c Birkhäuser Verlag, Basel 2005 GAFA Geometric And Functional Analysis

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1 GAFA, Geom. funct. anal. Vol X/5/ DOI 1.17/s y ONLINE FIRST: August 25 c Birkhäuser Verlag, Basel 25 GAFA Geometric And Functional Analysis KÄHLER RICCI SOLITONS ON COPACT COPLEX ANIFOLDS WITH C 1 > Huai-Dong Cao, Gang Tian Xiaohua Zhu Abstract. In this paper, we discuss the relation between the existence of Kähler Ricci solitons a certain functional associated to some complex onge Ampère equation on compact complex manifolds with positive first Chern class. In particular, we obtain a strong inequality of oser Trudinger type on a compact complex manifold admitting a Kähler Ricci soliton. Introduction In this paper, we study the existence of Kähler Ricci solitons by using properness of a certain functional. Our approach is similar to that of [T] also see [TZ1] for Kähler Einstein metrics. A Kähler metric g on a compact complex manifold with first Chern class c 1 > is called a homothetically shrinking Kähler Ricci soliton if there is a holomorphic vector field X on such that the Kähler form ω g of g satisfies Ricω g ω g = L X ω g, where Ricω g denotes the Ricci form of ω g L X is the Lie derivative operator along X. In particular, if X =, suchag is a Kähler Einstein metric. So Kähler Ricci solitons can be regarded as a generalization of Kähler Einstein metrics of positive scalar curvature. Ricci solitons have been studied extensively in the recent years [K], [H], [C], [2], [T], [TZ2,3], [WZ], etc.. One motivation is that they are very closely related to the singular behavior of limit solutions of certain PDEs which arise from geometric analysis, such as Hamilton s Ricci flow [H] certain complex onge Ampère equations associated to Kähler Einstein metrics [T]. Kähler Ricci solitons are special Ricci solitons. It was proved recently in [TZ2] [TZ3] that there exists at most one Kähler Ricci soliton on any compact G.T. partially supported by a NSF grant the Simon fund. X.Z. partially supported by the NSFC Grant the Hou Y.D fund.

2 698 HUAI-DONG CAO, GANG TIAN AND XIAOHUA ZHU GAFA Kähler manifold with positive first Chern class, modulo holomorphic automorphisms. This extends the uniqueness theorem of Bo abuchi for Kähler Einstein metrics with positive scalar curvature [B]. Let g be a Kähler metric on with its Kähler form ω g = 1 2π g ij dz i dz j, i,j representing c 1. Then there is a smooth real-valued function h g such that Ricω g ω g = 1 2π h g. Suppose that X is a holomorphic vector field on so that the integral curve K X of the imaginary part ImX ofx consists of isometries of g. By the Hodge decomposition theorem, there exists a unique smooth real-valued function θ X = θ X ω g on such that i X ω g = 1 2π θ X, eθ X ωg n = ωn g = V. Set X ω g = ψ C } ω ψ = ω g + 1 2π ψ >, ImXψ =. The following functional on X ω g was introduced in [TZ3], F ωg ψ = J ωg ψ 1 ψe θ X 1 ωg n log e hg ψ ωg n, V V where J ωg ψ = n n Cn 1 k tst k 1 st n 1 k e θx+stxψ dt ds 2πV k= ψ ψ ω k ψ ωn 1 k g. Let K K X be a maximal compact subgroup of the automorphisms group of such that σ η = η σ for any η K any σ K X.Thena Kähler Ricci soliton with respect to X on must be K -invariant by the uniqueness theorem in [TZ2]. We introduce Definition.1. The functional Fωg is said to be proper with respect to X if for any sequence ψ i } of K -invariant functions in X ω g, F ωg ψ i =+, lim i whenever lim i I ωg ψ i =+. Here I ωg ψ i = 1 ψ i ωg n ωψ n V i.

3 Vol. 15, 25 KÄHLER RICCI SOLITONS ON COPACT COPLEX ANIFOLDS 699 We note that the above definition is independent of the choice of ω g since functional Fωg satisfies a co-cycle condition cf. section 1. The following was essentially proved in [TZ2]. Theorem.1 [TZ2]. Suppose that F ωg is proper with respect to a holomorphic vector field X on. Then there is a Kähler Ricci soliton with respect to X on. This theorem says that the properness of F ωg is a sufficient condition for the existence of Kähler Ricci solitons. Since we did not state this result as a theorem in [TZ2], we will include its proof here following [TZ2]. In [TZ3], we proved that F ωg is bounded from below if there is a Kähler Ricci soliton on. The main purpose of this paper is to show that F ωg is actually proper wherever admits a Kähler Ricci soliton. This shows that F ωg is also a necessary condition for the existence of Kähler Ricci solitons cf. section 5. Suppose that admits a Kähler Ricci soliton ω KS with respect to some holomorphic vector field X. We define a weighted inner product on C by φ, ψ = φψe θ X ω KS ωks n, denote by Λ 1,ω KS = u C ωks u + Xu = u }. Theorem.2. Let be a compact complex manifold which admits a Kähler Ricci soliton ω KS with respect to some holomorphic vector field X G K X be a compact subgroup of K with σ η = η σ for any σ K X η G. Suppose that any G-invariant smooth function on is perpendicular to the space Λ 1,ω KS with respect to the weighted inner product defined above. Then there are two positive numbers c C such that for any G-invariant ψ in X ω g, F ωks ψ ci ωks ψ 1 4n+5 C..1 In particular, F ωks is proper under the same assumption. The proof of Theorem.2 is inspired by [T] [TZ1], where the authors proved a fully non-linear inequality of oser Trudinger type for Kähler Einstein manifolds. In fact, inequality.1 in Theorem.2 is equivalent to the following inequality, e ψ ωks n C exp J ωks ψ c J ωks ψ 1 4n+5 1 V ψωks n }..2

