Ancient solutions of the homogeneous Ricci flow on flag manifolds
aa r X i v : . [ m a t h . DG ] S e p ANCIENT SOLUTIONS OF THE HOMOGENOUS RICCI FLOW ON FLAGMANIFOLDS
STAVROS ANASTASSIOU AND IOANNIS CHRYSIKOS
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 solutionspass through an invariant Einstein metric on M , and by [B¨oLS17] they must develop a Type Isingularity in their extinction finite time, and also to the past. To illustrate the situation weengage ourselves with the global study of the dynamical system induced by the unnormalized Ricciflow on any flag manifold M “ G { K with second Betti number b p M q “
1, for a generic initialinvariant metric. We describe the corresponding dynamical systems and present non-collapsedancient solutions, whose α -limit set consists of fixed points at infinity of M G . We show that thesefixed points correspond to invariant Einstein metrics and based on the Poincar´e compactificationmethod, we study their stability properties, illuminating thus the structure of the system’s phasespace. Contents
Introduction 11. Preliminaries 41.1. Homogeneous Ricci flow 41.2. Flag manifolds 52. The homogeneous Ricci flow on flag manifolds 82.1. Invariant ancient solutions 82.2. The Poincar´e compactification procedure 123. Global study of HRF on flag spaces M “ G { K with b p M q “ M “ G { K with b p M q “ Introduction
Given a Riemannian manifold p M n , g q , recall that the unnormalized Ricci flow is the geometricflow defined by(0.1) B g p t qB t “ ´ Ric g p t q , g p q “ g , where Ric g p t q denotes the Ricci tensor of the one-parameter family g p t q . The above system consistsof non-linear second order partial differential equations on the open convex cone M of Riemannianmetrics on M . A smooth family t g p t q : t P r , T q Ă R u P M defined for some 0 ă T ď 8 , is said tobe a solution of the Ricci flow with initial metric g , if it satisfies the system (0.1) for any x P M and t P r , T q . The Ricci flow was introduced in the celebrated work of Hamilton [H82] and nowadaysis the essential tool in the proof of the famous Poincar´e conjecture and
Thurston’s geometrizaionconjecture , due to the seminal works [P02, P03] of G. Perelman.
In general, and since a system of partial differential equations is involved, it is hard to produceexplicit 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 [ChZ06]),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 p t q of (0.1) is also invariant. As a result, insome cases it is possible to solve the system explicitly and proceed to a study of their asymptoticproperties, or even specify analytical properties related to different type of singularities and deducecurvature estimates, see [B¨oW07, GM09, AnC11, GM12, Buz14, Laf15, B¨o15, B¨oLS17, AbN16,GMPSS20], and the articles quoted therein. Especially for the non-compact case, note that duringthe last decade the Ricci flow for homogeneous , or cohomogeneity-one metrics, together with theso-called bracket flow play a key role in the study of the Alekseevsky conjecture , see [Pn10, L11a,L11b, L13, LL14a, LL14b, B¨oL18a, B¨oL18b].In this work we examine the Ricci flow, on compact homogeneous spaces with simple spectrumof isotropy representation , in terms of Graev [Gr06, Gr13], or of monotypic isotropy representation ,in terms of Buzano [Buz14], or Pulemotov and Rubinstein [PR19]. Nowadays, such spaces areof special interest due to their rich applications in the theory of homogeneous Einstein metrics,prescribed Ricci curvature, Ricci iteration, Ricci flow and other (see [A ℓ P86, S99, B¨o04, B¨oWZ04,Gr06, AC10, AC11, Buz14, Gr13, CS14, B¨o15, PP15, PR19, GMPSS20]). Here, we focus on flagmanifolds M “ G { K of a compact simple Lie group G and examine the dynamical system inducedby the vector field corresponding to the homogeneous Ricci flow equation. On such cosets (even for G semisimple), the homogeneous Ricci flow cannot posses fixed (stationary) points, since by thetheorem of Alekseevsky and Kimel’fel’d [A ℓ K75] invariant Ricci flat metrics must be flat and sothey cannot exist. However, by a result of Lafuente [Laf15, Cor. 4.3], and since any flag manifold M “ G { K is compact and simply connected, and carries invariant Einstein metrics (e.g. invariantK¨ahler-Einstein metrics always exist), it follows that the homogeneous Ricci flow on M “ G { K must admit ancient invariant solutions with finite extinction time (see also the work of B¨ohm[B¨o15]).To be more specific, recall that a solution g p t q of the Ricci flow is called ancient if it has asinterval of the definition the open set p´8 , T q , for some T ă 8 . Such solutions are important,because they arise as limits of blow ups of singular solutions to the Ricci flow near finite timesingularities, see [EMT11]. For our case, by [Laf15], 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 flowsolution g p t q with initial condition g p q “ g has an interval of definition of the form p t a , T q , with ´8 ď t a ă T ă 8 . Indeed, for any flag space below we will provide explicit solutions ofthis type, which are ancient; They arise by using an invariant K¨ahler-Einstein metric, which alwaysexists, or any other possible existent invariant Einstein metric g , and they are defined on openintervals of the form p´8 , T q , where T “ λ “ n Scal p g q ´ , with λ “ Scal p g q n ą t Ñ T , in the sense that g p t q Ñ
0, i.e. they tend to 0. As ancientsolutions, they have positive scalar curvature
Scal p g p t qq ([Ch09]) and the asymptotic behaviour of Scal p g p t qq , as least in our case, can be easily treated, see Proposition 2.4 (and Examples 3.3, 3.4),which forms a specification of [Laf15, Thm. 1.1] on flag manifolds. By [B¨oLS17] it also follows thatthese ancient solutions develop a Type I singularity. This means (see [EMT11, Buz14, B¨o15, LW17])lim t Ñ´8 ´ | t | ¨ sup x P M } Rm p g p t qq} g p t q p x, t q ¯ ă 8 , NCIENT SOLUTIONS OF THE HOMOGENEOUS RICCI FLOW 3 where Rm p g p t qq denotes the curvature tensor of p M “ G { K, g p t qq , or equivalently that there is aconstant 0 ă C g ă 8 such that p T ´ t q ¨ sup M } Rm p g p t qq} g p t q ď C g , for any t P p´8 , T “ λ q .Finally, by [B¨oLS17] we also deduce that the given ancient solutions are non-collapsed (Corollary2.8). In this point we should mention that not any compact homogeneous space M “ G { K of acompact (semi)simple Lie group G admits Einstein metrics (see for instance [WZ86, PS97]). So,even assuming that the universal covering of M n “ G { K is not diffeomorphic to R n (which for thecompact case is equivalent to say that M n is not a n -torus), the predicted solutions of Lafuentecan be in general hard to be specified. However, as we explained above, for flag manifolds, thereare always solutions of this type which can be described explicitly. For an illustration of theaformentioned results, we restrict our attention to the class of flag manifolds M “ G { K withsecond Betti number equal to one, i.e. b p M q “ M “ G { K of a compact simple Lie group G , the symmetric space ofinvariant metrics M G is the phase space of the homogeneous Ricci flow and is flat, i.e. M G – R r ` for some r ě
1. Therefore, the dynamical system of the homogeneous Ricci flow can be convertedto a qualitative equivalent dynamical system of homogeneous polynomial equations and the well-known
Poincar´e compactification ([P1891]) strongly applies. The main idea back of this methodis to identify R n with the northern and southern hemispheres through central projections, andthen extend X to a vector field p p X q on S n (see Section 2.2). Here, for any (non-symmetric) flagmanifold M “ G { K of a compact simple Lie group G with b p M q “
1, we present the global studyof the dynamical system induced by the vector field corresponding to the unnormalized Ricci flowfor an initial invariant metric, which is generic. In particular, another contribution of this work isthe description via the Poincar´e compactification method, of the fixed points of the homogeneousRicci flow at the so-called infinity of M G (see Definition 2.11 and Remark 2.13). Based on thismethod we can study the stability properties of such fixed points, which we prove that are inbijective correspondence with the existent invariant Einstein metrics on M “ G { K , and moreoverthat coincide with the α -limit set of an invariant line, i.e. a solution of the homogeneous Ricci flowwhich has as trace a line of M G . It turns out that such solutions are ancient and non-collapsed,and develop Type I singularities. Note that through the compactification procedure of Poincar´e,we are able to distinguish the unique invariant K¨ahler-Einstein metric from the other invariantEinstein metrics in terms of (un)stable manifolds, while for the case r “ ω -limitof any solution of the homogeneous Ricci flow, for details see Theorem 3.1. Since we are interestedin the unnormalized Ricci flow on flag manifolds with b p M q “
1, we should finally mention thatthis dynamical system has been very recently examined by [GMPSS20], for flag spaces with threeisotropy summands (however via a different method of the Poincar´e compactification), while arelated study of certain examples of flag manifolds with two isotropy is given in [GM12]. Notefinally that for flag manifolds with r “
2, this work is complementary to [Buz14], in the sensethat there were studied homogeneous ancient solutions on compact homogeneous spaces with twoisotropy summands; However, the specific class of flag manifolds with r “ M “ G { K of a compact simple Lie group G with b p M q “
1, where a proof of Theorem 3.1 is presented.
STAVROS ANASTASSIOU AND IOANNIS CHRYSIKOS
Acknowledgements:
I.C. is grateful to Y. Sakane (Osaka) for helpful discussions. He acknowl-edges support by Czech Science Foundation, via the project GA ˇCR no. 19-14466Y. S. A. thanksthe University of Hradec Kr´alov´e for hospitality along a research stay in September 2019.1.
Preliminaries
We begin by recalling preliminaries of the homogeneous Ricci flow. After that we will refreshuseful notions of the structure and geometry of generalized flag manifolds.1.1.
Homogeneous Ricci flow.
