , c
5
23 = 2 , c

6
24 = 2 , c

6
33 = 2 ,

and the components rici of the Ricci tensor Ricg corresponding to g are given



124 s. anastassiou, i. chrysikos

by

ric1 =
1

2x1
−
c211
2 d1

x2
x12

+
c312
2 d1

( x1
x2x3

−
x2
x1x3

−
x3
x1x2

)
+
c413
2 d1

( x1
x3x4

−
x3
x1x4

−
x4
x1x3

) c514
2 d1

( x1
x4x5

−
x4
x1x5

−
x5
x1x4

)
+
c615
2 d1

( x1
x5x6

−
x5
x1x6

−
x6
x1x5

)
,

ric2 =
1

2x2
+
c211
4 d2

( x2
x12
−

2

x2

)
−
c422
2 d2

x4
x22

+
c312
2 d2

( x2
x1x3

−
x1
x2x3

−
x3
x2x1

)
+
c523
2 d2

( x2
x3x5

−
x3
x2x5

−
x5
x2x3

)
+
c624
2 d2

( x2
x4x6

−
x4
x2x6

−
x6
x2x4

)
,

ric3 =
1

2x3
−
c633
2 d3

x6
x32

+
c312
2 d3

( x3
x1x2

−
x2
x3x1

−
x1
x3x2

)
+
c413
2 d3

( x3
x1x4

−
x1
x3x4

−
x4
x1x3

)
+
c523
2 d3

( x3
x2x5

−
x2
x3x5

−
x5
x3x2

)
,

ric4 =
1

2x4
+
c422
4 d4

( x4
x22
−

2

x4

)
+
c413
2 d4

( x4
x1x3

−
x1
x3x4

−
x3
x4x1

)
+
c514
2 d4

( x4
x1x5

−
x1
x4x5

−
x5
x1x4

)
+
c624
2 d4

( x4
x2x6

−
x2
x4x6

−
x6
x2x4

)
,

ric5 =
1

2x5
+
c514
2 d5

( x5
x1x4

−
x1
x4x5

−
x4
x1x5

)
+
c523
2 d5

( x5
x2x3

−
x2
x3x5

−
x3
x2x5

)
+
c615
2 d5

( x5
x1x6

−
x1
x5x6

−
x6
x1x5

)
,

ric6 =
1

2x6
+
c633
4 d6

( x6
x32
−

2

x6

)
+
c615
2 d6

( x6
x1x5

−
x1
x5x6

−
x5
x1x6

)
+
c624
2 d6

( x6
x2x4

−
x2
x4x6

−
x4
x2x6

)
.

3.2. The main theorem. Let M = G/K be a flag manifold with
b2(M) = 1 and 2 ≤ r ≤ 6. The system of the homogeneous Ricci flow is
given by {

ẋi = −2xi · rici : i = 1, . . . ,r
}
. (3.4)



ancient solutions of the homogeneous ricci flow 125

For any case separately, a direct computation shows that system (3.4) does
not possess fixed points in M G ∼= Rr+, i.e., points (x1, . . . ,xr) ∈ R

r
r satisfying

the system {ẋi = ẋ2 = · · · = ẋr = 0}, which verifies Lemma 2.2.
We now agree on the following notation: We denote by ej a fixed point of

the homogeneous Ricci flow (HRF) at infinity of M G, as defined before. We
shall write N for the number of all such fixed points. We will also denote by
dunstbj (respectively d

stb
j ) the dimension of the unstable manifold (respectively

stable manifold) in M G (respectively in the infinity of M G), corresponding
to ej. In this terms we obtain the following

Theorem 3.1. Let M = G/K be a non-symmetric flag manifold with
b2(M) = 1, and let r (2 ≤ r ≤ 6) be the number of the corresponding isotropy
summands. Then, the following hold:

(1) The HRF admits exactly N fixed points ej at the infinity of M
G,

where for any coset G/K the number N is specified in Table 1. These fixed
points are in bijective correspondence with non-isometric invariant Einstein
metrics on M = G/K, and are specified explicitly in the proof.

(2) The dimensions of the stable/unstable manifolds corresponding to ej
are given in Table 1, where e1 represents the fixed point corresponding to the
unique invariant Kähler-Einstein metric on G/K. We see that

(i) For any M = G/K, the fixed point e1 has always an 1-dimensional
unstable manifold in M G, and a (r− 1)-dimensional stable manifold in
the infinity of M G.

(ii) Any other fixed point ek with 2 ≤ k ≤ N, has always a 2-dimensional
unstable manifold, while its stable manifold is (r − 2)-dimensional and
is contained in the infinity of M G, with the following three exceptions:

• the fixed point e5 for the space M∗ := E8 /U(1) × SU(3) × SO(10)
in the case with r = 4;

• the fixed point e5 in the case of M = E8 /U(1) × SU(4) × SU(5)
with r = 5;

• the fixed point e4 in the case of M = E8 /U(1) ×SU(2) ×SU(3) ×
SU(5) with r = 6.

These three exceptions have a 3-dimensional unstable manifold and a
(r − 3)-dimensional stable manifold. Similarly, the stable manifold for
these cases is contained entirely in the infinity of M G.



126 s. anastassiou, i. chrysikos

(3) Each unstable manifold of any ej, contains a non-collapsed ancient
solution, given by

gj :
(
−∞,

1

2λj

)
−→ M G , t 7−→ gj(t) = (1 − 2λjt) · ej , j = 1, . . . ,N ,

where λj is the Einstein constant of the corresponding Einstein metric gj(0),
j = 1, . . .N, on M = G/K (these are also specified below). All such solutions
gj(t) tend to 0 when t → T = 12λj > 0, and M = G/K shrinks to a point in
finite time.

(4) When r = 2, any other possible solution of the Ricci flow with initial
condition in M G has {0} as its ω–limit set.

conditions dunstbj d
stb
j

r for M = G/K N dunstb1 d
stb
1 (2 ≤ j ≤ N) (2 ≤ j ≤ N)

2 - 2 1 1 2 0

3 - 3 1 2 2 1

4 M � M∗ 3 1 3 2 2
4 M ∼= M∗ 5 1 3 (for j 6= 5) 2 (for j 6= 5) 2

(for j = 5) 3 (for j = 5) 1

5 - 6 1 4 (for j 6= 5) 2 (for j 6= 5) 3
(for j = 5) 3 (for j = 5) 2

6 - 5 1 5 (for j 6= 4) 2 (for j 6= 4) 4
(for j = 4) 3 (for j = 4) 3

Table 1: The exact number N of the fixed points ek of HRF at infinity of
M G for any non-symmetric flag space M = G/K with b2(M) = 1, and the
dimensions dstbk , d

unstb
k , for any 1 ≤ k ≤ N.

