ON THE STABILITY OF HARMONIC MAPS UNDER THE HOMOGENEOUS RICCI FLOW RAFAELAF.DOPRADOANDLINOGRAMA Abstract. In this work we study properties of stability and non-stability of harmonicmapsunderthehomogeneousRicciflow. Weprovideexampleswhere thestability(non-stability)ispreservedundertheRicciflowandanexample 7 wheretheRicciflowdoesnotpreservethestabilityofanharmonicmap. 1 0 2 n a 1. Introduction J Let (M,g) be a Riemannianmanifold. The Ricci flow is a1-parameter family of 0 2 metrics g(t) in M with initial metric g that satisfies the Ricci flow equation ] G ∂ (1.1) g(t)=−2Ric(g(t)). D ∂t . TheRicciflowwasfirstintroducedbyHamiltonbasedontheworkofEells-Sampson h as pointed out by him in [11]. One of the main ideas is to start with any metric t a g of strictly positive Ricci curvature and try to improve it by means of a heat m equation. Similar methods were used by Eells-Sampson in the context of harmonic [ maps(see[7]). Inthesamework[11]HamiltonshowedthatpositiveRiccicurvature is preserved by (1.1) on closed 3-manifolds. Hamilton also proved that the same 2 v results hold for positive isotropic curvature in closed 4-manifolds [12]. However, 6 some curvatures conditions may not be preserved by the Ricci flow. For example, 9 B¨ohm and Wilking [3] exhibited homogeneous metrics with sec > 0 that develop 3 mixedRiccicurvatureindimension12,andmixedsectionalcurvatureindimension 5 6. Abiev and Nikonorov [1] proved that, for all Wallach spaces, the normalized 0 . Ricci flow evolves all generic invariant Riemannian metrics with positive sectional 1 curvature into metrics with mixed sectional curvature and, more recently, Bettiol 0 andKrishnan[2]exhibitexamplesofclosed4-manifoldswithnonnegativesectional 7 1 curvature that develop mixed curvature under Ricci flow. There are several recent : papers about Ricci flow (and other geometric flows) in homogeneous spaces, for v i example, [1], [8], [9], [15], [16] and references therein. X Inthispaper,weareinterestedinstudyingthestabilityofharmonicmapsunder r the homogeneous Ricci flow. More specific, we want to know if the Ricci flow pre- a servesstabilityofaclassofharmonicmapsfromRiemannsurfacestohomogeneous spaces. Since harmonic maps are critical points of the energy functional, we are interested in whether the second variation of the energy of these maps are positive or non-negative a certain variation. In this sense, we say that a harmonic map is stable if the second variation of the energy of this map is non-negative for every variation. The harmonic maps we are going to consider are the so called generalized holomorhic-horizontal. They were first introduced by Bryant [5]. These maps LGissupportedbyFapespgrantno. 2014/17337-0and2012/18780-0. RPissupportedbyCNPq grant142259/2015-2. 