Recall that a homogeneous Riemannian manifold is a homoge-neous space M “ G { K (see [KN69, CE75, A ℓ VL91] for details on homogeneous spaces) endowedwith a G -invariant metric g , that is τ ˚ a g “ g for any a P G , where τ : G ˆ G { K Ñ G { K denotesthe transitive G -action. Equivalently, is a Riemannian manifold p M, g q endowed with a transitiveaction of its isometry group Iso p M, g q . If M is connected, then each closed subgroup G Ď Iso p M, g q which is transitive on M induces a presentation of p M, g q as a homogeneous space, i.e. M “ G { K ,where K Ă G is the stabilizer of some point o P 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 alsotransitive on M ). Usually, to emphasize on the transitive group G , we say that p M, g q is a G -homogeneous Riemannian manifold. However, note that may exist many subgroups of Iso p M, g q acting transitively on p M, g q . Next we shall work with connected homogeneous manifolds.As it is well-known, the geometric properties of a homogeneous space can be examined byrestricting our attention to a point. Set o “ eK for the identity coset of p M n “ G { K, g q and let T o G { K be the corresponding tangent space. Since we assume the existence of a G -invariant metric g , K Ă G can be identified with a closed subgroup of O p n q ” O p T o G { K q (or of SO p n q if G { K is oriented), so K is compact and hence any homogeneous Riemannian manifold p M n “ G { K, g q is a reductive homogeneous space . This means that there is a complement m of the Lie algebra k “ Lie p K q of the stabilizer K inside the Lie algebra g “ Lie p G q of G , which is Ad G p K q -invariant, i.e. g “ k ‘ m and Ad G p K q m Ă m , where Ad G ” Ad : G Ñ Aut p g q denotes the adjoint representationof G . Note that in general the reductive complement m may not be unique, and for a generalhomogeneous space G { K a sufficient condition for its existence is the compactness of K . On theother hand, once such a decomposition has been fixed, there is always a natural identification of m with the tangent space T o G { K “ g { k , given by X P m ÐÑ X ˚ o “ ddt ˇˇ t “ τ exp tX p o q P T o G { K , where exp tX is the one-parameter subgroup of G generated by X . Under the linear isomorphism m “ T o G { K , the isotropy representation χ : K Ñ Aut p m q , defined by χ p k q : “ p dτ k q o for any k P K ,is equivalent with the representation Ad G | K : K ˆ m Ñ m . Hence, χ p k q X “ Ad G p k q X for any k P K and X P m . In terms of Lie algebras we have χ ˚ p Y q X “ r Y, X s m for any Y P k and X P m ,or in other words χ ˚ p Y q “ ad p Y q| m .The homogeneous spaces M “ G { K that we will examine below (with K compact), are assumedto be almost effective, which means that the kernel Ker p τ q (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 wemay identify k with the Lie algebra χ ˚ p k q “ Lie p χ p K qq of the linear isotropy group χ p K q Ă Aut p m q .When only the identity element e P 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 examplethat G Ď Iso p M, g q is a closed subgroup, then the action of G to G { K is effective. Note that analmost effective action of G on G { K gives rise to an effective action of the group G “ G { Ker p τ q (of the same dimension with G ), so we will not worry much for the effectiveness of M “ G { K . NCIENT SOLUTIONS OF THE HOMOGENEOUS RICCI FLOW 5
Recall that the space of G -invariant symmetric covariant 2-tensors on a (almost) effective homo-geneous space M “ G { K with a reductive decomposition g “ k ‘ m , is naturally isomorphic withthe 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 coincideswith the space of inner products x , y on m satisfying x X, Y y “ x χ p k q X, χ p k q Y y “ x Ad G p k q X, Ad G p k q Y y , for any k P K and X, Y P m . The correspondence is given by x X, Y y “ g p X, Y q o . Moreover,when K is compact and m “ h K with respect to the Killing form B of G , then one can extendthe above correspondence between elements g “ x , y P M G and Ad G p K q -invariant B -selfadjointpositive-definite endomorphisms L : m Ñ m of m , i.e. x X, Y y “ B p LX, Y q , for any X, Y P m . Tosimplify the text, whenever is possible next we shall relax the notation Ad G p K q to Ad p K q andscalar products on m as above, will be just referred to as Ad p K q -invariant scalar products. Notethat the Ad p K q -invariance of x , y implies its ad p k q -invariance, which means that the endomorphismad p Z q| m : m Ñ m is skew-symmetric with respect to x , y , for any Z P k . When K is connected,the inclusions Ad p k q m Ă m and r k , m s Ă m are equivalent and hence one can pass from the Ad p K q -invariance to ad p k q -invariance and conversely. From now on will denote by P p m q Ad p K q the space ofall Ad p K q -invariant inner products on m .Given a Riemannian manifold p M n , g q , a solution of the Ricci flow is a family of Riemannianmetrics t g t u P M satisfying the system (0.1). If the initial metric g “ g p q P M is a G -invariantmetric with respect to some closed subgroup G Ď Iso p M, g q , i.e. p M “ G { K, g q is a homogeneousRiemannian manifold and so g P M G , then the solution t g p t qu is called homogeneous , i.e. t g p t qu P M G . Indeed, the isometries of g are isometries for any other evolved metric, and by [Kot10] itis known that the isometry group is preserved under the Ricci flow. Thus, after considering areductive decomposition g “ k ‘ m of p M “ G { K, g q , the homogeneity of g allows us to reduce theRicci flow to a system of ODEs for a curve of Ad p K q -invariant inner products on P p m q Ad p K q , where m – T o G { K is a reductive complement. In particular, due to the identification M G – P Ad p K q p m q we may write g p t q “ x , y t and then (0.1) takes the form ddt x , y t “ ´ Ric x , y t , x , y ” x , y “ g , where Ric x , y t denotes the Ad p K q -invariant bilinear form on m , corresponding to the Ricci tensorof g p t q . Note that since g “ g p q is an invariant metric, the solution g p t q of (0.1) must be uniqueamong complete, bounded curvature metrics (see [ChZ06]). Remark 1.1.
When one is interested in more general homogeneous spaces M “ G { K and areductive decomposition may not exist, the above setting can be appropriately transferred to g { k – T o G { K . However, the “reductive setting” serves well the goals of this paper and it is sufficient forour subsequent computations and description.1.2. Flag manifolds.
Let G be a compact semisimple Lie group with Lie algebra Lie p G q “ g . A flag manifold is an adjoint orbit of G , i.e. M “ Ad p G q w “ t Ad p G q w : g P G u Ă g for some left-invariant vector field w P g . Let K “ t g P G : Ad p g q w “ w u Ă G be the isotropysubgroup of w and let k “ Lie p K q be the corresponding Lie algebra. Since G acts on M transitively, M is diffeomorphic to the (compact) homogeneous space G { K , that is Ad p G q w “ G { K . In partic-ular, k “ t X P g : r X, w s “ u “ ker ad p w q , where ad : g Ñ End p g q is the adjoint representationof g . Moreover, the set S w “ t exp p tw q : t P R u is a torus in G and the isotropy subgroup K is Also called complex flag manifold , or generalized flag manifold . STAVROS ANASTASSIOU AND IOANNIS CHRYSIKOS identified with the centralizer in G of S w , i.e. K “ C p S w q . Hence rank G “ rank K and K isconnected. Thus, equivalently a flag manifold is a homogeneous space of the form G { K , where K “ C p S q “ t g P G : ghg ´ “ h for all h P S u is the centralizer of a torus S in G . When K “ C p T q “ 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, deRham irreducible homogeneous K¨ahler manifolds (see for example [A ℓ P86, AC10, C12, A ℓ C19] forfurther details).So, for any flag manifold M “ G { K we may work with G simply connected (if for instance aflag manifold of SO p n q is given, one can always pass to its universal covering by using the doublecovering Spin p n q ). 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 [KN69]. Hence, in the following we can alwaysassume that M “ G { K satisfies these conditions, and denote by B g the Killing form of the Liealgebra g . The Ad p G q -invariant inner product B : “ ´ B g induces a bi-invariant metric on G byleft translations, and we may fix, once and for all, a B -orthogonal Ad p K q -invariant decomposition g “ k ‘ m . G -invariant Riemannian metrics on G { K will be identified with Ad p K q -invariant innerproducts x , y on the reductive complement m “ T o G { K . Note that the restriction B ˇˇ m inducesthe so-called Killing metric g B P M G , which is the unique invariant metric for which the naturalprojection π : p G, B q ÝÑ p G { K, g B q 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 g C “ h C ‘ ř α P R g C α be the usual root spacedecomposition of the complexification g C of g , with respect to a Cartan subalgebra h C of g C , where R Ă p h C q ˚ is the root system of g C . Via the Killing form of g C we identify p h C q ˚ with h C . LetΠ “ t α , . . . , α ℓ u p dim h C “ ℓ q be a fundamental system of R and choose a subset Π K of Π. Wedenote by R K “ t β P R : β “ ř α i P Π K k i α i u the closed subsystem spanned by Π K . Then, the Liesubalgebra k C “ h C ‘ ř β P R K g C β is a reductive subalgebra of g C , i.e. it admits a decompositionof the form k C “ Z p k C q ‘ k C ss , where Z p k C q is its center and k C ss “ r k C , k C s the semisimple part of k C . In particular, R K is the root system of k C ss , and thus Π K can be considered as the associatedfundamental system. Let K be the connected Lie subgroup of G generated by k “ k C X g . Thenthe homogeneous manifold M “ G { K is a flag manifold, and any flag manifold is defined in thisway, i.e., by the choise of a triple p g C , Π , Π K q , see also [A ℓ P86, AC10, C12, A ℓ C19].Set Π M “ Π z Π K and R M “ R z R K , such that Π “ Π K \ Π M , and R “ R K \ R M , respec-tively. Roots in R M are called complementary roots . Let Γ “ Γ p Π q be the Dynkin diagram of thefundamental system Π. Definition 1.2.
Let M “ G { K be a flag manifold. By painting black the nodes of Γ correspondingto Π M , we obtain the painted Dynkin diagram of G { K (PDD in short). In this diagram thesubsystem Π K 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 theunique real form g of the complex simple Lie algebra g C (up to inner automorphisms of g C ), whichis reconstructed by the underlying Dynkin diagram. Moreover, the connected Lie subgroup K Ă G is defined by using the encoded by the PDD splitting Π “ Π K \ Π M ; The semisimple part of K isobtained from the (not necessarily connected) subdiagram of white simple roots, while each blackroot, i.e. each root in Π M , gives rise to a U p q -summand. Thus, the PDD determines the isotropygroup K and the space M “ G { K completely. By using certain rules to determine whether differentPDDs define isomorphic flag manifolds (see [A ℓ P86]), one can obtain all flag manifolds G { K of acompact simple Lie group G (see for example the tables in [A ℓ C19]).
