STABILITY OF SOLUTIONS OF CERTAIN EXTENDED RICCI FLOW SYSTEMS MICHAELBRADFORDWILLIAMS 3 1 0 Abstract. WeconsiderfourextendedRicciflowsystems—thatis,Ricciflow 2 coupledwithothergeometricflows—andprovedynamicalstabilityforcertain classes of stationary solutions of these flows. The systems include Ricci flow n coupledwithharmonicmapflow(studiedabstractlyandinthecontextofRicci a flow on warped products), Ricci flow coupled with both harmonic map flow J and Yang-Mills flow, and Ricci flow coupled with heat flow for the torsion of 6 a metric-compatible connection. The methods used to prove stability follow 1 a program outlined by Guenther, Isenberg, and Knopf, which uses maximal regularitytheoryforquasilinearparabolicsystemsandaresultofSimonett. ] G D . h Contents t a 1. Introduction 1 m 2. Harmonic-Ricci flow 5 [ 2.1. Setup and examples 5 1 2.2. Stability 6 v 3. Ricci flow on warped products 10 5 3.1. Setup 10 4 3.2. Estimates 11 9 3.3. Stability 13 3 . 4. Locally RN-invariant Ricci flow 16 1 4.1. Setup 16 0 3 4.2. Stability 16 1 5. Connection Ricci flow 20 : 5.1. Setup 20 v i 5.2. Stability 20 X Appendix A. Stability Theorem 22 r References 23 a 1. Introduction A fundamental problem in the study of differential equations is to determine the asymptotic behavior of solutions. This problem is central to the application of differential equations to Riemannian geometry. For example, Eells and Sampson demonstratedtheexistenceofharmonicmapsbyprovingthatsolutionsofthehar- monic map flow converge [6]. Hamilton placed strong restrictions on the topology 2010 Mathematics Subject Classification. 53C25,53C44. 1 2 M.B.WILLIAMS of three-manifolds admitting positive Ricci curvature by proving that solutions of Ricci flow converge [8] (see also [2,9]). One way to phrase this problem is in terms of stabilty of stationary solutions to thesystemofequationsinquestion:dosolutionswithinitialdatanearafixedpoint converge to that fixed point? For Ricci flow, whose fixed points include Einstein and Ricci-flat metrics, there are many stability results. To mention a few of these results in the compact case, Ye proved that Einstein metrics with certain curvature pinching properties are stable [27]; Guenther, Isen- berg and Knopf proved that certain flat and Ricci-flat metrics are stable [7], and some of these results were improved by Sˇeˇsum [19]; using the results of Sˇeˇsum, Dai, Wang, and Wei proved that K¨ahler-Einstein metrics with non-positive scalar curvature are stable [5]; Knopf and Young proved that hyperbolic space forms are stable [11]; Wu proved that complex hyperbolic space is stable [26]. We should note that these authors use various techniques and obtain stabilty relative to vari- ous topologies on the space of metrics. Thepurposeofthispaperistodescribestabilityofsolutionsofcertainextended Ricci flow systems, which often arise from natural geometric contexts involving Riemannian manifolds with additional structure. First, we consider Ricci flow coupled with the harmonic map flow; see Section 2. If φ : (M,g) → (N,γ) is a map of Riemannian manifolds, the harmonic-Ricci flow is the coupled system ∂ g =−2Rc+2c∇φ⊗∇φ t (1.1) ∂ φ=τ φ t g,γ where τ φ is the harmonic map Laplacian (or tension field) of φ, dφ⊗dφ=φ∗γ, g,γ andcisa(possiblytime-dependent)couplingconstant. Thiswasintroducedinthe caseN =Rin[13],andthegeneralcasewasaddressedin[17]. Thisflowalsoarises naturally in certain contexts; see Section 2.1 for examples. For this flow we demonstrate the stability of fixed points (g,φ), where g is an Einstein metric of negative curvature, and φ is constant. Theorem 1.2. Let φ : (Mn,g) → (N,γ) be a map of Riemannian manifolds. Suppose that M is compact and orientable, g is Einstein with negative sectional curvature (assumed to be constant when n = 2), and φ is constant. Then for any ρ∈(0,1), there exists θ ∈(ρ,1) such that the following holds. There exists a (1 + θ) little-H¨older neighborhood U of (g,φ) such that for all initialdata(g(cid:101)(0),φ(cid:101)(0))∈U,theuniquesolution(g(cid:101)(t),φ(cid:101)(t))ofcurvature-normalized harmonic-Ricci flow (2.2) exists for all t ≥ 0 and converges exponentially fast in the (2+ρ)-H¨older norm to a limit (g ,φ ). In this limit, φ is constant, g =g ∞ ∞ ∞ ∞ when n≥3, and g has constant negative sectional curvature when n=2. ∞ WenextconsiderRicciflowonwarpedproducts;seeSection3. Warpedproducts R×Fm,whereFm isapositively-curvedEinsteinmanifold,wereusedbySimonin a construction of metrics with pinching singularities [20]. More recently, Lott and Sˇeˇsum studied Ricci flow on compact warped products M3 =B2×S1 and proved several stability-type results [16]. Tran has also considered Ricci flow on warped products Bn×Fm where F is Ricci-flat [24]. Here, we consider fibers (Fm,γ) such that γ is µ-Einstein. The resulting flow equations on (B×F,g+e2φγ) are ∂ g =−2Rc+2mdφ⊗dφ t (1.3) ∂ φ=∆φ−µe−2φ t STABILITY OF SOLUTIONS OF CERTAIN EXTENDED RICCI FLOW SYSTEMS 3 This flow is a modified version of (1.1) with a one-dimensional target. Not surpris- ingly, the behavior here depends strongly on the sign of µ. When µ≤0, we obtain a result similar to Theorem 1.2 above. Theorem 1.4. Let g = g+e2φγ be a warped product metric on M = Bn×Fm, where B is orientable and compact, and (F,γ) is µ-Einstein with µ ≤ 0. Suppose that g is Einstein with negative sectional curvature (assumed to be constant when n=2). Thenfor anyρ∈(0,1), there existsθ ∈(ρ,1) suchthat thefollowingholds. There exists a (1+θ) little-H¨older neighborhood U of g such that for all initial data g(0) ∈ U, the unique solution g(τ) of curvature-normalized warped product (cid:101) (cid:101) Ricci flow (3.19) exists for all t≥0 and converges exponentially fast in the (2+ρ)- H¨older norm to a limit metric g∞ = g∞+e2φ∞γ. In this limit, φ∞ = φ, g∞ = g when n≥3, and g has constant negative sectional curvature when n=2. ∞ Next, we consider the locally RN-invariant Ricci flow; see Section 4. This flow wasintroducedbyLottasameansforprovingthatif(M3,g(t))isacompact,Type √ III Ricci flow solution with diameter O( t), then the pull-back solution (M(cid:102),g(cid:101)(t)) on the universal cover converges to a homogeneous Ricci soliton [15]. For this, Lottconsideredaclassof“twisted”principalRN-bundles. Certainmetricsonsuch bundles RN (cid:44)→ Mn+N → Bn can be represented locally as g = (g,A,G), where g is a metric on the base, A is an RN-valued 1-form corresponding to a connection on M, and G is an inner product on the fibers. Ricci flow on these locally RN- invariant metrics decomposes into a Ricci flow-type equation for g, a Yang Mills flow-type equation for A, and a heat-type equation for G: 1 ∂ g =−2R + GikGj(cid:96)∇ G ∇ G +gγδG (dA)i (dA)j , t αβ αβ 2 α ij β k(cid:96) ij αγ βδ (1.5) ∂ Ai =−(δdA)i +Gij∇βG (dA)k t α α jk βα 1 ∂ G =∆G −Gk(cid:96)∇ G ∇αG − gαγgβδG G (dA)k (dA)(cid:96) . t ij ij α ik (cid:96)j 2 ik j(cid:96) αβ γδ An important ingredient in the proof of Lott’s theorem is a set of stability results forthissystem,provedbyKnopfinthecasesN =1orn=1[10]. Weextendsome ofthoseresultstoarbitrarydimensions,andtoamoregeneralclassoffixedpoints. Theorem 1.6. Letg=(g,A,G)bealocallyRN-invariantmetricoftheform(4.1) on a product RN×Bn, where B is compact and orientable. Suppose that A vanishes andGisconstant,andthatg isEinsteinwithnegativesectionalcurvature(assumed to be constant when n = 2). Then for any ρ ∈ (0,1), there exists θ ∈ (ρ,1) such that the following holds. There exists a (1+θ) little-H¨older neighborhood U of g such that for all ini- tial data g(0) ∈ U, the unique solution g(τ) of curvature-normalized locally RN- (cid:101) (cid:101) invariant Ricci flow (4.2) exists for all t≥0 and converges exponentially fast in the (2+ρ)-H¨older norm to a limit metric g =(g ,A ,G ). In this limit, A van- ∞ ∞ ∞ ∞ ∞ ishes, G is constant, g =g when n≥3, and g has constant negative sectional ∞ ∞ ∞ curvature when n=2. Finally, we consider the connection Ricci flow; see Section 5. This flow was introduced by Streets as a geometric interpretation of renormalization group flow on (Mn,g) with B-field included, which takes the form of Ricci flow coupled with heat flow for a closed three-form [22] (see also [18,23]). Here, n ≥ 3. One can intepret the B-field strength as the torsion τ of a metric compatible connection, 4 M.B.WILLIAMS and Ricci flow in this setting becomes Ricci flow for g coupled with heat flow for the torsion: 1 ∂ g =−2Rc+ H (1.7) t 2 ∂ τ =∆τ t where H =gpqgrsτ τ . There are also certain other assumptions on τ that we ij ipr jqs will make precise in Section 5.1. We show that the flow is stable when the metric g is Einstein with negative sectional curvature and the connection is the Levi-Civita connection of g, that is, the torsion vanishes. Theorem 1.8. Let (Mn,g) be a compact, orientable Riemannian manifold with n≥3. Consider a metric-compatible connection ∇ on M with torsion τ. Suppose that g is Einstein with negative sectional curvature and that τ = 0. Then for any ρ∈(0,1), there exists θ ∈(ρ,1) such that the following holds. There exists a (1 + θ) little-H¨older neighborhood U of (g,τ) such that for all initialdata(g(0),τ(0))∈U, theuniquesolution(g(t),τ(t))ofcurvature-normalized (cid:101) (cid:101) (cid:101) (cid:101) connection Ricci flow (5.1) exists for all t ≥ 0 and converges exponentially fast in the (2+ρ)-H¨older norm to (g,τ)=(g,0). The proofs of these theorems follow the same general outline. (1) Modify