1 2 RAFAELAF.DOPRADOANDLINOGRAMA are equiharmonic, that is, harmonic with respect to any invariant metric. Equihar- monicmapswereintroducedbyNegreirosin[19]andseveralresultsaboutstability and non-stability of those kind of maps were proved in [18]. In [9] and [8] we have a study on the behavior of the homogeneous Ricci flow of left-invariant metrics on three types of homogeneous manifolds (1.2) SO(2(m+k)+1)/(U(k)×SO(2m+1)), (1.3) Sp(m+k)/(U(m)×Sp(k)), and (1.4) SU(3)/T2 by a dynamical system point of view. We are interested in studying stability and non-stability of generalized holomorphic-horizontal maps in these three classes of homogeneous manifolds. The homogeneous spaces described in (1.2), (1.3) and (1.4) belongs to a large class of homogeneous spaces called generalized flag manifolds and these spaces ap- pear in several well known situations. For example: the family (1.2) includes the non-symmetric complex homogeneous space CP2n+1 =Sp(n+1)/(Sp(n)×U(1)) - the total space ofa twistor fibration over HPn; the family (1.3) includes theCalabi twistorspaceSO(2n+1)/U(n)usedintheconstructionofharmonicmapsfromS2 to S2n; and the Wallach flag manifold SU(3)/T2 is a 6-dimensional homogeneous space that admits invariant metric with positive sectional curvature. By analyzing the dynamics of the homogeneous Ricci flow together with the re- sultsconcerningstability/unstabilityofequiharmonicmaps,weprovethefollowing results: Theorem A The homogeneous Ricci flow preserves the stability (respectively non- stability) of a generalized holomorphic-horizontal map on the homogeneous spaces SO(2(m+k)+1)/(U(k)×SO(2m+1)) and Sp(m+k)/(U(m)×Sp(k)). Theorem B The homogeneous Ricci flow does not preserve stability of a gen- eralized holomorphic-horizontal map on the homogeneous space SU(3)/T2. Thispaperisorganizedasfollows. Insection1, werecallthemainresultsabout the geometry of flag manifolds. In section 2, we review some of the theory of holomorphicmapsonflagmanifolds,includingtheresultsonwhetherageneralized holomorphic-horizontal map is stable or unstable. In section 3, we first recall the homogeneousRicciflowofinvariantmetriconSO(2(m+k)+1)/(U(k)×SO(2m+1)), Sp(m+k)/(U(m)×Sp(k)) and SU(3)/T2 and then prove our results. 2. The geometry of generalized flag manifolds 2.1. Generalizedflagmanifolds. LetgbeacomplexsemisimpleLiealgebraand G the correspondent Lie group. Let h be a Cartan subalgebra of g and denote by Π the set of roots of (g,h). Then (cid:88) g=h⊕ g , α α∈Π where g = {X ∈ g; ∀H ∈ h, [H,X] = α(H)X} denotes the root space (complex α 1-dimensional). Let(·,·)betheCartan-KillingformoftheLiealgebragandfixaWeylbasisofg, that is, choose vectors X ∈ g such that (X ,X ) = 1, [X ,X ] = m X , α α α −α α β α,β α+β ON THE STABILITY OF HARMONIC MAPS UNDER THE HOMOGENEOUS RICCI FLOW 3 where m ∈ R satisfying m = −m and m = 0 if α+β ∈/ Π (see α,β −α,−β α,β α,β Helgason [13] for details). Givenα∈h∗, defineH byα(·)=(H ,·)(remembertheCartan-Killingformis α α nondegenerate on h) and denote hR the real subspace generated by Hα, α∈Π. In the same way, h∗ denotes the real subspace of the dual g∗ generated by the roots. R DenotebyΠ+ thesetofpositiverootsandΣsetofsimpleroots. IfΘisasubset ofsimpleroots,denoteby(cid:104)Θ(cid:105)thesetofrootsgeneratedbyΘand(cid:104)Θ(cid:105)± =(cid:104)Θ(cid:105)∩Π±. Therefore, (cid:88) (cid:88) (cid:88) (cid:88) g=h⊕ g ⊕ g ⊕ g ⊕ g . α −α β −β α∈(cid:104)Θ(cid:105)+ α∈(cid:104)Θ(cid:105)+ β∈Π+\(cid:104)Θ(cid:105) β∈Π+\(cid:104)Θ(cid:105) The parabolic sub-algebra determined by Θ is given by (cid:88) (cid:88) p =h⊕ g ⊕ g Θ α α α∈(cid:104)Θ(cid:105)− α∈Π+ Define (cid:88) q = g , Θ −β β∈Π+\(cid:104)Θ(cid:105) and therefore