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Turk J Math (2015) 39: 490 – 500 Turkish Journal of Mathematics http://journals.tubitak.gov.tr/math/ c TÜBİTAK ⃝ doi:10.3906/mat-1406-4 Research Article Stability of compact Ricci solitons under Ricci flow Mina VAGHEF, Asadollah RAZAVI∗ Faculty of Mathematics and Computer Science, Amirkabir University of Technology, Tehran, Iran Received: 03.06.2014 • Accepted/Published Online: 07.05.2015 • Printed: 30.07.2015 Abstract: In this paper we establish stability results for Ricci solitons under the Ricci flow, i.e. small perturbations of the Ricci soliton result in small variations in the solution under Ricci flow. Key words: Stability, Ricci flow, Ricci soliton, compact manifolds 1. Introduction and preliminaries Differential equations are interesting mathematical topics employed throughout the sciences for modeling dynamic processes. When differential equations are difficult to be solved, we try to obtain qualitative information about the long-term or asymptotic behavior of solutions. Ricci flow is a partial differential equation that evolves a Riemannian metric ḡ on a manifold M under the following equation: ∂ g(t) = −2Ric(g(t)), ∂t g(0) = ḡ. (1.1) It was introduced by Hamilton in his seminal paper [14] in order to study the geometry and topology of manifolds. Ricci flow has been developed in the past several decades. In addition to being applied as a useful tool in geometry, it also has some applications in other fields such as computer science [28] and physics [18]. Therefore, it is important to study the equation of Ricci flow. There are some interesting questions about this equation, such as stability. The term “stable” means that a stated property is not destroyed when certain perturbations are made. The stability of solutions of differential equations is a quite difficult property to determine. Even though various kinds of stability may be discussed, the one we study here is dynamical stability. g̃ is dynamically stable if for ḡ belonging to a neighborhood of g̃ and sufficiently close to g̃ , the solution g(t) of Ricci flow with initial value ḡ stays near g̃ forever. Stability under the Ricci flow was first discussed by Ye [29]. He replaced the Ricci flow equation with normalized Ricci flow ∂ 2 g(t) = −2Ric(g(t)) + rg(t), ∂t n where r = ∫ ∫Rdµ dµ , and investigated stability of constant nonzero sectional curvature metrics. After him, Guenther et al. [11] using maximal regularity theory [8] and center manifold analysis [9] studied stability of flat and Ricciflat metrics. The behavior of Ricci flow close to Ricci-flat metrics was also investigated in [16, 17, 26]. Knopf ∗Correspondence: arazavi@aut.ac.ir 2010 AMS Mathematics Subject Classification: 53C44, 35B35. 