NCIENT SOLUTIONS OF THE HOMOGENEOUS RICCI FLOW 7
Proposition 1.4. ([BH58, A ℓ P86])
The second Betti number of a flag manifold M “ G { K equalsto the totality of black nodes in the corresponding PDD, i.e. the cardinality of the set Π M . Note that for any flag manifold M “ G { K , the B -orthogonal reductive complement m decom-poses into a direct sum of Ad p K q -inequivalent and irreducible submodules, which we call isotropysummands , see [A ℓ P86, AC10, A ℓ C19] and the references therein. This means that when m – T o M is viewed as a K -module, then there is always a B -orthogonal Ad p K q -invariant decomposition(1.1) m “ m ‘ ¨ ¨ ¨ ‘ m r , for some r ě
1, such that ‚ K acts (via the isotropy representation) irreducibly on any m , . . . , m r , and ‚ m i fl m j are inequivalent as Ad p K q -representations for any i ‰ j .In fact, a decomposition as in (1.1) satisfying the given conditions must be unique, up to a per-mutation of the isotropy summands, see for example [PR19]. When r “ M “ G { K is anisotropy irreducible compact Hermitian symmetric space (HSS in short), and all these cosets canbe viewed as flag manifolds with b p M q “
1. Note that from the class of full flag manifolds only C P – S “ SU p q{ U p q is an irreducible HSS, and hence a flag manifold with b p M q “
1. In thistext we are mainly interested in non-symmetric flag manifolds M “ G { K with b p M q “
1, and inthis case r is bounded by the inequalities 2 ď r ď Lemma 1.5. ([A ℓ P86, AC10])
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 ě .Moreover, any G -invariant metric on M “ G { K is given by (1.2) g “ x , y “ r ÿ i “ x i ¨ B | m i , where x i P R ` are positive real numbers for any i “ , . . . , r . Thus, M G coincides with the openconvex cone R r ` “ tp x , . . . , x r q P R r : x i ą for any i “ . . . r u . Invariant metrics as in (1.2) are called diagonal (see [WZ86] for details). By Schur’s Lemma,the Ricci tensor
Ric g of such a diagonal invariant metric g , needs to preserve the splitting (1.1)and consequently, Ric g is also diagonal, i.e. Ric g p m i , m j q “
0, whenever i ‰ j . As before, Ric g isdetermined by a symmetric Ad p K q -invariant bilinear form on m , although not necessarily positivedefinite, and hence it has the expression Ric g ” Ric x , y “ r ÿ i “ y i ¨ B ˇˇ m i “ r ÿ i “ p x i ¨ ric i q ¨ B ˇˇ m i , for some y i “ x i ¨ ric i P R , where x , y P P Ad p K q p m q is the Ad p K q -invariant inner product corre-sponding to g P M G and ric i are the so-called Ricci components . There is a simple description of ric i , and hence of Ric g (and also of the scalar curvature Scal g “ tr Ric g q , in terms of the metricparameters x i , the dimensions d i “ dim R m i and the so-called structure constants of G { K withrespect to the decomposition (1.1), c kij ” „ ki j “ ÿ α,β,γ B pr X α , Y β s , Z γ q , i, j, k P t , . . . , r u , where t X α u , t Y β u , t Z γ u are B -orthonormal bases of m i , m j , m k , respectively. These non-negativequantities were introduced in [WZ86] and they have a long tradition in the theory of (compact)homogeneous Einstein spaces, see for example [PS97, S99, B¨o04, B¨oWZ04, CS14]. Following [Gr13],next we shall refer to rational polynomials depending on some real variables x , x ´ , . . . , x m , x ´ m for some positive integer m , by the term Laurent polynomials . If such a rational polynomial is
STAVROS ANASTASSIOU AND IOANNIS CHRYSIKOS homogeneous, then it will be called a homogeneous Laurent polynomial . Now, as a conclusion ofmost general results presented by [WZ86, PS97], one obtains the following
Proposition 1.6. ([WZ86, PS97])
Let M “ G { K be a flag manifold of a compact simple Lie group G . Then,1) The components ric k of the Ricci tensor Ric g corresponding to g “ ř ri “ x i ¨ B | m i P M G arehomogeneous Laurent polynomials in x , x ´ , . . . , x r , x ´ r of degree -1, given by ric k “ x k ` d k r ÿ i,j “ x k x i x j „ ki j ´ d k r ÿ i,j “ x j x k x i „ jk i , p k “ , . . . , r q .
2) The scalar curvature
Scal g “ ř ri “ d i ¨ ric i corresponding to g “ ř ri “ x i ¨ B | m i P M G is ahomogeneous Laurent polynomial in x , x ´ , . . . , x r , x ´ r of degree -1, given by Scal g “ r ÿ i “ d i x i ´ r ÿ i,j,m “ „ mi j x m x i x j . Note that
Scal g : M Ñ R is a constant function. The homogeneous Ricci flow on flag manifolds
In this section we fix a flag manifold M “ G { K of a compact, simply connected, simple Liegroup G , whose isotropy representation m decomposes as in (1.1). Since M G endowed with the L -metric coincides with the open convex cone R r ` , for the study of the Ricci flow, as an initialinvariant metric we fix the general invariant metric g “ ř ri “ x i ¨ B | m i ; This often will be denotedby g “ p x , . . . , x r q P R r ` – M G . Then, the Ricci flow equation (0.1) with initial condition g p q “ g descents to the following system of ODEs (in component form):(2.1) ! x k “ ´ x k ¨ ric k p x , . . . , x r q , ď k ď r ) . Since g P 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 p t q “ r ÿ i “ x i p t q ¨ B | m i , g p q “ g “ x , y , where the smooth functions x i p t q are positive on the same maximal interval t P p t a , t b q for which g p t q is defined. Usually, solutions have a maximal interval of definition p t a , t b q , where 0 P p t a , t b q , ´8 ď t a ă t b ě `8 . Next we are mainly interested in ancient solutions .2.1. Invariant ancient solutions.
Recall that (see for example [Buz14, Laf15]).
Definition 2.1.
A solution g p t q of (2.1) which is defined on an interval of the form p´8 , t b q with t b ă `8 , is called ancient .Note that ancient solutions typically arise as singularity models of the Ricci flow and it iswell-known that all ancient solutions have non-negative scalar curvature, see for instance [Ch09,Cor. 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 possessfixed points in M G – R r ` . In other words, considering the associated flow to (2.1) , i.e. the map ψ t : M G Ñ M G , g ÞÝÑ g p t q where t P p´ ǫ, ǫ q for some ǫ P p , , there is no g P M G – R r ` such that ψ t p g q “ g for all t . NCIENT SOLUTIONS OF THE HOMOGENEOUS RICCI FLOW 9
Proof.
Obviously, fixed points of the homogeneous Ricci flow need to correspond to invariant Ricci-flat metrics , and conversely. Since M “ G { K is compact, according to Alekseevsky and Kimel’fel’d[A ℓ K75] such a metric must be necessarily flat. But then M must be a torus, a contradiction. (cid:3) However, the system (2.1) can admit more general solutions, different than stationary points.Indeed, any flag manifold M n “ G { K is compact and simply connected, and hence its universalcovering is not diffeomorphic with an Euclidean space. Hence, by [Laf15, Cor. 4.3] it follows that Proposition 2.3. ([Laf15])
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 p t q withinitial condition g p q “ g , has an interval of definition of the form p t a , t b q , with ´8 ď t a ă and t b ă 8 . In the following section, for all flag manifolds M “ G { K with b p M q “ p t a , t b q and study their asymp-totic behaviour at the infinity of the corresponding phase space M G (in terms of the Poincar´ecompactification, see Definition 2.11). These solutions are induced by invariant Einstein metricsand are shrinking type solutions of (2.1), in terms of [CLN06, p. 98] for instance. Moreover, theyare ancient and hence the scalar curvature along such solutions is positive, while they extinct infinite time. In fact, for any flag manifold M “ G { K we obtain the following Proposition 2.4.