g=q ⊕p . Θ Θ The generalized flag manifold F (associated to p ) is the homogeneous space Θ Θ G F = , Θ P Θ where the subgroup P is the normalizer of p in G. Θ Θ Recall the compact real form of g is the real subalgebra given by u=spanR{ihR,Aα,iSα :α∈Π}, where A =X −X and S =X +X . α α −α α α −α Let U = expu be the corresponding compact real form of G and put K = Θ P ∩U. TheLiegroupU actstransitivelyonthegeneralizedflagmanifoldF with Θ Θ isotropy subgroup K . Therefore we have also F =U/K Θ Θ Θ Let k be the Lie algebra of K and denote by kC its complexification. Thus, Θ Θ Θ k =u∩p and Θ Θ C (cid:88) k =h⊕ g . Θ α α∈(cid:104)Θ(cid:105) Let o = eK be the origin (trivial coset) of F . Then the tangent space T F Θ Θ o Θ identifies with the orthogonal complement of k in u, that is, Θ (cid:88) T F =m =span {A ,iS :α∈/ (cid:104)Θ(cid:105)}= u , o Θ Θ R α α α α∈Π\(cid:104)Θ(cid:105) where u = (g ⊕g )∩u = span {A ,iS }. By complexifying m , we obtain α α −α R α α Θ the complex tangent space of TCF , which can be identified with o Θ C (cid:88) m =q = g . Θ Θ β β∈Π\(cid:104)Θ(cid:105) 2.2. Almost complex structures. An K -invariantalmost complexstructure J Θ on F is completely determined by its value at the origin, that is, by J :m →m Θ Θ Θ inthetangentspaceofF attheorigin. ThemapJ satisfiesJ2 =−1andcommutes Θ with the adjoint action of K on m . We also denote by J its complexification to Θ Θ q . Θ The invariance of J entails that J(g ) = g for all σ ∈ Π(Θ). The eigenvalues √ σ σ of J are ± −1 and the eigenvectors in q are X , α ∈ Π\(cid:104)Θ(cid:105). Hence, in each Θ α 4 RAFAELAF.DOPRADOANDLINOGRAMA √ irreducible component, J = −1(cid:15) Id, where (cid:15) = ±1 and (cid:15) = −(cid:15) . An U- σ σ −σ σ invariantstructureonF isthencompletelydescribedbyasetofsigns{(cid:15) } Θ σ σ∈Π(Θ) with (cid:15) =−(cid:15) . −σ σ √ The eigenvectors associated to −1 are of type (1,0) while the eigenvectors √ associated to − −1 are of type (0,1). Thus, the (1,0) vectors at the origin are multiples of X , (cid:15) = 1, and the (0,1) vectors are also multiples of X , (cid:15) = −1, α σ α σ where α∈σ. Also, (2.1) T F(1,0) = (cid:88) E T F0,1 = (cid:88) E . x Θ (cid:15)σσ x Θ (cid:15)−σσ σ∈Π(Θ)+ σ∈Π(Θ)+ Since F is a homogeneous space of a complex Lie group, it has a natural struc- Θ tureofacomplexmanifold. TheassociatedintegrablealmostcomplexstructureJ c is given by (cid:15) =1 if the roots in σ are all negative. The conjugate structure −J is σ c also integrable. 2.3. Isotropy representation. The adjoint representations of k and K leave Θ Θ m invariant,sothatwegetawell-definedrepresentationofbothk andK inm . Θ Θ Θ Θ Analogously, thecomplextangentspaceq isinvariantundertheadjointrepresen- Θ tation of kC and we can define the complexification of the isotropy representation Θ from kC to Aut(mC). Since the representation is semissimple, we can decompose Θ Θ it into irreducible components, where each irreducible component is a sum of root spaces. We will denote an irreducible component of mC = q by g , where σ is the Θ Θ σ subset of roots such that (cid:88) g = g , σ α α∈σ and we write Π(Θ) for the set of σ’s. Then, we have (cid:88) q = g . Θ σ σ∈Π(Θ) The roots in each irreducible component σ ∈ Π(Θ) are either all positive or all negative,sowewriteΠ(Θ)+ andΠ(Θ)− forthesetofthoseirreduciblecomponents containing only positive roots and negative roots, respectively. Denote by Σ(Θ) the set of σ ∈Π(Θ) that has height 1 module (cid:104)Θ(cid:105), i.e, Σ(Θ)={σ ∈Π(Θ): the height of σ ∈Π(Θ) is 1}. Since Ad(k)(g )=g , for each σ ∈Π(Θ), we have a well defined complex plane σ σ field on F given by Θ E (k·o)=k (g ) σ ∗ σ and, for any x∈F , we have Θ TCF = (cid:88) E (x). x Θ σ σ∈Π(Θ) 2.4. Invarianmetrics. Thereisa1-1correspondencebetweenU-invariantmetrics gonF andAd(K )-invariantscalarproductsBonm (seeforinstance[14]). Any Θ Θ Θ B can be written as B(X,Y)=(cid:104)X,Y(cid:105) =−(ΛX,Y), Λ with X,Y ∈ m , where Λ is an Ad(K )-invariant positive symmetric operator on Θ Θ m with respect to the Cartan-Killing form. The scalar product B(.,.)=(cid:104)X,Y(cid:105) Θ Λ admits a natural extension to a symmetric bilinear form on mC =q . We will use Θ Θ the same notation for this extension. ON THE STABILITY OF HARMONIC MAPS UNDER THE HOMOGENEOUS RICCI FLOW 5 As a consequence of Schur’s Lemma, in each irreducible component of q we Θ have Λ=λ , with λ =λ >0. σ −σ σ Remark 2.1. Inthenextsectionsweabusenotationanddenoteaninvariantmetric g on F just by Λ=(λ ) , that is, a n-uple of positive real numbers indexed Θ σ σ∈Π(Θ) by the irreducible components of mC =q . Θ Θ 3. Stability of equiharmonic maps on generalized flag manifolds 3.1. J-holomorphic curves on F . If M = M2 is a Riemann surface and φ : Θ M →F isadifferentiablemap, weletdCφbethecomplexificationofthedifferen- Θ tial of φ. We endow F with a complex structure J and, as usual, decompose dCφ Θ into ∂φ(p) : TM(1,0) → TF(1,0) and ∂φ : TM(0,1) → TF(0,1), which are identified ∂z Θ ∂z Θ withvectorsinthecomplextangentspace. WeusethedecompositionofTCF into Θ irreducible components. By (2.1), we have ∂φ (cid:88) ∂φ (cid:88) (p)= φ (p) (p)= φ (p), ∂z (cid:15)σσ ∂z (cid:15)−σσ σ∈Π(Θ)+ σ∈Π(Θ)+ where, for each σ ∈ Π(Θ), the function φ : M2 → E takes value in E (φ(p)), σ σ σ p∈M. GivenanalmostcomplexstructureonF ,amapφ:M2 →F isJ-holomorphic Θ Θ if, for all p∈M, holds ∂φ (cid:88) (p)= φ (p)=0. ∂z (cid:15)−σσ σ∈Π(Θ)+ 3.2. Stability of equiharmonic maps on F . In[18]wereprovedseveralresults Θ about stability and non-stability of equiharmonic maps (maps that are harmonic with respect to any invariant metric) in a generalized flag manifold F . Θ Let us recall some of the results in [18]. Consider M2 a compact Riemann surface equipped with a metric g, let (N,h) be a compact Riemannian manifold and φ:(M2,g)→(N,h) a differentiable map. The energy of φ is given by 1(cid:90) E(φ)= |dφ|2ω , 2 g M where ω is the volume measure defined by the metric g and |dφ| is the Hilbert- g Schmidt norm of dφ. The differentiable map φ is harmonic if it is a critical point of the energy functional. Let us restrict ourselves to harmonic maps from compact Riemann surfaces to a generalized flag manifold F . Given a harmonic map φ : (M2,g) → (F ,ds2), we Θ Θ Λ consider perturbed maps φt(p) given by φt(p)=exp(tq(p))·φ(p), whereq :M →uisasmoothmap. Thesecondvariationoftheenergyofφ,denoted by Iφ(q), is given by Λ Iφ(q)= d2 (cid:12)(cid:12)(cid:12) E(φt). Λ dt2(cid:12) t=0 