490 VAGHEF and RAZAVI/Turk J Math [21] obtained stability of locally RN -invariant solutions of Ricci flow. Schnürer et al. showed the stability of Euclidean space [23] and hyperbolic space [24]. The objective of this paper is to investigate dynamical stability of the Ricci soliton g̃ on compact manifolds under the following modified Ricci flow: ∂ g(t) = −2Ric(g(t)) − LX g − 2ρg, ∂t g(0) = ḡ. (1.2) Definition 1.1 A Ricci soliton is a fixed complete Riemannian manifold (M n , g̃) that satisfies the equation −2Ric(g̃) = LX g̃ + 2ρg̃ (1.3) for some constant ρ and some complete vector field X on M n , where LX g̃ denotes the usual Lie derivative in the direction of the field X. A Ricci soliton is said to be a gradient Ricci soliton if its vector field X can be written as the gradient of some function f : M n → R. The function f is called a potential function for g̃ . Equation (1.3) then becomes −Ric(g̃) = ∇∇f + ρg̃. g̃ is called shrinking, steady, or expanding if ρ < 0 , ρ = 0 , or ρ > 0, respectively. When either the vector field X is trivial, or the potential functional f is constant, g̃ is an Einstein metric. Thus, Ricci solitons are natural generalizations of Einstein metrics. They were introduced by Hamilton [15]. Indeed, they are equivalent to the self-similar solution of the Ricci flow [14]. That is, these solutions evolve by rescalings and diffeomorphisms of the initial metric, and so they can be regarded as fixed points of the Ricci flow on the space of Riemannian metrics modulo diffeomorphisms and scalings. They are also of great importance owing to their relationship with singularities of the Ricci flow (see [2, 15, 25]). Ricci solitons also are discussed in string theory in physics (see [1, 7, 10]). Theorem 1.2 (Perelman) Every compact Ricci soliton is a gradient Ricci soliton. For more details on Ricci solitons we refer the reader to [3]. Ricci solitons are stationary points of the modified Ricci flow (1.2) and so remain unchanged throughout the flow. For analytic reasons, we study the following flow, which is similar to Ricci–DeTurck flow: ∂ g(t) = −2Ric(g(t)) + ∇i Wj + ∇j Wi − LX g − 2ρg, ∂t g(0) = ḡ, (1.4) ˜ to denote respectively the Christoffel symbol and the covariant where Wi = gik (Γkrs − Γ̃krs )g rs . We use Γ̃, ∇ derivative with respect to g̃ . Here g̃ denotes the Ricci soliton. Variational stability of gradient Ricci solitons was studied for the first time by Cao et al. [4]. Since Ricci solitons are critical points of Perelman’s λ-entropy and ν -entropy, they displayed the second variation of λ and ν functionals and, according to the second variation, explored the linear stability of some examples. For more results on stability of Ricci solitons with respect to the second variation of Perelman’s ν -functional see [6, 5, 13]. Kröncke [22] considered a modified τ -flow with X(t) = −gradg(t) fg(t) where fg(t) is a smooth function produced by minimizing Perelman’s entropy functional and