the flow so that the fixed points are more easily studied. This involves pulling back the flow by diffeomorphisms, and the fixed points include Einstein metrics together with other objects relevant to the flow in question (e.g., maps or connections). (2) Compute the linearization of the modified flow, and prove linear stability at the fixed points. This actually involves a second modification (by a DeTurck trick) to make the flow strictly parabolic. (3) Prove dynamical stability by setting up the appropriate H¨older spaces and applyingatheoremofSimonentt. (SeeAppendixAforthestatmentofthe theorem.) The technique described here was introduced by Guenther, Isenberg, and Knopf [7], and has subsequently been used to prove several other results. For example, as mentioned before, Knopf proves stabilty for certain solutions of locally invariant RN-invariant Ricci flow [10]. Young considers Ricci flow coupled with Yang-Mills flow and proves stability of certain solutions [28]. Wu, as cited above, uses these methodstoprovethatcomplexhyperbolicspaceisstableunderRicciflow[26]. He also adapts the method for use in the non-compact setting. Step (3) in this technique relies on the maximal regularity theory of Da Prato and Grisvard [4], which exploits the smoothing properties of quasilinear parabolic operators. The actual stability then follows from a (quite general) theorem of Simonett, which is based on this maximal regularity theory [21]. The theorem also has the feature of giving stability even in the presense of center manifolds. The first analysis of center manifolds in problems relating to Ricci flow appeared in [7]. Remark 1.9. All of these theorems are true when the Einstein manifold is replaced by a two-dimensional sphere with constant positive sectional curvature. Unfortu- nately,thetechniquesusedheredonotgeneralizetohigherdimensionsforpositively curved manifolds. See [10, Remark 2]. STABILITY OF SOLUTIONS OF CERTAIN EXTENDED RICCI FLOW SYSTEMS 5 Acknowledgement. TheauthorwishestothankDanKnopfforhishelpfulcomments and suggestions, and Peter Petersen for many enlightening discussions. 2. Harmonic-Ricci flow 2.1. Setup and examples. Letusprovidebackgroundforthecoupledflow (1.1). Let(M,g)beaclosedRiemannianmanifold, with(N,γ)aclosed targetmanifold. Let φ : M → N be a smooth map. The Levi-Civita covariant derivative ∇TM of the metric g on M induces a covariant derivative ∇T∗M on the cotangent bundle, which satisfies ∇T∗Mω(Y)=X(cid:0)ω(Y)(cid:1)−ω(cid:0)∇TMY(cid:1). X X By requiring a product rule and compatibility with the metric, we also have con- variant derivatives on all tensor bundles Tp(M)=(T∗M)⊗p⊗(TM)⊗q. q TheLevi-Civitacovariantderivative∇TN ofthemetricγ onN inducesacovariant derivative ∇φ∗TN on the pull-back bundle φ∗TN →M, given by ∇φ∗TNY =∇TN Y, X φ∗X for X ∈C∞(TM) and Y ∈C∞(TN). As before, we get a covariant derivative on all tensor bundles over M of the form Tp(M)⊗Tr(φ∗N)=(T∗M)⊗p⊗(TM)⊗q⊗(φ∗T∗N)⊗r⊗(φ∗TN)⊗s. q s We refer to them simply as ∇. Related quantities are decorated with the metric name, if necessary, e.g., g∇. In local coordinates (xi) on M and (yλ) on N, ∇φ=φ =∂ φλdxi⊗∂ | ∈C∞(T∗M⊗φ∗TN). ∗ i λ φ Similarly, we have ∇2φ=(cid:0)∂ ∂ φλ−gΓk∂ φλ+(γΓ◦φ)λ ∂ φµ∂ φν(cid:1)dxi⊗dxj ⊗∂ | i j ij k µν i j λ φ ∈C∞(T∗M⊗T∗M⊗φ∗TN). The harmonic map Laplacian (or tension field) of φ with respect to g and h is τ φ=tr ∇2φ g,γ g (cid:16) (cid:17) (2.1) =gij ∂ ∂ φλ−gΓk∂ φλ+(γΓ◦φ)λ ∂ φµ∂ φν ∂ | i j ij k µν i j λ φ ∈C∞(φ∗TN). Additionally, ∇φ⊗∇φ=γ ∂ φλ∂ φµdxi⊗dxj =φ∗h λµ i j is a symmetric (2,0)-tensor on M, and we define S=Rc−c∇φ⊗∇φ where c=c(t)≥0 is a coupling function. Now we recall the flow (1.1): if φ : (M,g) → (N,γ) is a map of Riemannian manifolds, the harmonic-Ricci flow is the coupled system ∂ g =−2S=−2Rc+2c∇φ⊗∇φ t ∂ φ=τ φ t g,γ 6 M.B.WILLIAMS We will call this hrf for short, although it is also sometimes called the (RH) c flow. We will assume that c(t) is non-increasing. As mentioned above, this flow was introduced in [17] and is a generalization of one studied in [13]. Nowweconsidersomeexampesoftheflow. InstudyingexpandingRiccisolitons onhomogeneousspaces,Lottconsideredasamodelaspecialtypeofvectorbundle. LetMbeanRN-vectorbundlewithflatconnection,flatmetricGonthefibers,and Riemannian base (Bn,g). Assume that the connection preserves fiberwise volume forms. Lott showed that the soliton equation becomes a pair of equations. One is a soliton-like equation for g. The other is an equation for G, which says that, interpreted as a map G:B →(SL(N,R)/SO(N),γ), G is harmonic. Here, γ is the natural metric induced by G; see (4.3). In fact, more is true. Ricci flow on such bundles is the coupled flow 1 ∂ g =−2Rc+ ∇G⊗∇G t 4 ∂ G=τ G t g,γ which is the harmonic-Ricci flow on (B,g) with map G and c=1/8; see [25]. This is in fact a special case of the coupled system considered in Section 4. All known 3D and 4D homogeneous spaces admitting expanding Ricci solitons have this bundle structure, so the corresponding Ricci flow solutions are harmonic- Ricci flow solutions. Ricci flow on a warped product (Mn × S1,g + e2udθ), after modification by diffeomorphisms, takes a special form: ∂ g =−2Rc+2du⊗du t ∂ u=∆u t This is harmonic-Ricci flow with target R and c=1, and was studied by Lott and Sˇeˇsum [16]. It is also a special case of the system considered in Section 3. 2.2. Stability. In this section we follow the outline given in the introduction to prove Theorem 1.2. We transform the system into one whose fixed points include pairs (g,φ) with g Einstein and φ constant. Suppose that (g(t),φ(t)) is a solution of (1.1) (with time parameter t). Let s(t) be a function with positive anti-derivative σ, and consider (cid:0) (cid:1) (cid:0) (cid:1) g(t),φ(t) (cid:55)→ g(t),φ(t) , where (cid:90) t dr g =σ−1g, φ=φ, t= σ(r) t0 for t ∈ I, the interval of existence of the solution. A straightforward calculation 0 shows that this transformation results in the modified flow ∂ g =−2Rc+2c∇φ⊗∇φ−sg t (2.2) ∂ φ=τ φ t g,γ We want to describe the fixed points of this system, with the proper choice of s. Lemma 2.3. Suppose that (Mn,g ) is compact and Einstein, with Rc(g )=λg . 