On the simply connectedness of non-negatively curved K\"ahler manifolds and applications

We study complete noncompact long time solutions $(M, g(t))$ to the K\"ahler-Ricci flow with uniformly bounded nonnegative holomorphic bisectional curvature. We will show that when the Ricci curvature is positive and uniformly pinched, i.e. $ R_\ijb \ge cRg_\ijb$ at $(p,t)$ for all $t$ for some $c>0$, then there always exists a local gradient K\"ahler Ricci soliton limit around $p$ after possibly rescaling $g(t)$ along some sequence $t_i \to \infty$. We will show as an immediate corollary that the injectivity radius of $g(t)$ along $t_i$ is uniformly bounded from below along $t_i$, and thus $M$ must in fact be simply connected. Additional results concerning the uniformization of $M$ and fixed points of the holomorphic isometry group will also be established. We will then consider removing the condition of positive Ricci for $(M, g(t))$. Combining our results with Cao's splitting for K\"ahler Ricci flow \cite{Cao04} and techniques of Ni-Tam \cite{NiTam03}, we show that when the positive eigenvalues of the Ricci curvature are uniformly pinched at some point $p \in M$, then $M$ has a special holomorphic fiber bundle structure. We will treat a special cases, complete K\"ahler manifolds with non-negative holomorphic bisectional and average quadratic curvature decay as well as the case of steady gradient K\"ahler Ricci solitons.

Abstract. We study complete noncompact long time solutions (M, g(t)) to the Kähler-Ricci flow with uniformly bounded nonnegative holomorphic bisectional curvature. We will show that when the Ricci curvature is positive and uniformly pinched, i.e. R i ≥ cRg i at (p, t) for all t for some c > 0, then there always exists a local gradient Kähler-Ricci soliton limit around p after possibly rescaling g(t) along some sequence t i → ∞. We will show as an immediate corollary that the injectivity radius of g(t) along t i is uniformly bounded from below along t i , and thus M must in fact be simply connected. Additional results concerning the uniformization of M and fixed points of the holomorphic isometry group will also be established. We will then consider removing the condition of positive Ricci for (M, g(t)). Combining our results with Cao's splitting for Kähler-Ricci flow [7] and techniques of Ni-Tam [25], we show that when the positive eigenvalues of the Ricci curvature are uniformly pinched at some point p ∈ M , then M has a special holomorphic fiber bundle structure. We will treat as special cases, complete Kähler manifolds with non-negative holomorphic bisectional curvature and average quadratic curvature decay aswell as the case of steady gradient Kähler-Ricci solitons.

introduction
In this paper we study the class of complete noncompact Kähler manifolds (M n , g) of complex dimension n with bounded and nonnegative holomorphic bisectional curvature. Let R be the scalar curvature and let (1.1) k(r, x) = 1 V x (r) Bx(r) R 1 Research partially supported by NSERC grant no. #327637-06. 2 Research partially supported by Earmarked Grant of Hong Kong #CUHK403005. 1 be the average of the scalar curvature over the geodesic ball with radius r and center at x. In [11], [12] and [13] the authors obtained, among other things, the following uniformization results: (1) If k(r, x) ≤ C/(1 + r 2 ) for some C for all r and x, then the universal cover of M is biholomorphic to C n .
(2) Under a weaker assumption that k(r, x) ≤ C/(1 + r) for some C for all r and x, the universal cover of M is homeomorphic to R 2n and is biholomorphic a pseudoconvex domain in C n .
The results support the following conjecture due to Yau [33]: If (M, g) has positive holomorphic bisectional curvature then M is biholomorphic to C n .
