Plurisubharmonic functions and the Kahler-Ricci flow

Transcription

1 Plurisubharmonic functions and the Kahler-Ricci flow Lei Ni, Luen-fai Tam American Journal of athematics, Volume 125, Number 3, June 23, pp (Article) Published by The Johns Hopkins University Press DOI: /ajm For additional information about this article Access provided by Penn State Univ Libraries (29 Apr :12 GT)

2 PLURISUBHARONIC FUNCTIONS AND THE KÄHLER-RICCI FLOW By LEI NI and LUEN-FAI TA Abstract. In this paper, a sharp linear trace Li-Yau-Hamilton inequality for Kähler-Ricci flow is proved. The new inequality extends the previous trace Harnack inequality obtained by H.-D. Cao. We also establish sharp gradient estimates for the positive solution of the time-dependent heat equation for some cases. Finally, we apply this new linear trace Li-Yau-Hamilton inequality to study the Liouville properties of the plurisubharmonic functions on complete Kähler manifolds with bounded nonnegative holomorphic bisectional curvature.. Introduction. Consider the Kähler-Ricci flow on a Kähler manifold (, g α β(x)): (.1) t g α β = R α β, g α β (x,)=g α β (x). In this work, (, g α β(x)) will be assumed to be complete and noncompact with bounded nonnegative holomorphic bisectional curvature. Solutions of (.1) on complete noncompact Kähler manifolds with bounded nonnegative bisectional curvature were extensively studied in a series of paper of Shi [Sh1 3]. Important properties and applications have also been obtained, see [Sh2 3], [C-Z], [C-T-Z]. In [N-T], the authors studied the Kähler-Ricci flow from another point of view. Namely, solutions of (.1) are investigated by using the solution to the Poincaré-Lelong equation obtained in [-S-Y], [N-S-T1]. ore precisely, it was proved in [N-S-T1], under some mild average assumptions on scalar curvature R (x) of the initial metric g α β(x), one can solve the Poincaré- Lelong equation: 1 u = Ric, (.2) where Ric is the Ricci form of the initial metric. We should mention that (.2) was solved by ok-siu-yau [-S-Y] and ok [1] in the case when has maximal volume growth and the scalar curvature has quadratical pointwise decay. Using the solution of (.2), one can easily find a function u(x, t) so that anuscript received October 21, 21. Research of the first author supported in part by NSF Grant DS and DS-2323; research of the second author supported in part by an Earmarked Grant of Hong Kong #CUHK4217/99P. American Journal of athematics 125 (23),

3 624 LEI NI AND LUEN-FAI TA 1 u(x, t) = Ric(x, t) where Ric is the Ricci form of the metric g(t). oreover u(x, t) satisfies the time-dependent heat equation: ( ) t u = (.3) u(x,)=u (x). Hence one can study (.1) by using (.3). For example, we gave a simple proof of the long time existence of (.1) under some growth conditions on R in [N-T]. Note that in this case u and u(, t) are both plurisubharmonic because g(t) has nonnegative holomorphic bisectional curvature [Sh2 3]. This motivates us to study (.3) for general plurisubharmonic initial data u. Under a rather mild assumption on the growth rate of u, we can prove that (.3) has a long time solution in the sense that if the Kähler-Ricci flow (.1) has a solution up to time T, then (.3) also has a unique solution up to time T. The next important question is whether or not under the flow (.3) the plurisubharmonicity will be preserved. In order to study this problem, we have to study the complex Hessian u α β of u. One can show that u α β satisfies the complex Lichnerowicz-Laplacian heat equation (see (1.2) for details). We shall prove that if the u is of at most exponential growth, then plurisubharmonicity will be preserved. Here is the Laplacian of the initial metric. In fact, we shall prove the result for more general Hermitian symmetric (1,1) tensors which satisfy the Lichnerowicz-Laplacian heat equation. See Proposition 1.1. In case u is the solution of (.2), the assumption on the rate of growth of u is the same as the assumption on the rate of growth of the scalar curvature R, which is assumed to be bounded in [Sh2 3]. There are many important differential Harnack type inequalities for Ricci flow and curvature flows obtained by various people, see [L-Y], [H3], [Cw1 2], [Co1 2], [A] for examples. Works in this area can be traced back to the fundamental works of Li-Yau [L-Y] and Hamilton [H3]. For this reason, in this paper we shall call this kind of inequalities to be Li-Yau-Hamilton type inequalities, or LYH inequalities for short. In [C-H], Chow and Hamilton obtained a linear trace LYH inequality for a symmetric two-tensor on a Riemannian manifold with a family of metric g(t) satisfying the Ricci flow equation so that the initial metric has nonnegative curvature operator. The two-tensor is assumed to satisfy the real Lichnerowicz-Laplacian heat equation. In this paper, using the results of Cao [Co1 2], we shall prove a complex version of Chow-Hamilton s result. ore precisely, suppose (.1) has a solution on [, T] so that the initial metric has nonnegative bounded holomorphic bisectional curvature. Let h α β(x, t) bea solution of the complex Lichnerowicz-Laplacian heat equation so that h α β(x,) and h α β(x, t) satisfies some growth conditions. Then on (, T], we have Z = 1 2 [gα β β div (h) α + g γ δ γ div (h) δ] + g α βg γ δ[r α δh γ β + γ h α δv β + βh α δv γ + h α δv βv γ ]+ H t

4 PLURISUBHARONIC FUNCTIONS, KÄHLER-RICCI FLOW 625 where div (h) α = g γ δ γ h α δ, div (h) β = g γ δ δh γ β, H is the trace of h α β with respect to g α β(x, t), and V is any vector field of type (1,). See Theorem 1.1 for details. In case h α β is the complex Hessian of a plurisubharmonic solution of (.3) so that u is not harmonic, then our result implies that w = u t satisfies w t w 2 w + w t which extends Cao s trace LYH type inequality for the scalar curvature [Co1 2]. Unlike [C-H], which mainly considers compact manifolds and is not very specific on noncompact manifolds, we need the growth conditions on h α β so that one can apply the maximum principle in [N-T]. As an application of the above results on the study of (.3) and the linear trace LYH type inequality, we shall study Liouville properties for plurisubharmonic functions on (, g α β(x)). Suppose (.1) has long time solution. Then we have the following: THEORE 3.1. Let (, g α β(x)) be a complete noncompact manifold with bounded nonnegative holomorphic bisectional curvature so that the (.1) has long time solution. Suppose u is a plurisubharmonic function such that (i) u is bounded; and (ii) u (x) exp (a(1 + r (x)) for some constant a >. Then u must be constant. In C m, a plurisubharmonic function with sub-logarithmic growth must be constant. It is conjectured that this is still true for complete noncompact Kähler manifolds with nonnegative Ricci curvature. In this paper, we shall also prove that in some cases, the condition that u is bounded in the above theorem can be relaxed. For example, one can prove that if the scalar curvature has quadratic decay in the average sense, then Theorem 3.1 is still true if the condition (i) is replaced by the condition that u has sub-logarithmic growth. This is a special case of a more general result, see Theorem 3.2. In particular, when u (x) is the solution of (.2), the Liouville result mentioned above implies the gap theorem proved in [C-Z] and [N-T]. For previous results of Liouville properties of plurisubharmonic functions, please see [N-S-T1 2], [N-T]. As a by-product of our argument, we also prove a Li-Yau type differential inequality for the positive plurisubharmonic solution u(x, t) of (.3) (see Theorem 2.2). Namely, we have u t u u 2 u 2 + m t exactly as in [L-Y] for a fixed metric. Hopefully, this differential inequality will have applications to the study of the plurisubharmonic functions, the Kähler-Ricci flow and other problems. Here is the organization of the paper. In 1, we shall prove the preservation of nonnegativity of (1,1) tensors and the linear trace LYH type inequality for