examined the stability of compact shrinking Ricci solitons under the flow. Guenther et al. [12] demonstrated linear stability of some nongradient homogeneous expanding 491 VAGHEF and RAZAVI/Turk J Math Ricci solitons. Jablonski et al. studied the linear stability of algebric Ricci solitons on simply connected solvable Lie groups [19] and expanding Ricci solitons with bounded curvature [20]. In the linear stability of Ricci solitons after normalizing Ricci flow, one applies the DeTurck trick and linearizes the flow at a fixed point: ∂ h = Lh := ∆L h + 2λh + LX h, ∂t where ∆L is the Lichnerowicz Laplacian. A stationary solution of the previous equation is strictly (resp. weakly) linearly stable if the operator L has a negative (resp. nonpositive) spectrum. Following Guenther et al. [11], one applies Simonett’s stability theorem to deduce dynamical stability from linear stability. 2. Equivalency of flows In this section we show that (1.1) and (1.2) are equivalent. Lemma 2.1 Let (M n , ḡ(t̄))t̄∈[0,T̄ ) be a solution to (1.2). Define (M n , g(t))t∈[0,T ) by g(t) = (1 + 2ρt)φ∗t (ḡ (t̄)) , where φt denotes the diffeomorphisms generated by the family of vector fields Yt (x) = t̄(t) = Proof ln(1+2ρt) 2ρ , and T = e2ρT̄ −1 2ρ 1 1+2ρt X(x) on M n , . Then g(t) is a solution of the Ricci flow. We prove it by calculating. ∂ ∂ g(t) = 2ρφ∗t (ḡ(t̄)) + (1 + 2ρt) φ∗t (ḡ(t̄)) ∂t ∂t = 2ρφ∗t (ḡ(t̄)) + (1 + 2ρt)φ∗t (LYt ḡ) ( dt̄ ) +(1 + 2ρt)φ∗t (−2Ric(ḡ) − LX ḡ − 2ρḡ) dt = 2ρφ∗t (ḡ(t̄)) + φ∗t (LX ḡ) − 2Ric(g) − φ∗t (LX ḡ) − 2ρφ∗t (ḡ(t̄)) = −2Ric(g(t)), 2 and hence g(t) is a solution of the Ricci flow. Lemma 2.2 Let (M n , ḡ(t̄))t̄∈[0,T̄ ) be a solution to (1.1). Define (M n , g(t))t∈[0,T ) by g(t) = e−2ρt φ∗t (ḡ(t̄)), where t̄(t) = e2ρt −1 2ρ , T = ln(1+2ρT̄ ) 2ρ , and φt denotes the 1-parameter family of maps φt : M n −→ M n satisfying ∂ φt (p) = (φt )∗ Xp , ∂t for all p ∈ M n . Then g(t) solves the modified Ricci flow. 492 φ0 (p) = p VAGHEF and RAZAVI/Turk J Math Proof By calculating we have ∂ g(t) = ∂t −2ρe−2ρt φ∗t (ḡ(t̄)) + e−2ρt ∂ ∗ φ (ḡ(t̄)) ∂t t = −2ρe−2ρt φ∗t (ḡ(t̄)) + e−2ρt L[(φ−1 )∗ (φt )∗ X] (φ∗t ḡ(t̄)) + e−2ρt φ∗t (e2ρt (−2Ric(ḡ(t̄))) t = −2ρg(t) + LX g(t) − 2Ric(g(t)). We show how to go from a solution of (1.4) back to a solution of the modified Ricci flow. 2 Lemma 2.3 If g(t) is a solution of (1.4), we claim that ḡ(t) = φ∗t g(t) is a solution to the modified Ricci flow, where φt : M n −→ M n defined by ∂ φt (p) = −W (φt (p)) + Xφt (p) − φt∗ (Xp ), ∂t φ0 (p) = p is a 1-parameter family of maps. Proof We have ∂ ḡ ∂t = ∂ ∗ φ g(t) ∂t t = φ∗t (−2Ric(g(t)) + LW (t) g(t) − LX g − 2ρg) +L[(φ−1 )∗ (−W (t)+X−φt t ∗ X)] (φ∗t g(t)) = −2Ric(ḡ(t)) + φ∗t (LW (t) g(t)) − φ∗t (LX g) − 2ρḡ − L(φ−1 )∗ W (t) (φ∗t g(t)) t +L(φ−1 )∗ X (φ∗t g(t)) t − L(φ−1 )∗ (φt )∗ X (φ∗t g(t)) t = −2Ric(ḡ(t)) − LX ḡ − 2ρḡ. 2 3. Stability and main results In