In this paper we want to discuss the structure of M itself. In light of the above results, a reasonable approach is as follows. First one studies more closely the structure of the universal coverM . Then one would like to study properties of the first fundamental group, and hopefully these together will provide information about the structure of M. As in the previous works, we will study the Kähler-Ricci flow equation on (M, g): By the works [26] and [28] of Shi, the Kähler-Ricci flow has a long time solution g(t) with uniformly bounded curvature under assumption (2) (and hence (1)) above. By the splitting result of Cao [7], (M, g(t)), the pull back of g(t) to the universal coverM , splits for all t > 0 as M = C k ×Ñ and g(t) = g e × h(t) where g e is the standard metric on C k and h(t) has nonnegative holomorphic bisectional curvature and positive Ricci curvature. Hence we will begin our studies with long time solutions (M, g(t)) to Kähler-Ricci flow where g(t) has uniformly bounded nonnegative holomorphic bisectional curvature and positive Ricci curvature. We will prove that: If at some point p ∈ M, the eigenvalues of Ric(p, t) are uniformly pinched (i.e. the smallest eigenvalue of Ric(p, t) is at least C times the largest eigenvalue for some C > 0 independent of t), then M is simply connected and is in fact biholomorphic to a pseudoconvex domain in C n . See Theorems 2.1, 2.2 and 2.3. A particular case is when (M, g) has average quadratic curvature decay and positive Ricci in which case it is shown in Corollary 2.1 that M is biholomrophic to C n . This generalizes previous results in [12] where the same result, in particular Corollary 2.1, is proved assuming in addition either that (1.2) has an eternal solution (i.e. g(t) exists for −∞ < t < ∞) or that tR is uniformly bounded in spacetime. Note that either conditions will imply that eigenvalues of Ric(p, t) are uniformly pinched, see [12] for example.
More generally, if we only assume the positive eigenvalues of Ric(p, t) are uniformly pinched, then using the above result on simple connectedness, we prove that: M is a holomorphic and Riemannian fiber bundle over C k /Γ where Γ is a discrete subgroup of the holomorphic isometry group of C k , with fiber either C n−k or a pseudoconvex domain in C n−k .
See Theorem 3.1. In particular, the fiber is C n−k when (M, g) has average quadratic curvature decay, which we show in Corollary 3.1. Similar results under different conditions have been obtained by Takayama [30], Ni-Tam and Zheng [25].
Our results are related to the second part of Yau's conjecture in [33]: If the holomorphic bisectional curvature is only non-negative then M is biholomorphic to a complex vector bundle over a compact Hermitian symmetric space. In fact, we prove that if n − k = 1 in the above situation, then M is actually a line bundle over C k /Γ. In order to prove this, we have to study more carefully about the isometry group of a complete noncompact Kähler manifold (M, g) with nonnegative holomorphic bisectional curvature and positive Ricci curvature. We prove that if there is a long time solution g(t) to Kähler-Ricci flow with uniformly bounded non-negative holomorphic bisectional curvature and positive uniformly pinched Ricci curvature, then any finite set of isometries of M has a fixed point. Under the stronger assumption that (1.2) has an eternal solution or tR(p, t) is uniformly bounded for all time, then one can prove the stronger result that there is a fixed point for all isometries of (M, g). See Theorems 2.4 and 2.5. One might want to compare this with a theorem of Cartan which says that a compact subgroup of isometries of a Cartan-Hadamard manifold has a fixed point.
A special case of the above is when g(t) is actually a gradient Kähler-Ricci soliton. Namely, there is a real valued function f with f ij = 0 such that R i = f i (steady type) or after rescaling −R i + g i = f i (expanding type). In the case of expanding type, it is known that M is biholomorphic to C n by [10]. For the case of steady type, under an additional assumption that the scalar curvature attains its maximum at some point, we prove that in most cases, M is a holomorphic vector bundle or C k /Γ. More precisely, we prove that this is the case if the scalar curvature R is maximal at p ∈ M say, and the positive eigenvalues λ 1 , . . . , λ l of Ric at p satisfy the Diophantine condition l i=1 m i λ i = λ j for all j and for all nonnegative integers m i such that l i=1 m i ≥ 2. See Theorem 4.1. The proof relies on the result of Bryant [2] on the existence of Poincaré coordinates. This is related to a result of Yang [32]: Suppose M is a noncompact Kähler-Ricci soliton with nonnegative Ricci curvature such that its scalar curvature attains a positive maximum at a compact complex submanifold K with codimension 1 and the Ricci curvature is positive away from K, then M is a holomorphic line bundle over K. proves that We now describe the organization of the paper. In §2 we study long time solutions to Kähler-Ricci flow with bounded non-negative holomorphic bisectional curvature and positive Ricci curvature. We will prove the existence of gradient Kähler-Ricci soliton limits and consequences of this concerning the injectivity radius along the flow and the simple connectedness of M. We will also prove results on the existence of fixed point of isometries. In §3 will prove the fiber bundle structure and line bundle of M. Finally, in §4 we prove vector bundle structure on gradient Kähler-Ricci solitons. We will also prove a result on the volume growth of expanding gradient Ricci solitons in this section which is related to the work of Ni [24] 1 .