5 626 LEI NI AND LUEN-FAI TA (1,1) tensors. In 2, we shall study the initial value problem (.3) and prove the Li-Yau type inequality for the positive solution to (.3). In 3, we shall study Liouville properties of plurisubharmonic functions. Acknowledgments. The authors would like to thank Ben Chow for his interest and discussions and also Huai-Dong Cao for discussions. 1. A Li-Yau-Hamilton inequality. In this section we shall prove a linear trace Li-Yau-Hamilton inequality which is the Kähler version of the one obtained by Chow and Hamilton in [C-H]. Just as Chow-Hamilton s LYH inequality extends the trace Li-Yau-Hamilton inequality of Hamilton [H3] our differential inequality extends the trace LYH inequality of Cao [Co1] for the scalar curvature. Applications of this new inequality will be given in the following sections. Let ( m, g α β(x)) be a complete noncompact Kähler manifold with bounded nonnegative holomorphic bisectional curvature. Because of the results in [Sh1], in this section we always assume that solution of the following Ricci-Kähler flow exists on [, T] { t g α β = R α β (1.1) such that on [, T], g α β(x,)=g α β(x) (i) g α β(x, t) is nonincreasing in t and is uniformly equivalent to g α β(x, ); (ii) the curvature tensors of g α β(x, t) are uniformly bounded; (iii) there exists a constant C such that Rm (x, t) C ; and t 1 2 (iv) g α β(x, t) has nonnegative holomorphic bisectional curvature. Sufficient conditions that (1.1) has long time existence are given in [Sh2 3], [N-T], see also [C-T-Z] for the surfaces case. In this work, we will use the maximum principle of the authors [N-T, Theorem 1.2] from time to time. For the convenience of the readers, we include the statement of this maximum principle here. THEORE 1.1. Let g ij (x, t) be a smooth family of complete Riemannian metrics defined on with t T for some T > such that for any T t 2 t 1 Cg ij (x, t 1 ) g ij (x, t 2 ) g ij (x, t 1 ) for some constant C > for all x and let f (x, t) be a smooth function such

6 PLURISUBHARONIC FUNCTIONS, KÄHLER-RICCI FLOW 627 that ( t )f (x, t) whenever f (x, t). Assume that exp ( ar 2 (x))f+ 2 (x, s) dv ds < for some a >, where r (x) is the distance function to a fixed point o with respect to g ij (x,). Suppose f (x,) for all x. Then f (x, t) for all (x, t) [, T]. In the following let h α β(x, t) be a Hermitian symmetric tensor defined on [, T], which is also deformed by the complex Lichnerowicz-Laplacian heat equation: ( ) (1.2) t h γ δ = R βᾱγ δh α β 1 2 (R γ ph p δ + R p δh γ p ). We shall obtain a LYH inequality for h α β provided h α β is nonnegative and does not grow very fast on [, T]. In application, usually we only know that h α β is nonnegative initially. Hence we shall discuss conditions on h α β so that nonnegativity is preserved under the flow. The following lemma is basically from [H3, Lemma 5.1]. LEA 1.1. For any a > and C > there exists a positive function φ(x, t) and b > such that exp (b(r (x) + 1)) φ(x, t) exp (a(r (x) + 1)) and φ Cφ on [, T], where r (x) is the distance from a fixed point o with respect to the initial metric g(). Proof. By Lemma 5.1 in [H3], there is a smooth function f (x) and a constant C 1 > on such that C1 1 (1 + r (x)) f (x) C 1 (1 + r (x)), f + 2 f C 1. As in [H3], we can choose φ(x, t) = exp (At+αf (x)) for suitable positive constants A and α, then φ will be the required function. To simplify notations, in the rest of the section, let h be the norm of h with respect to g α β(x, t), (1.3) Φ = h 2 Ψ = h 2 = αβγ ( γh α β 2 + γ h α β 2 ) Λ = h 2 = αβγδ ( δ γ h α β 2 + δ γ h α β 2 ).

7 628 LEI NI AND LUEN-FAI TA Suppose h satisfies (1.2), then direct computations show (see [H1] for example): (1.4) (1.5) Φ = Ψ + A, Ψ = Λ + B, where A and B satisfy the following conditions: There exists a constant C > such that A CΦ and t B C(Φ + Ψ) on [, T]. Here we have used properties (ii) and (iii) of g α β. oreover, in normal coordinates Φ 2 = α Φ α Φᾱ = α ξ,τ h ξ τ,α h ξτ + h ξ τ h ξτ,α ξ,τ h ξ τ,ᾱ h ξτ + h ξ τ h ξτ,ᾱ 4 h 2 α ξ,τ h ξ τ,α 2 2 h ξ τ,ᾱ 2 ξ,τ 2 h 2 ( h ξ τ,α 2 + h ξ τ,ᾱ 2 ). α,ξτ Hence (1.6) Φ 2 2ΦΨ. Similarly, (1.7) Ψ 2 2ΨΛ. LEA 1.2. Let h α β be a tensor satisfying (1.2). Suppose (1.8) and (1.9) T h α β(x,) exp (a(1 + r (x)) exp ( br(x)) h 2 2 (x, t) dv t dt < for some positive constants a and b. Then there exists a positive constant c > such that (1.1) h α β(x, t) exp (c(1 + r (x)) on [, T].