Let M “ G { K be a flag manifold of a compact, connected, simply connected,simple Lie group G , with M G – R r ` , for some r ě . Let g be any G -invariant Einstein metricon M “ G { K with Einstein constant λ and consider the 1-parameter family g p t q “ p ´ λt q g .Then,(1) g p t q is an ancient solution of (2.1) defined on the open interval p´8 , λ q with g p t q Ñ as t Ñ λ (from below). Hence, its scalar curvature Scal p g p t qq is a monotonically increasing functionwith the same interval of definition, and satisfies lim t Ñ´8
Scal p g p t qq “ , lim t Ñ λ Scal p g p t qq “ `8 . In particular,
Scal p g p t qq ą for any t P p´8 , λ q .(2) Similarly, the Ricci components ric ti ” ric i p g p t qq corresponding to the solution g p t q “ p ´ λt q g ,satisfy the following asymptotic properties: lim t Ñ´8 ric ti “ , lim t Ñ λ ric ti “ `8 . In particular, ric ti ą , for any t P p´8 , λ q .Proof. (1) Let g be a G -invariant Einstein metric on M “ G { K (e.g. one can fix as g an invariantK¨ahler-Einstein metric, which always exists). Set c p t q “ p ´ λt q and note that c p t q “ p ´ λt q ą t P p´8 , λ q . Then, since g p t q “ c p t q g “ c p t q g is a 1-parameter family of invariant metrics,we have g p t q “ r ÿ i “ p ´ λt q x i ¨ B | m i “ r ÿ i “ c p t q x i ¨ B | m i “ r ÿ i “ x i p t q ¨ B | m i , where x i p t q : “ c p t q x i , for any 1 ď i ď r with x i p q “ x i , where without loss of generality weassume that g “ p x , . . . , x r q for some x i P R r ` and some N Q r ě
1. As it is well-known from thegeneral theory of Ricci flow, and it is trivial to see, g p t q is a solution of (2.1) which has as intervalof definition the open set p´8 , λ q . Hence it is an ancient solution, since 0 ă λ ă `8 (recall that λ ą and moreover lim t Ñ´8 g p t q “ `8 , lim t Ñ λ g p t q “ . On the other hand, by Proposition 1.6, the scalar curvature
Scal g ” Scal p g q is a homogeneousLaurent polynomial of degree ´
1. Hence,
Scal p c p t q g q “ c p t q ´ Scal p g q “ p ´ λt q Scal p g q . Consequently
Scal p g p t qq “ p ´ λt q Scal p g q ą
0, for any t P p´8 , λ q , since Scal p g q ą
0. Obviously,
Scal p g p t qq “ λ Scal p g qp ´ λt q ą , for any p´8 , λ q and when t tends to λ from below, we see that Scal p g p t qq Ñ `8 . Since Scal p g p t qq ,as a smooth function of t , is defined only for t P p´8 , λ q , we conclude. Moreover, for t Ñ ´8 weget
Scal p g p t qq Ñ ric i ” ric i p g q “ ric i p ď i ď r q of the Riccitensor of g are homogeneous Laurent polynomials of degree -1. Hence, ric i p c p t q g q “ c p t q ric i p g q “ c p t q ric i , i.e. ric i p g p t qq “ c p t q ´ ric i . The conclusion now easily follows, since g is Einstein and so ric i “ λ ,which is independent of t , for any 1 ď i ď r . (cid:3) Remark 2.5. p i q The conclusions for the asymptotic behaviour of scalar curvature for the solutions g p t q verify a more general statement for the limit behaviour of the scalar curvature of homogeneousancient solutions, obtained by Lafuente [Laf15, Thm. 1.1, (i)] in terms of the so-called bracket flow . Later, in Section 3 we shall illustrate Proposition 2.4 by certain examples (see Examples 3.3, 3.4). p ii q Recall by [CLN06, p. 545] that
Ric t ” Ric p g p t qq “ Ric p g p qq “ λg “ λ p ´ λt q g p t q . Thus, thesolutions g p t q given in Proposition 2.4, being homothetic to invariant Einstein metrics, they alsosatisfy the Einstein equation (and therefore the Ricci flow equation), and hence lim t Ñ´8
Ric t “ λg “ lim t Ñp λ q ´ Ric t . The Ad p K q -invariant g p t q -self-adjoint operator r g p t q ” r t : m Ñ m (Ricciendomorphism) corresponding to the Ricci tensor Ric t is defined by Ric t p X, Y q “ g t p r t p X q , Y q forany X, Y P m . It satisfies the relation r t “ λc p t q A t “ λA “ r , for any t P p´8 , λ q , where A “ ř ri “ x i ¨ Id m i is the positive definite Ad p K q -invariant g -self-adjoint operator correspondingto the diagonal metric g . The endomorphism of the 1-parameter family g p t q “ c p t q g , given by A t “ c p t q A , is also positive definite for any t P p´8 , λ q , and the same satisfies r t . p iii q Obviously, Proposition 2.4 and the above conclusions can be extended to any homogeneousspace G { K of a compact semisimple Lie group G modulo a compact subgroup K Ă G , with amonotypic isotropy representation admitting an invariant Einstein metric g . A simple exampleis given below. On the other side, there are examples of effective compact homogeneous spaces,with non-monotypic isotropy representation, i.e. m i – m j for some 1 ď i ‰ j ď r , for whichthe invariant (Einstein) metrics are still diagonal. To take a taste, consider the Stiefel manifold V p R ℓ ` q “ G { K “ SO p ℓ ` q{ SO p ℓ ´ q and assume for simplicity that ℓ ‰
3. This is a p ℓ ´ q -dimensional compact homogeneous space, admitting a U p q -fibration over the GrassmannianGr ` p R ℓ ` q “ SO p ℓ ` q{ SO p ℓ ´ q ˆ SO p q . Let so p ℓ ` q “ so p ℓ ´ q ‘ m be a B -orthogonalreductive decomposition. Then, it is not hard to see that m “ m ‘ m ‘ m , where m is 1-dimensional and m – m are two irreducible submodules of dimension ℓ ´
1, both isomorphic to The solutions g p t q described above have as maximal interval of definition the open set p´8 , λ i q , so the second partof [Laf15, Thm. 1.1] does not apply in our situation. NCIENT SOLUTIONS OF THE HOMOGENEOUS RICCI FLOW 11 the standard representation of SO p ℓ ´ q . Hence, the isotropy representation of V p R ℓ ` q is not monotypic. However, the invariant metrics on V p R ℓ ` q can be shown that are still diagonal. This isbased on the action of the generalized Weyl group (gauge group) N G p K q{ K on the space P p m q Ad p K q of Ad p K q -invariant inner products on m (see for example [NRS07, AB15]). For the specific case of V p R ℓ ` q , the group N G p K q{ K is isomorphic to a circle and this action was used in [Ker98, p. 121]to eliminate the off-diagonal components of the invariant metrics (note that for ℓ ‰ V p R ℓ ` q admits a unique SO p ℓ ` q -invariant Einstein metric). Hence, the results discussed above can alsobe extended in that more general case. Example 2.6.
Let M “ G { K be an isotropy irreducible homogeneous space of a compact simpleLie group G . Consider a B -orthogonal reductive decomposition g “ k ‘ m . Then, M G “ R ` and the Killing form g B “ B | m is the unique invariant Einstein metric (up to a scalar). Hence, g p t q “ p ´ λ B t q g B is a homogeneous ancient solution of the corresponding homogeneous Ricciflow, defined on the open interval p´8 , λ B q , where λ B ą g B . Thisapplies 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 “ Definition 2.7.
A homogeneous ancient solution of (2.1) is called non-collapsed , if the correspond-ing curvature normalized metrics have a uniform lower injectivity radius bound.Non-collapsed homogeneous ancient solutions of the Ricci flow on compact homogeneous spacehave been recently studied in [B¨oLS17]. In this work, among other results, the authors proved that: ‚ p α q If G, K are connected and does not exist some intermediate group K Ă L Ă G suchthat L { K is a torus, i.e. if G { K is not a homogeneous torus bundle G { K Ñ G { L over G { L ,then any ancient solution on G { K is non-collapsed ([B¨oLS17, Rem 5.3]). ‚ p β q Non-trivial homogeneous ancient solutions of Ricci flow must develop a
Type I singu-larity close to their extinction time and also to the past (see [B¨oLS17], 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 ancient solutionof the form g p t q “ p ´ λt q g , where g P M G is an invariant Einstein metric on M with Einsteinconstant λ , is non-collapsed and develops a Type I singularity close to its extinction time (and alsoto the past). Therefore, as t Ñ λ , the volume of M “ G { K with respect to g p t q tends to , Vol g p t q p G { K q Ñ , 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 Ă L Ă G such that L { K is torus T s for some s ě
1. Then obviously, it must be rank L ą rank K . But rank K “ rank G and then the inclusion L Ă G gives a contradiction. Hence, M “ G { K cannotbe homogeneous torus bundle. Thus, according to p α q any possible homogeneous ancient solutionof (2.1) on M “ G { K must be non-collapsed. The second claim and the assertions for Type Ibehaviour, follow now by p β q . (cid:3) Remark 2.9.
Recall that on a compact homogeneous space M “ G { K the total scalar curvaturefunctional S p g q “ ż M Scal p g q dV g , restricted on the set M G of G -invariant metrics of volume 1, coincides with the smooth function M G Q g ÞÝÑ
Scal p g q ” Scal g , since the scalar curvature Scal g 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 ofthe restriction S | M G . A G -invariant Einstein metric is called unstable if is not a local maximum of S | M G . By [B¨oLS17, Lem. 5.4] it also known that for any unstable homogeneous Einstein metric on acompact homogeneous space G { K , there exists a non-collapsed invariant ancient solution emanatingfrom it . For general flag manifolds, up to our knowledge, is an open question if all possible existentinvariant Einstein metrics are unstable or not. However, by [AC11, Thm. 1.2] it is known thaton a flag manifold with r “
2, i.e. with two isotropy summands, there exist two non-isometricinvariant Einstein metrics which are both local minima of S | M G , and hence unstable in the abovesense. Thus, for r “ The Poincar´e compactification procedure.