0 0 0 Let φ : (M,g) → (N,γ) be a constant map. Setting s = −2λ makes (g ,φ ) a 0 0 0 stationary solution of (2.2) STABILITY OF SOLUTIONS OF CERTAIN EXTENDED RICCI FLOW SYSTEMS 7 With these choices, we call (2.2) the curvature-normalized harmonic-Ricci flow system. Consider a fixed point (g ,φ ) of the flow (2.2) on Mn. From Lemma (2.3), 0 0 assume that g is Einstein with Rc(g ) = λg and φ is constant. To analyze the 0 0 0 0 stability near this fixed point, we must compute the linearization of the flow. Let (cid:0) (cid:1) g(cid:101)((cid:15)),φ(cid:101)((cid:15)) be a variation of (g0,φ0) such that (cid:12) (2.4) g(cid:101)(0)=g0, ∂(cid:15)(cid:12)(cid:15)=0g(cid:101)((cid:15))=h, (cid:12) φ(cid:101)(0)=φ0, ∂(cid:15)(cid:12)(cid:15)=0φ(cid:101)((cid:15))=ψ, for some symmetric (2,0)-tensor h and variational vector field ψ ∈ C∞(φ∗TN). More explicitly, φ(cid:101)(x,(cid:15)) = expφ(x)((cid:15)ψ(x)). Let ∆(cid:96) denote the Lichnerowicz Lapla- cian acting on symmetric (2,0)-tensor fields. Its components are ∆ h =∆h +2R hpq−Rkh −Rkh . (cid:96) ij ij ipqj i kj j ik Lemma 2.5. The linearization of (2.2) at a fixed point (g ,φ ) with Rc(g )=λg 0 0 0 0 and φ constant acts on (h,ψ) by 0 (2.6a) ∂ h =∆ h +∇ (δh) +∇ (δh) +∇ ∇ tr h+2λh , t ij (cid:96) ij i j i j i j g0 ij (2.6b) ∂ ψα =∆ψα t where∆ istheLichnorowiczLaplacianand∆istheLaplacianactingonfunctions. (cid:96) Proof. With a variation as in (2.4), we must compute (cid:12) (cid:16) (cid:17) (cid:12) (cid:16) (cid:17) ∂(cid:15)(cid:12)(cid:15)=0 ∂tg(cid:101)((cid:15)) , ∂(cid:15)(cid:12)(cid:15)=0 ∂tφ(cid:101)((cid:15)) . Such computations involve standard variational formulas for geometric objects like g−1,Γ,Rc,andR. See[3,Section3.1],forexample. Thefirstequationisconsidered in [10, Lemma 3]. For the second equation, using the coordinate expression for the tension field from (2.1), it is easy to see that ∂(cid:15)(cid:12)(cid:12)(cid:15)=0(cid:16)∂tφ(cid:101)((cid:15))(cid:17)α =∂(cid:15)(cid:12)(cid:12)(cid:15)=0τg(cid:101)((cid:15)),hφ(cid:101)((cid:15))α =gij(∂ ∂ ψα−gΓk∂ ψα) i j ij k =∆ψα, as desired. (cid:3) We next use the DeTurck trick to make the linearized system (2.2) strictly par- abolic. That is, we pull back by diffeomorphisms generated by carefully chosen vector fields, which has the effect of subtracting a Lie derivative term from both equations in (2.2). To this end, define a vector field depending on g(t) by (2.7) Wk =gij(Γk −g0Γk), k =1,...,n. ij ij Let F be diffeomorphisms generated by W(t), with initial condition F =id. The t 0 one-parameter family (cid:0)F∗g(t),F∗φ(t)(cid:1) is the solution of the curvature-normalized t t harmonic-Ricci DeTurck flow. A stationary solution (g ,φ ) of (2.2) with Rc = 0 0 λg and φ constant is then also a stationary solution of the curvature-normalized 0 0 harmonic-Ricci DeTurck flow. 8 M.B.WILLIAMS Lemma 2.8. The linearization of the curvature-normalized harmonic-Ricci De- Turck flow at a fixed point (g ,φ ) with Rc = λg is the autonomous, self-adjoint, 0 0 0 strictly parabolic system (2.9a) ∂ h=L h=∆ h+2λh t 0 (cid:96) (2.9b) ∂ ψ =L ψ =∆ψ, t 1 where L =∆ satisfies (∆ψ)α =∆(ψα). 