long time solutions and local limits
In this section we study limits of long time solutions of the Kähler-Ricci flow (2.1) on complete noncompact Kähler manifolds with uniformly bounded nonnegative holomorphic bisectional curvature and positive Ricci curvature. We will investigate when such a solution subconverges, after rescaling, to a gradient Kähler-Ricci soliton limit, and we will derive a number of consequences from this. In Corollary 2.1 we treat the particular case of a complete non-compact Kähler manifold having bounded non-negative holomorphic bisectional curvature and positive Ricci curvature which decays quadratically in the average sense. Our study of the limiting behavior of Kähler-Ricci flow in this section can be viewed as an extension of that in [11] and [12]. We begin with the following Basic Assumption 1: Let M n be a noncompact complex manifold of complex dimension n. Let g(t) be a complete solution of the Kähler-Ricci flow with uniformly bounded nonnegative holomorphic bisectional curvature in spacetime and with positive Ricci curvature. Let R(x, t) be the scalar curvature of g(t) at x. Let p ∈ M be a fixed point. Then we have the following possibilities: (1) R(p, t) ≥ c for some constant c > for all t; or (2) inf t∈[0,∞) R(p, t) = 0. In the second case, since tR(p, t) is nondecreasing in time by a Li-Yau-Hamilton (LYH) type differential inequality of Cao [4], we have two subcases: (2a) tR(p, t) ≤ c for some constant c > 0; and (2b) lim t→∞ tR(p, t) = ∞. We address these cases separately in Theorems 2.1, 2.2 and 2.3 below.
First, we have the following lemma.
Replacing 1 4 t by T , we have for some constant C 2 independent of x, y, t, ρ, α 0 , α 1 . From this the lemma follows.
We will also need the following result which is basically from [29,11] (see also [12]).
with the following properties: We may now state our first Theorem 2.1. Suppose that at some p ∈ M we have Ric(p, t) ≥ cg(p, t) for all t for some c > 0. Then the following are true.
(i) The injectivity radius of g(t) at p satisfies inj t (p) ≥ a for some a > 0 for all t. (ii) M n is homeomorphic to the Euclidean space and is biholomorphic to a pseudoconvex domain in C n .
(iii) For any t i → ∞ we can find a subsequence t i k such that (M, g(t+ t i k ), p) converges to a complete eternal solution (M ∞ , g ∞ (t), o) of (2.1) with positive Ricci curvature and bounded non-negative holomorphic bisectional curvature, and M ∞ is biholomorphic to converges to a steady gradient Kähler-Ricci soliton with positive Ricci curvature and bounded nonnegative holomorphic bisectional curvature which is biholomprhic to C n .
Proof. Assume (i) is true. Since the curvature, and its covariant derivatives, of g(t) are uniformly bounded by [28] for t ≥ 1, for each t ≥ 1 there is a biholomorphism Φ t from D(r) → M satisfying the conditions in Lemma 2.2 with r and C being independent of t. Since Ric > 0 for all t, it is easy to see that Φ t (D(r)) exhausts M as t → ∞ by the eigenvalue inequality for Rc(t) in [13] (Theorem 6.1). Hence M is homeomorphic to the Euclidean space by [1]. This proves the first part of (ii). The second part of (ii) then follows from Theorem 1.2 in [13].
(iii) follows from (i), the standard result on compactness of solutions of Ricci flow [22] and Theorem 1.4 in [12].
