Gap theorems for Kähler-Ricci solitons

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1 Gap theorems for Kähler-Ricci solitons Haozhao Li arxiv: v [math.dg] 30 Jun 2009 Abstract In this paper, we prove that a gradient shrinking compact Kähler-Ricci soliton cannot have too large Ricci curvature unless it is Kähler-Einstein. Introduction The main purpose of this paper is to give two gap theorems on gradient shrinking Kähler-Ricci solitons with positive Ricci curvature on a compact Kähler manifold with c () > 0. Here we only discuss compact manifolds. Since Ricci flow was introduced by R. Hamilton in [4], Ricci solitons have been studied extensively. One interesting question is to classify all the Ricci solitons with some curvature conditions, especially with positive curvature operator. It is well known that there is no expanding or steady Ricci solitons on a compact Riemannian manifold of any dimension, and no shrinking Ricci solitons of dimension 2 and 3 (cf. [5][6]). Recently Böhm-Wilking in [] extended Hamilton s maximum principles, and they essentially proved that there is no nontrivial shrinking Ricci solitons with positive curvature operator on a compact manifold. For the Kähler case, by Siu-Yau s result in [0] a Kähler-Ricci soliton with positive holomorphic bisectional curvature is Kähler-Einstein. It is still very interesting to know how to prove this result via Kähler-Ricci flow (cf. [4]), or via complex onge-ampére equations. Rotationally symmetric Kähler-Ricci solitons have been constructed by Koiso [7] and Cao [2]. The existence and uniqueness of Kähler-Ricci solitons have been extensively studied in literature (cf. [3][2][3][5]). Let be a compact Kähler manifold with c () > 0. A Kähler metric g with its Kähler form ω is called a Kähler-Ricci solition with respect to a holomorphic vector field X if the equation Ric(ω) ω = L X ω is satisfied. Since ω is closed, we may write L X ω = u for some functionuwithu ij = u ī j = 0. Then the Futaki invariant is given by f X = Xuω n = u 2 ω n > 0.

2 Note thatf X depends only on the Kähler class [ω] and the holomorphic vector field X. The following is the first result in this paper: Theorem.. Let ω be a Kähler-Ricci soliton with a holomorphic vector field X. If thenω is Kähler-Einstein. Ric(ω) ω < f X + fx 2 +4f X, (.) 2 Remark.2. Theorem. tells us that the Ricci curvature of a Kähler-Ricci soliton can not be too close to the Kähler form, so it gives a gap between Kähler-Ricci solitons and Kähler-Einstein metrics. In [], G. Tian proposed a conjecture on the solution ω t of complex onge-ampére equations on a Kähler manifold withc () > 0 (ω + ϕ) n = e hω tϕ ω n, ω + ϕ > 0, (.2) whereh ω is the Ricci potential with respect to the metricω. It is known that admits a Kähler- Einstein metric if and only if (.2) is solvable for t [0,]. On the other hand, if admits no Kähler-Einstein metrics, then (.2) is solvable for t [0,t 0 )(t 0 ). Tian conjectured that when this case occurs, (,ω t ) converges to a space (,ω ), which might be a Kähler-Ricci soliton after certain normalization. Observe that the metricω t of (.2) has Ric(ω t ) > ( t)ω t, the following theorem tells us that if Tian s conjecture is true,t 0 might not be close to. Theorem.3. Let ω be a Kähler-Ricci soliton with a holomorphic vector field X. There exists a constantǫ > 0 depending only on the Futaki invariantf X such that if thenω is Kähler-Einstein. Ric(ω) > ( ǫ)ω, Remark.4. In Theorem.3, ǫ can be expressed explicitly from the proof. Acknowledgements: I would like to thank Professors F. Pacard for the support during the course of the work, and Professor X. X. Chen, W. Y. Ding and X. H. Zhu for their help and enlightening discussions. 2 Proof of Theorem. In this section, we prove Theorem.. Let g be a Kähler-Ricci soliton with Ric ω = u. By the definition of the Futaki invariant, f X = X(u)ω n = 2 u 2 ω n > 0.