1 Proof. The curvature-normalized Ricci-DeTurck flow is obtained by what amounts to subtracting a Lie derivative from the right side of (4.2): ∂ g =−2Rc+2c∇φ⊗∇φ−L g, t W ∂ φ=τ φ−L φ, t g,h W with intial data (g(0),φ(0)), so we must compute the linearization of this Lie de- rivative, as in Lemma 4.8. Take a variation (g(cid:101)((cid:15)),φ(cid:101)((cid:15))) as before. It is well-known that (cid:12) ∂(cid:15)(cid:12)(cid:15)=0(LW(cid:102)g(cid:101))ij =∇i(δh)j +∇j(δh)i+∇i∇jtrg0h. Subtracting this from (2.6a) gives (2.9a). For the second equation, L φ=∇φ(W)=∂ φαW ∂ | , W i i α φ and is it easy to see that W(g(0))=0, so (cid:101) ∂(cid:15)(cid:12)(cid:12)(cid:15)=0∂iφ(cid:101)αW(cid:102)i =0, giving (2.9b). (cid:3) Now assume that (g ,φ ) is a fixed point of the curvature-normalized harmonic- 0 0 Ricci-DeTurck flow with φ constant and g a λ-Einstein metric with negative 0 0 sectional curvature. RecallthatalinearoperatorLisweakly(strictly)stableifitsspectrumisconfined tothehalfplaneRez ≤0(andisuniformlyboundedawayfromtheimaginaryaxis). We thank Peter Petersen for the idea behind the following estimate. Lemma 2.10. Suppose that g is λ-Einstein, and that there exists K <0 such that sec≤K. Then if Lh:=∆ h+2λh, we have (Lh,h)≤K(n−2)(cid:107)h(cid:107)2 <0. (cid:96) Proof. First, write a symmetric 2-tensor h as h = g((cid:101)h·,·) and let {ei} be an or- thonormal basis of eigenvectors for (cid:101)h. That is, (cid:101)h(ei)=λiei. Now, we write part of (cid:104)Lh,h(cid:105) in components with respect this basis. Using that g is λ-Einstein, we have (cid:88) (cid:88) (cid:88) R hi(cid:96)hjk− Rjhkhi +2λ h hij ijk(cid:96) i j k ij i,j,k,(cid:96) i,j,k i,j (cid:88) (cid:88) = R λ λ +λ h hij ijji i j ij i,j i,j (cid:88) (cid:88) = sec(e ,e )λ λ + sec(e ,e )λ2 i j i j i j i i,j i,j (cid:88) (cid:88) (cid:88) = 1 sec(e ,e )2λ λ + 1 sec(e ,e )λ2+ 1 sec(e ,e )λ2 2 i j i j 2 i j i 2 i j j i,j i,j i,j (cid:88) = 1 sec(e ,e )(λ +λ )2. 2 i j i j i,j STABILITY OF SOLUTIONS OF CERTAIN EXTENDED RICCI FLOW SYSTEMS 9 Now, integrating by parts and using Koiso’s Bochner formula [12] together with the above equation, (cid:90) (cid:90) (cid:90) (cid:88) (Lh,h)=−(cid:107)∇h(cid:107)2+ R hijhkl− Rjhkhi + 1 sec(e ,e )(λ +λ )2 ijk(cid:96) i j k 2 i j i j i,j =−1(cid:107)T(cid:107)2−(cid:107)δh(cid:107)2+K(n−2)(cid:107)h(cid:107)2+K(cid:107)trh(cid:107)2 2 ≤K(n−2)(cid:107)h(cid:107)2, as desired. (cid:3) Lemma 2.11. Let (g ,φ ) be such that φ is constant and g is Einstein with 0 0 0 0 negative sectional curvature. Then the linear system (2.9a)-(2.9b) has the following linear stability properties: If n = 2, then the operator L is weakly stable. On an orientable surface B of 0 genus γ ≥ 2, the null eigenspace is the (6γ−6)-dimensional space of holomorphic quadratic differentials. If n≥3, the operator L is strictly stable. 