Proof. We split the proof in cases, depending on the possible values of r.
Case r = 2. In this case the system (3.4) reduces to:{

ẋ1 = −
(d1 + 4d2)x1 −d2x2

(d1 + 4d2)x1
, ẋ2 = −

8d2x
2
1 + d1x

2
2

2(d1 + 4d2)x
2
1

}
. (3.5)

To search for fixed points of the HRF at infinity of M G, we first multiply the
right-hand side of these equations with the positive factor 2(d1 + 4d2)x

2
1. This



ancient solutions of the homogeneous ricci flow 127

multiplication does not qualitatively affect system’s behaviour, and we result
with the following equivalent system:{

ẋ1 = RF1(x1,x2) = −2(d1 + 4d2)x21 + 2d2x1x2 ,

ẋ2 = RF2(x1,x2) = −
(
8d2x

2
1 + d1x

2
2

)}
,

where the right-hand side consists of two homogeneous polynomials of degree
2. This is the maximal degree d of the system, as discussed above. We can
therefore apply the Poincaré compactification procedure to study its behaviour
at infinity. For the formulas given in Example 2.14, i.e.,{

ẋ1 = x
2
2

(
−x1RF1(x1,x2) + RF2(x1,x2)

)
, ẋ2 = x

2
2

(
−x2RF1(x1,x2)

)}
,

we compute

RF1(x1,x2) = RF1
( 1
x2
,
x1
x2

)
= −

2(d1 + 4d2 −d2x1)
x22

,

RF2(x1,x2) = RF2
( 1
x2
,
x1
x2

)
= −

(8d2 + d1x
2
1)

x22
.

Hence finally we result with the system{
ẋ1 = −(x1 − 2)(−4d2 + d1x1 + 2d2x1) , ẋ2 = 2(d1 + 4d2 −d2x1)x2

}
,

which is the desired expression in the U chart. To study the behaviour of this
system at infinity of R2+, we set x2 = 0. Then, the second equation becomes
ẋ2 = 0, confirming that infinity remains invariant under the flow. To locate
fixed points, we solve the equation

ẋ1 = h(x1) = −(x1 − 2)(−4d2 + d1x1 + 2d2x1) = 0 .

We obtain two exactly solutions, namely xa1 = 2 and x
b
1 = 4d2/(d1 + 2d2),

and since h′(xa1) = −2d1, h
′(xb1) = 2d1, both fixed points are hyperbolic. In

particular, h′(xa1) < 0 and so x
a
1 is always an attracting node with eigenvalue

equal to −2d1. On the other hand, h′(xb1) > 0 and hence x
b
1 is a repelling

node, with eigenvalue equal to 2d1.
Recall now that the central projection F maps the sphere to the y1 = 1

plane. Therefore, the fixed points xa1, x
b
1 represent the points e1 = (1,x

a
1) and

e2 = (1,x
b
1) in R

2
+, which correspond to the invariant Einstein metrics

gKE = 1 ·B|m1 + 2 ·B|m2 , gE = 1 ·B|m1 + 4d2/(d1 + 2d2) ·B|m2 ,



128 s. anastassiou, i. chrysikos

respectively (see also [9]). To locate the ancient solutions, let ĝ(t) =
t
(
x1(0),x2(0)

)
be a straight line in M G. At the point (a,b), belonging in

the trace of ĝ(t), the vector normal to the straight line is the vector (−b,a).
The straight line ĝ(t) must be tangent to the vector field X(x1,x2) defined by
the homogeneous Ricci flow, hence the following equation should hold:

(−b,a) ·X(a,b) = 0 .

This gives us two solutions, namely (a1,b1) = (1, 2) and (a2,b2) = (1, 4d2/(d1+
2d2)), which define the lines γi(t) = t(ai,bi), i = 1, 2. The solutions g1(t),
g2(t) of system (3.5) corresponding to the lines γ1(t) and γ2(t), are determined
by the following equations

g1(t) = (1 − 2λ1t) · (1, 2) = (1 − 2λ1t) · e1 ,

g2(t) = (1 − 2λ2t) ·
(

1,
4d2

d1 + 2d2

)
= (1 − 2λ2t) · e2 ,

where λ1,λ2 are the Einstein constants of gKE = g1(0) and gE = g2(0),
respectively, given by

λ1 =
d1 + 2d2

2(d1 + 4d2)
, λ2 =

d21 + 6d1d2 + 4d
2
2

2(d1 + 2d2)(d1 + 4d2)
.

Obviously, these are both ancient solutions since are defined on the open set
(−∞, 1

2λi
) (see also Proposition 2.4), in particular gi(t) → 0 when t → 12λi ,

i = 1, 2. The assertion that gi(t) are non-collapsed follows by Corollary 2.8.
Based on the definitions of the central projections f and F given before,

we verify that

F(f(γ1(t))) = (1, 2) and F(f(γ2(t))) =
(

1,
4d2

d1 + 2d2

)
.

Thus, we have that

lim
t→ 1

2λi

γi(t) = 0 and lim
t→−∞

γi(t) = ei , ∀ i = 1, 2,

which proves the claim that these ancient solitons belong to the unstable
manifolds of the fixed points located at infinity.

To verify the claim in (4), we use the function V2(x1,x2) = x
2
1 + x

2
2 as a

Lyapunov function for the system (3.5). We compute that

dV

dt
(x1(t),x2(t)) = −

2d2x
2
1(4x1 + 3x2) + d1

(
2x31 + x

3
2

)
2(d1 + 4d2)x

2
1

,



ancient solutions of the homogeneous ricci flow 129

which for x1,x2 > 0 is always negative. Hence, if we denote as (ta, tb), ta, tb ∈
R∪{±∞} the domain of definition of the solution curve g(t) = (x1(t),x2(t)),
we conclude that g(t) tends to the origin, as t → tb. This completes the proof
for r = 2.

Case r = 3. In this case, to reduce system (3.4) in a polynomial dynami-
cal system, we must multiply with the positive factor 2d1d2(d1+4d2+9d3)x

2yz.
This gives

ẋ1 = RF1(x1,x2,x3) = −2d2x1
(
d1d3x

3
1 + d2d3x

3
1 −d1d3x1x

2
2 −d2d3x1x

2
2

+ d21x1x2x3 + 4d1d2x1x2x3 + 9d1d3x1x2x3 −d1d2x
2
2x3 − 2d1d3x

2
2x3

+ d2d3x
2
2x3 −d1d3x1x

2
3 −d2d3x1x

2
3

)
,

ẋ2 = RF2(x1,x2,x3) = −d1x2
(
− 2d1d3x31 − 2d2d3x

3
1 + 2d1d3x1x

2
2 (3.6)

+ 2d2d3x1x
2
2 + 8d

2
2x

2
1x3 − 4d1d3x

2
1x3 + 20d2d3x

2
1x3 + d1d2x

2
2x3

+ 2d1d3x
2
2x3 −d2d3x

2
2x3 − 2d1d3x1x

2
3 − 2d2d3x1x

2
3

)
ẋ3 = RF3(x1,x2,x3) = 2d1d2x1x3

(
d1x

2
1 + d2x

2
1 −d1x1x2 − 4d2x1x2

− 9d3x1x2 + d1x22 + d2x
2
2 −d1x

2
3 −d2x

2
3

)
.

The right-hand side of the system above consists of homogeneous polynomials
of degree 4. Let us apply the Poincaré compactification procedure and set
x3 = 0, to obtain the equations governing the behaviour of the system at
infinity of M G. By Example 2.14 we deduce that in the U-chart, these must
be given as follows:

ẋ1 = x1
(
2d21d3 + 4d1d2d3 + 2d

2
2d3 − 2d

2
1d3x

2
1 − 4d1d2d3x

2
1 − 2d

2
2d3x

2
1

− 8d1d22x2 + 4d
2
1d3x2 − 20d1d2d3x2 + 2d

2
1d2x1x2 + 8d1d

2
2x1x2

+ 18d1d2d3x1x2 −d21d2x
2
1x2 − 2d1d

2
2x

2
1x2 − 2d

2
1d3x

2
1x2

− 3d1d2d3x21x2 + 2d
2
2d3x

2
1x2 + 2d

2
1d3x

2
2 − 2d

2
2d3x

2
2

)
,

ẋ2 = −2d2x2
(
−d21 −d1d2 −d1d3 −d2d3 + d

2
1x1 + 4d1d2x1 + 9d1d3x1

−d21x
2
1 −d1d2x

2
1 + d1d3x

2
1 + d2d3x

2
1 −d

2
1x1x2 − 4d1d2x1x2

− 9d1d3x1x2 + d1d2x21x2 + 2d1d3x
2
1x2 −d2d3x

2
1x2 + d

2
1x

2
2 + d1d2x

2
2

+ d1d3x
2
2 + d2d3x

2
2

)
,

ẋ3 = 0 .