Definition 3.1. Let φ : (M2,g) → (F ,ds2) be an arbitrary harmonic map. We Θ Λ say that φ is stable if Iφ(q) ≥ 0 for any variation q : M2 → g. Otherwise, we say Λ that φ is unstable. We are interested in the following situation: let Λ be a invariant metric on F 0 Θ andΛ anotherinvariantmetricobtainedfromΛ byaspecialkindofperturbation 1 0 (defined bellow). Suppose that the map φ:M2 →F is harmonic with respect to Θ 6 RAFAELAF.DOPRADOANDLINOGRAMA both invariant metrics Λ and Λ . One of the main contribution of [18] is provide 0 1 the understanding of the behavior of Iφ in therms of Iφ . Λ1 Λ0 Definition 3.2. LetP beasubsetofΠ(Θ). AninvariantmetricΛ# =(λ#) σ σ∈Π(Θ) is called a P-perturbation of Λ if the following holds: (1) λ# =λ for all σ ∈P; σ σ (2) λ# =λ +ξ >0, ξ ∈R, for all σ ∈Π(Θ)\P. σ σ σ σ Since we need to consider maps that are harmonic with respect to the invariant metricΛandtheperturbedmetricΛ#,itisnaturaltoconsiderequiharmonicmaps. Definition 3.3. A map φ:M2 →F is called equihamonic map if it is harmonic Θ with respect any invariant metric on F . Θ Examplesofequiharmonicmapsarethesocalledgeneralizedholomorphic-horizontal maps, whose definition is given bellow. Definition 3.4. A map φ : M2 → (F ,J) is called generalized holomorphic- Θ horizontal if it is J-holomorphic and satisfies φ =0 if σ ∈Π(Θ)\Σ(Θ). σ Remark 3.5. Here we are using the same terminology of Bryant [5]. In [6], these maps are called super-horizontal maps. Observe that, when working with generalized holomorphic-horizontal maps, we are taking P = Σ(Θ). Also, the following result guarantees that those maps are equiharmonic maps. Theorem 3.6. [10] If φ is a generalized holomorphic-horizontal map, then φ is equiharmonic. Thefollowingtheoremsarecrucialintheunderstandingofthestabilityofgener- alized holomorphic maps under the Ricci flow of a perturbed invariant metric. We start with a classical result of Lichnerowicz: Theorem 3.7. [17] Let φ : (M2,J,g) → (F ,J,ds2) be a J-holomorphic map, Θ Λ where (F ,J,ds2) is a K¨ahler structure. Then, φ is harmonic and stable. Θ Λ NowweconsidersomespecialP-perturbationofainvariantK¨ahlerstructureon F in order to construct several examples of stable/unstable harmonic maps. Θ Theorem 3.8. [18] Let φ : M2 → F be a generalized holomorphic-horizontal Θ map and (F ,J,ds2) a K¨ahler structure. Consider a P-perturbation Λ# of this Θ Λ structure, with ξ ≥ 0 for all σ ∈ Π(Θ)\P. Then φ : (M2,g) → (F ,ds2 ) is σ Θ Λ# stable. Theorem 3.9. [18] Let φ : M2 → F be a generalized holomorphic-horizontal Θ map and take a K¨ahler structure (F ,J,ds2) with Λ = (λ ) . Suppose that Θ Λ σ σ∈Π(Θ) Λ# = (λ#) is another metric such that, for some σ ∈ Π(Θ)\Σ(Θ), the σ σ∈Π(Θ) 0 inequality ξ =λ# −λ <0 holds. Then, φ is unstable with respect to Λ#. σ0 σ0 σ0 4. Stability on F under the Ricci Flow Θ Inthissectionwestudythestabilityofageneralizedholomorphic-horizontalmap φ:M2 →F underthehomogeneousRicciflowofaperturbedinvariantmetricfor Θ threetypesofflagmanifolds: SO(2n+1)/(U(k)×SO(2n+1)),Sp(n)/(U(m)×Sp(k)) and SU(3)/T2. For more details about this topic, see [8] and [9]. 