4 7 HUAI-DONG CAO, GANG TIAN AND XIAOHUA ZHU GAFA Inequality.2 generalizes the corresponding one in [T] [TZ1]. Recall that on S 2 with the stard Riemannian metric such an inequality is just a kind of the stronger version of oser Trudinger inequality [A], [T]. The organization of this paper is as follows. In section 1, we give a proof of Theorem.1 following [TZ2]. In section 2, we give an estimate for a certain heat kernel on manifolds with modified positive Ricci curvature. In section 3, a C -estimate for certain complex onge Ampère equations is obtained. In section 4, we prove a smoothing lemma by using Hamilton s Ricci flow. Theorem.2 will be proved in section 5. Acknowledgment. The third author would like to thank Professor T. abuchi for valuable discussions during his visit at Osaka University in the spring of An Analytic Criterion for Kähler Ricci Solitons Let,g beann-dimensional compact Kähler manifold with the first Chern class c 1 >. Denote by Aut the connected component of the automorphism group of. LetK be a maximal compact subgroup of Aut, then the Chevalley decomposition allows us to write Aut as a semi-direct decomposition [F], Aut =Aut r R u, where Aut r Aut is a reductive algebraic subgroup the complexification of K, R u is the unipotent radical of Aut. Let η r be the Lie subalgebra of Aut r. Let X η r such that the one-parameter subgroup K X generated by ImX is contained in K, whereimx denotes the imaginary part of X. Choose a K X -invariant Kähler metric g on with the Kähler form ω g. Then by the Hodge decomposition theorem, there is a unique smooth realvalued function θ X = θ X ω g of such that i X ω g = 1 2π θ X, eθ X ωg n = ωn g = V, where ωg n = ω g... ω g. The first relation above implies L X ω g = di X ω g = 1 2π θ X. We consider the following complex onge Ampère equations with parameter t [, 1]: detg ij + ϕ ij =detg ij exp h θ X Xϕ tϕ }, g ij + ϕ ij >, 1.1

5 Vol. 15, 25 KÄHLER RICCI SOLITONS ON COPACT COPLEX ANIFOLDS 71 where h = h g is a smooth real-valued function on defined by Ricω g ω g = 1 2π h, eh ωg n = ωn g. Then one can check that ω φ = ω g + 1 2π φ is a Kähler Ricci soliton with respect to X, i.e. ω φ satisfies Ricω φ ω φ = L X ω φ, if only if φ modulo a constant is a solution of equation 1.1 t for t =1. In fact, 1.1 t is equivalent to Ricω φt L X ω φt =tω φt +1 tω g, 1.2 where φ t is a solution of 1.1 t. Since d e θ X +Xφ t ωφ n dt t = φt + X φ t e θ X+Xφ t ωφ n t =, we have e θ X+Xφ t ωφ n t = e θ X ωg n = V. 1.3 Integrating 1.1 t after multiplying by e θ X+Xφ t, it follows that e h tφt ωg n = V. Therefore, differentiating the above identity, we get φ t e h tφt ωg n = 1 φ t e h tφt ωg n t. 1.4 Set ω g = φ C } ω φ = ω g + 1 2π φ > X ω g = φ ω g } ImXφ =. We define the following two functionals on X ω g cf.[tz2]: Ĩ ωg φ = 1 φ e θ X ωg n V eθx+xφ ωφ n J ωg φ = 1 V 1 φ s e θ X ω n g e θ X+Xφ s ω n φ s ds, 1.5 where φ s s 1 is a path from to φ in X ω g. Since J ωg φ is independent of the choice of a path, by choosing φ s = sφ, one can show that J ωg φ defined here coincides with one in the introduction. It is known that there are two uniform positive constants c 1 c 2 such that c 1 I ωg φ Ĩω g φ J ωg φ c 2 I ωg φ, 1.6