130 s. anastassiou, i. chrysikos

As before, the last equation confirms that infinity remains invariant under
the flow of the system. To locate fixed points, we have to solve the system of
equations {ẋ1 = ẋ2 = 0}. For this, it is sufficient to study each case separately
and replace the dimensions di. For any case we get exactly three fixed points,
which we list as follows:

• E8 /E6 ×SU(2) × U(1) : In this case, we have d1 = 108, d2 = 54, d3 = 4
and the fixed points at infinity are located at:

(2, 3) , (0.914286, 1.54198) , (1.0049, 0.129681) .

• E8 /SU(8)×U(1) : In this case, we have d1 = 112, d2 = 56, d3 = 16 and
the fixed points at infinity are located at:

(2, 3) , (0.717586, 1.25432) , (1.06853, 0.473177) .

• E7 /SU(5)×SU(3)×U(1) : In this case, we have d1 = 60, d2 = 30, d3 = 8
and the fixed points at infinity are located at:

(2, 3) , (0.733552, 1.27681) , (1.06029, 0.443559) .

• E7 /SU(6)×SU(2)×U(1) : In this case, we have d1 = 60, d2 = 30, d3 = 4
and the fixed points at infinity are located at:

(2, 3) , (0.85368, 1.45259) , (1.01573, 0.229231) .

• E6 /SU(3)×SU(3)×SU(2)×U(1) : In this case, we have d1 = 36, d2 = 18,
d3 = 4 and the fixed points at infinity are located at:

(2, 3) , (0.771752, 1.33186) , (1.04268, 0.373467) .

• F4 /SU(3)×SU(2)×U(1) : In this case, we have d1 = 24, d2 = 12, d3 = 4
and the fixed points at infinity are located at:

(2, 3) , (0.678535, 1.20122) , (1.09057, 0.546045) .

• G2 /U(2) : In this case, we have d1 = 4, d2 = 2, d3 = 4 and the fixed
points at infinity are located at:

(2, 3) , (1.67467, 2.05238) , (0.186894, 0.981478) .



ancient solutions of the homogeneous ricci flow 131

The projections of these solutions via F : U → R3≥, give us the points ei,
i = 1, 2, 3. These are the points obtained from the coordinates of the fixed
points given above, with one extra coordinate equal to 1, in the first entry.

According to (3.1), the indicated fixed point e1 = (1, 2, 3) corresponds to
the unique invariant Kähler-Einstein metric, and simple eigenvalue calcula-
tions show that it possesses a 2-dimensional stable manifold at the infinity
of M G, and a 1-dimensional unstable manifold, which is contained in M G.
All the other fixed points, correspond to non-Kähler non-isometric invariant
Einstein metrics (see [35, 7]), and have a 2-dimensional unstable manifold and
a 1-dimensional stable manifold, contained at the infinity of M G.

Let us now consider an invariant line of system (3.6), of the form

ĝ(t) = t
(
x1(0),x2(0),x3(0)

)
.

At a point (a,b,c) belonging to this line, the normal vectors are: (−b,a, 0),
(0,−c,b), thus ĝ(t) is tangent to the vector field X(x1,x2,x3) defined by the
homogeneous Ricci flow if

(−b,a, 0) ·X(a,b,c) = 0 , (0,−c,b) ·X(a,b,c) = 0 .

These equations possess three solutions, with respect to (a,b,c). One of them
is always (1, 2, 3), while the other two can be obtained after numerically solving
equations above for every value of d1,d2,d3. These solutions give us the three
non-collapsed ancient solutions gi(t) = (1 − 2λit) · ei with t ∈ (−∞, 12λi ), for
any i = 1, 2, 3, where the Einstein constants λi for the cases i = 2, 3 can be
computed easily by replacing the corresponding Einstein metrics g2(0), g3(0)
in the Ricci components, while for g1(t), λ1 is specified as follows:

λ1 =
d1 + 2d2 + 3d3

2d1 + 8d2 + 18d3
.

Moreover, it is easy to show, taking limits, that the α-limit set of each of these
solutions is one of the located fixed points at infinity of M G, while the ω-limit
set, for all of them, is {0}. Finally, for any i we have gi(t) → 0 as t → 12λi ,
where T = 1

2λi
depends on the dimensions di, i = 1, 2, 3, and so on the flag

manifold M = G/K, while again the assertion that gi(t) are non-collapsed,
for any i = 1, 2, 3, is a consequence of Corollary 2.8.

Case r = 4. The proof follows the lines of the previous cases r = 2, 3.
Thus, we avoid to present similar arguments. Consider for example the case
of the flag manifold M = G/K with b2(M) = 1 and r = 4, corresponding



132 s. anastassiou, i. chrysikos

to M∗ := E8 /U(1) × SU(3) × SO(10). After multiplication with the positive
term 60x21x

2
2x3x4, the system (3.4) of HRF equation turns into the following

polynomial system:

ẋ1 = x1x2
(
− 5x4x31 + x

2
4x1x2 −x

3
1x2 + 5x4x1x

2
2 − 60x4x1x2x3

+ 10x4x
2
2x3 + 5x4x1x

2
3 + x1x2x

2
3

)
,

ẋ2 = 2x2x4
(
4x31x2 − 4x1x

3
2 + x4x

2
1x3 − 22x

2
1x2x3 − 4x

3
2x3 + 4x1x2x

2
3

)
,

ẋ3 = 3x1x2x3
(
5x4x

2
1 + x

2
4x2 − 20x4x1x2 + x

2
1x2 + 5x4x

2
2

− 5x4x23 −x2x
2
3

)
,

ẋ4 = −2x1x4
(
8x24x

2
2 − 8x

2
1x

2
2 + 5x

2
4x1x3 + 20x1x

2
2x3 − 8x

2
2x

2
3

)
.

Note that the right-hand side of the equations above are all homogeneous
polynomials of degree 6. Hence we can apply Proposition 2.12 to study the
behaviour of this system at infinity of M G. In the U chart and by setting
x4 = 0, we obtain

ẋ1 = −x1
(
−x21 + x

2
1x

2
2 − 13x1x3 + 13x

3
1x3 + 44x1x2x3 − 60x

2
1x2x3

+ 18x31x2x3 − 3x1x
2
2x3 + x

2
1x

2
3 − 2x2x

2
3

)
,

ẋ2 = −2x1x2
(
− 2x1 + 2x1x22 − 10x3 + 30x1x3 − 5x

2
1x3 − 30x1x2x3

+ 5x21x2x3 + 10x
2
2x3 −x1x

2
3

)
,

ẋ3 = −x3
(
− 17x21 + 40x

2
1x2 − 15x

2
1x

2
2 − 5x1x3 + 5x

3
1x3 − 60x

2
1x2x3

+ 10x31x2x3 + 5x1x
2
2x3 + 17x

2
1x

2
3 + 10x2x

2
3

)
,

ẋ4 = 0 .