4.1. HomogeneousRicciFlow. Wewillbeginbyreviewingtheglobalbehaviour ofthehomogeneousRicciflowonSO(2n+1)/(U(k)×SO(2n+1)), Sp(n)/(U(m)× Sp(k)) and SU(3)/T2. ON THE STABILITY OF HARMONIC MAPS UNDER THE HOMOGENEOUS RICCI FLOW 7 4.1.1. Ricci flow on SO(2(m+k)+1)/(U(k)×SO(2m+1)) and Sp(m+k)/(U(m)× Sp(k)). The isotropy representation of the families SO(2(m + k) + 1)/(U(k) × SO(2m+1)) and Sp(m+k)/(U(m)×Sp(k)) decompose into two non-equivalent isotropy summands, that is, m = m ⊕m . We keep our notation and denote an Θ 1 2 invariant metric just by Λ = (λ ,λ ). The Ricci tensor of an invariant metric Λ 1 2 is again an invariant tensor, and therefore completely determined by its value at the origin of the homogeneous space and constant in each irreducible component of the isotropy representation. In the case of SO(2n+1)/(U(k)×SO(2n+1)) and Sp(n)/(U(m)×Sp(k)), the components of the Ricci tensor are given, respectively, by  2(m−1) 1+2k λ2 r1 =− 2n−1 − 2(2n−1)λ122, r2 =−2nn+−k1 − 2(m2n−−11)(λ22−(λλ1λ−λ2)2), 1 2 and  2+2m) 2k λ2 r1 =− 2n+2 − 4n+4λ122, r2 =−4m4+n+4k4+3 + 146mn++126λλ1. 2 The Ricci flow equation on the manifold M is defined by ∂g(t) (4.1) =−2Ric(g(t)), ∂t where Ric(g) is the Ricci tensor of the Riemannian metric g. The solution of this equation, the so called Ricci flow, is a 1-parameter family of metrics g(t) in M. The homogeneous Ricci flow equation for invariant metrics is given by the fol- lowing systems of ODEs:  2(m−1) 1+2k x2 x˙ = 2n−1 + 2(2n−1)y2, (4.2) y˙ = n+k + m−1 (y2−(x−y)2), 2n−1 2(2n−1) xy for SO(2n+1) , n=m+k, m>1 and k (cid:54)=1, U(k)×SO(2m+1) and  2m+2) 2k x2 x˙ = 2n+2 + 4n+4y2, (4.3) y˙ = 4m+4k+3 − 4m+2 x, 4n+4 16n+16y for Sp(n) , n=m+k, m≥1 and k ≥1. U(m)×Sp(k) In [9], using the Poincar´e compactification on the space of invariant Riemann- ian metrics, the dynamics of (4.2) and (4.3) was completely described as follows, respectively. 8 RAFAELAF.DOPRADOANDLINOGRAMA (cid:18) (cid:19) 2(m−1) γ (t)= t,t , γ (t)=(2t,t), in the case (4.2) 1 m+2k 2 and (cid:18) (cid:19) (cid:18) (cid:19) 14k+2m+1 1 γ (t)= t, t , γ (t)= t, t , in the case (4.3). 1 4 m+1 2 2 The global behavior of the Ricci flow on both generalized flag manifolds is de- scribed using its phase portrait (see Figure 4.1.1). Figure 1. Phase portrait of the Ricci flow on SO(2n+1)/(U(k)×SO(2n+1)) and on Sp(n)/(U(m)×Sp(k)). Onecandescribepreciselythe“asymptoticbehavior”oftheflowslineoftheRicci flow. Let g be an invariant metric and g(t) the Ricci flow with initial condition 0 g . We will denote by g the limit lim g(t). 0 ∞ t→∞ Theorem4.1. [9]Letg beaninvariantmetriconSO(2n+1)/(U(k)×SO(2n+1)) 0 or Sp(n)/(U(m)×Sp(k)) and R , R , R , γ and γ as described in Figure 4.1.1. 1 2 3 1 2 We have: a) if g ∈R ∪R ∪γ then g is the Einsten (non-K¨ahler) metric; 0 1 2 1 ∞ b) if g ∈γ then g is the Ka¨hler-Einstein metric. 0 2 ∞ c) if g ∈R , consider the natural fibration from a flag manifold in a symmet- 0 3 ric space G/K →G/H. Then, the Ricci flow g(t) with g(0)=g evolves in 0 such a way that the diameter of the base of this fibration converges to zero when t→∞. 