0 The operator L is weakly stable. Its null eigenspace is the space of constant 1 variations ψ ∈C∞(φ∗TN), whose dimension is equal to dimN. Proof. The statements about L in dimension 2 follows from [10, Lemma 5], and 0 the statement in dimension n ≥ 3 follows from Lemma 2.10. That the operator L =∆ is weakly stable follows from integrating by parts. (cid:3) 1 We now turn to the proof of the the main theorem. See the appendix for the statement of Simonett’s theorem, [10, Section 2] for a more detailed description of its application as used here, and [21] for the original statement. If V → M is a vector bundle, let hr+ρ(V) denote the completion of the vector spaceC∞(V)withrespecttother+ρlittle-H¨oldernorm. Forfixed0<σ <ρ<1, consider the following densely and continuously embedded spaces: E :=h0+σ(S2M)×h0+σ(φ∗TN) 0 ∪ X :=h0+ρ(S2M)×h0+ρ(φ∗TN) 0 ∪ E :=h2+σ(S2M)×h2+σ(φ∗TN) 1 ∪ X :=h2+ρ(S2M)×h2+ρ(φ∗TN) 1 For fixed 1/2≤β <α<1, define the continuous interpolation spaces X :=(X ,X ) , X :=(X ,X ) . β 0 1 β α 0 1 α For fixed 0 < (cid:15) (cid:28) 1, let G be the open (cid:15)-ball around (g ,φ ) in X , and define β 0 0 β G :=G ∩X . α β α Proof of Theorem 1.2. We follow the proof of Theorem 1 in [10], which consists of four steps. First, one must show that the complexification of the operator in (2.9a)-(2.9b) is sectorial. This holds exactly as in [10]. With this established, once checks that conditions (1)-(7) of Simonett’s theorem hold, and this follows exactly 10 M.B.WILLIAMS as in [10, Lemmas 1 and 2]. The second step is then to apply Simonett’s theorem (Theorem A.1). Third,inthecasesofweaklinearstability,oneprovestheuniquenessofasmooth centermanifoldconsistingoffixedpointsoftheflow(2.2). Ourcaseisinsomesense aspecialcaseofthatconsideredin[10]. Inparticular,then=2caseforthemetric g is handled. Note that fixed points of this flow still coincide with those of the curvature-normalized Ricci-DeTurck flow. Finally, one proves convergence of the metric. Again, the arguments of [10] carry through without modification. (cid:3) 3. Ricci flow on warped products 3.1. Setup. Consider a warped product metric g = g+e2φγ on a manifold M = Bn×Fm, where φ ∈ C∞(B). Let H,V ⊂ TM denote the horizontal and vertical distributions, respectively. The Ricci curvature of g is gRc| =gRc−mHess(φ)−mdφ⊗dφ H×H (3.1) gRc|H×V =0 gRc| =γRc−e2ψ(∆φ+m|dφ|2)γ. V×V See [1], for example. We wish to describe the Ricci flow equation on the warped product(M,g)intermsoftheevolutionofgandthewarpingfunctionφ. Asimilar approach was taken in [16,24]. Proposition 3.2. Let (M = Bn×Fm,g(t)) be a solution to Ricci flow, with M closed and g(0)=g(0)+e2φ(0)γ(0) a warped product. If (F,h) is µ-Einstein, then γ(0) = γ is constant under the flow, g(t) is a warped product, and the evolutions of g and φ are given by ∂ g =−2gRc+2mHess(φ)+2mdφ⊗dφ t (3.3) ∂ φ=∆φ+m|dφ|2−µe−2φ t Proof. Define g(t)=g(t)+e2φ(t)h, where g(t) and φ(t) are solutions of (3.3) with (cid:101) g(0)=g and φ(0)=φ . Using (3.1), we see that the evolution of g(t) is 0 0 (cid:101) ∂ g(t)=∂ g(t)+∂ (e2φ(t)γ) t(cid:101) t t =−2gRc+2mHess(φ)+2mdφ⊗dφ+2e2φ(t)∂ φ(t)γ t =−2g(cid:101)Rc|H×H+2e2φ(t)(∆φ+m|dφ|2−µe−2φ)γ =−2g(cid:101)Rc|H×H−µh+2e2φ(t)(∆φ+m|dφ|2)γ =−2g(cid:101)Rc|H×H−2gRc|V×V =−2g(cid:101)Rc, since −2µγ = −2γRc. This means g(cid:101)(t) solves Ricci flow with g(cid:101)(0) = g0 +e2φ0γ. By uniqueness of solutions of Ricci flow, for any solution g(t) of Ricci flow with g(0) = g0 +e2φ0h, we must have g(cid:101)(t) = g(t). This means the warped product structure is preserved and the components of g(t) = g(t)+e2φ(t)γ must satsify (3.3). (cid:3)