8 PLURISUBHARONIC FUNCTIONS, KÄHLER-RICCI FLOW 629 (1.11) Proof. By (1.4) and (1.8), it is easy to see that [e C 1t (1 + Φ) 1 2 ] for some constant C 1 >. By Lemma 1.1, there exists a function φ(x, t) and constant c > such that exp (c(r (x) + 1)) φ(x, t) exp (a(r (x) + 1)) and φ Cφ with C >. By (1.9), (1.1) and (1.11), we have φ +1 (1+Φ) 1 2 by the maximum principle Theorem 1.1. The lemma follows by choice of an even larger c. Next we shall prove that nonnegativity of h will be preserved by the flow under certain conditions. PROPOSITION 1.1. Suppose h α β satisfy (1.2) and the conditions (1.8) and (1.9) of Lemma 1.2. Suppose also that h α β(x,). Then h α β(x, t) for t >. Proof. By Lemma 1.2, there exists a constant c > such that (1.12) h (x, t) exp (c(1 + r (x))). By Lemma 1.1, for any C >, there exists a function φ such that (1.13) and (1.14) exp (c (1 + r (x))) φ exp (2c(1 + r (x))) φ>c φ. It is enough to show that h α β(x, t) +ɛφg α β(x, t) >, for any ɛ>. Now we calculate ( ) (1.15) t (h α β + ɛφg α β) =R α βγ δ(h γδ + ɛφg γδ ) 1 2 R α p(h p β + ɛφg p β) 1 2 R p β (h α p + ɛφg α p ) + ɛ(φ t φ)g α β ɛφr α β. Here we have used (1.2) and the Ricci flow equation. By (1.12), (1.13) and the fact that at t =,h α β + ɛφg α β >, if h α β(x, t) +ɛφg α β(x, t) > fails to hold at some t >, then there is (x, t ) and unit (1,) vector at x with t > such

9 63 LEI NI AND LUEN-FAI TA that (h α β(x, t )+ɛφg α β(x, t))v α v β = and t is the first time that happens. As in [Cw3], we can extend v in a neighborhood in space-time of (x, t ) such that v and v = at (x, t ) with respect to the metric g(t ) and such that v is independent of time. Hence at (x, t ) we have = ( ) t [(h α β + ɛφg α β)v α v β ] [( ) ] t (h α β + ɛφg α β) v α v β = R α βγ δ(h γδ + ɛφg γδ )v α v β 1 2 R β α p(h p β + ɛφg p β)v α v 1 2 R p β (h α p + ɛφg α p )v α v β + ɛ(φ t φ)g α βv α v β ɛφr α βv α v β. Since v minimizes h α β + ɛφg α β among all (1,) unit vectors at x, first variation gives (h α β + ɛφg α β)v α =(h α β + ɛφg α β) v β =. Using also the fact that, g(t ) has nonnegative holomorphic bisectional curvature, we conclude that ɛ(φ t φ) ɛφr α βv a v β > for sufficiently large C, since Rm is bounded. This is a contradiction. We should remark that the result is still true if is compact. In this case, there is no need to impose a growth condition on h α β. oreover if h α β(x,) is positive at some point, then h α β(x, t) will be positive for all t >. In order to apply the maximum principle we also need the following estimates. LEA 1.3. Let h α β as in Lemma 1.2. Then for any a >, (1.16) (1.17) and (1.18) e ar2 (x) Ψ(x, t) dv t dt <, te ar2 (x) Λ(x, t) dv t dt < te ar2 (x) Ψ 2 (x, t) dv t dt <. Proof. Let f (x) be a smooth function such that f 1, f =1onB (o, R), f = outside B (o,2r) and f C/R for some constant C independent of R.

10 PLURISUBHARONIC FUNCTIONS, KÄHLER-RICCI FLOW 631 Here B (o, R) is the geodesic ball with center at o and radius R with respect to g(). ultiply (1.4) by f 2 and integrating by parts, we have: ( ) T f 2 Ψ dv t dt f 2 t Φ dv t dt + C 1 f 2 Φ dv t dt f 2 Φ dv dt +2 f f Φ dv t dt + C 1 f 2 Φ dv t dt for some constant C 1. Here we have used the fact that dv t is nonincreasing. Using (1.6) and Schwarz inequality, we have [ f 2 Ψ dv t dt C 2 ] f 2 Φ dv dt + ( f 2 + f 2 )Φ dv dt, for some constant C 2, where we have used the fact that g(t) and g() are equivalent in [, T]. Hence (1.19) [ Ψ dv t dt C 3 B (o,r) B (o,2r) ] Φ dv dt + Φ dv dt, B (o,2r) for some constant C 3, where we assume R 1. By Lemma 1.1, h is at most of exponential growth, hence it is easy to see (1.16) is true because g() has nonnegative Ricci curvature. To prove (1.17), multiplying (1.5) by tf 2 and integrating by parts we have for R 1, tf 2 Λ dv t dt ( ) T tf 2 t Ψ dv t dt + C 4 f 2 (Φ + Ψ) dv t dt f 2 Ψ dv t dt +2 tf f Ψ dv t dt + C 4 f 2 (Φ + Ψ) dv t dt C 5 ( f 2 + f 2 )(Φ + Ψ) dv t dt + 1 tf 2 Λ dv t dt 2 for some constants C 4, C 5, where we have used (1.7). Hence if R is large (1.2) B (o,r) tλ dv t dt 3C 5 B (o,2r) (Φ + Ψ) dv t dt. Combining (1.19) and (1.2), we can conclude that (1.17) is true.

11 632 LEI NI AND LUEN-FAI TA To prove (1.18), multiplying (1.4) by tf 2 Ψ and integrating by parts, we have tf 2 Ψ 2 dv t dt C 6 C 6 T + C 7 T +2 ( ) T tf 2 Ψ t Φ dv t dt + T f 2 ΨΦ dv t dt + T f 2 ΨΦ dv t dt + tf 2 ΦBdV t dt tf 2 t tf 2 ΨAdV t dt (Φ t Ψ + Ψ Φ ) dv t dt tf 2 (Φ Ψ + Ψ Φ) dv t dt f 2 (Ψ + Φ)Φ dv t dt +2 tf 2 Φ Ψ dv t dt tf f (Φ Ψ + Ψ Φ ) dv t dt for some constants C 6, C 7, where we have used (1.5). Apply (1.6) and (1.7) to Φ and Ψ respectively, and use Schwarz inequality, we have, ( tf 2 Ψ 2 dv t dt C 8 ) ( f 2 + f 2 )(Φ + Ψ)Φ dv t dt + tf 2 ΛΦ dv t dt for some constant C 8. Combining this with (1.16) and (1.17) and the fact that Φ grows at most exponentially, we can conclude that (1.18) is true. Remark 1.1. (1.17) and (1.18) imply that for any ɛ>, ɛ e ar2 (x) Λ(x, t) dv t dt <, and ɛ e ar2 (x) Ψ 2 (x, t) dv t dt <. Now we are ready to prove a LYH inequality. Let div (h) α = g γ δ γ h α δ and div (h) β = g γ δ δh γ β. Consider the quantity (1.21) Z = g α βg γ δ [ 1 2 ( β γ + γ β )h α δ + R α δ h γ β