4.1.2. Ricci flow on the space SU(3)/T2. The isotropy representation of the space SU(3)/T2 decomposes into three non-equivalent irreducible components, that is, m = m ⊕m ⊕m . So, we keep our notation and denote an invariant metric Θ 1 2 3 by (λ ,λ ,λ ). In the case of the space SU(3)/T2, the components of the Ricci 12 13 23 tensor are given by  1 1 (cid:18) λ λ λ (cid:19) r12 = 2λ112 + 112(cid:18)λ1λ31λ223 − λ1λ21λ323 − λ1λ22λ313(cid:19), (4.4) r = + 13 − 12 − 23 , 13 2λ 12 λ λ λ λ λ λ r23 = 2λ113 + 112(cid:18)λ1λ22λ323 − λ1λ31λ323 − λ1λ21λ213(cid:19). 23 12 13 23 12 23 13 ON THE STABILITY OF HARMONIC MAPS UNDER THE HOMOGENEOUS RICCI FLOW 9 The Ricci flow equation system for a left-invariant metric on SU(3)/T2 is given by (4.5) λ˙ =−2r , 1≤i<j ≤3. ij ij Theinvariantlinesof(4.5)canbedescribedasfollows: defineγ (t):=tp(cid:48),where √ j j p(cid:48) =(1/ρ ,(1/ρ )(2+2 2),1/ρ ), 1 1 1 1 p(cid:48) =(1/ρ ,1/ρ ,1/ρ ), 2 2 2 √ 2 √ p(cid:48) =(ρ (−1/2+ 2/2),ρ (−1/2+ 2/2),ρ ), 3 3 √ 3 3 p(cid:48) =((1/ρ )(2+2 2),1/ρ ,1/ρ ), 4 1 1 1 (cid:113) √ (cid:113) √ √ 1/ρ = 1+2(−1/2+1/2 2)2, ρ = 2+(2+ 2)2 and ρ = 3. 3 1 3 We remark that γ ,γ ,γ and γ are solutions for (4.4). It is well know that 1 2 3 4 the manifold SU(3)/T2 admits four invariant Einstein metrics: three are K¨ahler- Einstein metrics, represented by γ , γ , γ , and the normal Einstein metric (non- 1 3 4 K¨ahler), represented by γ . 2 Usingtechniquesfromdynamicalsystems(Poincar´ecompactifications,Lyapunov exponents), the behavior of the Ricci flow near the normal Einstein metric is de- scribed as follows. Theorem 4.2. [8] Consider ε > 0 sufficiently small and Λ = (λ ,λ ,λ ) with 12 13 23 λ > 0, (cid:107) Λ (cid:107)> δ for δ > 1/2 and d(Λ,γ ) < ε. Let g be the Ricci flow with g ij 2 t 0 the metric defined by (λ ,λ ,λ ). Then g is a normal (Einstein) metric. In 12 13 23 ∞ particular, if Λ∈/ γ , then g is left-invariant and g is bi-invariant. 2 0 ∞ 4.2. Stability on SO(2(m+k)+1)/(U(k)×SO(2m+1)), Sp(m+k)/(U(m)× Sp(k)) and SU(3)/T2 under the homogeneous Ricci flow. Now we are going toestablishhowthestabilityofageneralizedholomorphic-horizontalmapφ:M2 → F behaves under the Ricci flow of a perturbed invariant metric of SO(2(m+k)+ Θ 1)/(U(k)×SO(2m+1)) and Sp(m+k)/(U(m)×Sp(k)). In the next Theorem, F Θ will denote one of those two types of flags. Theorem 4.3. Let φ:M2 →F be a generalized holomorphic-horizontal map and Θ (F ,J,g ) a K¨ahler structure with g = Λ = (λ ,λ ). Suppose that g# = Λ# = Θ 0 0 1 2 0 (λ ,λ#) is any P-perturbation of Λ. The following holds: 1 2 (1) If 0<λ# <λ , then φ is unstable with respect to g# and remains unstable 2 2 0 under the Ricci flow, g#, with initial condition g#; t 0 (2) If 0 < λ < λ#, then φ is stable with respect to g# and remains stable 2 2 0 under the Ricci flow, g#, with initial condition g# and for t<∞. t 0 Proof. Borel [4] described the K¨aler structures on a generalized flag manifold F . Θ In fact, (F ,J,g ) is K¨ahler if and only if J is integrable and, if α∈σ ∈Π(Θ) can Θ 0 be written as k (cid:88) α= n α i i i=1 k (cid:88) with α ∈σ ∈Σ(Θ), then λ = n α , where n ≥0 if α is positive. i i α i i i i=1 Inourcase,bothSO(2(m+k)+1)/(U(k)×SO(2m+1))andSp(m+k)/(U(m)× Sp(k)) have only two isotropy summands, i.e., Π(Θ)={σ ,σ }. Here, σ is the set 1 2 1 of roots whose height is one module (cid:104)Θ(cid:105), that is, Σ(Θ)=σ . If α∈σ , then α can 1 2 bewrittenasα=α +α ,whereα ,α ∈σ . Then,λ =2λ andg =Λ=(1,2) 1 2 1 2 1 σ2 σ1 is the only K¨ahler metric in F (up to scale). Θ Let g = (λ ,2λ ). Theorem 3.7 states that φ is stable with respect to g and, 0 1 1 0 by theorem 3.8, we have that φ is also stable with respect to the perturbed metric 10 RAFAELAF.DOPRADOANDLINOGRAMA g# =(λ ,2λ +ξ ), where ξ ≥0. Also, theorem 3.9 says that φ is unstable with 0,1 1 1 1 1 respecttotheperturbedmetricg# =(λ ,2λ −ξ ),whereξ <0and2λ −ξ >0. 