Again, last equation confirms that infinity remains invariant under the flow
of the system. Solving the system

{
ẋ1 = ẋ2 = ẋ3 = 0

}
, we get exactly three

fixed points, given by

(2, 3, 4) , (1.09705, 0.770347, 1.29696) , (1.15607, 1.01783, 0.214618)

(0.649612, 1.10943, 1.06103) , (0.763357, 1.00902, 0.191009) .

These solutions give us, through the central projection F of the U chart on R4≥,
the five fixed points e1,e2,e3,e4,e5. The fixed point e1 = (1, 2, 3, 4) corresponds
to the unique invariant Kähler-Einstein metric. Consider the Jacobian matrix



ancient solutions of the homogeneous ricci flow 133

corresponding to the system in U chart, that is

Jac :=




∂f1
∂x1

∂f1
∂x2

∂f1
∂x3

∂f2
∂x1

∂f2
∂x2

∂f2
∂x3

∂f3
∂x1

∂f3
∂x2

∂f3
∂x3


 ,

where

f1 := −x1
(
−x21 + x

2
1x

2
2 − 13x1x3 + 13x

3
1x3 + 44x1x2x3 − 60x

2
1x2x3

+ 18x31x2x3 − 3x1x
2
2x3 + x

2
1x

2
3 − 2x2x

2
3

)
,

f2 := −2x1x2
(
− 2x1 + 2x1x22 − 10x3 + 30x1x3 − 5x

2
1x3 − 30x1x2x3

+ 5x21x2x3 + 10x
2
2x3 −x1x

2
3

)
,

f3 := −x3
(
− 17x21 + 40x

2
1x2 − 15x

2
1x

2
2 − 5x1x3 + 5x

3
1x3 − 60x

2
1x2x3

+ 10x31x2x3 + 5x1x
2
2x3 + 17x

2
1x

2
3 + 10x2x

2
3

)
.

Calculating the Jacobian matrix at the corresponding fixed point (2,3,4), we
find it to be equal to:

Jac(2,3,4) =


 −1600 368 32960 −1248 192

320 1760 −1696


 .

This has 3 negative eigenvalues, thus we conclude that the fixed point e1
possesses a 3-dimensional stable manifold, contained in the infinity of M G,
while the straight line joining e1 with the origin of M

G corresponds to its
1-dimensional unstable manifold.

Computing the Jacobian matrix at the other fixed points and calculating
their eigenvalues, we conclude that all the other fixed points, correspond-
ing to non-Kähler, non-isometric invariant Einstein metrics (see [8]), have
a 2-dimensional stable manifold, located in the infinity of M G, and a 2-
dimensional unstable manifold, one direction of which is the straight line in
M G tending towards the origin, with the exception of fixed point e5. The
Jacobian matrix there becomes

Jac(0.763357,1.00902,0.191009) =


 1.33763 −1.13506 0.2514940.0597364 −4.80309 0.447799
−0.13328 −0.320307 3.98402


 ,



134 s. anastassiou, i. chrysikos

which has one negative eigenvalue and two positive ones. Thus, e5 possesses
a 1-dimensional stable manifold, contained in the infinity of M G, and a 3-
dimensional unstable manifold, one direction of which is the straight line em-
anating from e5 and tending to the origin of M

G. Invariant lines, correspond-
ing to non-collapsed ancient solutions gi(t) = (1 − λ2it) · ei, t ∈ (−∞, 12λi ),
can be found as in the previous cases, confirming once again that their ω-limit
sets are equal to {0}, while the α-limit set is the corresponding fixed point at
infinity of M G. This proves our claims. The Ricci flow equations on the rest
three homogeneous spaces of that type can be treated similarly.

Case r = 5. After multiplication with the positive term 60x21x
2
2x3x4x5,

system (3.4) turns into the following polynomial system:

ẋ1 = −x1x2
(
x31x2x3 −x1x2x3x

2
4 + 3x

3
1x2x5 − 3x1x2x

2
3x5

+ 6x31x4x5 − 6x1x
2
2x4x5 + +60x1x2x3x4x5 − 9x2

2x3x4x5

− 6x1x23x4x5 − 3x1x2x
2
4x5 −x1x2x3x

2
5

)
,

ẋ2 = 2x2x4
(
−x21x

3
2 + x

2
1x2x

2
3 + 4x

3
1x2x5 − 4x1x

3
2x5 − 24x

2
1x2x3x5

− 3x32x3x5 + 4x1x2x
2
3x5 + 2x

2
1x3x4x5 + x

2
1x2x

2
5

)
,

ẋ3 = 3x1x2x3
(
x1x

2
2x4 −x1x

2
3x4 + 2x

2
1x2x5 − 2x2x

2
3x5 + 4x

2
1x4x5

− 20x1x2x4x5 + 4x22x4x5 − 4x
2
3x4x5 + 2x2x

2
4x5 + x1x4x

2
5

)
,

ẋ4 = −2x1x4
(
− 2x21x

2
2x3 + 2x

2
2x3x

2
4 − 6x

2
1x

2
2x5 + 24x1x

2
2x3x5

− 6x22x
2
3x5 + 6x

2
2x

2
4x5 + 3x1x3x

2
4x5 − 2x

2
2x3x

2
5

)
,

ẋ5 = 5x1x2x5
(
2x21x2x3 + 3x1x

2
2x4 − 12x1x2x3x4 + 3x1x

2
3x4

+ 2x2x3x
2
4 − 2x2x3x

2
5 − 3x1x4x

2
5

)
.

Note that the right-hand side consists of homogeneous polynomials of degree
7 and we can use Proposition 2.12 to study the behaviour of this system at
infinity of M G. Using the expressions given above, the system at infinity,
written in the U chart and setting x5 = 0, reads as follows:

ẋ1 = −x1
(
−x21x2 + 2x

3
1x3 − 2x1x

2
2x3 + x

2
1x2x

2
3 − 3x

2
1x4 + 3x

2
1x

2
2x4

− 14x1x3x4 + 14x31x3x4 + 48x1x2x3x4 − 60x
2
1x2x3x4 + 15x

3
1x2x3x4

− 2x1x22x3x4 + 3x
2
1x

2
3x4 − 4x2x

2
3x4 + x

2
1x2x

2
4 − 2x1x3x

2
4

)
,



ancient solutions of the homogeneous ricci flow 135

ẋ2 = −x1x2
(
−x1x2 − 3x21x3 + 3x

2
2x3 + x1x2x

2
3 − 9x1x4 + 9x1x

2
2x4

− 18x3x4 + 60x1x3x4 − 6x21x3x4 − 60x1x2x3x4 + 9x
2
1x2x3x4

+ 18x22x3x4 − 3x1x
2
3x4 + x1x2x

2
4 − 3x3x

2
4

)
,

ẋ3 = −x3
(
− 5x21x2 + 5x

2
1x2x

2
3 − 15x

2
1x4 + 48x

2
1x2x4 − 9x

2
1x

2
2x4

− 6x1x3x4 + 6x31x3x4 − 60x
2
1x2x3x4 + 9x

3
1x2x3x4

+ 6x1x
2
2x3x4 + 15x

2
1x

2
3x4 + 6x2x

2
3x4 − 3x

2
1x2x

2
4

)
,

ẋ4 = −x1x4
(
− 11x1x2 − 15x21x3 + 60x1x2x3 − 15x

2
2x3 − 9x1x2x

2
3

− 3x1x4 + 3x1x22x4 − 6x3x4 + 6x
2
1x3x4 − 60x1x2x3x4 + 9x

2
1x2x3x4

+ 6x22x3x4 + 3x1x
2
3x4 + 11x1x2x

2
4 + 15x3x

2
4

)
,

ẋ5 = 0 .