12 PLURISUBHARONIC FUNCTIONS, KÄHLER-RICCI FLOW 633 +( γ h α δv β + βh α δv γ )+h α δv βv γ ] + H t = 1 2 [gα β β div (h) α + g γ δ γ div (h) δ] + g α βg γ δ[r α δh γ β + γ h α δv β + βh α δv γ + h α δv βv γ ]+ H t, where H is the trace of h α β with respect to g α β(x, t). THEORE 1.2. Let h α β be a Hermitian symmetric tensor satisfying (1.2) on [, T]. Suppose h α β(x,) and satisfies (1.8) and (1.9) in Lemma 1.2. Then Z on (, T] for any smooth vector field V of type (1, ). In order to prove the theorem, we need to compute t )Z. As in [C-H], we need to calculate t )Z. We break up the computations into several lemmas. LEA 1.4. Under normal coordinates at a point, ( ) (1.22) t ( div (h) α )=R s t t h α s + α R s t h st 1 2 R α t ( div (h) t ). Proof. Direct calculation shows (1.23) t (gγ δ γ h α δ) = t [gγ δ( γ h α δ Γ p αγh p δ)] = g γ t R s t g s δ γ h α δ +g γ δ γ t h α δ ) ( ) g γ δ t Γp αγ h p δ = R s t t h α s + γ ( h α γ +R α γs t h st 1 2 R α th t γ 1 2 R t γh α t ) + γ R α p h p γ Now we calculate γ ( h α γ ). By definition, = R s t t h α s + α R s t h st + R α γs t γ h st γr α t h t γ 1 2 R α t γ h t γ 1 2 trh α t 1 2 R t γ γ h α t + γ ( h α γ ). On the other hand, γ ( h α γ )= 1 2 γ( s s + s s )h α γ. γ s s h α γ = s γ s h α γ = s [ s γ h α γ R α pγ s h p γ + R p γγ s h α p ] = s s γ h α γ γ R α p h p γ R α pγ s s h p γ + p Rh α p + R p s s h α p.

13 634 LEI NI AND LUEN-FAI TA Similarly, γ s s h α γ = s γ s h α γ + R p γγ s s h α p R s pγ s p h α γ R α pγ s s h p γ = s s γ h α γ + R p s s h α p R γ p p h α γ R α pγ s s h p γ. Combining the above three we have γ ( h α γ )= ( γ h α γ ) 1 2 γr α p h p γ R α pγ s s h p γ prh α p + R p s s h α p 1 2 R γ p p h α γ. Plugging the above into (1.23), the lemma is proved. LEA 1.5. Under normal coordinates at a point, (1.24) t ) (g α β β div (h) α )=R sᾱ s div (h) α + ᾱr s t t h α s + α R s t ᾱh st + R s t ᾱ t h α s +( ᾱ α R s t )h st. Proof. Direct calculation shows that β t (gα β div (h) α )=g α t R s t g s β β div (h) α + ᾱ α) t div (h) = R sᾱ s ( div (h) α ) + ᾱ [ div (h) α + R s t t h α s + α R s t h st 1 ] 2 R α t div (h) t, by Lemma 1.4. Therefore we have that (1.25) β t (gα β div (h) α )= ᾱ( div (h) α )+ 1 2 R sᾱ s ( div (h) α ) + ᾱr s t t h α s + α R s t ᾱh st + R s t ᾱ t h α s + ᾱ α R s t h st 1 2 t R( div (h) t ). Now we calculate ᾱ( div (h) α ). By definition ᾱ( div (h) α )= 1 2 ᾱ s s div (h) α ᾱ s s div (h) α.

14 PLURISUBHARONIC FUNCTIONS, KÄHLER-RICCI FLOW 635 On the other hand and ᾱ s s div (h) α = s ᾱ s div (h) α = s [ s ᾱ div (h) α + R α psᾱ div (h) p ] = s s ᾱ div (h) α +( s R)( div (h) s )+R s p s div (h) p ᾱ s s div (h) α = s ᾱ s div (h) α + R ps s div (h) p R pᾱ p div (h) α = s s ᾱ div (h) α + R ps s div (h) p R pᾱ p div (h) α. Combining the above three we have that ᾱ( div (h) α )= ( ᾱ div (h) α )+ 1 2 sr( div (h) s )+ 1 2 R s p s div (h) p. Plugging into (1.25), this completes the proof of Lemma 1.3. Taking the conjugation we will have the following lemmas. LEA 1.4. Under normal coordinates at a point, ( ) (1.26) t ( div (h) β) =R s p s h p β + βr p γ h γ p 1 2 R t β div (h) t. (1.27) LEA 1.5. Under normal coordinates at a point, (g βᾱ β div (h)ᾱ) =R α p p div (h)ᾱ + α R s p s h pᾱ + ᾱr p γ α h γ p + R s p α s h pᾱ +( α ᾱr p γ )h γ p. Now we are ready to calculate t )Z. By Proposition 1.1, h is nonnegative. However h may be zero somewhere, we consider Ẑ instead, where: Ẑ =I+II+III+IV+V, where I= 1 2 [gα β β div (h) α )+g γ δ γ div (h) δ], II = g α βg γ δ[r α δh γ β + ɛr],

15 636 LEI NI AND LUEN-FAI TA III = g α β div (h) α V β + g γ δ div (h) δv γ, IV = g α βg γ δ h α δv βv α, V= H + ɛm, t and where ɛ > is a fixed constant, R is the scalar curvature, and h α β = h α β + ɛg α β. We calculate them one by one. In the following, we always do computations in normal coordinates at a point because the final result will not depend on the choice of coordinates. From Lemma 1.5, Lemma 1.5, and the second Bianchi identity we have that (1.28) I= 1 2 [R α p p div (h)ᾱ + R pᾱ p div (h) α ] [R s p α s h pᾱ + R s p ᾱ p h α s ] + R s t h st + tr sᾱ t h α s + t R sᾱ th sα. (1.29) II = (g α δg γ βr α βh γ δ + ɛr) (( ) ) =2R β γ R α βh γᾱ + t R α β h βᾱ ( h βᾱ ) + R α β s R α β s h βᾱ s R α β s h βᾱ + ɛ R α β 2 =2R α βs t R st h βᾱ s R α β s h βᾱ s R α β s h βᾱ + ɛ R α β 2. Here we have used (1.2) and the equation satisfies by the Ricci form [Sh3]: ( ) (1.3) t R α β = R α βγ δr γδ R α s R s β. Using Lemma 1.4, Lemma 1.4 and the second Bianchi identity we have that (1.31) III = [g α β( div (h) α V β + div (h) βv α )] = Rᾱβ div (h) α V β + Rᾱβ div (h) βv α + div (h) α (( ) ) (( ) ) t Vᾱ + div (h)ᾱ t V α