To study the asymptotic behaviour of homoge-neous Ricci flow solutions, even of more general abstract solutions than these given in Proposition2.4, one can successfully use the compactification method of Poincar´e which we refresh below,adapted to our setting (see also [GM09, AnC11, GM12]). Indeed, the Poincar´e compactificationprocedure is used to study the behaviour of a polynomial system of ordinary differential equationsin a neighbourhood of infinity, see [G69, VG03] for an explicit description. We describe it here, ina form suitable for our purposes.Let p M “ G { K, g q be a flag manifold with M G – R r ` for some r ě
1. By using the expressions ofProposition 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 homogeneouspolynomial equations of positive degree. Let us denote this system obtained from the homogeneousRicci flow via this procedure, by(2.2) ! x k “ RF k p x , . . . , x r q : k “ , . . . , r ) . So, for any 1 ď k ď r , RF k p x , .., x r q are homogeneous polynomials, the maximum degree of whichis 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 p x , .., x r q : “ ` RF p x , . . . , x r q , . . . , RF r p x , . . . , x r q ˘ and referred to as the homogeneous vector field associated tothe homogeneous Ricci flow on p M “ G { K, g q . Consider the subset of the unit sphere S r Ă R r ` , containing all points having non-negativecoordinates, i.e. S r ě “ ! y P R r ` : } y } “ , y i ě , i “ , . . . , r ` ) Ă S r . It is also convenient toidentify R r ě with the subset of R r ` defined by T : “ ! p y , . . . , y r , y r ` q P R r ` : y r ` “ , y i ě , @ i “ , . . . , r ) – R r ě . Consider now the central projection f : R r ě – T Ñ S r ě , assigning to every p P T the point T p p q P S r ě ,defined as follows: T p p q is the intersection of the straight line joining the initial point p with theorigin of R r ` . The explicit form of f is given by f p y , . . . , y r , q “ }p y , . . . , y r , q} p y , . . . , y r , q . Through this projection, R r ě can be identified with the subset of S r ě with y i ` ą
0. Moreover, theequator of S r ě , that is S r ´ ě “ ! y P S r ě : y r ` “ ) , is identified as the infinity of R r ě . To be more explicit, let us state this as a definition. Definition 2.11. A point at infinity of M G – R r ě is understood to be point of the equator S r ´ ě . NCIENT SOLUTIONS OF THE HOMOGENEOUS RICCI FLOW 13
Note that S r ě is diffeomorphic with the standard r –dimensional simplex∆ r “ ! p y , . . . , y r ` q P R r ` : ÿ i y i “ , @ y i ě , i “ , .., r ` ) and thus with the so-called non-negative part of the real projective space R P r ě “ p R r ` ě zt uq{ R ` (see [Fu ℓ
93, Ch. 4]). Via push-forward, the central projection carries the vector field X onto S r ě .The vector field obtained by this procedure, i.e. the vector field p p X qp y q : “ y d ´ r ` f ˚ X p y q is called the Poincar´e compactification of X , and this is an analytic vector field defined on all S r ě .Actually, in order to perform computations with p p X q one needs its expressions in a chart. Thus,consider the chart U “ ! y P S r : y ą ) of S r , and project it to the plane ! p y , . . . , y r ` q P R r ` : y “ q ) – R r ě . Then, via the corre-sponding central projection, assign to every point of U the point of intersection of the straight linejoining the origin with the original point. This second central projection denoted by F : U Ñ R r ě ,has obviously the form F p y , y , . . . , y r ` q “ p , y y , . . . , y r ` y q . We can now compute the local ex-pression of p p X q in the chart U ; As above, let us denote by x i the coordinates on R r ě . Then, weobtain Proposition 2.12.
The local expression of the Poincar´e compactification p p X q of the homogeneousvector field X associated to the Ricci flow on p M “ G { K, g q , reads as (2.3) x i “ x dr p´ x i RF ` RF i ` q for any i “ , . . . , r ´ , and x r “ x dr p´ x r RF q , where RF i p x , . . . , x r q : “ RF i p x r , x x r , . . . , x r ´ x r q . Remark 2.13.
In the expressions given by (2.3), the factor x dr is canceled by the dominators of therational polynomials RF i p x , . . . , x r q . By locating then the fixed points of the resulting dynamicalsystem, under the condition x r “
0, we obtain the so-called fixed points at the infinity of M G – R r ` of the homogeneous Ricci flow on p M “ G { K, g q . Example 2.14.
For r “ p p X q in the local chart U is given by x “ x d p´ x RF ` RF q , x “ x d p´ x RF q , where RF i p x , x q “ RF i p x , x x q for any i “ ,
2. We may omit the term }p x ,x , q} , by using a timereparametrization. Fixed points of the homogeneous Ricci flow on p M “ G { K, g q at infinity of M G “ R ` , can be studied by setting x “
0. For r “ p p X q in the local chart U reads by x “ x d p´ x RF ` RF q , x “ x d p´ x RF ` RF q , x “ x d p´ x RF q , where RF i “ RF i p x , x x , x x q , for any i “ , ,
3. In an analogous way, fixed points at infinity canbe studied by setting x “
0, while similarly are treated cases with r ą Global study of HRF on flag spaces M “ G { K with b p M q “ M “ G { K with b p M q “
1. Again we can workwith G simple. Let us begin first with a few details about this specific class of flag spaces. Here, we have omitted the term }p x ,...,x r , q} , since it does not affect the qualitative behaviour of the system. Flag manifolds M “ G { K with b p M q “ . According to Proposition 1.4, flag manifolds M “ G { K of a compact simple Lie group G with b p M q “ G only a simple root, i.e. Π M “ t a i o u for some a i o P Π. The number of theisotropy summands of a flag space M “ G { K with b p M q “ Dynkin marks of the simple roots; These are the positiveintegers coefficients appearing in the expression of the highest root of G as a linear combination ofsimple roots. For flag manifolds with b p M q “
1, i.e. Π M “ Π { Π K “ t a i o : 1 ď i o ď ℓ “ rank G u we have the relation r “ Dynk p α i o q (see [C12, CS14]), where r is the integer appearing in (1.1). Inother words, a flag manifold M “ G { K with b p M q “ r isotropy summands is obtained bypainting black a simple root α i o with Dynkin mark r , and conversely. Since for a compact simpleLie group G the maximal Dynkin mark equals to 6 (and occurs for G “ E only), we result withthe bound 1 ď r ď M “ G { K is a flag manifold as above with b p M q “ ď r ď finiteness conjecture of B¨ohm-Wang-Ziller ([B¨oWZ04]). Notethat M “ G { K admits a unique invariant complex structure, and thus a unique invariant K¨ahler-Einstein metric, with explicit form (see [BH58, CS14])(3.1) g KE “ r ÿ i i ¨ B | m i “ B | m ` B | m ` . . . ` rB | m r . The isotropy summands satisfy the relations r k , m i s Ă m i , r m i , m i s Ă k ` m i , r m i , m j s Ă m i ` j ` m | i ´ j | p i ‰ j q . Hence, the non-zero structure constants are listed as follows: r non-zero structure constants c kij c ([AC11])3 c , c ([Kim90, AnC11])4 c , c , c , c ([AC10])5 c , c , c , c , c , c ([CS14])6 c , c , c , c , c , c , c , c , c ([CS14])For r “ ,
3, the use of the K¨ahler-Einstein metric is sufficient for an explicit computation of c kij ,and this yields a general expression of them in terms of d i “ dim m i (see [Kim90, AC11, AnC11]).For 4 ď r ď c kij , which dependon the specific coset (see [AC10, CS14]). Case r “ : All flag manifolds M “ G { K with G simple and r “ b p M q “
1, see [AC11]for details. We recall that: ric “ x ´ c x d x , ric “ x ` c d ´ x x ´ x ¯ , c “ d d d ` d , Scal g “ ÿ i “ d i ¨ ric i “ ´ d x ` d x ¯ ´ c ´ x x ` x ¯ . (3.2) Case r “ : Let M “ G { K be a flag manifold with b p M q “ r “
3. Such flag spaceshave been classified by [Kim90], see also [AnC11, GMPSS20]. They all correspond to exceptionalcompact simple Lie groups, but the correspondence is not a bijection. One should mention thatnot all flag manifolds with r “ NCIENT SOLUTIONS OF THE HOMOGENEOUS RICCI FLOW 15 that still one can construct flag spaces with r “ b p M q “
2. We recall that c “ d d ` d d ´ d d d ` d ` d , c “ d p d ` d q d ` d ` d , ric “ x ´ c x d x ` c d ´ x x x ´ x x x ´ x x x ¯ , ric “ x ` c d ´ x x ´ x ¯ ` c d ´ x x x ´ x x x ´ x x x ¯ , ric “ x ` c d ´ x x x ´ x x x ´ x x x ¯ , Scal g “ ´ d x ` d x ` d x ¯ ´ c ´ x x ` x ¯ ´ c ´ x x x ` x x x ` x x x ¯ . (3.3) Case r “ : Let M “ G { K be a flag manifold with G simple, r “ b p M q “
1. Thereexist 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 thecorresponding PDD and Dynkin marks. As for the case r “
3, one should be aware that still existflag manifold with r “
4, but b p M q “ M “ G { K with b p M q “ r “ { SU p q ˆ SU p q ˆ U p q ❝ α ❝ α ą s α ❝ α E { SU p q ˆ SU p q ˆ SU p q ˆ U p q ❝ α ❝ α ❝ α s α ❝ α ❝ α ❝ α E { SO p q ˆ SU p q ˆ U p q ❝ α ❝ α s α ❝ α ❝ α ❝ α ❝ α ❝ α E { SU p q ˆ SU p q ˆ U p q ❝ α ❝ α ❝ α ❝ α ❝ α ❝ α s α ❝ α We also recall that ric “ x ´ c d x x ` c d ´ x x x ´ x x x ´ x x x ¯ ` c d ´ x x x ´ x x x ´ x x x ¯ , ric “ x ´ c d x x ` c d ´ x x ´ x ¯ ` c d ´ x x x ´ x x x ´ x x x ¯ , ric “ x ` c d ´ x x x ´ x x x ´ x x x ¯ ` c d ´ x x x ´ x x x ´ x x x ¯ , ric “ x ` c d ´ x x ´ x ¯ ` c d ´ x x x ´ x x x ´ x x x ¯ , Scal g “ ÿ i “ d i x i ´ c p x x x ` x x x ` x x x q ´ c p x x x ` x x x ` x x x q´ c p x x ` x q ´ c p x x ` x q . Let us finally present the values of c kij and the corresponding dimensions: M “ G { K c c c c F { SU p q ˆ SU p q ˆ U p q { E { SU p q ˆ SU p q ˆ SU p q ˆ U p q { E { SO p q ˆ SU p q ˆ U p q { E { SU p q ˆ SU p q ˆ U p q { { d d d d F { SU p q ˆ SU p q ˆ U p q
12 18 4 6 E { SU p q ˆ SU p q ˆ SU p q ˆ U p q
48 36 16 6 E { SO p q ˆ SU p q ˆ U p q
96 60 32 6 E { SU p q ˆ SU p q ˆ U p q
84 70 28 14
Case r “ : According to [CS14], there is only one flag manifold M “ G { K with G simple, b p M q “ r “
5; This is the coset space M “ G { K “ E { U p q ˆ SU p q ˆ SU p q and is determined by painting black the simple root α of E , i.e. Π M “ t α u , with Dynk p α q “ r “ ric i are given by ric “ x ´ c d x x ` c d ˆ x x x ´ x x x ´ x x x ˙ ` c d ˆ x x x ´ x x x ´ x x x ˙ ` c d ˆ x x x ´ x x x ´ x x x ˙ , ric “ x ` c d ˆ x x ´ x ˙ ´ c d x x ` c d ˆ x x x ´ x x x ´ x x x ˙ ` c d ˆ x x x ´ x x x ´ x x x ˙ , ric “ x ` c d ˆ x x x ´ x x x ´ x x x ˙ ` c d ˆ x x x ´ x x x ´ x x x ˙ ` c d ˆ x x x ´ x x x ´ x x x ˙ , ric “ x ` c d ˆ x x ´ x ˙ ` c d ˆ x x x ´ x x x ´ x x x ˙ ` c d ˆ x x x ´ x x x ´ x x x ˙ , ric “ x ` c d ˆ x x x ´ x x x ´ x x x ˙ ` c d ˆ x x x ´ x x x ´ x x x ˙ . The non-zero structure constants have been computed in [CS14, Prop. 6] and it is useful to recallthem: c “ , c “ , c “ , c “ { , c “ , c “ . Moreover, d “ d “ d “ d “
20 and d “ Case r “ : By [CS14] it is known that there is also only one flag manifold M “ G { K with G simple, b p M q “ r “
6. This is isometric to the homogeneous space M “ G { K “ E { U p q ˆ SU p q ˆ SU p q ˆ SU p q , which is determined by painting black the simple root α of E , i.e. Π M “ t α u , with Dynk p α q “ d “ d “ d “ d “ d “
12 and d “
10. The values of the non-zerostructure constants are given by (see [CS14, Prop. 12]) c “ , c “ , c “ , c “ , c “ , c “ , c “ , c “ , c “ , NCIENT SOLUTIONS OF THE HOMOGENEOUS RICCI FLOW 17 and the components ric i of the Ricci tensor Ric g corresponding to g are given by ric “ x ´ c d x x ` c d ˆ x x x ´ x x x ´ x x x ˙ ` c d ˆ x x x ´ x x x ´ x x x ˙ ` c d ˆ x x x ´ x x x ´ x x x ˙ ` c d ˆ x x x ´ x x x ´ x x x ˙ , ric “ x ` c d ˆ x x ´ x ˙ ´ c d x x ` c d ˆ x x x ´ x x x ´ x x x ˙ ` c d ˆ x x x ´ x x x ´ x x x ˙ ` c d ˆ x x x ´ x x x ´ x x x ˙ , ric “ x ´ c d x x ` c d ˆ x x x ´ x x x ´ x x x ˙ ` c d ˆ x x x ´ x x x ´ x x x ˙ ` c d ˆ x x x ´ x x x ´ x x x ˙ , ric “ x ` c d ˆ x x ´ x ˙ ` c d ˆ x x x ´ x x x ´ x x x ˙ ` c d ˆ x x x ´ x x x ´ x x x ˙ ` c d ˆ x x x ´ x x x ´ x x x ˙ , ric “ x ` c d ˆ x x x ´ x x x ´ x x x ˙ ` c d ˆ x x x ´ x x x ´ x x x ˙ ` c d ˆ x x x ´ x x x ´ x x x ˙ , ric “ x ` c d ˆ x x ´ x ˙ ` c d ˆ x x x ´ x x x ´ x x x ˙ ` c d ˆ x x x ´ x x x ´ x x x ˙ . The main theorem.