6 72 HUAI-DONG CAO, GANG TIAN AND XIAOHUA ZHU GAFA where I ωg φ = 1 V Let φωn g ωφ n. ˆF ωg φ = J ωg φ 1 φe θ X ωg n V = 1 1 φ s e θ X +Xφ s ωφ n V s ds. By simple computations, one can show that for any two functions φ ψ in X ω g, the following co-cycle condition is satisfied, ˆF ωg ψ = ˆF ωg φ+ ˆF ωφ ψ φ, where ˆF ωφ ψ φ = 1 1 φ V s e θ X+Xφ s ωφ n s ds φ s is a path from toψ φ in X ω g. Proposition 1.1. Let φ s be a solution of 1.1 s for all s t. Then t ˆF ωg φ t = 1 Ĩωg φ s t J ωg φ s ds. Proof. Differentiating 1.1 s on s, wehave φs + X φ s = s φ s + φ s. 1.7 Then by using , we get d φ s J ωg φ s = 1 d φ s e θ X +Xφ s ω n φ dsĩωg V ds s = 1 V φ s φs + X φ s e θ X+Xφ s ω n φ s = 1 V φ s s φ s + φ s e θ X+Xφ s ω n φ s = 1 φ s s V φ s + φ s e h sφs ωg n = 1 d φ s e h sφs ωg n + 1 V ds V = 1 d s φ s e h sφs ωg n sv ds = 1 d s φ s e θ X +Xφ s ωφ n sv ds s. It follows that s Ĩ ωg φ s J ωg φ s Ĩ ωg φ s J ωg φ s d ds = 1 V φ s e h sφs ω n g d s φ s e θ Xφ s ωφ n ds s, 1.8

7 Vol. 15, 25 KÄHLER RICCI SOLITONS ON COPACT COPLEX ANIFOLDS 73 consequently, Remark 1.1. ˆF ωg φ t = 1 V = 1 t t φe θ X+Xφ t ω n t Ĩ ωg φ t J ωg φ t Ĩωg φ s J ωg φ s ds. It was proved in Lemma 3.2 in [TZ2] that d dsĩωg φ s J ωg φ s. Recall that the functional Fωg is defined as F ωg ψ = ˆF 1 ωg ψ log e h ψ ωg n V = J ωg ψ 1 ψe θ X 1 ωg n log V V e h ψ ω n g F ωg is called proper with respect to X if for any sequence ψ i } of K -invariant functions in X ω g, lim i Fωg ψ i =+, whenever lim i I ωg ψ i =+, wherek K X is a maximal compact subgroup of automorphisms group Aut of such that σ η = η σ for any σ K X η K. Proof of Theorem.1. We use arguments from [TZ2]. Assuming the functional Fωg is proper, we shall prove the existence of a Kähler Ricci soliton with respect to X on. Thisisequivalenttoprovingthatthereis asolutionof1.1 t for t =1. ItsufficestoprovethatI ωg φ t is uniformly bounded for any solution of 1.1 t for t 1. This is because C 3 -norm of φ t can be uniformly bounded by I ωg φ t the set of parameter t for which there exists a smooth solution of 1.1 t is non-empty open cf. [TZ2]. By the implicit function theorem, one can show the solution of 1.1 t varies smoothly with t<1. Without the loss of generality, we may assume that the Kähler form ω g is K -invariant. Then all solutions φ t are all K -invariant. By Proposition 1.1, we have F ωg φ t = 1 t Ĩωg φ s t J ωg φ s 1 ds log e h φt ωg n V 1 log e h φt ωg n. 1.9 V,

8 74 HUAI-DONG CAO, GANG TIAN AND XIAOHUA ZHU GAFA On the other h, by using 1.1 t concavity of the logarithmic function, one can deduce 1 log e h φt ωg n = 1 t φ t e θ X+Xφ t ωφ n V V t 1 t φ t e h tφt ωg n C. 1.1 V Combining , we get F ωg φ t C. Therefore, the assumption of properness on F ωg implies I ωg φ t C, consequently, there is a Kähler Ricci soliton on with respect to X. 2 A Heat Kernel Estimate In this section, we give an estimate on the heat kernel of a linear elliptic operator P associated to a Kähler form ω a holomorphic vector field X on, where P = P ω = + X is defined on the space N X = u C ImXu = }. As a consequence, we derive a lower bound of the Green function of P. The method here follows that of T. abuchi in [1] with modifications which in turn were inspired by Li Yau [LY]. Note that P is a self-adjoint elliptic operator on N X with respect to the inner product, φ, ψ = φψe θ X ω n. Lemma 2.1. Let ω be a Kähler form on X a holomorphic vector field on with Ricω L X ω, θ X k, for some positive number k, whereθ X = θ X ω is defined as in section 1 for Kähler form ω. Letvx, t be a positive smooth solution on, of equation P v =. Suppose that lim sup t v 2 v, v v 1 v t 2n, t t} where v t = v/. Then there is a positive number C depending only on m 1 = max θ X m 2 = min θ X

9 Vol. 15, 25 KÄHLER RICCI SOLITONS ON COPACT COPLEX ANIFOLDS 75 such that vx, t 1 vy,t 2 t 2 t1 n C exp t 2 t 1 1 rx, y 2 /2+C 1 kt 2 t 1 }, t 1 <t 2, 2.1 where rx, y denotes the distance between x y associated to metric ω. Proof. Letf =lnv ˆF = t f, f f t. Then P ˆF + tp f t = t tr ω f, f + tr ω f, f X f, f + f 2 + f 2. For simplicity, we choose a local holomorphic coordinate system z 1,...,z n near each point p such that g ij p =δ ij f ij p =δ ij f ii p. By a direct computation, one sees that f i f ijj = f i f jji + f l R il, f i f ijj = f i f jji, X i f j f j i + f i f j X ji = f i X j f j i + f i X j f j i. Inserting the above identities into 2.2, we obtain Since we have P ˆF t f, Pf + Pf, f tp f t + t n f2 =2t Pf, f tp f t + t n f2. Pf f t = f, f, ˆF = tp f, 2.3 ˆF 1 ˆF t = tp f t. Hence by 2.3, we get P ˆF 2 1 ˆF, f ˆF t + t 1 ˆF 2 n t + Xf. 2.4 Let m 1 = sup x θ X x m 2 = inf x θ X x. As in [1], we define a monotone-increasing function η on [m 1,m 2 ]by s 1 ηs =exp m b e y 1 dy, where b = e m n. Then η is a solution of the ODE on [m 1,m 2 ], η η η η =2 η 2 η. oreover, one can check that the number C defined by C = min ηs ηs 1 ηs n 2 s [m 1,m 2 ]