Similarly with before, last equation confirms that infinity remains invariant
under the flow of the system. Now, by solving the system {ẋ1 = ẋ2 = ẋ3 =
ẋ4 = 0}, we get exactly six fixed points at the infinity of M G, given by:

(2, 3, 4, 5) , (0.599785, 1.08371, 0.901823, 1.22291) ,

(1.02137, 0.546007, 1.05352, 1.10879) , (1.08294, 1.04088, 0.532615, 1.10351) ,

(0.720713, 1.02546, 0.475234, 1.07095) , (1.03732, 1.04718, 1.03082, 0.29862) .

As before, these fixed points induce via the central projection F the explicit
presentations ei, i = 1, . . . , 6. The fixed point represented by e1 = (1, 2, 3, 4, 5)
corresponds to the unique invariant Kähler-Einstein metric on

M = E8 /U(1) × SU(4) × SU(5) and eigenvalues calculations show that it
possesses a 4-dimensional stable manifold, contained in the infinity of M G,
and a 1-dimensional unstable manifold which coincide with a straight tending
to the origin. The other three fixed points, corresponding to non-Kähler, non-
isometric invariant Einstein metrics (see [24]), have a 3-dimensional stable
manifold, located in the infinity of M G, and a 2-dimensional unstable mani-
fold, one direction of which is the straight line in M G tending to the origin,
with the exception of the fixed point e5, which possesses a 2-dimensional sta-
ble manifold, contained in the infinity of M G, and a 3-dimensional unstable
manifold. The ancient solutions gi(t) = (1 − 2λit) · ei, i = 1, . . . , 6 are defined
on the open interval (−∞, 1/2λi), where the corresponding Einstein constant



136 s. anastassiou, i. chrysikos

λi is given by

λ1 = 11/60 for e1 = (1, 2, 3, 4, 5) ,

λ2 = 0.37877 for e2 = (1, 0.599785, 1.08371, 0.901823, 1.22291) ,

λ3 = 0.365507 for e3 = (1, 1.02137, 0.546007, 1.05352, 1.10879) ,

λ4 = 0.339394 for e4 = (1, 1.08294, 1.04088, 0.532615, 1.10351) ,

λ5 = 0.386982 for e5 = (1, 0.720713, 1.02546, 0.475234, 1.07095) ,

λ6 = 0.337271 for e6 = (1, 1.03732, 1.04718, 1.03082, 0.29862) .

Using these constants and by taking limits, as above, we obtain the rest claims
for r = 5.

Case r = 6. To reduce the system (3.4) to a polynomial dynamical
system, a short computation shows that we must multiply with the positive
term 60x21x

2
2x

2
3x4x5x6. This gives the following:

ẋ1 = −x1x2x3
(
x31x2x3x4 −x1x2x3x4x

2
5 + 2x

3
1x2x3x6 − 2x1x2x3x

2
4x6

+ 4x31x2x5x6 − 4x1x2x
2
3x5x6 + 6x

3
1x4x5x6 − 6x1x

2
2x4x5x6

+ 60x1x2x3x4x5x6 − 8x22x3x4x5x6 − 6x1x
2
3x4x5x6 − 4x1x2x

2
4x5x6

− 2x1x2x3x25x6 −x1x2x3x4x
2
6

)
,

ẋ2 = 2x2x3
(
−x21x

3
2x3x5 + x

2
1x2x3x

2
4x5 −x

2
1x

3
2x4x6 + x

2
1x2x

2
3x4x6

+ 3x31x2x4x5x6 − 3x1x
3
2x4x5x6 − 26x

2
1x2x3x4x5x6 − 2x

3
2x3x4x5x6

+ 3x1x2x
2
3x4x5x6 + 3x

2
1x3x

2
4x5x6 + x

2
1x2x4x

2
5x6 + x

2
1x2x3x5x

2
6

)
,

ẋ3 = 3x1x2x3x6
(
x1x

2
2x3x4 −x1x

3
3x4 + 2x

2
1x2x3x5 − 2x2x

3
3x5

+ 3x21x3x4x5 − 20x1x2x3x4x5 + 3x
2
2x3x4x5 − 3x

3
3x4x5 + 2x2x3x

2
4x5

+ x1x3x4x
2
5 + x1x2x4x5x6

)
,

ẋ4 = 2x1x3x4
(
2x1x

3
2x3x5 − 2x1x2x3x

2
4x5 + 2x

2
1x

2
2x3x6 − 2x

2
2x3x

2
4x6

+ 4x21x
2
2x5x6 − 24x1x

2
2x3x5x6 + 4x

2
2x

2
3x5x6 − 4x

2
2x

2
4x5x6

− 3x1x3x24x5x6 + 2x
2
2x3x

2
5x6 + 2x1x2x3x5x

2
6

)
,



ancient solutions of the homogeneous ricci flow 137

ẋ5 = 5x1x2x3x5
(
x21x2x3x4 −x2x3x4x

2
5 + 2x

2
1x2x3x6 + 2x1x

2
2x4x6

− 12x1x2x3x4x6 + 2x1x23x4x6 + 2x2x3x
2
4x6 − 2x2x3x

2
5x6

− 2x1x4x25x6 + x2x3x4x
2
6

)
,

ẋ6 = 6x1x2x6
(
x21x2x

2
3x4 + 2x1x

2
2x

2
3x5 − 8x1x2x

2
3x4x5 + 2x1x

2
3x

2
4x5

+ x2x
2
3x4x

2
5 −x2x

2
3x4x

2
6 − 2x1x

2
3x5x

2
6 −x1x2x4x5x

2
6

)
.

Note that the right-hand side consists of homogeneous polynomials of degree
9. Hence again we can apply the Poincaré compactification procedure to study
the behaviour of this system at infinity of M G. Using the expressions given
above, the system at infinity, written in the U chart and setting x6 = 0, reads
as follows:

ẋ1 = −x1x2
(
−x21x2x3 + 2x

3
1x2x4 − 2x1x2x

2
3x4 + x

2
1x2x3x

2
4 − 2x

2
1x2x5

+ 2x31x3x5 − 2x1x
2
2x3x5 + 2x

2
1x2x

2
3x5 − 4x

2
1x4x5 + 4x

2
1x

2
2x4x5

− 12x1x3x4x5 + 12x31x3x4x5 + 52x1x2x3x4x5 − 60x
2
1x2x3x4x5

+ 12x31x2x3x4x5 + 4x
2
1x

2
3x4x5 − 6x2x

2
3x4x5 + 2x

2
1x2x

2
4x5

− 2x1x3x24x5 + x
2
1x2x3x

2
5 − 2x1x2x4x

2
5

)
,

ẋ2 = −x1x2
(
−x1x22x3 + x1x

2
2x3x

2
4 − 2x1x

2
2x5 − 3x

2
1x2x3x5 + 3x

3
2x3x5

+ 2x1x
2
2x

2
3x5 − 10x1x2x4x5 + 10x1x

3
2x4x5 −15x2x3x4x5 + 60x1x2x3x4x5

− 3x21x2x3x4x5 − 60x1x
2
2x3x4x5 + 8x

2
1x

2
2x3x4x5 + 15x

3
2x3x4x5

− 2x1x2x23x4x5 + 2x1x
2
2x

2
4x5 − 3x2x3x

2
4x5 + x1x

2
2x3x

2
5 − 3x1x3x4x

2
5

)
,

ẋ3 = −x2x3
(
−x21x2x3 − 4x

3
1x2x4 + 4x1x2x

2
3x4 + x

2
1x2x3x

2
4 − 6x

2
1x2x5

+ 6x21x2x
2
3x5 − 12x

2
1x4x5 + 48x

2
1x2x4x5 − 4x

2
1x

2
2x4x5 − 6x1x3x4x5

+ 6x31x3x4x5 − 60x
2
1x2x3x4x5 + 8x

3
1x2x3x4x5 + 6x1x

2
2x3x4x5

+ 12x21x
2
3x4x5 + 6x2x

2
3x4x5 − 2x

2
1x2x

2
4x5 + x

2
1x2x3x

2
5 − 4x1x2x4x

2
5

)
,

ẋ4 = −2x1x2x4
(
− 3x1x2x3 + 3x1x2x3x24 − 6x1x2x5 − 5x

2
1x3x5

+ 30x1x2x3x5 − 5x22x3x5 − 4x1x2x
2
3x5 − 2x1x4x5 + 2x1x

2
2x4x5

− 3x3x4x5 + 3x21x3x4x5 − 30x1x2x3x4x5 + 4x
2
1x2x3x4x5 + 3x

2
2x3x4x5

+ 2x1x
2
3x4x5 + 6x1x2x

2
4x5 + 5x3x

2
4x5 − 2x1x2x3x

2
5

)
,



138 s. anastassiou, i. chrysikos

ẋ5 = −x1x5
(
− 7x1x22x3 − 12x

2
1x

2
2x4 + 48x1x

2
2x3x4 − 12x

2
2x

2
3x4

− 5x1x22x3x
2
4 − 2x1x

2
2x5 + 2x1x

2
2x

2
3x5 − 4x1x2x4x5 + 4x1x

3
2x4x5

− 6x2x3x4x5 + 6x21x2x3x4x5 − 60x1x
2
2x3x4x5 + 8x

2
1x

2
2x3x4x5

+ 6x32x3x4x5 + 4x1x2x
2
3x4x5 + 2x1x

2
2x

2
4x5 + 7x1x

2
2x3x

2
5

+ 12x22x4x
2
5 + 6x1x3x4x

2
5

)
,

ẋ6 = 0 .

By the last equation one deduces that the infinity of M G remains invariant
under the flow of the system. Solving now the system {ẋ1 = ẋ2 = ẋ3 = ẋ4 =
ẋ5 = 0}, we get exactly five fixed points given by:

(2, 3, 4, 5, 6) , (0.823084, 1.1467, 1.17377, 1.42664, 1.46519) ,

(0.986536, 0.636844, 1.06853, 1.13323, 0.921127) ,

(0.90422, 0.778283, 0.927483, 1.03408, 0.359949) ,

(0.954875, 0.965321, 1.00534, 0.290091, 1.01965) .

These solutions, projected through the central projection F, induce the fixed
points ei at infinity of M

G, namely

e1 = (1, 2, 3, 4, 5, 6) ,

e2 = (1, 0.823084, 1.1467, 1.17377, 1.42664, 1.46519) ,

e3 = (1, 0.986536, 0.636844, 1.06853, 1.13323, 0.921127) ,

e4 = (1, 0.90422, 0.778283, 0.927483, 1.03408, 0.359949) ,

e5 = (1, 0.954875, 0.965321, 1.00534, 0.290091, 1.01965) .

These points are in bijective correspondence with non-isometric invariant Ein-
stein metrics on M = G/K = E8 /U(1) × SU(2) × SU(3) × SU(5) (see [24]),
and determine the non-collapsed ancient solutions gi(t) = (1−2λit)·ei, where
the corresponding Einstein constant λi has the form

λ1 = 3/20 , λ2 = 0.313933 , λ3 = 0.348603 , λ4 = 0.367518 , λ5 = 0.349296 .

All the other conclusions are obtained in an analogous way as before. This
completes the proof.

Remark 3.2. For r ≥ 3 a statement of ω-limits of general solutions of (3.4),
as in the assertion (4) of Theorem 3.1, is not presented, since for these cases
a Lyapunov function is hard to be computed.



ancient solutions of the homogeneous ricci flow 139

Let us now take a view of the limit behaviour of the scalar curvature for the
solutions gi(t) given in Theorem 3.1, and verify the statements of Proposition
2.4. We do this by treating examples with r = 2, 3.

Example 3.3. Let M = G/K be a flag manifold with r = 2. Then, an
application of (3.2) shows that both Scal(gi(t)) are positive hyperbolas, given
by

Scal(g1(t)) =
(d1 + d2)(d1 + 2d2)

2(d1 + 4d2) − 2t(d1 + 2d2)
,

Scal(g2(t)) =
(d1 + d2)(d

2
1 + 6d1d2 + 4d

2
2)

2
(
(d21 + 6d1d2 + 8d

2
2) − t(d

2
1 + 6d1d2 + 4d

2
2)
) ,

respectively. Therefore, they both are increasing in
(
−∞, 1

2λi

)
, which is the

open interval which are defined, i.e., Scal′(g1(t)) and Scal
′(g2(t)) are strictly

positive. The limit of Scal(gi(t)) as t → 12λi must be considered only from
below, and it is direct to check that

lim
t→ 1

2λi

Scal(gi(t)) = +∞ , lim
t→−∞

Scal(gi(t)) = 0 , ∀ i = 1, 2 .

Hence, Scal(gi(t)) > 0, for any t ∈
(
−∞, 1

2λi

)
and for any i = 1, 2, as it

should be for ancient solutions in combination with the non-existence of Ricci
flat metrics ([37, 38]). Let us present the graphs of Scal(g1(t)) and Scal(g2(t))
for the flag spaces G2 /U(2) and F4 /Sp(3) × U(1), both with r = 2. We
list all the related details below, together with the graphs of Scal(gi(t)) for
the corresponding intervals of definition

(
−∞, 1

2λi

)
(see Figure 1 and 2, for

i = 1, 2, respectively).

M = G/K d1 d2 1/2λ1 1/2λ2 Scal(g1(t)) Scal(g2(t))

G2 /U(2) 8 2 4/3 12/11
120

32 − 24t
1760

384 − 362t

F4 /Sp(3) ×U(1) 28 2 9/8 72/71
960

72 − 64t
34080

2304 − 2272t

The graphs now of the ricci components of the solutions gj(t), which we
denote by

ricij(t) := rici(gj(t)) , 1 ≤ i ≤ r , 1 ≤ j ≤ N ,



140 s. anastassiou, i. chrysikos

where again N represents the number of fixed points of HRF at infinity of
M G, are very similar. For example, for any flag manifold M = G/K with
r = 2 and for the first ancient solution g1(t) passing through the invariant
Kähler-Einstein metric g1(0) = gKE, we compute

ric11(t) = ric1(g1(t)) = ric21(t) = ric2(g1(t))

=
d1 + 2d2

2(d1 + 4d2) − 2(d1 + 2d2)t
.