this section we shall focus on investigating the stability of Ricci solitons. Lemma 3.1 Let g(t) be a solution to (1.4). The evolution equations for gij and g ij in coordinate form satisfy the following equation: ∂ gij ∂t ˜ α∇ ˜ β gij − g αβ gip g̃ pq R̃jαqβ − g αβ gjp g̃ pq R̃iαqβ = g αβ ∇ 1 ˜ i gpα · ∇ ˜ j gqβ + 2∇ ˜ α gjp · ∇ ˜ q giβ − 2∇ ˜ α gjp · ∇ ˜ β giq + g αβ g pq (∇ 2 ˜ j gpα · ∇ ˜ β giq − 2∇ ˜ i gpα · ∇ ˜ β gjq ) −2∇ ˜ i Xj − ∇ ˜ j Xi + g αβ (∇ ˜ i gjβ + ∇ ˜ j giβ − ∇ ˜ β gij )Xα − 2ρgij . −∇ (3.1) 493 VAGHEF and RAZAVI/Turk J Math By this equation we also have ∂ ij g ∂t ˜ α∇ ˜ β g ij + g αβ g ik g jl gkp g̃ pq R̃lαqβ = g αβ ∇ ˜ α g jq · ∇ ˜ β gpq + g αβ g jq ∇ ˜ α g ip · ∇ ˜ β gpq +g αβ g ik g jl gpl g̃ pq R̃kαqβ + g αβ g ip ∇ 1 ˜ α gpl · ∇ ˜ β gqk + 2∇ ˜ l gpα · ∇ ˜ β gqk + 2∇ ˜ k gpα · ∇ ˜ β gql + g αβ g pq g ik g jl · (2∇ 2 ˜ α gpl · ∇ ˜ q gβk − ∇ ˜ k gpα · ∇ ˜ l gqβ ) −2∇ ˜ k Xl + ∇ ˜ l Xk ) + g ik g jl g αβ (∇ ˜ β gkl − ∇ ˜ k glβ − ∇ ˜ l gkβ )Xα + ρg ij . +g ik g jl (∇ Proof In Shi’s paper [27], the evolution equation was obtained for solutions to Ricci–DeTurck flow in coordinate form as follows. ∂ ij g ∂t = ˜ α∇ ˜ β gij − g αβ gip g̃ pq R̃jαqβ − g αβ gjp g̃ pq R̃iαqβ g αβ ∇ 1 ˜ i gpα · ∇ ˜ j gqβ + 2∇ ˜ α gjp · ∇ ˜ q giβ − 2∇ ˜ α gjp · ∇ ˜ β giq + g αβ g pq (∇ 2 ˜ j gpα · ∇ ˜ β giq − 2∇ ˜ i gpα · ∇ ˜ β gjq ) −2∇ 2 Using Shi’s results, the lemma is clear. Remark 3.2 One writes A < B for symmetric 2 -tensor A and B if B − A is a nonnegative definite quadratic form, that is, if (B − A)(V, V ) ≥ 0 for all vectors V . Definition 3.3 Let g be a metric on M n . Let ε > 0. Then we say that g is ε-close to ḡ if (1 + ε)−1 ḡ ≤ g ≤ (1 + ε)ḡ. We prove the following theorem. Theorem 3.4 Suppose gij (x, t) > 0 is a solution of (1.4). Then for any δ > 0 there exists a constant T such that (1 − δ)ḡij (x) ≤ gij (x, t) ≤ (1 + δ)ḡij (x), x ∈ M, 0 ≤ t ≤ T. First, we prove the following lemmas. We may always choose a local coordinate around a fixed point p , so that at p we have ḡij (p) = δij , gij (p) = δij λi (p). (3.2) Lemma 3.5 Suppose gij (x, t) is a solution of (1.4). Then for any δ > 0 there exists a constant T such that gij (x, t) ≥ (1 − δ)ḡij (x), 494 x ∈ M, 0 ≤ t ≤ T. VAGHEF and RAZAVI/Turk J Math Proof Define φ(x, t) = g α1 β1 ḡβ1 α2 g α2 β2 ḡβ2 α3 g α3 β3 ḡβ3 α4 . . . g αm βm ḡβm α1 . In the selected local coordinate we have n ∑ 1 ( )m , λk φ(x, t) = k=1 and thus ∂φ 1 ∂ = m( )m−1 g ii . ∂t λi ∂t Using Lemma 3.1 and as in [27] Lemma 2.2, we see that ( ) ∂φ ˜ α∇ ˜ β φ + 2m g̃ ip R̃iqpq + m( 1 )m−1 ( 1 )2 (LX g)ii + 1 ρ . ≤ g αβ ∇ ∂t λm λi λi λi i λq Since M is compact, there is a constant c such that | 2m ip 1 g̃ R̃iqpq + m( )(LX g)ii + mρ| ≤ c. λq λi It is easy to see that ∂φ ˜ α∇ ˜ β φ + cφ. ≤ g αβ ∇ ∂t We define φ(t) = max φ(x, t). x∈M Using the maximal principle, we get dφ ≤ cφ, dt φ(0) = n. Thus, we have φ(x, t) ≤ nect ∀x ∈ M. ∑n 1 m 1 1/m If 0 ≤ t ≤ 1c ln 2 , then φ(x, t) ≤ 2n ; that is, ≤ 2n . Then ( λ1k )m ≤ 2n ∀k and λk ≥ ( 2n ) ∀k . k=1 ( λk ) According to the selected coordinate it follows that gij (x, t) ≥ ( Let m be an integer so that log 2n log (1/(1−δ)) ≤m≤ 1 1/m ) ḡij (x). 