16 Using (1.2) we have ( ) (1.32) t IV = PLURISUBHARONIC FUNCTIONS, KÄHLER-RICCI FLOW (R s t t h α s + s R α t h st 1 ) 2 R α t div (h) t Vᾱ ( + R s t s h tᾱ + s R tᾱ h s t 1 ) 2 R tᾱ div (h) t V α s div (h) α s Vᾱ s div (h) α s Vᾱ s div (h)ᾱ s V α s div (h)ᾱ s V α. (g α βg γ δ h α δv γ V β) = R α βs t h st V β Vᾱ R α sh s γ V γ Vᾱ h α sr s γ V γ Vᾱ + h α γ ( V γ ) Vᾱ + h α γ V γ ( Vᾱ s h α γ s (V γ Vᾱ) s h α γ s (V γ Vᾱ) h α γ [ s V γ s Vᾱ + s V γ s Vᾱ + ɛr α γ VᾱV γ ]. Taking trace on (1.2) one can have ( ) t V= R α sh sᾱ (1.33) H + ɛm t t 2. Now combining them together we have that ( ) (1.34) t Ẑ = Y [R α p p div (h)ᾱ + R pᾱ p div (h) α ] [R s p α s h pᾱ + R s p ᾱ p h α s ] + R α βs t Rᾱβ h t s + Rᾱβ div (h) α V β + Rᾱβ div (h) βv α (( ) ) (( ) ) + div (h) α t Vᾱ + div (h)ᾱ t V α s div (h) α s Vᾱ s div (h) α s Vᾱ s div (h)ᾱ s V α s div (h)ᾱ s V α + R s t t h α s Vᾱ 1 2 R α t div (h) t Vᾱ + R s t s h tᾱ V α ) 1 2 R tᾱ div (h) tv α R α sh s γ V γ Vᾱ h α sr s γ V γ Vᾱ

17 638 LEI NI AND LUEN-FAI TA + h α γ ( V γ ) Vᾱ + h α γ V γ ( Vᾱ s h α γ s (V γ Vᾱ) s h α γ s (V γ Vᾱ) h α γ [ s V γ s Vᾱ + s V γ s Vᾱ] H + ɛm t 2 + ɛ R α β 2 + ɛr α γ VᾱV γ, ) where (1.35) [ Y 1 = R s t + R s tα βrᾱβ + α R s t Vᾱ + ᾱr s t V α + R s tα βvᾱv β + R ] s t t h st. By Proposition 1.1, h α β on [, T]. Hence by Cao s LYH inequality [Co1 2] and the fact that (, g α β(x, t)) has nonnegative holomorphic bisectional curvature, the two factors in Y 1 are all nonnegative tensors. Therefore Y 1. Since h α β ɛg α β, for each (x, t), Ẑ attains minimum for some V. Then by the first variation we have (1.36) div (h) α + h α γ V γ = and div (h)ᾱ + h γᾱ V γ =. Direct calculation also shows that (1.37) R pᾱ p div (h) α + R α p p div (h)ᾱ = R s p α s h pᾱ + R s p ᾱ p h α s +2R α p R pᾱs γ h γ s 2R α p R p s h sᾱ. Combining (1.34) (1.37) we have that (1.38) Ẑ = Y 1 +[R α p p div (h)ᾱ + R pᾱ p div (h) α ]+R α p R p s h sᾱ Differentiate (1.36) and we have s div (h) α s Vᾱ s div (h) α s Vᾱ s div (h)ᾱ s V α s div (h)ᾱ s V α + R s t t h α s Vᾱ + R s t s h tᾱ V α s h α γ s (V γ Vᾱ) s h α γ s (V γ Vᾱ) h α γ [ s V γ s Vᾱ + s V γ s Vᾱ] H + ɛm t 2. (1.39) s div (h) α +( s h α γ )V γ + h α γ s V γ =, s div (h)ᾱ +( s h γᾱ )V γ + h γᾱ s V γ =,

18 PLURISUBHARONIC FUNCTIONS, KÄHLER-RICCI FLOW 639 s div (h) α +( s h α γ )V γ + h α γ s V γ =, s div (h)ᾱ +( s h γᾱ )V γ + h γᾱ s V γ =. Plugging the above into (1.34) we have that (1.4) Ẑ = Y 1 + R α p R p s h sᾱ R α p h γᾱ p V γ R pᾱ h α γ p V γ + h γᾱ s V γ s V α + h γᾱ s V γ s V α H + ɛm t 2 + ɛ R α β 2. Let (1.41) Y 2 = h γᾱ [ p V γ R p γ 1 ][ t g p γ p V α R α p 1 ] t g pα + h γᾱ p V γ p V α. By Proposition 1.1 again, Y 2. (1.42) ( ) t Ẑ = Y 1 + Y 2 1 [ t h γᾱ α V γ h γᾱ ᾱv γ +2R α γ h γᾱ + 2(H + ɛm) t ] +2ɛR. Using (1.36) we also know that (1.43) Ẑ = R α βhᾱβ 1 2 h α β ᾱv β 1 2 h βᾱ α V β + H + ɛm t + ɛr. Plugging into (1.42) and using the fact that Y 1 and Y 2, we have (1.44) Ẑ 2Ẑ t, where V is the smooth vector field given by (1.36). Note that both sides of (1.44) do not depend on the choice of coordinates. Proof of Theorem 1.2. Since h α β ɛg α β on [, T], by (1.36) and (1.39), we have V C 1 h,

19 64 LEI NI AND LUEN-FAI TA and V C 2 ( h + h 2 ), for some constants C 1 and C 2. Combining this with (1.44), we have (1.45) t 2Ẑ 2 C 3 t(φ + Φ(Ψ 2 + Λ)+1) for some constant C 3. By (1.44), the corresponding t2ẑ satisfies ( ) (1.46) t (t 2Ẑ) for the vector field which minimizes Ẑ. By Lemma 1.2, Lemma 1.3, and (1.45), we have exp ( ar 2 (x))(t 2Ẑ)2 dv t dt < for any a >. By the maximum principle Theorem 1.1, we have t 2Ẑ because it is obvious that t2ẑ =att =. Since this is true for the vector field V minimizing Ẑ, we have Ẑ for any (1,) vector field. Let ɛ and the proof of the theorem is completed. Remark 1.1. (i) The theorem is still true for the case that is compact with positive holomorphic bisectional curvature because of the result in [Co1]. (ii) When h α β = R α β, it is known that the Ricci tensor satisfies (1.2). Therefore we can apply Theorem 1.1 to this case. Since we have div (h) α = γ R α γ = α R and div (h) δ = ᾱr α δ = δr (1.47) Z = R + R α βrᾱβ + α RVᾱ + ᾱrv α + R α βvᾱv β + R t. It is the trace of the LYH inequality proved by Cao in [Co1 2]. Hence Theorem 1.1 can be considered as a generalization of the LYH inequality of Cao for the scalar curvature. However, we should emphasize that Cao s result has been used in the proof of Theorem Deforming plurisubharmonic functions. Let ( m, g α β(x, t)) be a complete noncompact Kähler manifold with bounded nonnegative holomorphic bisectional curvature deformed by the Kähler-Ricci flow (1.1). As in the previous section we assume that (1.1) has solution on [, T] which satisfies condi-