0,2 1 1 2 2 1 2 Observethat,accordingdofigure4.1.1,g# ∈R ∪γ andg# ∈R ∪R ∪γ . Ifg# 0,1 3 2 0,2 1 2 1 t,1 and g# are the homogeneous Ricci flow for the metrics g# and g# , respectively, t,2 0,1 0,2 then φ is stable with respect to g# for finite t and is unstable with respect to g# t,2 t,1 for all t. (cid:3) Corollary 4.4. The homogeneous Ricci flow preserves the stability/non-stability of a generalized holomorphic-horizontal map on the homogeneous spaces SO(2(m+ k)+1)/(U(k)×SO(2m+1)) and Sp(m+k)/(U(m)×Sp(k)). Let us consider now the homogeneous space SU(3)/T2 in order to produce an example where the Ricci flow do not preserve the stability of harmonic maps. The following Lemma tell us about the non-stability of the normal metric on full flag manifolds. Lemma 4.5. [18] Consider (SU(3)/T2,g) equipped with the normal metric. Let φ:M2 →SU(3)/T2 beanarbitrarygeneralizedholomorphic-horizontalmap. Then, φ is unstable with respect to g. We have that the complexification of su(3) is the Lie algebra sl(3,C). The root spacedecompositionofsl(3,C)isgivenasfollows. ConsidertheCartansubalgebra hgivenbydiagonalmatricesoftracezero. Then,therootsystemofsl(3,C)relative to h is composed by α :=ε −ε , 1≤i(cid:54)=j ≤3, where ε is the functional given ij i j i by ε :diag{a ,a ,a }→a . A simple system of roots is Σ={α ,α }. i 1 2 3 i 12 23 Recall the SU(3)-invariant Einstein metrics are described as follow: the K¨ahler- Einstein metrics parametrized by Λ = (1,1,2), Λ = (1,2,1), Λ = (2,1,1); the 1 2 3 normal Einstein (non-K¨ahler) metric parametrized by Λ =(1,1,1). 4 In the proposition bellow, we choose a P-perturbation being P = {α } and 12 show that the equiharmonic 2-sphere φ : S2 → SU(3)/T2 subordinated to P is unstable under the Ricci flow. Theorem 4.6. Let φ : S2 → SU(3)/T2 be a J-holomorphic map subordinated to P = {α } and (SU(3)/T2,J,g ) a K¨ahler-Einstein structure with g = Λ = 12 0 0 (2,1,1). Consider ε sufficiently small and a P-perturbation of g given by g# = 0 0 Λ# =(2,2−ε,2−ε). Denote by g# the Ricci flow with initial condition g#. Then, t 0 φ is stable with respect to g# and is unstable with respect to g#. 0 ∞ Proof. Since g is a K¨ahler-Einstein metric, φ is stable with respect to g , by 0 0 Theorem 3.7. We also remark (using Theorem 3.8) that φ is stable with respect to the P-perturbed metric g#. 0 Also, (cid:107) Λ (cid:107)> 1/2. The invariant line represented by the normal metric on the phase portrait of the Ricci flow is given by √ √ √ γ (t)=(t/ 3,t/ 3,t/ 3). 2 Then, d(g#,γ )<ε. 0 2 By Theorem 4.2, g# is a normal Einstein and the result follows from Lemma 4.5. ∞ (cid:3) Remark 4.7. According to [6] the harmonic 2-sphere subordinated to a simple root {α } described in Theorem 4.6 is a generator of the second homotopy group 12 π (SU(3)/T2)≈Z⊕Z (each generator is represented by a simple root of sl(3,C)). 2