Let M “ G { K be a flag manifold with b p M q “ ď r ď
6. Thesystem of the homogeneous Ricci flow is given by(3.4) ! x i “ ´ x i ¨ ric i : i “ , . . . , r ) . For any case separately, a direct computation shows that system (3.4) does not possess fixed pointsin M G – R r ` , i.e. points p x , . . . , x r q P R rr satisfying the system t x i “ x “ ¨ ¨ ¨ “ x r “ u , whichverifies Lemma 2.2.We agree now on the following notation: We denote by e j a fixed point of the homogeneous Ricciflow (HRF) at infinity of M G , as defined before. We shall write N for the number of all such fixedpoints. We will also denote by d unstb j (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 e j . Inthis terms we obtain the following Theorem 3.1.
Let M “ G { K be a non-symmetric flag manifold with b p M q “ , and let r ( ď r ď ) be the number of the corresponding isotropy summands. Then, the following hold:(1)The HRF admits exactly N fixed points e j 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 e j are given in Table 1, where e represents the fixed point corresponding to the unique invariant K¨ahler-Einstein metric on G { K .We see thati) For any M “ G { K , the fixed point e has always an 1-dimensional unstable manifold in M G ,and a p r ´ q -dimensional stable manifold in the infinity of M G . ii) Any other fixed point e k with ď k ď N , has always a 2-dimensional unstable manifold in M G ,and a p r ´ q -dimensional stable manifold in the infinity of M G .(3) Each unstable manifold of any e j , contains a non-collapsed ancient solution, given by g j : p´8 , λ j q ÝÑ M G , t ÞÝÑ g j p t q “ p ´ λ j t q ¨ e j , j “ , . . . , N , where λ j is the Einstein constant of the corresponding Einstein metric g j p q , j “ , . . . N , on M “ G { K (these are also specified below). All such solutions g j p t q tend to when t Ñ T “ λ j ą ,and M “ G { K shrinks to a point in finite time. Note that in Table 1, M ˚ denotes the homogeneousspace M ˚ : “ E { SO p q ˆ SU p q ˆ U p q , with r “ .(4) When r “ , any other possible solution of the Ricci flow with initial condition in M G has t u as its ω –limit set. r conditions for M “ G { K N d unstb1 d stb1 d unstb j p ď j ď N q d stb j p ď j ď N q M fl M ˚ M – M ˚ Table 1.
The exact number N of the fixed points e k of HRF at infinity of M G for any non-symmetric flag space M “ G { K with b p M q “
1, and the dimensions d stb k , d unstb k , for any 1 ď k ď N . Proof.
We split the proof in cases, depending on the possible values of r . Case r “ . In this case the system (3.4) reduces to:(3.5) ! x “ ´ p d ` d q x ´ d x p d ` d q x , x “ ´ d x ` d x p d ` d q x ) . To search for fixed points of the HRF at infinity of M G , we first multiply the right-hand side ofthese equations with the positive factor 2 p d ` d q x . This multiplication does not qualitativelyaffect system’s behaviour, and we result with the following equivalent system: ! x “ RF p x , x q “ ´ p d ` d q x ` d x x , x “ RF p x , x q “ ´p d x ` d x q ) , where the right-hand side consists of two homogeneous polynomials of degree 2. This is the maximaldegree d of the system, as discussed above. We can therefore apply the Poincar´e compactificationprocedure to study its behaviour at infinity. For the formulas given in Example 2.14, i.e. ! x “ x ` ´ x RF p x , x q ` RF p x , x q ˘ , x “ x ` ´ x RF p x , x q ˘) , we compute RF p x , x q “ RF p x , x x q “ ´ p d ` d ´ d x q x , RF p x , x q “ RF p x , x x q “ ´ p d ` d x q x . Hence finally we result with the system ! x “ ´p x ´ qp´ d ` d x ` d x q , x “ p d ` d ´ d x q x ) , NCIENT SOLUTIONS OF THE HOMOGENEOUS RICCI FLOW 19 which is the desired expression in the U chart. To study the behaviour of this system at infinityof R ` , we set x “
0. Then, the second equation becomes x “
0, confirming that infinity remainsinvariant under the flow. To locate fixed points, we solve the equation x “ h p x q “ ´p x ´ qp´ d ` d x ` d x q “ . We obtain exactly two solutions, namely x a “ x b “ d {p d ` d q , and since h p x a q “ ´ d , h p x b q “ d , both fixed points are hyperbolic. In particular, h p x a q ă x a is always anattracting node with eigenvalue equal to ´ d . On the other hand, h p x b q ą x b is arepelling node, with eigenvalue equal to 2 d .Recall now that the central projection F maps the sphere to the y “ x a , x b represent the points e “ p , x a q and e “ p , x b q in R ` , which correspond tothe invariant Einstein metrics g KE “ ¨ B | m ` ¨ B | m , g E “ ¨ B | m ` d {p d ` d q ¨ B | m , respectively (see also [AC11]). To locate the ancient solutions, let ˆ g p t q “ t p x p q , x p qq be a straightline in M G . At the point p a, b q , belonging in the trace of ˆ g p t q , the vector normal to the straight lineis the vector p´ b, a q . The straight line ˆ g p t q must be tangent to the vector field X p x , x q definedby the homogeneous Ricci flow, hence the following equation should hold: p´ b, a q ¨ X p a, b q “ . This gives us two solutions, namely p a , b q “ p , q and p a , b q “ p , d {p d ` d qq , which definethe lines γ i p t q “ t p a i , b i q , i “ ,
2. The solutions g p t q , g p t q of system (3.5) corresponding to thelines γ p t q and γ p t q , are determined by the following equations g p t q “ p ´ λ t q ¨ p , q “ p ´ λ t q ¨ e ,g p t q “ p ´ λ t q ¨ p , d d ` d q “ p ´ λ t q ¨ e , where λ , λ are the Einstein constants of g KE “ g p q and g E “ g p q , respectively, given by λ “ d ` d p d ` d q , λ “ d ` d d ` d p d ` d qp d ` d q . Obviously, these are both ancient solutions since are defined on the open set p´8 , λ i q (see alsoProposition 2.4), in particular g i p t q Ñ t Ñ λ i , i “ ,
2. The assertion that g i p t q arenon-collapsed follows by Corollary 2.8.Based on the definitions of the central projections f and F given before, we verify that F p f p γ p t qqq “ p , q , and F p f p γ p t qqq “ p , d d ` d q . Thus, we have that lim t Ñ λi γ i p t q “ , and lim t Ñ´8 γ i p t q “ e i , @ i “ , , which proves the claim that these ancient solitons belong to the unstable manifolds of the fixedpoints located at infinity.To verify the claim in (4), we use the function V p x , x q “ x ` x as a Lyapunov function forthe system (3.5). We compute that dVdt p x p t q , x p t qq “ ´ d x p x ` x q ` d p x ` x q p d ` d q x , which for x , x ą p t a , t b q , t a , t b P R Yt˘8u the domainof definition of the solution curve g p t q “ p x p t q , x p t qq , we conclude that g p t q tends to the origin,as t Ñ t b . This completes the proof for r “ Case r “ . In this case, to reduce system (3.4) in a polynomial dynamical system, we mustmultiply with the positive factor 2 d d p d ` d ` d q x yz . This gives x “ RF p x , x , x q “ ´ d x p d d x ` d d x ´ d d x x ´ d d x x `` d x x x ` d d x x x ` d d x x x ´ d d x x ´ d d x x `` d d x x ´ d d x x ´ d d x x q , x “ RF p x , x , x q “ ´ d x p´ d d x ´ d d x ` d d x x ` d d x x ` (3.6) ` d x x ´ d d x x ` d d x x ` d d x x ` d d x x ´ d d x x ´´ d d x x ´ d d x x q , x “ RF p x , x , x q “ d d x x p d x ` d x ´ d x x ´ d x x ´´ d x x ` d x ` d x ´ d x ´ d x q . The right-hand side of the system above consists of homogeneous polynomials of degree 4. Let usapply the Poincar´e compactification procedure and set x “
0, to obtain the equations governingthe 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: x “ x p d d ` d d d ` d d ´ d d x ´ d d d x ´ d d x ´ d d x ` d d x ´´ d d d x ` d d x x ` d d x x ` d d d x x ´ d d x x ´ d d x x ´´ d d x x ´ d d d x x ` d d x x ` d d x ´ d d x q , x “ ´ d x p´ d ´ d d ´ d d ´ d d ` d x ` d d x ` d d x ´ d x ´ d d x `` d d x ` d d x ´ d x x ´ d d x x ´ d d x x ` d d x x ` d d x x ´´ d d x x ` d x ` d d x ` d d x ` d d x q , x “ . 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 t x “ x “ u . For this, it issufficient to study each case separately and replace the dimensions d i . For any case we get exactlythree fixed points, which we list as follows: ‚ E { E ˆ SU p q ˆ U p q : In this case, we have d “ , d “ , d “ p , q , p . , . q , p . , . q . ‚ E { SU p q ˆ U p q : In this case, we have d “ , d “ , d “