2n log 2n log (1/(1−δ)) gij (x, t) ≥ (1 − δ)ḡij (x), + 1 . Therefore, (1/2n)1/m ≥ 1 − δ and x ∈ M, 0 ≤ t ≤ 1 ln 2. c 2 Lemma 3.6 Suppose gij (x, t) is a solution of (1.4). Then for any δ > 0 there exists a constant T such that gij (x, t) ≤ (1 + δ)ḡij (x), x ∈ M, 0 ≤ t ≤ T. 495 VAGHEF and RAZAVI/Turk J Math Proof Define ψ(x, t) = ḡ α1 β1 gβ1 α2 ḡ α2 β2 gβ2 α3 ḡ α3 β3 gβ3 α4 . . . ḡ αm βm gβm α1 . By (3.2) we have ψ(x, t) = ∂ψ ∂t ∑n m k=1 (λk ) . Using Lemma 3.1, ∂ gii ∂t ( ˜ α∇ ˜ β gii − 2 λi g̃ pq R̃iαqα + 1 (∇ ˜ i gpq · ∇ ˜ i gpq + 2 = m(λi )m−1 g αβ ∇ λα 2λq λp = m(λi )m−1 ˜ q gip · ∇ ˜ p giq − 2∇ ˜ q gip · ∇ ˜ q gip − 4∇ ˜ i gpq · ∇ ˜ q gip ) − (λi )2 (LX g)ii − 2ρλi ∇ ) and then ∂ψ ∂t [ ] ˜ α∇ ˜ β ψ − m (∇ ˜ α gij )2 (λ)m−2 + (λi )m−3 (λj ) + · · · + (λj )m−2 = g αβ ∇ i λα ( m m ˜ i gpq · ∇ ˜ i gpq + 2∇ ˜ q gip · ∇ ˜ p giq +(λi )m − 2 g̃ pq R̃iαqα + (∇ λα 2λi λq λp ) ˜ q gip · ∇ ˜ q gip − 4∇ ˜ i gpq · ∇ ˜ q gip ) − λi (LX g)ii − 2ρ . −2∇ Since M is compact, there exists a constant c such that ∂ψ ˜ α∇ ˜ β ψ + cψ, ≤ g αβ ∇ ∂t ψ(x, 0) = n. Using the maximal principle, it follows that ψ(x, t) ≤ nect . If we let 0 ≤ t ≤ 1 c ln 2, we have ψ(x, t) ≤ 2n; that is, ∑n m k=1 (λk ) ≤ 2n. Thus, λk ≤ (2n)1/m ∀k . By (3.2) we have gij (x, t) ≤ ( Let m be an integer so that log 2n log (1/(1+δ)) ≤m≤ 1 1/m ) ḡij (x). 2n log 2n log (1/(1+δ)) gij (x, t) ≤ (1 + δ)ḡij (x), + 1 . Therefore, (1/2n)1/m ≤ 1 + δ and x ∈ M, 0 ≤ t ≤ A combination of Lemmas 3.5 and 3.6 easily gives Theorem 3.4. 1 ln 2. c 2 Lemma 3.7 Let g ∈ M∞ (M n , [0, T )), 0 < T < ∞ , be a solution to (1.4), which is ε -close to the g̃ . If ε is sufficiently small, then ∂ ˜ α∇ ˜ β |g − g̃|2 + C|g − g̃|2 . |g − g̃|2 ≤ g αβ ∇ ∂t 496 VAGHEF and RAZAVI/Turk J Math Proof We choose a local coordinate around a fixed point p , so that at p we have g̃ij (p) = δij , gij (p) = δij λi (p). (3.3) |g − g̃|2 = (gij − g̃ij )(gkl − g̃kl )g̃ ik g̃ jl = (gii − g̃ii )2 . (3.4) Then by (3.3) we find Lemma 3.1 yields ∂ |g − g̃|2 ∂t = 2 ∑ ∂ (gii − g̃ii )( gii ) ∂t i = 2 ( ∑ ˜ α∇ ˜ β gii − 2 λi R̃iqiq + 1 (∇ ˜ i gpq · ∇ ˜ i gpq (gii − g̃ii ) g αβ ∇ λq 2λq λp i ˜ q gip · ∇ ˜ p giq − 2∇ ˜ q gip · ∇ ˜ q gip − 4∇ ˜ i gpq · ∇ ˜ q gip ) +2∇ ) −(LX g)ii − 2ρλi . (3.5) From (3.4) it follows that ˜ α∇ ˜ β |g − g̃|2 = 2(gii − g̃ii )∇ ˜ α∇ ˜ β (gii − g̃ii ) + 2∇ ˜ α (gii − g̃ii )∇ ˜ β (gii − g̃ii ). ∇ (3.6) We use ∗ as in [14]. Substituting (3.6) and (3.3) into (3.5) gives ∂ |g − g̃|2 ∂t ˜ α∇ ˜ β |g − g̃|2 − 2 1 ∇ ˜ α (gii − g̃ii )∇ ˜ α (gii − g̃ii ) = g αβ ∇ λα ( ) 1 ˜ ∗ ∇g) ˜ ii +λi (λi − 1) − 4 R̃iqiq − λi (LX g)ii − 2ρ + (λi − 1)(∇g λq ( ) ˜ α∇ ˜ β |g − g̃|2 + (λi − 1)2 − 4 1 R̃iqiq − λi (LX g)ii − 2ρ ≤ g αβ ∇ λq ( ) 1 +(λi − 1) − 4 R̃iqiq − λi (LX g)ii − 2ρ λq ˜ α∇ ˜ β |g − g̃|2 + (λi − 1)2 | − 4 1 R̃iqiq − λi (LX g)ii − 2ρ| ≤ g αβ ∇ λq +c1 (λi − 1)2 | − 4 1 R̃iqiq − λi (LX g)ii − 2ρ|, λq where c1 is a fixed enough big number. Then we have ∂ ˜ α∇ ˜ β |g − g̃|2 + C|g − g̃|2 , |g − g̃|2 ≤ g αβ ∇ ∂t where C is a constant. 