20 PLURISUBHARONIC FUNCTIONS, KÄHLER-RICCI FLOW 641 tions (i) (iv) in that section. In this section we shall study the plurisubharmonic functions deformed by the time-dependent heat equation: ( ) t u(x, t) =, (2.1) u(x,)=u (x) where = g α β(x, t) 2 z α z β and u (x) is a plurisubharmonic function on. First, we shall consider the more general case and drop the assumption that u is plurisubharmonic. We have the following existence result. PROPOSITION 2.1. Let u be a continuous function such that u (x) exp (a (r (x)+1)) for all x for some positive constant a >. Then there is a unique solution of (2.1) on [, T] such that (2.2) u(x, t) exp (b(r (x) + 1)) on [, T] for some positive constant b. Proof. By Lemma 1.1, there exists a function ϕ(x) such that exp (b(r (x) + 1)) ϕ(x) exp (a(r (x) + 1)) for some positive constant and b for all (x, t) [, T], and ϕ. Using ϕ and ϕ as barriers, the existence part of the proposition follows. Uniqueness follows from the maximum principle Theorem 1.1. Next we shall study properties of the solution u obtained in the proposition. We need the following lemma. LEA 2.1. Let u(x, t) be a solution of (2.1). Then u α β satisfies the complex Lichnerowicz heat equation (1.2). Proof. Differentiate (2.1) and we have (2.3) (u t ) γ δ = R βᾱγ δu α β + g α βu α βγ δ. By definition u α β = 1 2 (u α β,γ γ + u α β, γγ ), in normal coordinates at a point. We need to calculate the difference between the partial derivative u α βγ δ and the covariant derivative u α β,γ δ. Direct computations show that, for normal coordinates at a point, (2.4) u γ δ,α β = u γ δα β + u s δr α βγ s.

21 642 LEI NI AND LUEN-FAI TA Using the fact that (2.5) u γ δ,αᾱ = u γ δ,ᾱα + R γ p u p δ R p δu γ p and (2.4) we have (2.6) u γ δ = 1 2 (u γ δ,αᾱ + u γ δ,ᾱα ) = u γ δαᾱ (R γ pu p δ + R p δu γ p ). Combining with (2.3), we conclude that u α β satisfies (1.2). In the following, and denote the covariant derivative and the Laplacian with respect to the initial metric. PROPOSITION 2.2. Let u be a smooth function such that u (x) exp (a(r (x)+ 1)) for all x for some positive constant a >. Let u(x, t) be the solution of (2.1) obtained in Proposition 2.1. We have the following: (i) For any b > (2.7) exp ( br 2 (x))( u 2 (x, t)+t u α β 2 (x, t)) dv t dt <, where u α β 2 = g α δg γ βu α βu γ δ. (ii) If in addition, B (o,r) u 2 dv exp (a (1 + r)) for some a >, where B (o, r) is the geodesic ball with center at o and radius r with respect to the initial metric g(), then (2.8) exp ( br 2 (x)) u α β 2 (x, t) dv t dt <, (iii) If in addition u 2 C 1 on then (2.9) u 2 C 1 and (2.1) u α β 2 (x, t) C 2 t for some constant C 2 on [, T]. Proof. By Proposition 2.1, there exist a positive constant and c such that (2.11) u(x, t) exp (c(r (x) + 1))

22 PLURISUBHARONIC FUNCTIONS, KÄHLER-RICCI FLOW 643 on [, T]. Since u 2 = u 2, we can proceed as in the proof of Lemma 1.1 to conclude that for any b > (2.12) exp ( br 2 (x)) u 2 dv t dt <. Direct computations show (see [N-T, Lemma 1.1] for example) ( ) (2.13) t u 2 = u αβ 2 u α β 2. Combining with (2.12), one can also proceed as in the proof of Lemma 1.1 and conclude that (2.7) is true. In case u 2 satisfies the condition in (ii), then one can prove (2.8) similarly. By (2.13), it is easy to see that t u )( 2 +1). Suppose u 2 C 1 on, then by (2.7) we can apply Theorem 1.1 to conclude that (2.9) is true. Since u α β satisfies (1.2), as in the proof of (1.11) we have (1 + tφ) 1 2 C3 Φ on [, T] for some constant C 3 >, where Φ = u α β 2. Hence on [, T], (C 3 u 2 +(1+tΦ) 1 2 ) where we have used (2.13). By (2.8) and (2.9), we can apply the maximum principle in [N-T] and conclude that sup (C 3 u 2 +(1+tΦ) 1 2 ) C3 C 1 +1 [,T] where we have used the fact that u 2 C 1. From this (2.1) follows. Next, we shall study the properties of u(x, t) in case the initial value u is plurisubharmonic. THEORE 2.1. Let u (x) be a smooth function on such that (a) u is plurisubharmonic; and (b) there exists a > such that u (x) exp (a(1 + r (x)) and u exp (a(1 + r (x)). Let u be the solution of (2.1) obtained in Proposition 2.1. We have the following: (i) u(x, t) is plurisubharmonic for t >.

23 644 LEI NI AND LUEN-FAI TA (ii) If u is not harmonic, then w = u t > for t >, and we have the following differential inequality: w t w 2 w + w t for t >. If in addition, sup u 2 C 1 < for some constant C 1, then u α β satisfies (2.1) for some constant C 2. Proof. Let f = u. By assumptions, u (x) exp (a(1 + r (x)) and f (x) exp (a(1 + r (x)). It is easy to see that B (o,r) u 2 dv exp (a (1 + r (x)) for some a >. Hence u α β satisfies (2.8) by Proposition 2.2. Since u is plurisubharmonic, we also have u α β 2 (x,) exp (a (1 + r (x)) for some a >. By (i), Proposition 1.1 and Lemma 2.1, we conclude that u is plurisubharmonic for t >. Since u α β satisfies (1.2) by Lemma 2.1 and w = u t = u, taking trace of (1.2), we have ( ) (2.14) t w = R α βu βᾱ. If w(x, t) = for some x and t >, then by the strong maximum principle (see [Cw3, Proposition 3.6]), we have u =on. Hence if u is not harmonic, then w > for t >. Since u α β satisfies the conditions in Theorem 1.2, if we let h α β, in Theorem 1.2 to be u α β, then in normal coordinates and div (h) α = γ u α γ = α (u t ) and div (h) δ = ᾱu α δ = δ(u t ), Z = (u t )+R α βuᾱβ + ᾱ(u t )V α + α (u t )Vᾱ + u α βvᾱv β + u t t for any (1, ) vector field V. Combining this with (2.14), we have w t + ᾱwv α + α wvᾱ + u α βvᾱv β + w t. Choosing V α = αw w we conclude that (ii) is true. The last assertion follows from Proposition 2.2 immediately.