For the solution g2(t) passing from the non-Kähler invariant Einstein metric
g2(0) = gE we obtain

ric12(t) = ric1(g2(t)) = ric22(t) = ric2(g2(t))

=
d21 + 6d1d2 + 4d

2
2

2(d21 + 6d1d2 + 8d
2
2) − 2(4d

2
2 + d

2
1 + 6d1d2)t

.

Note that the equalities ric11(t) = ric21(t) and ric12(t) = ric22(t) occur since
gi(t) (i = 1, 2) are both 1-parameter families of invariant Einstein metrics on
M = G/K (as we mentioned in Section 2). For instance, for G2 /U2 with
r = 2, the above formulas reduce to

ric11(t) = ric21(t) =
12

32 − 24t
, ric12(t) = ric22(t) =

88

192 − 176t
,

with ric11(0) = ric21(0) = λ1 = 3/8 and ric12(0) = ric22(0) = λ2 = 11/24,
respectively. The corresponding graphs are given in Figure 3.

t=
4

3

- 10 - 8 - 6 - 4 - 2
t

1

2

3

4

5

Scal(g1(t))

t=
9

8

- 10 - 8 - 6 - 4 - 2
t

5

10

15

Scal(g1(t))

Figure 1: The graph of Scal(g1(t)) for G2 /U(2) (left) and F4 /Sp(3) × U(1)
(right).



ancient solutions of the homogeneous ricci flow 141

t=
12

11

- 10 - 8 - 6 - 4 - 2
t

1

2

3

4

5

Scal(g2(t))

t=
72

71

- 10 - 8 - 6 - 4 - 2
t

5

10

15

Scal(g2(t))

Figure 2: The graph of Scal(g2(t)) for G2 /U(2) (left) and F4 /Sp(3) × U(1)
(right).

t=
4

3

ric11(0)= �1

- 10 - 8 - 6 - 4 - 2
t

0.1

0.2

0.3

0.4

0.5

ric11(t)

t=
12

11

ric22(0)= �2

- 10 - 8 - 6 - 4 - 2
t

0.1

0.2

0.3

0.4

0.5

ric22(t)

Figure 3: The graphs of ric11(t) = ric21(t) and ric12(t) = ric22(t) for G2 /U(2)
with m = m1 ⊕m2.

Example 3.4. Let M = G/K be a flag manifold with b2(M) = 1 and
r = 3. Let us describe the asymptotic properties of the scalar curvature
Scal(g1(t)), related to the solution g1(t) = (1 − 2λ1t) · e1 only, where e1 =
(1, 2, 3) is the fixed point corresponding to the invariant Kähler-Einstein met-
ric g1(0) ≡ gKE. By (3.3) we see that Scal(g1(t)) is the positive hyperbola
given by

Scal(g1(t)) =
(d1 + d2 + d3)(d1 + 2d2 + 3d3)

2(d1 + 4d2 + 9d3) + 2t(d1 + 4d2 + 6d3)
.

Thus, Scal(g1(t)) increases on the open interval (−∞, 12λ1 ), where g1(t) is
dedined. Note that the value

Scal(g(0)) =
(d1 + d2 + d3)(d1 + 2d2 + 3d3)

2(d1 + 4d2 + 9d3)

equals to the scalar curvature of the Kähler-Einstein metric g1(0). The limit
of Scal(g1(t)) as t → 12λ1 must be considered only from below, and it follows



142 s. anastassiou, i. chrysikos

that
lim
t→ 1

2λ1

Scal(g1(t)) = +∞ , lim
t→−∞

Scal(g1(t)) = 0 .

For example for G2 /U2 and r = 3, we compute λ1 = 5/24, so

Scal(g1(t)) =
25

12 − 5t
, t ∈

(
−∞,

12

5

)
, Scal(g1(0)) =

25

12

and the graph of Scal(g1(t)) is given by

Scal(g1(0))

t=
12

5

- 10 - 8 - 6 - 4 - 2 2
t

1

2

3

4

5

Scal(g1(t))

Figure 4: The graph of Scal(g1(t)) for G2 /U(2) with m = m1 ⊕m2 ⊕m3.

Acknowledgements

The authors are grateful to C. Böhm, R. Lafuente and Y. Sakane
for insightful comments. They also thank J. Lauret and C. Will for
their remarks and pointing out a mistake in the main Theorem 3.1 in
a previous draft of this manuscript. I. Chrysikos acknowledges support
by Czech Science Foundation, via the project GAČR no. 19-14466Y.
S. Anastassiou thanks the University of Hradec Králové for hospitality
along a research stay in September 2019.

References

[1] N.A. Abiev, Y.G. Nikonorov, The evolution of positively curved invariant
Riemannian metrics on the Wallach spaces under the Ricci flow, Ann. Global
Anal. Geom. 50 (1) (2016), 65 – 84.

[2] D.V. Alekseevksy, I. Chrysikos, Spin structures on compact homo-
geneous pseudo-Riemannian manifolds, Transform. Groups, 24 (3), (2019),
659 – 689.



ancient solutions of the homogeneous ricci flow 143

[3] D.V. Alekseevsky, B.N. Kimel’fel’d, Structure of homogeneous Rie-
mannian spaces with zero Ricci curvature, Funct. Anal. Appl. 9 (2) (1975),
5 – 11.

[4] D.V. Alekseevsky, A. Perelomov, Invariant Kähler-Einstein metrics on
compact homogeneous spaces, Funct. Anal. Appl. 20 (3) (1986), 171 – 182.

[5] D.V. Alekseevsky, A.M. Vinogradov, V.V. Lychagin, Basic Ideas
and Concepts of Differential Geometry, in “ Geometry I ”, Encyclopaedia
Math. Sci., 28, Springer, Berlin, 1991, 1 – 264.

[6] M. Alexandrino, R.G. Bettiol, “ Lie Groups and Geometric Aspects of
Isometric Actions ”, Springer, Cham, Switzerland, 2015.

[7] S. Anastassiou, I. Chrysikos, The Ricci flow approach to homogeneous
Einstein metrics on flag manifolds, J. Geom. Phys. 61 (2011), 1587 – 1600.

[8] A. Arvanitoyeorgos, I. Chrysikos, Invariant Einstein metrics on flag
manifolds with four isotropy summands, Ann. Global Anal. Geom. 37 (2)
(2010), 185 – 219.

[9] A. Arvanitoyeorgos, I. Chrysikos, Invariant Einstein metrics on gener-
alized flag manifolds with two isotropy summands, J. Aust. Math. Soc. 90 (2)
(2011), 237 – 251.

[10] C. Böhm, Homogeneous Einstein metrics and simplicial complexes, J. Differ-
ential Geom. 67 (1) (2004), 79 – 165.

[11] C. Böhm, On the long time behavior of homogeneous Ricci flows, Comment.
Math. Helv. 90 (2015), 543 – 571.

[12] C. Böhm, R.A. Lafuente, Immortal homogeneous Ricci flows, Invent.
Math. 212 (2018), 461 – 529.

[13] C. Böhm, R.A. Lafuente, M. Simon, Optimal curvature estimates for
homogeneous Ricci flows, Int. Math. Res. Not. IMRN 2019 (14) (2019),
4431 – 4468.

[14] C. Böhm, R.A. Lafuente, Homogeneous Einstein metrics on Euclidean
spaces are Einstein solvmanifolds. arXiv:1811.12594

[15] C. Böhm, M. Wang, W. Ziller, A variational approach for com-
pact homogeneous Einstein manifolds, Geom. Funct. Anal. 14 (4) (2004),
681 – 733.