10 76 HUAI-DONG CAO, GANG TIAN AND XIAOHUA ZHU GAFA is positive, < η η 1 n. Let F = η θ X ˆF. Then by 2.4, one can show that P F 2 F, f 1 t F + η η η η 2 η 2 η X 2 F + η 1 η 2XF F θ X By the assumption in the lemma, we have P + n 1 t 1 F 2 η η 1 η n 2 + t n η [ 1 t η 1 F 1 nη 1 η 2 Xf ] 2. F 2 F, f 2η 1 η XF 1 t + k n F + C nt F Applying the maximal principle to the function F x, t on,t], we get from 2.5, F x, t C 1 2n + kt, for any x, t,t], consequently, vx, t 2 vx, t, vx, t vx, t 1 v t x, t C 1 2n t + k, for any x, t,. Now by integrating the above estimate as in [LY], we can immediately obtain 2.1. Let H = Hx, y, t be a fundamental solution on [, of equation P vx, y, t =, 2.6 i.e. H is a smooth solution of 2.6 satisfying Hx, y.t =Hy,x,t, Hx, y, t = Hx, z, t shz,y.seθ X ω n, lim t Hx, y, t =δ x y. By using the asymptotic behavior of H, foreachfixedx, one sees that lim sup t t} t H 2 Hx,,t, Hx,,t > H 1 H t 2n. oreover, following an argument by Li Yau cf. [LY, Lemma 3.2], we can deduce Lemma 2.2. Let Z 1 Z 2 be two measurable subset of. Let T,δ,τ be three positive numbers with τ < 1 + 2δT. For each x, y

11 Vol. 15, 25 KÄHLER RICCI SOLITONS ON COPACT COPLEX ANIFOLDS 77 <t τ, denote F x,t y,t = Hy,,tHx,,Te θ X ω n. Z 1 Then F x,t,t 2 e θ X rx, ω n Z1 2 exp Z δT + Rx, Z 2 2 } F x,t x, T, δT 2t where rx, Z 1 =inf z Z1 rx, z Rx, Z 2 =sup z Z2 rx, z. Proposition 2.1. Let Hx, y, t be a fundamental solution on [, of equation 2.6. Suppose Ricω L X ω, θ X k, for some positive number k. Then for any δ> we have ṽol B x t 1/2ṽol B y t 1/2 Hx, y, t 1+δ 4n/C exp rx, y t} 2 + 4t1+3δ+2δ 2 + tδ2+δc 1 k + } 3 4δ1+δ + 1, 2δ 2.7 where ṽolb y t = B y t eθ X ω n, } rx, y t + =max,rx, y t }. Proof. First applying Lemma 2.1 to the function F x,t y,t witht 1,t 2 = T,τ =1+δT, we have F x,t x, T F x,t y,τ1 + δ 2n T 1 δ 1 rx, y 2 } C exp + C 1 ktδ Let Z 1 = B y tz 2 = B x t. Then integrating the square of the above inequality over all y Z 2 using Lemma 2.2, it follows that ṽol B x t F x,t x, T δ 4n/C exp consequently, ṽol B x t Z 1 Hx,,T 2 e θ X ω n rx, Z δT +2C 1 ktδ + 3t } F x,t x, T, 2Tδ = ṽol B x t F x,t x, T rx, 1 + δ 4n/C Z1 2 exp δT +2C 1 ktδ + 3t }. 2Tδ 2.9

12 78 HUAI-DONG CAO, GANG TIAN AND XIAOHUA ZHU GAFA On the other h, applying Lemma 2.1 to the function Hx, z, t inz with t 1,t 2 =t, T =1+δt, we have for any x, y, z, Hx, y, t δ 4n/C Hx, z, T 2 exp T 1 δ 1 ry,z 2 +2 C 1 ktδ }. Integrating this inequality over all z Z 1, using 2.9, we can get 2.7. Theorem 2.1. Let φ X ω g. Suppose that Ricω φ L X ω φ λω φ, 2.1 θ X ω φ k, for some positive numbers λ k. Then there is a uniform constant C depending only on λ k such that sup φ 1 φe θ X ω φ ωφ n + C V Proof. Letµ i µ =,i =, 1,...,be the increasing sequence of eigenvalues of operator P = ωφ + X associated to metric ω φ. Then by using the stard Bochner technique, one can obtain µ 1 λ cf. [TZ2]. Let Gx, y be the Green function with Gx, eθ X ω φ ωφ n = associated to the operator P.Then Since we have Gx, y = Hx, y, t 1 V dt. H x, y, t =Hx, y, t 1 V = e µ it f i xf i y, i=1 H x, x, t + t e µ 1t H x, x, t, 2.12 for any t,t >, where f i x denote the eigenfunctions of µ i. In [2], it was proved under the condition 2.1 that there is a uniform constant C 1 such that Diam,ω φ C 1. λ Choose t = 1 4 Diam,ω φ 2. Then by Proposition 2.1 we have H x, x, t C 2, 2.13 for some uniform constant C 2 depending only on λ k. By using , we get t 1 Gx, y V dt e µ 1t t H x, x, t H y,y,t 1/2 dt C 3. t