16 and the fixed points atinfinity are located at: p , q , p . , . q , p . , . q . ‚ E { SU p q ˆ SU p q ˆ U p q : In this case, we have d “ , d “ , d “ p , q , p . , . q , p . , . q . ‚ E { SU p q ˆ SU p q ˆ U p q : In this case, we have d “ , d “ , d “ p , q , p . , . q , p . , . q . NCIENT SOLUTIONS OF THE HOMOGENEOUS RICCI FLOW 21 ‚ E { SU p q ˆ SU p q ˆ SU p q ˆ U p q : In this case, we have d “ , d “ , d “ p , q , p . , . q , p . , . q . ‚ F { SU p q ˆ SU p q ˆ U p q : In this case, we have d “ , d “ , d “ p , q , p . , . q , p . , . q . ‚ G { U p q : In this case, we have d “ , d “ , d “ p , q , p . , . q , p . , . q . The projections of these solutions via F : U Ñ R ě , give us the points e i , i “ , ,
3. These are thepoints obtained from the coordinates of the fixed points given above, with one extra coordinateequal to 1, in the first entry.According to (3.1), the indicated fixed point e “ p , , q corresponds to the unique invariantK¨ahler-Einstein metric, and simple eigenvalue calculations show that it possesses a 2-dimensionalstable 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-K¨ahler non-isometric invariant Einstein metrics(see [Kim90, AnC11]), and have a 2-dimensional unstable manifold and a 1-dimensional stablemanifold, contained at the infinity of M G .Let us now consider an invariant line of system (3.6), of the form ˆ g p t q “ t p x p q , x p q , x p qq . At apoint p a, b, c q belonging to this line, the normal vectors are: p´ b, a, q , p , ´ c, b q , thus ˆ g p t q is tangentto the vector field X p x , x , x q defined by the homogeneous Ricci flow if p´ b, a, q ¨ X p a, b, c q “ , p , ´ c, b q ¨ X p a, b, c q “ . These equations possess three solutions, with respect to p a, b, c q ; One of them is always p , , q ,while the other two can be obtained after numerically solving equations above for every value of d , d , d . These solutions give us the three non-collapsed ancient solutions g i p t q “ p ´ λ i t q ¨ e i with t P p´8 , λ i q , for any i “ , ,
3, where the Einstein constants λ i for the cases i “ , g p q , g p q in the Riccicomponents, while for g p t q , λ is specified as follows: λ “ d ` d ` d d ` d ` d . Moreover, it is easy to show, taking limits, that the α -limit set of each of these solutions is one ofthe located fixed points at infinity of M G , while the ω -limit set, for all of them, is t u . Finally, forany i we have g i p t q Ñ t Ñ λ i , where T “ λ i depends on the dimensions d i , i “ , , M “ G { K , while again the assertion that g i p t q are non-collapsed, for any i “ , , , follows by Corollary 2.8. Case r “ . The proof follows the lines of the previous cases r “ ,
3. Thus, we avoid to presentsimilar arguments. Consider for example the case of the flag manifold M “ G { K with b p M q “ r “
4, corresponding to F . After multiplication with the positive term 36 x x x x , the system(3.4) of Ricci flow equation turns into the following polynomial system: x “ x x p´ x x ` x x x ´ x x ` x x x ´ x x x x `` x x x ` x x x ` x x x q , x “ x x p x x ´ x x ` x x x ´ x x x ´ x x ` x x x q , x “ x x x p x x ` x x ´ x x x ` x x ` x x ´ x x ´ x x q , x “ ´ x x p x x ´ x x ` x x x ` x x x ´ x x q . 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 . Inthe U chart and by setting x “
0, we obtain x “ ´ x p´ x ` x x ´ x x ` x x ` x x x ´ x x x ` x x x ´´ x x x ` x x ´ x x q , x “ ´ x x p´ x ` x x ´ x ` x x ´ x x ´ x x x ` x x x `` x x ´ x x q , x “ ´ x p´ x ` x x ´ x x ´ x x ` x x ´ x x x ` x x x `` x x x ` x x ` x x q , x “ . Again, last equation confirms that infinity remains invariant under the flow of the system. Solvingthe system t x “ x “ x “ u , we get exactly three fixed points, given by p , , q , p . , . , . q , p . , . , . q . These solutions give us, through the central projection F of the U chart on R ě , the three fixedpoints e , e , e . The fixed point e “ p , , , q corresponds to the unique invariant K¨ahler-Einsteinmetric and simple eigenvalues calculations show that is possesses a 3-dimensional stable manifold,contained in the infinity of M G , and a 1-dimensional unstable manifold. The other two fixedpoints, corresponding to non-K¨ahler, non-isometric invariant Einstein metrics (see [AC10]), havea 2-dimensional stable manifold, located in the infinity of M G , and a 2-dimensional unstablemanifold, one direction of which is the straight line in M G tending towards the origin. Invariantlines, corresponding to non-collapsed ancient solutions g i p t q “ p ´ λ i t q ¨ e i , t P p´8 , λ i q , can befound as in the previous cases, confirming once again that their ω –limit sets are equal to t u , whilethe α –limit set is the corresponding fixed point at infinity of M G . This proves our claims. TheRicci flow equations on the rest three homogeneous spaces of that type can be treated similarly. Case r “ . After multiplication with the positive term 60 x x x x x , system (3.4) turns into thefollowing polynomial system: x “ ´ x x p x x x ´ x x x x ` x x x ´ x x x x ` x x x ´ x x x x `` x x x x x ´ x x x x ´ x x x x ´ x x x x ´ x x x x q , x “ x x p´ x x ` x x x ` x x x ´ x x x ´ x x x x ´ x x x ` x x x x `` x x x x ` x x x q , x “ x x x p x x x ´ x x x ` x x x ´ x x x ` x x x ´ x x x x ` x x x ´´ x x x ` x x x ` x x x q , x “ ´ x x p´ x x x ` x x x ´ x x x ` x x x x ´ x x x ` x x x `` x x x x ´ x x x q , x “ x x x p x x x ` x x x ´ x x x x ` x x x ` x x x ´ x x x ´ x x x q . Note that the right-hand side consists of homogeneous polynomials of degree 7 and we can useProposition 2.12 to study the behaviour of this system at infinity of M G . Using the expressions NCIENT SOLUTIONS OF THE HOMOGENEOUS RICCI FLOW 23 given above, the system at infinity, written in the U chart and setting x “
0, reads as follows: x “ ´ x p´ x x ` x x ´ x x x ` x x x ´ x x ` x x x ´ x x x ` x x x `` x x x x ´ x x x x ` x x x x ´ x x x x ` x x x ´ x x x `` x x x ´ x x x q , x “ ´ x x p´ x x ´ x x ` x x ` x x x ´ x x ` x x x ´ x x ` x x x ´´ x x x ´ x x x x ` x x x x ` x x x ´ x x x ` x x x ´ x x q , x “ ´ x p´ x x ` x x x ´ x x ` x x x ´ x x x ´ x x x ` x x x ´´ x x x x ` x x x x ` x x x x ` x x x ` x x x ´ x x x q , x “ ´ x x p´ x x ´ x x ` x x x ´ x x ´ x x x ´ x x `` x x x ´ x x ` x x x ´ x x x x ` x x x x ` x x x ` x x x `` x x x ` x x q , x “ . Similarly with before, last equation confirms that infinity remains invariant under the flow of thesystem. Now, solving the system t x “ x “ x “ x “ u , we get exactly six fixed points at theinfinity of M G , given by: p , , , q , p . , . , . , . q , p . , . , . , . q , p . , . , . , . q , p . , . , . , . q , p . , . , . , . q . As before, these fixed points induce via the central projection F the explicit presentations e i , i “ , . . . ,