2 497 VAGHEF and RAZAVI/Turk J Math Theorem 3.8 Let g ∈ M∞ (M n , [0, T )) , 0 < T < ∞ , be a solution to (1.4). Let δ > 0 . Then there exists ε = ε(n, T, δ) such that supM n |ḡ − g̃| ≤ ε implies sup M n ×[0,T ) |g − g̃| ≤ δ. Proof Let g ∈ M∞ (M n , [0, T )), T > 0, be a solution to (1.4). By Lemma 3.4 and the maximum principle we have sup|g(t) − g̃|2 ≤ |ḡ − g̃|eCt . Fix ε = δe−CT where C is the constant in Lemma 3.7, and then sup|g(t) − g̃|2 eC(T −t) ≤ sup|ḡ − g̃|2 eCT ≤ εeCT = δ. 2 Theorem 3.9 Let g ∈ M∞ (M n , [0, T )) , T > 0 , be a solution to (1.4). If sup |g − g̃| < ˜ + divX we have 8|R̃m | + 8|Ric| ∫ 1 3 then for a = ∫ |g − g̃|2 dµg̃ ≤ eat |g(0) − g̃|2 dµg̃ . Therefore, for a < 0 we have L2 -norm stability. Proof ∂ ∂t ∫ ∫ |g − g̃| dµg̃ ≤ 2 ˜ α∇ ˜ β |g − g̃|2 − (2 − 1 )|∇(g ˜ − g̃)|2 − 4(gii − g̃ii )g αβ gip g̃ pq R̃iαqβ g αβ ∇ 3 −2(gii − g̃ii )((LX g)ii + 2ρgii )dµg̃ ∫ ˜ α g αβ ∇ ˜ β |g − g̃|2 − (2 − 1 )|∇(g ˜ − g̃)|2 − 4(gii − g̃ii )g αβ gip g̃ pq R̃iαqβ ≤ − ∇ 3 −2(gii − g̃ii )((LX (g − g̃))ii + (LX g̃)ii + 2ρg̃ii + 2ρ(gii − g̃ii ))dµg̃ ∫ 1 ˜ ≤ −(2 − 3( ))|∇(g − g̃)|2 − 4(gii − g̃ii )g αβ gip g̃ pq R̃iαqβ + 4(gii − g̃ii )R̃ii 3 −4ρ|g − g̃|2 − 2(gii − g̃ii )(LX (g − g̃))ii dµg̃ , 2 ˜ ˜ 2 , which is valid whenever |h| ̸= 0 . Furthermore, we used where we applied Kato’s inequality |∇|h|| ≤ |∇h| ˜ α g αβ ∇ ˜ β |h|2 | ≤ 2( 1 )|∇h| ˜ 2 and that g̃ satisfies equation (1.3). Let h = g − g̃ , for the last term in previous that |∇ 3 inequality; we have ∫ ∫ (LX h)ii hii dµg̃ = 1 (− divX|h|2 − 2R̃ik hkj hij − 2ρ|h|2 )dµg̃ 2 and then ∂ ∂t 498 ∫ ∫ |h| dµg̃ ≤ 2 −4hii g αβ gip g̃ pq R̃iαqβ + 4hii R̃ii − 4ρ|h|2 + divX|h|2 + 4R̃ik hkj hij + 4ρ|h|2 . VAGHEF and RAZAVI/Turk J Math In the local coordinate mentioned in Lemma 3.7 we get −(gii − g̃ii )g αβ gip g̃ pq R̃iαqβ = − λi (λi − 1)R̃iαiα λα = λi (λi − 1)(1 − 1 )R̃iαiα − λi (λi − 1)R̃iαiα λα = λi (λi − 1)(1 − 1 )R̃iαiα − (λi − 1)2 R̃iαiα − (λi − 1)R̃iαiα λα λi ˜ − hii R̃ii (λi − 1)(λα − 1)R̃iαiα + |h|2 |Ric| λα = ˜ − hii R̃ii ≤ 2|R̃m ||h|2 + |h|2 |Ric| and then ∂ ∂t ∫ ∫ |h| dµg̃ ≤ 2 ˜ + divX)|h|2 , (8|R̃m | + 8|Ric| and hence ∥g − g̃∥L2 ≤ eat ∥g(0) − g̃∥L2 . 2 References [1] Akbar MM, Woolgar E. Ricci solitons and Einstein-scalar field theory. Class Quantum Grav 2009; 26: 055015. [2] Cao HD. Limits of solutions to the Kähler-Ricci flow. J Diff Geom 1997; 45: 257272. [3] Cao HD. Recent progress on Ricci solitons. 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