13 Vol. 15, 25 KÄHLER RICCI SOLITONS ON COPACT COPLEX ANIFOLDS 79 Note that ωφ φ n sup Xψ C c 2.14 ψ X ω g for some uniform constant c = cω g,x cf. Lemma 5.1 in [Z]. Therefore applying the Green formula to function φ, weprove sup φ = 1 φe θ X ω φ ωφ n V inf P φ Gx, e θ X ω φ ωφ n = 1 φe θ X ω φ ωφ n V inf ωφ φ Xφ Gx, e θ X ω φ ωφ n 1 φe θ X ω φ ωφ n V + C 3V n + Xφ C 1 φe θ X ω φ ωφ n V + C. 3 A C -Estimate for Solutions of 1.1 t Lemma 3.1. There are two positive numbers c 1 c 2 < 1 such that for any φ X ω g, c 1 Ĩ ωg φ Ĩ ωg φ J ωg φ c 2 Ĩ ωg φ. 3.1 Proof. Letω sφ = ω + s 1 2π φ. Then one can compute Ĩ ωg φ = 1 1 d φ V ds eθ X +sxφ ωsφ n ds = n 1 1 ds φ φe θ X +sxφ ω n 1 2πV sφ = n n Cn 1 k s k 1 s n 1 k e θx+sxφ ds 2πV J ωg φ = n 1 2πV k= φ φ ωφ k ωn 1 k g n n 1 1 2πV C 1 Cn 1 φ k φ ωφ k ωn 1 k g. 1 dt 1 k= ds t φ φe θx+stxφ ω n 1 stφ

14 71 HUAI-DONG CAO, GANG TIAN AND XIAOHUA ZHU GAFA = n 1 2πV n 1 Cn 1 k k= 1 1 tst k 1 st n 1 k e θx+stxφ dt ds φ φ ωφ k ωn 1 k g 3.3 n n 1 1 2πV C 1 Cn 1 φ k φ ωφ k ωn 1 k g. k= Combining , we get J ω φ C 1 C 1 Ĩ ω φ, consequently, prove the second inequality of 3.1. On the other h, we have cf. [TZ2], Ĩ ωg φ J ωg φ = n 1 2πV n 1 Cn 1 k k= 1 s k+1 1 s n 1 k e θx+sxφ ds φ φ ωφ k ωn 1 k g 3.4 n n 1 1 2πV C 2 Cn 1 φ k φ ωφ k ωn 1 k g. k= Hence, combining , we also prove the first inequality of 3.1. Proposition 3.1. Let φ = φ t t t > be a solution of equation 1.1 t. Then there are two uniform constants C 1 C 2 depending only on X t such that osc φ C 1 φωg n ωn φ +C Proof. Letθ X = θ Xω φ. First we note θ X = θ X Xh ω φ +c, for some constant c. Clearly c osc θ X,sinceθ X changes the sign. By using the fact cf. the relation 1.2, h ωφ = θ X 1 tφ +const., we have θ X = θ X X ωφ +1 txφ+c 2osc θ X +3 Xφ C 1, for some uniform constant C 1. Applying Theorem 2.1, we see that there is some uniform constant C 2 depending only on X t such that sup φ 1 φe θ X ω n V φ + C

15 Vol. 15, 25 KÄHLER RICCI SOLITONS ON COPACT COPLEX ANIFOLDS 711 On the other h, by using the Green formula, we have sup φ 1 φe θ X ωg n + C V for some uniform constant C 3 cf. Lemma 5.3 in [TZ3]. Hence, combing , we get osc φ 1 φe θ X ωg n V eθ X ω n φ +C 2 + C 3. By using , we prove A Smoothing Lemma In this section, following an approach in [T], we will prove a smoothing lemma by using Hamilton s Ricci flow. This lemma will be used in the proof of Theorem.2. Let ω be any Kähler form in c 1 > such that Ricω L X ω 1 ɛω, Xhω θ X ω 4.1 ɛc1, for some constant c 1 <ɛ<1. We consider the heat equation u =log ω+ 1 2π u n ω + u h n ω + θ X ω, 4.2 u t= =. Note that eq. 4.2 is the scalar version of the modified Kähler Ricci flow ω t = Ricω t +ω t + L X ω t. 1 2π u t,u t = ux,. Denote by h t = h ωt θ t = Here ω t = ω + θ X ω t, then it follows from the above equation maximum principle that h t θ t = u + c t where c t depends on t only. Also, u = hence c =. Differentiat- We list a few basic estimates for the solution ux, t. ing 4.2, we get u = +X u Applying the maximum principle, we have Lemma 4.1. Let u t be a solution of 4.2. Then u t C e t h ω θ X ω C, u C e t hω θ X ω C. u