6. The fixed point represented by e “ p , , , , q corresponds to the unique invariantK¨ahler-Einstein metric and eigenvalues calculations show that it possesses a 4-dimensional stablemanifold, contained in the infinity of M G , and a 1-dimensional unstable manifold which coincideswith a straight line tending to the origin. The other three fixed points, corresponding to non-K¨ahler,non-isometric invariant Einstein metrics (see [CS14]), have a 3-dimensional stable manifold, locatedin the infinity of M G , and a 2-dimensional unstable manifold, one direction of which is the straightline in M G tending to the origin. The ancient solutions g i p t q “ p ´ λ i t q ¨ e i , i “ , . . . , p´8 , { λ i q , where the corresponding Einstein constant λ i is given by λ “ { , for e “ p , , , , q ,λ “ . , for e “ p , . , . , . , . q ,λ “ . , for e “ p , . , . , . , . q ,λ “ . , for e “ p , . , . , . , . q ,λ “ . , for e “ p , . , . , . , . q ,λ “ . , for e “ p , . , . , . , . q . Using these constants and by taking limits, as above, we obtain the rest claims for r “ Case r “ . To reduce the system (3.4) to a polynomial dynamical system, a short computation shows that we must multiply with the positive term 60 x x x x x x . This gives the following: x “ ´ x x x p x x x x ´ x x x x x ` x x x x ´ x x x x x `` x x x x ´ x x x x x ` x x x x ´ x x x x x ` x x x x x x ´´ x x x x x ´ x x x x x ´ x x x x x ´ x x x x x ´ x x x x x q , x “ x x p´ x x x x ` x x x x x ´ x x x x ` x x x x x `` x x x x x ´ x x x x x ´ x x x x x x ´ x x x x x `` x x x x x x ` x x x x x ` x x x x x ` x x x x x q , x “ x x x x p x x x x ´ x x x ` x x x x ´ x x x ` x x x x ´´ x x x x x ` x x x x ´ x x x ` x x x x ` x x x x ` x x x x x q , x “ x x x p x x x x ´ x x x x x ` x x x x ´ x x x x `` x x x x ´ x x x x x ` x x x x ´ x x x x ´ x x x x x `` x x x x ` x x x x x q , x “ x x x x p x x x x ´ x x x x ` x x x x ` x x x x ´´ x x x x x ` x x x x ` x x x x ´ x x x x ´ x x x x ` x x x x q , x “ x x x p x x x x ` x x x x ´ x x x x x ` x x x x `` x x x x ´ x x x x ´ x x x x ´ x x x x x q . Note that the right-hand side consists of homogeneous polynomials of degree 9. Hence again wecan apply the Poincar´e compactification procedure to study the behaviour of this system at infinityof M G . Using the expressions given above, the system at infinity, written in the U chart andsetting x “
0, reads as follows: x “ ´ x x p´ x x x ` x x x ´ x x x x ` x x x x ´ x x x ` x x x ´´ x x x x ` x x x x ´ x x x ` x x x x ´ x x x x `` x x x x ` x x x x x ´ x x x x x ` x x x x x ` x x x x ´´ x x x x ` x x x x ´ x x x x ` x x x x ´ x x x x q , x “ ´ x x p´ x x x ` x x x x ´ x x x ´ x x x x ` x x x `` x x x x ´ x x x x ` x x x x ´ x x x x ` x x x x x ´´ x x x x x ´ x x x x x ` x x x x x ` x x x x ´´ x x x x x ` x x x x ´ x x x x ` x x x x ´ x x x x q , x “ ´ x x p´ x x x ´ x x x ` x x x x ` x x x x ´ x x x `` x x x x ´ x x x ` x x x x ´ x x x x ´ x x x x `` x x x x ´ x x x x x ` x x x x x ` x x x x x `` x x x x ` x x x x ´ x x x x ` x x x x ´ x x x x q , x “ ´ x x x p´ x x x ` x x x x ´ x x x ´ x x x ` x x x x ´´ x x x ´ x x x x ´ x x x ` x x x x ´ x x x ` x x x x ´´ x x x x x ` x x x x x ` x x x x ` x x x x ` x x x x `` x x x ´ x x x x q , x “ ´ x x p´ x x x ´ x x x ` x x x x ´ x x x ´ x x x x ´´ x x x ` x x x x ´ x x x x ` x x x x ´´ x x x x ` x x x x x ´ x x x x x ` x x x x x `` x x x x ` x x x x x ` x x x x ` x x x x `` x x x ` x x x x q , x “ . NCIENT SOLUTIONS OF THE HOMOGENEOUS RICCI FLOW 25
By the last equation one deduces that the infinity of M G remains invariant under the flow of thesystem. Solving now the system t x “ x “ x “ x “ x “ u , we get exactly five fixed pointsgiven by: p , , , , q , p . , . , . , . , . q , p . , . , . , . , . q , p . , . , . , . , . q , p . , . , . , . , . q . These solutions, projected through the central projection F , induce the fixed points e i at infinityof M G , namely e “ p , , , , , q , e “ p , . , . , . , . , . q , e “ p , . , . , . , . , . q , e “ p , . , . , . , . , . q , e “ p , . , . , . , . , . q . These points are in bijective correspondence with non-isometric invariant Einstein metrics on M “ G { K (see [CS14]), and determine the non-collapsed ancient solutions g i p t q “ p ´ λ i t q ¨ e i , wherein this case the corresponding Einstein constant λ i has the form λ “ { , λ “ . , λ “ . , λ “ . , λ “ . . All the other conclusions are obtained in an analogous way as before. This completes the proof. (cid:3)
Remark 3.2.
For r ě 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.Let us now take a view of the limit behaviour of the scalar curvature for the solutions g i p t q givenin Theorem 3.1, and verify the statements of Proposition 2.4. We do this by treating exampleswith r “ , Example 3.3.
Let M “ G { K be a flag manifold with r “
2. Then, an application of (3.2) showsthat both
Scal p g i p t qq are positive hyperbolas , given by Scal p g p t qq “ p d ` d qp d ` d q p d ` d q ´ t p d ` d q , Scal p g p t qq “ p d ` d qp d ` d d ` d q ` p d ` d d ` d q ´ t p d ` d d ` d q ˘ , respectively. Therefore, they both are increasing in p´8 , λ i q , which is the open interval which aredefined, i.e. Scal p g p t qq and Scal p g p t qq are strictly positive. The limit of Scal p g i p t qq as t Ñ λ i must be considered only from below, and it is direct to check thatlim t Ñ λi Scal p g i p t qq “ `8 , lim t Ñ´8
Scal p g i p t qq “ , @ i “ , . Hence,
Scal p g i p t qq ą
0, for any t P p´8 , λ i q and for any i “ ,
2, as it should be for ancientsolutions in combination with the non-existence of Ricci flat metrics ([Kot10, Laf15]). Let uspresent the graphs of
Scal p g p t qq and Scal p g p t qq for the flag spaces G { U p q and F { Sp p q ˆ U p q ,both with r “
2. We list all the related details below, together with the graphs of
Scal p g i p t qq forthe corresponding intervals of definition p´8 , λ i q (see Figure 1 and 2, for i “ ,
2, respectively). M “ G { K d d { λ { λ Scal p g p t qq Scal p g p t qq G { U p q ´ t ´ t F { Sp p q ˆ U p q
28 2 9/8 72/71 96072 ´ t ´ t t = - - - - - t ( g ( t )) t = - - - - - t ( g ( t )) Figure 1.
The graph of
Scal p g p t qq for G { U p q (left) and F { Sp p q ˆ U p q (right). t = - - - - - t ( g ( t )) t = - - - - - t ( g ( t )) Figure 2.
The graph of
Scal p g p t qq for G { U p q (left) and F { Sp p q ˆ U p q (right).The graphs now of the ricci components of the solutions g j p t q ,which we denote by ric ij p t q : “ ric i p g j p t qq , ď i ď r , ď j ď N, 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 “ g p t q passing through the invariant K¨ahler-Einstein metric g p q “ g KE , we compute ric p t q “ ric p g p t qq “ ric p t q “ ric p g p t qq “ d ` d p d ` d q ´ p d ` d q t . For the solution g p t q passing from the non-K¨ahler invariant Einstein metric g p q “ g E we obtain ric p t q “ ric p g p t qq “ ric p t q “ ric p g p t qq “ d ` d d ` d p d ` d d ` d q ´ p d ` d ` d d q t . Note that the equalities ric p t q “ ric p t q and ric p t q “ ric p t q occur since g i p t q p i “ , q are both1-parameter families of invariant Einstein metrics on M “ G { K (as we mentioned in Section 2).For instance, for G { U with r “
2, the above formulas reduce to ric p t q “ ric p t q “ ´ t , ric p t q “ ric p t q “ ´ t , with ric p q “ ric p q “ λ “ { ric p q “ ric p q “ λ “ {
24, respectively. Thecorresponding graphs are given in Figure 3.
NCIENT SOLUTIONS OF THE HOMOGENEOUS RICCI FLOW 27 t = ric ( )= λ - - - - - t ( t ) t = ric ( )= λ - - - - - t ( t ) Figure 3.
The graphs of ric p t q “ ric p t q and ric p t q “ ric p t q for G { U p q with m “ m ‘ m . Example 3.4.
Let M “ G { K be a flag manifold with b p M q “ r “
3. Let us describe theasymptotic properties of the scalar curvature
Scal p g p t qq , related to the solution g p t q “ p ´ λ t q¨ e only, where e “ p , , q is the fixed point corresponding to the invariant K¨ahler-Einstein metric g p q ” g KE . By (3.3) we see that Scal p g p t qq is the positive hyperbola given by Scal p g p t qq “ p d ` d ` d qp d ` d ` d q p d ` d ` d q ` t p d ` d ` d q . Thus,
Scal p g p t qq increases on the open interval p´8 , λ q , where g p t q is dedined. Note that thevalue Scal p g p qq “ p d ` d ` d qp d ` d ` d q p d ` d ` d q equals to the scalar curvature of the K¨ahler-Einsteinmetric g p q . The limit of Scal p g p t qq as t Ñ λ must be considered only from below, and it followsthat lim t Ñ λ Scal p g p t qq “ `8 , lim t Ñ´8
Scal p g p t qq “ . For example for G { U and r “
3, we compute λ “ {
24, so
Scal p g p t qq “ ´ t , t P p´8 , q , Scal p g p qq “ Scal p g p t qq is given by Scal ( g ( )) t = - - - - - t ( g ( t )) Figure 4.
The graph of
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Center for Research and Applications of Nonlinear Systems (CRANS), Department of Mathemat-ics, University of Patras, Rion 26500, Greece
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