3 Define we have the following lemma: ǫ := max Ric ω, (2.) Lemma 2.. Let λ be the first eigenvalue of g, thenλ ǫ. Proof. Letf be an eigenfunction satisfying g f = λ f, then ( 0 f 2 ω n = ( g f) 2 Ric( f, f) ) ω n (λ 2 ( ǫ)λ ) Thus,λ ǫ and the lemma is proved. f 2 ω n. By Lemma 2., we have u 2 ω n ( ǫ) u 2 ω n = ( ǫ)f X. (2.2) In fact, we can writeu = i= c if i, where f i are the eigenfunctions of g such that g f i = λ i f i, fi 2 ω n =. Here0 < λ λ 2 λ m <. Then g u = i= c i λ i f i. Integrating by parts, we have u 2 ω n = ( g u) 2 ω n = i= c 2 iλ 2 i λ i= c 2 iλ i = u 2 ω n. Hence the inequality (2.2) holds. On the other hand, by (2.) we have u 2 ω n ǫ 2. Combining this together with (2.2), we have ǫ 2 ( ǫ)f X 0. Then The theorem is proved. ǫ f X + fx 2 +4f X. 2 3

4 3 Proof of Theorem.3 In this section, we prove Theorem.3. Let g be a Kähler Ricci soliton with Ric(ω) ω = u, whereu ij = u ī j = 0. We can normalizeusuch that u ω n = 0. (3.) Lemma 3.. u satisfies the following equation Proof. By direct calculation, we have g u+u u 2 = f X. (3.2) (u iī +u u i u ī ) j = u iīj +u j u i u īj = R j ku k +u j u i u īj = 0. Then g u + u u 2 is a constant. By (3.) and the definition of the Futaki invariant, (3.2) holds and the lemma is proved. Now we assumeric(ω) λω withλ > 0, so the scalar curvaturer nλ. By Lemma 3., u is bounded from below. In fact, u = f X + u 2 u = f X + u 2 +R n f X +nλ n. (3.3) Now we can prove the following lemma Lemma 3.2. The scalar curvature is uniformly bounded from above, i.e. R Λ(λ,f X ), whereλ(λ,f X ) is a constant depending only onλand f X. oreover, lim λ Λ(λ,f X ) is finite. Proof. By the Kähler-Ricci soliton equation (3.2), we have u 2 = g u+u+f X = n R+u+f X u+n+f X nλ. (3.4) LetB = f X nλ+n+, then by (3.3)u+B. Hence u 2 u+b u+b. Let p be a minimum point of u, by the normalization condition (3.) u(p) 0. Then for anyx, we have u c n π u+b(x) u+b(p) u+b diam(g) c nπ 2 u+b λ 2 λ, (3.5) 4

5 where diam(g) is the diameter of (,g) and c n is a constant depending only on n. It follows that u c2 n π2 B 4λ +c nπ λ. Therefore, R = n u = n+f X u 2 +u n+f X + c2 n π2 B 4λ +c nπ λ. Now we can finish the proof of Theorem.3. By Lemma 3.2 andr nλ, we have u 2 ω n = (R n) 2 ω n = {R>n}(R n) 2 ω n + (R n) 2 ω n {R<n} (Λ n) (R n)ω n +n 2 ( λ) 2. {R>n} On the other hand, Therefore, 0 = (R n)ω n + (R n)ω n. {R>n} {R<n} u 2 ω n = (Λ n) (n R)ω n +n 2 ( λ) 2 {R<n} (Λ n)n( λ)+n 2 ( λ) 2 0, (3.6) asλ. On the other hand, by the inequality (2.2) we have u 2 ω n λf X > 0, which contradicts (3.6) whenλis sufficiently close to. The theorem is proved. References [] C. Böhm, B. Wilking: anifolds with positive curvature operators are space forms, math.dg/ [2] H. D. Cao: Existence of gradient Kahler-Ricci soliton, Elliptic and parabolic methods in geometry, Eds. B.Chow, R.Gulliver, S.Levy, J.Sullivan, A K Peters, pp.-6,

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