16 712 HUAI-DONG CAO, GANG TIAN AND XIAOHUA ZHU GAFA Lemma Xh t θ t c 1 + nɛe t. Proof. From eq. 4.3, we have + X u = +X 2 u + +X u + X 2 u It follows from the maximum principle that + +X u. + Xh t θ t e t inf + X h ω θ X ω. On the other h, 4.1 implies that, at t =, Then the lemma follows directly. + Xh ω θ X ω c 1 + nɛ. u 2 Lemma 4.3. u 2 + t C u 2 e 2t hω θ C X ω 2. C Proof. By direct computations, we have Hence u u 2 = +X u u 2 = +X u 2 + t u 2 2 u 2 + t u 2 u 2 +2 u 2, + X u 2 +2 Lemma 4.3 follows from the maximum principle again. Set Lemma 4.4. v = h 1 θ 1 1 V 2 u 2 + u 2. u 2 + t u 2. h 1 θ 1 e θ 1 ω 1 n. v 2 L 2 2c 1 + ne 2 V λ 1 ɛ hω θ X ω C. Proof. By Lemma 4.2, we have + Xv +c 1 + neɛ. Itfollows that + Xv +c1 + neɛ e θ 1 ω n 1 = + Xv +c1 + neɛ e θ 1 ω 1 n =c 1 + neɛv.

17 Vol. 15, 25 KÄHLER RICCI SOLITONS ON COPACT COPLEX ANIFOLDS 713 Hence, by applying the Poincaré inequality Lemma 4.1, we have λ 1 v 2 e θ 1 ω n 1 1 v 2 e θ 1 ω1 n V V = 1 v + Xve θ 1 n ω 1 V = 1 v [ + Xv + c 1 eɛ ] e θ 1 n ω 1 V 1 V v C + Xv +c1 + neɛ e θ 1 n ω 1 2e 2 c 1 + nɛ hω θ X ω C. This shows the lemma is true. Lemma 4.5. We have v C Cn, c 1 a, λ 1 1+ h ω θ X ω C ɛ 1/2n+1, 4.4 provided that the following condition holds: there exists a constant a> such that for any x <r<1, Br x ar 2n 4.5 with respect to the metric ω 1. Proof. Pick r = ɛ 1/2n+1 cover by geodesic balls of radius r. For any x, wehavex B r x forsomex. Now inf B rx v 2 ɛ n 1 n+1 v 2 e θ 1 n ω 1 a Hence B rx 2c 1 + ne 2 V 2 aλ 1 ɛ h ω θ X ω C. inf v Cn, c 1,a,λ 1 ɛ 1/2n+1 hω θ X ω 1/2. B rx C Assuming inf Brx v = vx, then vx vx vx + vx r sup v + vx B rx e hω θ X ω C ɛ 1/2n+1 + Cɛ 1/2n+1 hω θ X ω 1/2 C C 1+ h ω θ X ω C ɛ 1/2n+1. This finishes the proof of Lemma 4.5. Proposition 4.1 Smoothing lemma. Let ω c 1 > be any Kähler metric satisfying 4.1. Then there is another Käher form ω = ω + such that 1 2π u

18 714 HUAI-DONG CAO, GANG TIAN AND XIAOHUA ZHU GAFA 1 u C e h ω θ X ω 2 h θ C 1/2 Cn, c 1,a,λ h ω θ X ω C ɛ 4n+1, where Cn, c 1,a,λ 1 is a constant depending only on the dimension n, the Poincare constant λ 1, constants c 1 a appeared in 4.5. Proof. We shall prove that ω 1 satisfies the above two conditions of the proposition under the assumption 4.5. By Lemma 4.1, it suffices to check the second condition only. Since 1 e h θ e θ 1 ω n 1 =1, V by 4.4 in Lemma 4.5, we have h θ C Cn, c 1,a,λ 1 1+ h ω θ X ω C ɛ 1/2n For any two points x, y in, if the distance dx, y ɛ 1/2n+1,then Lemma 4.3 implies that h θ e h ω θ X ω C hence h θ x h θ y e hω θ X ω dx, y C ɛ 1/4n+1. On the other h, if dx, y ɛ 1/2n+1 then 4.6 implies that h θ x h θ y dx, y Cn, c 1,a,λ 1 1+ h ω θ X ω C ɛ 1/4n+1. 1 This completes the proof of Proposition Properness of F ωks ψ We are now ready to prove Theorem.2 stated in the introduction. The method here is analogous to one in [T] for Kähler Einstein manifolds with positive scalar curvature. Proof of Theorem.2. Let ω KS be the Kähler form of Kähler Ricci soliton g KS ω g = ω ψ = ω KS + 1 2π ψ. We consider the complex onge Ampère equations with parameter t [, 1]: detg ij + ϕ ij =detg ij exp h g θ X Xϕ tϕ }, 5.1 g ij + ϕ ij >. Clearly, ψ modulo a constant is a solution of 5.1 t for t =1. Sinceψ is G- invariant, by the implicit function theorem, there are G-invariant solutions of 5.1 t for t sufficiently close to 1. In fact, in [TZ2], it was proved that there are G-invariant solutions of 5.1 t for any t [, 1]. This is because