24 PLURISUBHARONIC FUNCTIONS, KÄHLER-RICCI FLOW 645 Remark 2.1. If u (x) is a solution to the Poincaré-Lelong equation 1 u = Ric(x, ), by Theorem 1.3 of [N-T] we know that we have a solution u(x, t) to (2.1) in this case with 1 u(x, t) = Ric(x, t). Then (ii) in Theorem 2.1 is nothing but the differential LYH inequality of Cao on the scalar curvature since w(x, t) =R(x, t). Next we shall prove a Li-Yau type differential inequality for the positive plurisubharmonic solution of (2.1). The result will not be needed in the next section. THEORE 2.2. Let u(x, t) be a positive solution to (2.1) such that u(x, t) is plurisubharmonic for all t. Then we have the following differential inequality: (2.15) u t u u 2 u 2 + m t. Proof. As in Li-Yau [L-Y], we let v = log u. Then (2.16) v v t = v 2. Let G(x, t) =t( v 2 ηv t ), where η>1 is a constant. Direct calculation shows that in normal coordinates at a point: (2.17) v 2 = Rᾱβ v α v β + v αγ vᾱ γ + v α γ vᾱγ +( v) α vᾱ + v α ( v)ᾱ, (2.18) t v 2 = Rᾱβ v α v β +(v t ) α vᾱ + v α (v t )ᾱ and (2.19) v tt (v t )=Rᾱβ v α β + Rᾱβ v α v β +(v t ) α vᾱ + v α (v t )ᾱ. Combining (2.16) (2.19) we have that ( ) ( v 2 ηv t )=v αγ vᾱ γ + v α γ vᾱγ ( v 2 ηv t ) α vᾱ t Using the fact that v α ( v 2 ηv t )ᾱ + ηrᾱβ (v α β + v α v β). Rᾱβ (v α β + v α v β) = 1 u R ᾱβu α β

25 646 LEI NI AND LUEN-FAI TA we then have (2.2) ( ) G tv α γ vᾱγ 2 G, v G t t t m ( v)2 2 G, v G t = t m ( v 2 v t ) 2 2 G, v G t. Once we have (2.2), we can use the cut-off function argument as in [L-Y] to carry the interior estimates. For the sake of the completeness we include the argument here. Let ψ(s) be a cut-off function such that ψ 1, ψ(s) 1 for s [, 1] and ψ(s) for s 2. We also require that (2.21) ψ, ψ C 1 and ψ 2 ψ C 1 for some positive constant C 1. Now we let φ(x) = ψ(r t (x)/r). Let Φ = φg. Suppose Φ attains a positive maximum at (x, t ). Then we have at (x, t ): ( ) Φ and Φ =. t Note that φ may not be smooth at x in the space variable, but we can always use the trick of Calabi as in [L-Y]. φ may not be smooth in the t variable at t, but we can use the difference quotient so that the final result of the following computations is correct. Hence the above differential inequality together with (2.2) implies that at (x, t ) (2.22) ( t φ ) Φ t 1 m (t φ v 2 t φv t ) 2 2Gt φ 2 φ + t ( φ ) t φ G Gφ 2 +2 φ, v Gφt 1 m (t φ v 2 t φv t ) 2 G 2 φ φ 1/2 G( v φ1/2 t 1/2 )t 1/2. [ 2t φ 2 φ ( t φ ) ] t φ +1 Using (2.21) we have that (2.23) φ 2 φ C 2 R 2 and φ C 2 R 2.

26 PLURISUBHARONIC FUNCTIONS, KÄHLER-RICCI FLOW 647 Also Theorem 17.2 of [H4] implies that (2.24) t φ C 2 R. Here C 2 is a constant dependent of C 1, m and the upper bound of Rm (x, t). Combining (2.22) (2.24) we have, at the maximum of Φ over [, T], that (2.25) 1 m (y z)2 C ( 2 R (y ηz)y1/2 t 1/2 C2 t (y ηz) R 2 + C ) 2t R +1. Here y = t φ v 2 (x, t ), z = t φv t (x, t ). Using the trick of [L-Y], we write (y z) 2 = 1 η 2 (y ηz)2 +2 η 1 ( ) η 1 2 (y ηz)y + y 2. η η Using the ax 2 + bx b2 4a, for R 1 we have that 1 ( ) mη 2 (y C3 t ηz)2 (y ηz) R +1 for some constant C 3 independent of R. From which we have that ( sup t( v 2 ηv t ) mη 2 1+ C(m, η, T, sup [,T] Rm ) ). B (o,r) [,T] R Here we have used the fact that g(t) is nonincreasing so that B t (o, R) B (o, R). Letting R and then η 1 we have (2.15). 3. Liouville properties of plurisubharmonic functions. In this section, we shall discuss Liouville properties of plurisubharmonic functions using the LYH type inequality in 1 and the results of 2. In this section, we always assume that (, g α β(x)) is a complete noncompact Kähler manifold with bounded nonnegative holomorphic bisectional curvature. We also assume that for all x and r >, k(x, r) ɛ(r) for some nonincreasing function ɛ(r) with lim r ɛ(r) =, where (3.1) k(x, r) = B (x,r) R dv and R is the scalar curvature of, g α β(x). By [N-T], we know that (1.1) has a solution g α β(x, t) on [, ) such that for any < T <, g α β satisfies (i) (iv) in 1.

27 648 LEI NI AND LUEN-FAI TA (3.2) Define ( det (gα F(x, t) = log β(x, ) t)) det (g α β(x,. )) To illustrate the idea of the proof to a more general result, let us begin with the following particular case. In this case, what we need is to assume that (1.1) has long time solution g α β so that for any T <, conditions (i) (iv) in 1 are satisfied by g α β on [, T]. THEORE 3.1. With the above assumptions, suppose u is a plurisubharmonic function such that (i) u is bounded; and (ii) u (x) exp (a(1 + r (x)) for some constant a >. Then u must be constant. Proof. Let u = f, then f. Since u is bounded, by [N-S-T1, Corollary 2.1] we have ( ) s fdv ds C 1 B (x,s) for some constant C 1 independent of x. By [N-S-T1, Corollary 1.2], we know that sup u C 2. By Proposition 2.1, there is a unique solution u(x, t) with initial data u. oreover, by Proposition 2.1 and the maximum principle in [N-T, Theorem 1.2], we conclude that u is uniformly bounded. Since u (x) exp (a(1 + r (x)), by Theorem 2.1(i) we conclude that u(x, t) is plurisubharmonic for all t >. oreover, suppose u is not harmonic, then by Theorem 2.1(ii) w = u t > for t > and tw is nondecreasing in t. Hence u(x, t) u (x) = t w(x, s) ds t 1 w(x,1) 1 s ds = w(x, 1) log t. Since w(x,1) >, let t, the above inequality contradicts the fact that u is uniformly bounded. Hence u must be harmonic and is constant by [Y]. Next we shall generalize Theorem 3.1 by relaxing the condition that u is bounded. In the following, we always assume that u is a plurisubharmonic func-