[16] C. Böhm, B. Wilking, Nonnegatively curved manifolds with finite funda-
mental groups admit metrics with positive Ricci curvature, Geom. Funct.
Anal. 17 (2007), 665 – 681.

[17] A. Borel, F. Hirzebruch, Characteristics classes and homogeneous spaces
I, Amer. J. Math. 80 (1958), 458 – 538.

[18] M. Buzano, Ricci flow on homogeneous spaces with two isotropy summands,
Ann. Global Anal. Geom. 45 (2014), 25 – 45.

[19] J. Cheeger, D.G. Ebin, “ Comparison Theorems in Riemannian Geome-
try ”, North-Holland Publishing Co., Amsterdam-Oxford; American Elsevier
Publishing Co., Inc., New York, 1975.

[20] B.L. Chen, Strong uniqueness of the Ricci flow, J. Differential Geom. 82 (2)
(2009), 363 – 382.

[21] B.L. Chen, X.P. Zhu, Uniqueness of the Ricci flow on complete noncompact

arXiv:1811.12594


144 s. anastassiou, i. chrysikos

manifolds, J. Differential Geom. 74 (2006), 119 – 154.

[22] B. Chow, P. Lu, L. Ni, “ Hamilton’s Ricci Flow ”, Graduate Studies in
Mathematics, 77; American Mathematical Society, Providence, RI; Science
Press Beijing, New York, 2006.

[23] I. Chrysikos, Flag manifolds, symmetric t-triples and Einstein metrics, Dif-
ferential Geom. Appl. 30 (2012), 642 – 659.

[24] I. Chrysikos, Y. Sakane, The classification of homogeneous Einstein met-
rics on flag manifolds with b2(M) = 1, Bull. Sci. Math. 138 (2014), 665 – 692.

[25] J. Enders, R. Müller, P.M. Topping, On type-I singularities in Ricci
flow, Comm. Anal. Geom. 19 (5) (2011), 905 – 922.

[26] W. Fulton, “ Introduction to Toric Varieties ”, Princeton University Press,
Princeton, NJ, 1993.

[27] E.A. Gonzalez-Velasco, Generic properties of polynomial vector fields at
infinity, Trans. Amer. Math. Soc. 143 (1969), 201 – 222.

[28] L. Grama, R.M. Martins, The Ricci flow of left invariant metrics on full
flag manifold SU(3)/T from a dynamical systems point of view, Bull. Sci.
Math. 133 (2009), 463 – 469.

[29] L. Grama, R.M. Martins, Global behavior of the Ricci flow on generalized
flag manifolds with two isotropy summands, Indag. Math. (N.S.) 23 (1-2)
(2012), 95 – 104.

[30] L. Grama, R.M. Martins, M. Patrão, L. Seco, L.D. Sperança,
Global dynamics of the Ricci flow on flag manifolds with three isotropy sum-
mands. arXiv:2004.01511

[31] M. Graev, On the number of invariant Eistein metrics on a compact homo-
geneous space, Newton polytopes and contraction of Lie algebras, Int. J.
Geom. Methods Mod. Phys. 3 (5-6) (2006), 1047 – 1075.

[32] M. Graev, On the compactness of the set of invariant Einstein metrics, Ann.
Global Anal. Geom. 44 (2013), 471 – 500.

[33] R. Hamilton, Three-manifolds with positive Ricci curvature, J. Differential
Geometry 17 (1982), 255 – 306.

[34] M. Kerr, New examples of of homogeneous Einstein metrics, Michigan Math.
J. 45 (1998), 115 – 134.

[35] M. Kimura, Homogeneous Einstein metrics on certain Kähler C-spaces, in
“ Recent Topics in Differential and Analytic Geometry ”, Academic Press,
Boston, MA, 1990, 303 – 320.

[36] S. Kobayashi, K. Nomizu, “ Foundations of Differential Geometry, Vol. II ”,
John Wiley & Sons, Inc., New York-London-Sydney, 1969.

[37] B. Kotschwar, Backwards uniqueness for the Ricci flow, Int. Math. Res.
Not. IMRN 2010 21 (2010), 4064 – 4097.

[38] R.A. Lafuente, Scalar curvature behavior of homogeneous Ricci flows, J.
Geom. Anal. 25 (2015), 2313 – 2322.

[39] R.A. Lafuente, J. Lauret, On homogeneous Ricci solitons, Q. J. Math.
65 (2) (2014), 399 – 419.

[40] R.A. Lafuente, J. Lauret, Structure of homogeneous Ricci solitons and

arXiv:2004.01511


ancient solutions of the homogeneous ricci flow 145

the Alekseevskii conjecture, J. Differential Geom. 98 (2) (2014), 315 – 347.

[41] J. Lauret, The Ricci flow for simply connected nilmanifolds, Comm. Anal.
Geom. 19 (5) (2011), 831 – 854.

[42] J. Lauret, Ricci soliton solvmanifolds, J. Reine Angew. Math. 650 (2011),
1 – 21.

[43] J. Lauret, Ricci flow of homogeneous manifolds, Math. Z. 274 (2013),
373 – 403.

[44] P. Lu, Y.K. Wang, Ancient solutions of the Ricci flow on bundles, Adv.
Math. 318 (2017), 411 – 456.

[45] Yu.G. Nikonorov, E.D. Rodionov, V.V. Slavskii, Geometry of ho-
mogeneous Riemannian manifolds, J. Math. Sci. (N.Y.) 146 (6) (2007),
6313 – 6390.

[46] J.S. Park, Y. Sakane, Invariant Einstein metrics on certain homogeneous
spaces, Tokyo J. Math. 20 (1) (1997), 51 – 61.

[47] T.L. Payne, The Ricci flow for nilmanifolds, J. Mod. Dyn. 4 (1) (2010),
65 – 90.

[48] G. Perelman, The entropy formula for the Ricci flow and its geometric
applications. arXiv:math/0211159

[49] G. Perelman, Ricci flow with surgery on three-manifolds.
arXiv:math/0303109v1

[50] F. Panelli, F. Podestà, On the first eigenvalue of invariant Kähler metrics,
Math. Z. 281 (2015), 471 – 482.

[51] H. Poincaré, Sur l’ integration des équations différentielles du premier ordre
et du premier degré I, Rend. Circ. Mat. Palermo 5 (1891), 161 – 191.

[52] A. Pulemotov, Y.A. Rubinstein, Ricci iteration on homogeneous spaces,
Trans. Amer. Math. Soc. 371 (2019), 6257 – 6287.

[53] Y. Sakane, Homogeneous Einstein metrics on flag manifolds, Lobachevskii J.
Math. 4 (1999), 71 – 87.

[54] C. Vidal, P. Gomez, An extension of the Poincare compactification and a
geometric interpretation, Proyecciones 22 (3) (2003), 161 – 180.

[55] M. Wang, W. Ziller, Existence and nonexcistence of homogeneous Einstein
metrics, Invent. Math. 84 (1986), 177 – 194.

arXiv:math/0211159
arXiv:math/0303109v1

	Preliminaries
	Homogeneous Ricci flow.
	Flag manifolds.

	The homogeneous Ricci flow on flag manifolds
	Invariant ancient solutions.
	The Poincaré compactification procedure.

	Global study of HRF on flag spaces M=G/K with b2(M)=1
	Flag manifolds M=G/K with b2(M)=1.
	The main theorem.