19 Vol. 15, 25 KÄHLER RICCI SOLITONS ON COPACT COPLEX ANIFOLDS 715 Ĩ ωg ϕ t J ωg ϕ t is nondecreasing in t cf. Remark 1.1 in section 1, consequently the C 3 -norm of ϕ t can be uniformly bounded cf. [TZ2], [Y]. Put ω t = ω g + 1 2π ϕ t.thenω 1 = ω KS. oreover, by 1.2, we have h ωt θ X ω t = 1 tϕ t + c t, Ricω t L X ω t =tω t +1 tω tω t, where c t is determined by e 1 tϕt+ct e θ X ω t ωt n = V. In particular, c t 1 t ϕ t C, Xhωt θ X ω t = 1 txϕt hωt θ X ω t C 21 t ϕ t C, 1 t Xϕ t ϕ 1 + Xψ 1 tc 1 ω KS,X. Hence by applying Proposition 4.1 to each ω t, we obtain a modified Kähler form ω t = ω t + 1 2π u t satisfying u t C 2e1 t ϕ t C, h ω t C 1/2 Cn, c 1,a,λ tϕ t C 1 t 1/4n+1. As before, there are ψ t such that ω KS = ω t + 1 2π ψ t ω n KS =ω t n e h ω t θ Xω t X ψ t ψ t. It follows from the maximum principle that where µ t are constants with ϕ t = ϕ 1 ψ t + µ t, 5.2 µ t 2e + 11 t ϕ t C + c 1 ω KS,X. 5.3 Hence, ϕ t is uniformly equivalent to ϕ 1 as long as ψ t is uniformly bounded. Consider the operator Φ t : C 2,1/2 C,1/2 by Φ t ψ ωks 1 2π =log ψ n ωks n + h ω t θ X ω t X ψ ψ. Its linearization at ψ =is 1 X, so it is invertible in the space of G-invariant functions by the assumption of Theorem.2. Then by the implicit function theorem, there is a δ>, such that if the Hölder norm h ω t C 1/2 ω KS with respect to ω KS is less than δ, then there is a unique ψ t such that Φ t ψ t = ψ t C 2,1/2 Cδ.

20 716 HUAI-DONG CAO, GANG TIAN AND XIAOHUA ZHU GAFA We observe that λ 1,ω 2 n 1 λ 1,ωKS, a 1 2 2n a, whenever 1 2 ω KS ω 2ω KS,wherea is a constant appeared in 4.5 a is a constant such that a r 2n ωks Br x, x. Now we choose t such that 1 t δ/4c 4n+1 1 t ϕ t C 1 t 1/4n+1 = δ 4C, 5.4 where C = C n, c 1,a,λ 1,ωKS. Then by Proposition 4.1 the above argument, one can prove that for any t t,wehave u t C 2e1 t ϕ t C, ψ t C 1 4. Therefore, by using , we get ϕ 1 ϕ t C 6e1 t ϕ t C + c 2, ϕ 1 C c 2 ϕ t C 2 ϕ 1 C + c 3, 5.6 for some uniform constants c 2 c 3,aslongas1 t min1/12e, 1 t }. Since Ĩω g ϕ t J ωg ϕ t is nondecreasing, by 3.1, we have F ωks ψ = F ωg ϕ 1 1 = Ĩωg ϕ t J ωg ϕ t dt 1 t Ĩ ωg ϕ t J ωg ϕ t ε1 t J ωg ϕ t, where ε>isauniform constant. On the other h, by using the co-cycle condition of ˆF ωg, we have J ωg ϕ t = J ωg ϕ 1 1 ϕ 1 ϕ t e θ X ωg n V + ˆF ϕ1 ϕ t ϕ 1 J ωg ϕ 1 1 ϕ 1 ϕ t e θ X ωg n e θ X+Xϕ 1 ωϕ n V 1 J ωg ϕ 1 osc ϕ 1 ϕ t. It follows by , J ωg ϕ t ε I ωg ϕ 1 osc ϕ 1 ϕ t. Thus by , we get F ωks ψ εε 1 ti ωg ϕ 1 1 tosc ϕ t ϕ 1 εε 1 ti ωg ϕ 1 12e1 t 2 ϕ 1 C C 1 5.7