28 PLURISUBHARONIC FUNCTIONS, KÄHLER-RICCI FLOW 649 tion on such that there exists a constant a > such that u (x) a, (3.3) u (x) exp (a(1 + r (x)) for all x. Note that in the proof of Theorem 3.1, we know that if u is bounded, then u will satisfy the first inequality of (3.3). Because of (3.3), let u be the solution of (2.1) with initial data u constructed in Proposition 2.1. By Proposition 2.2, u is plurisubharmonic for all t. Let v(x, t) = u(x, t) u (x). Also, let m(t) = inf x F(x, t). Then m(t), nonincreasing, and is finite for fixed t by properties (ii) and (iv) in 1 and the fact that F(x, t) = t R(x, s) ds, where R(x, s) is the scalar curvature at time s. LEA 3.1. With the above assumptions and notations, we have (3.4) v e F v t u. Proof. As in [Shi, p. 156], using the fact that g α β is nonincreasing, we have u e F u = e F u t = e F v t. Hence v = u u e F v t u. The result follows. LEA 3.2. With the same assumptions and notations as in Lemma 3.1, there is a constant C such that for all (x, t) [, ), we have (3.5) v(x, t) Ct 1 2 ( m(2t)+1). Proof. First note that v(x, ) = and v t = u t = u. Hence v. We need a more refined estimate of (2.1). ore precisely, the Bochner formula on u α β 2 says that ( ) u t α β 2 u α βγ 2 + u α β γ 2 R(x, t) u α β 2. Using the LYH type inequality of H.-D. Cao as in [N-T] we have that tr(x, t) 2m(2t).

29 65 LEI NI AND LUEN-FAI TA Combining them we have that ( ) (1 + tφ) 1 2 ( m(2t)+1)φ. t Here Φ = u α β 2. Now we can proceed as in the proof of Proposition 2.2 (iii) to conclude that sup (( m(2t)+1) u 2 +(1+tΦ) 1 2 ) a 2 ( m(2t) + 1) + 1, [,T] which then implies u α β (x, t) C 1 t 1 2 ( m(2t)+1) for some constant C 1 depending only on m and sup u. Hence v(x, T) = = = T T T v t (x, t) dt u t (x, t) dt u(x, t) dt T C 1 t 1 2 ( m(2t)+1)dt C 2 T 1 2 ( m(2t)+1) for some constant C 2 independent of x and t. The proof of the lemma is completed. Using the method of proof of Theorem 2.1 in [N-T], we have: THEORE 3.2. Let u be a plurisubharmonic function on satisfying (3.3). Suppose (3.6) lim sup R sup B (o,r) u log t where R 2 = t 3 2 e m(2t) ( m(2t)+1), then u must be constant. Remarks. (a) It is easy to see that R if and only if t. (b) By [Sh2 3] and [N-T, Remark 2.2] if ɛ(r) at the beginning of this section satisfies ɛ(r) Cr 2 or more generally if r sɛ(s) ds C log (r+2), then m(t)

30 PLURISUBHARONIC FUNCTIONS, KÄHLER-RICCI FLOW 651 C log (t + 1), and the assumption (3.6) can be replaced by lim sup x u (x) log r (x). (c) Similarly, if ɛ(r) r θ for some θ>, then the assumption (3.6) can be replaced by lim sup x u (x) log log r (x). If r sɛ(s) ds Cr2 / log (2 + r), then the assumption (3.6) can be replaced by lim sup x u (x) log log log r (x), and so on. (d) It is easy to see that Theorem 3.1 is a particular case of Theorem 3.2. Proof of Theorem 3.2. Let u and v as in Lemmas 3.1 and 3.2. Let (x, T) (, ). For any R >, let G R be the positive Green s function with zero boundary value on B (x, R) with respect to the initial metric. By (3.4) (3.7) G R (x, y) v(y, t) dv dt B (x,r) T G R (x, y) u (y) dv B (x,r) + G R (x, y)e F(y,t) v t (y, t) dv dt B (x,r) T G R (x, y) u (y) dv B (x,r) + e m(t) G R (x, y)v(y, T) dv B (x,r) [ ( ) ] C 1 (m) T u (x ) + sup u +e m(t) R B 2 v(y, T) dv, (x, R 5 ) B (x,r) for some positive constant C 1 depending only on m, where we have used Theorem 2.1 in [N-S-T1], Lemma 2.2 in [N-T], and the fact that v, F t. On the other hand, by Green s formula and Lemma 3.2, we have that, for any < t < T, G R (x, y) v(y) dv = v(x, t) v G R (3.8) B (x,r) B (x,r) ν C 2 t 1 2 ( m(2t)+1)

31 652 LEI NI AND LUEN-FAI TA for some constant C 2 independent of (x, t). By (3.4) and the fact that v t, we have v u. Since v, by the generalized mean value inequality [N-T, Lemma 2.1], (3.7) and (3.8), we have v(x, T) C 3 v(y, T) dv + B (x, R 5 ) B (x, R 5 ) G R 5 (x, y) u (y) dv [ ) C 4 R 2 Te ( u m(2t) (x ) + sup u + T 1 2 ( m(2t)+1) B (o,2r) ] u (x ) + sup u B (o,2r) if R is large, for some constants C 3 and C 4 independent of (x, T) and R. Let R be such that (2R) 2 = T 3 2 (1 + T)e m(2t) ( m(2t) + 1), then by (3.6), we can conclude that (3.9) lim sup t v(x, t) log t We claim that u is harmonic. Suppose not, then as in the proof of Theorem 3.1, we have u(x, t) C log t for some constant C > for all t 1. This is impossible. By the definition of R in (3.6), it is easy to see that log R log t when t is large. Hence (3.6) implies that =. lim sup R sup B (o,r) u log R =. Since u is harmonic, it must be constant by [C-Y]. Since one can solve the Poincaré-Lelong equation for a (1,1) form on a complete noncompact manifold with nonnegative holomorphic bisectional curvature under rather weak assumptions on the (1,1) form (see [N-S-T1]), one can apply Theorem 3.2 (or Theorem 3.1) to obtain results on the flatness of the holomorphic line bundles. As an example, we have the following: COROLLARY 3.1. Let (, g α β(x)) be a complete nocompact Kähler manifold with bounded nonnegative holomorphic bisectional curvature satisfying the conditions in Theorem 3.1. Let (L, h ) be a holomorphic line bundle on with the Hermitian metric h. Suppose Ω(h ) and suppose its trace S = g α β(x)ω α β(h )(x) is bounded and (3.1) s S (y) dy ds C B (x,s) for some constant C > for all x. Then (L, h ) is flat.

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