21 Vol. 15, 25 KÄHLER RICCI SOLITONS ON COPACT COPLEX ANIFOLDS 717 = εε 1 ti ωks ψ 12e1 t 2 osc ψ C 1, for any t t. Therefore, in case osc ψ C 1+I ωke ψ for some uniform constant C, then by choosing t=t in 5.7 using 5.4, we see that there are two positive numbers C C such that F ωks ψ CI ωks ψ 1/4n+5 C, 5.8 consequently this would prove the theorem. In the general case, we shall use a trick in [TZ1]. First by Proposition 3.1, we have for any t 1/2, osc ϕ t ϕ 1 C 2 1+IωKS ϕ t ϕ 1. Set ψ = ϕ t ϕ 1. Then applying inequality 5.8 to function ψ,weget F ωg ϕ t F ωg ϕ 1 = F ωks ψ 5.9 C 3 I ωks ψ 1/4n+5 C 4. On the other h, by integrating 1.8 from t to 1, we have ˆF ωg ϕ 1 ˆF ωg ϕ t J ωg ϕ 1 1 ϕ 1 e θ X ωg n V t J ωg ϕ t 1 ϕ t e θ X ωg n V 1 t Ĩ ωg ϕ 1 J ωg ϕ C 5 1 ti ωg ϕ 1 = C 5 1 ti ωks ψ. By using the concavity of the logarithmic function 3.6, we also have 1 log e h ϕt ωg n 1 t ϕ t e θ Xω t ωϕ n V V t 1 t 5.11 V sup ϕ t +C 6 C 6. Hence combining , we get F ωg ϕ t F ωg ϕ 1 C 5 1 ti ωks ψ+c From , we deduce 1 ti ωks ψ c 3 osc ϕ t ϕ 1 1/4n+5 c 4. Then as in 5.7, we prove cf. [TZ1], F ωks ψ εε 1 ti ωks ψ 1 tosc ϕ t ϕ 1 εε 1 ti ωks ψ 1 tc 1 3 4n+5 1 ti ωks ψ+c 4 4n ci ωks ψ 1/4n+5 C, for some small positive number c large number C. Thus Theorem.2 is proved.

22 718 HUAI-DONG CAO, GANG TIAN AND XIAOHUA ZHU GAFA Remark 5.1. By , we see that the inequality 5.13 is equivalent to the following non-linear inequality of oser Trudinger type, e ψ ωks n C exp J ωks ψ c J ωks ψ 1/4n+5 1 } ψωks n V for some positive numbers c C. Remark 5.2. In view of results in [T] [TZ1], one should be able to generalize Theorem.2 by proving the following: F ωks ψ ci ωks ψ 1/4n+5 C holds for any ψ Λ 1,ω KS. In the proof of Theorem.2, we used a technical assumption on subgroup G K in order to apply the implicit function theorem. We also notice from Theorem.2 that 5.13 holds for any almost plurisubharmonic function on a Kähler Einstein manifold without any nontrivial holomorphic vector field. References [A] T. Aubin, Réduction du cas positif de i equation de onge Ampère sure les varieétés Kählerinnes compactes àladémonstration d un intégralité, J. Func. Anal , [B] S. Bo, T. abuchi, Uniqueness of Kähler Einstein metrics modulo connected group actions, Algebraic Geometry, Adv. Studies in Pure ath. Sendai, Japan, [C] H.D. Cao, Limits of solutions of Kähler Ricci flow, J. Diff. Geom , [F] A. Futaki, T. abuchi, Bilinear forms extremal Kählervector fields associated with Kähler classes, ath. Ann , [H] R.S. Hamilton, Formation of singularities in the Ricci-flow, Surveys in Diff. Geom. International Press, Boston, , [K] N. Koiso, On rationally symmetric Hamilton s equation for Kähler Einstein metrics, Algebraic Geometry, Adv. Studies in Pure ath., Sendai, Japan [LY] P. Li, S.T. Yau, On the parabolic kernel of the Schrödinger operator, Acta ath , [1] T. abuchi, Heat kernel estimates the Green functions on ultiplier hermitian manifolds, Tohöku ath. J , [2] T. abuchi, ultiplier hermitianstructuresonkählermanifolds, Nagoya ath. J , [T] G. Tian, Kähler Einstein metrics with positive scalar curvature, Invent. ath , 1 39.

23 Vol. 15, 25 KÄHLER RICCI SOLITONS ON COPACT COPLEX ANIFOLDS 719 [TZ1] G. Tian, X.H. Zhu, A nonlinear inequality of oser Trudinger type, Calc. Var. Partial Differential Equations 1:4 2, [TZ2] G. Tian, X.H. Zhu, Uniqueness of Kähler Ricci solitons, Acta ath , [TZ3] G. Tian, X.H. Zhu, A new holomorphic invariant uniqueness of Kähler Ricci solitons. Comm. ath. Helv , [WZ] X.J. Wang, X.H. Zhu, Kähler Ricci solitons on toric manifolds with positive first Chern class, Adv. in ath , [Y] S.T. Yau, On the Ricci curvature of a compact Kähler mainfold the complex onge Ampère equation, 1, Comm. Pure Appl. ath , [Z] X.H. Zhu,Kähler Ricci soliton typed equations on compact complex manifolds with C 1 >, J. Geometric Analysis 1 2, Huai-Dong Cao, Department of athematics, Texas A& University, College Station, TX 77843, USA cao@math.tamu.edu Gang Tian, Department of athematics, assachusetts Institute of Technology, Cambridge, A 2139, USA tian@math.mit.edu School of athematical Sciences, Peking University, Beijing, 1871, China Current address: Department of athematics, Princeton NJ USA Xiaohua Zhu, School of athematical Sciences, Peking University, Beijing, 1871 China xhzhu@math.pku.edu.cn Received: October 24 Revised: February 25 Accepted: February 25