(iv) will follows from (i) and the content in the proof of (i). It remains to prove (i). Suppose there exist t i → ∞ such that lim i→∞ inj t i (p) = 0. Since the curvature are uniformly bounded in spacetime, for any positive integer k ≥ 1 there is a positive constant C 1 which is independent of i, but possibly depending on k, such that 1 and for any t ∈ [t i , t i + k]. By volume comparison and [17] on the injectivity radius bound we have: for any k. On the other hand, since the scalar curvature of g(t) are uniformly bounded, for any i, there exists . By passing to a subsequence still denoted by t i , we can find k i → ∞ and By Lemma 2.2, there is r > 0 and C 3 > 0 independent of i such that for each i, there exists a a holomorphic map Φ i : D(r) → M corresponding to the metric g(s i ) satisfying the conditions in the lemma with C = C 3 in property (iv) in the lemma. Now by considering the sequence (D(r), g i (s) = Φ * i g(s i + s)), we may then obtain a local subsequence limit solution (D(r), h(z, t)) to (2.1) on an eternal time interval and satisfying ∂ ∂t R h (0, 0) = 0 where R h is the scalar curvature of h. By Theorem 2.1 in [12] we find that (D(r), h(0)) is a local steady gradient Kähler-Ricci soliton with positive Ricci curvature centered at the origin. Namely, there is a real valued function f on D(r) such that is a holomorphic vector field, and it can be shown that the gradient of f is zero at the origin (see [12] for example), which is an isolated zero because f i = R h i > 0. In fact by its construction in [12, §2] and [6], there exists 0 < r 1 < r such that Z is the unique solution of the equation Note that (2.7) has a unique solution Z ∈ T (0,1) on any Kähler manifold having positive Ricci curvature. By our assumptions on the positivity of Ricci curvature, we may then let W (i) ∈ T (0,1) M be the unique solutions to (2.7) on (M, g(s i )) for any k. In particular, for any i as Φ i is a local biholomorphism, V (i) = (Φ i ) * (W (i)) is a smooth (0, 1) vector field in D(r). By the definition of g(s i ), V (i) is just the unique solution of (2.7) in D(r 1 ) relative to the metric (Φ i ) * (g(s i )). By uniqueness and the fact that (Φ i ) * (g(s i )) → h, we conclude that: It is easy to see that the integral curves of −Z in D(r) will converge to the origin. Since V (i) converge to Z, one expects the integral curves of V (i) have similar behaviors if i is large. In fact, In real coordinates x α of D, the integral curves of −Z are given by the following ODE On the other hand, for any ǫ > 0 there exists i 0 and 0 < r 1 < r such that the integral curves of V (i) for i ≥ i 0 is given by where C 4 depends only on F , provided r 2 < c 2C 3 and |x| ≤ r 2 . Fix r 2 satisfying these conditions. Then if r 2 2 ≤ |x| ≤ r 2 we have for some z α inside D(r 2 ), provided r 2 and ǫ are sufficiently small, both depending only on the constant c. Hence In particular, if x(τ ) is an integral curve, and if τ 2 > τ 1 > 0, then . Hence x(τ ) will converge to a point x 0 ∈ D(r 2 ), and by (2.13), we see that any set in D(r 2 ) will contract to x 0 along integral curves. In fact, x 0 ∈ D( r 2 2 ) by (2.11). Moreover, x 0 is a fixed point of V (i).
We now claim that Φ i is injective on D(r 2 ) for i ≥ i 0 . Assume this is not the case and that for some Let γ 1 and γ 2 be two integral curves for V (i) starting at z 1 and z 2 respectively. Then by the construction of V (i), Φ i (γ 1 ) and Φ i (γ 2 ) are integral curves for W (i) starting from q. Hence by the uniqueness of integral curves we must have Φ i (γ 1 (τ )) = Φ i (γ 2 (τ )) for all τ . On the other hand, for all τ , we have γ 1 (τ ) = γ 2 (τ ) also by uniqueness of integral curves. But γ 1 (τ ) and γ 2 (τ ) both converge to a x 0 ∈D(r 2 ). It is readily seen that these facts contradict the fact that Φ i is a biholomorphism in some neighborhood of x 0 . Thus we have shown that Φ i is injective on D(r 2 ) for all i ≥ i 0 . Now It is easy to see that Φ i (D(r 2 )) is contained in some geodesic ball B s i (p, r 3 ) for some r 3 independent of i. Thus by the injectivity of Φ i shown above, and by the fact that Φ * i (g(s i )) is uniformly equivalent to the Euclidean metric by Lemma 2.2, it is easy to see that there exists b > 0 such that for i ≥ i 0 large enough B s i (p, r 3 ) has volume bounded below by b. By the result of Cheeger-Gromov-Taylor [17], inj s i (p) must be uniformly bounded away from zero. But this contradicts our assumption that inj s i (p) → 0. This completes the proof of the theorem by contradiction.
Next we consider the case that tR(p, t) is uniformly bounded.  We will need the following lemma.
for all x ∈ B t (p, ρ √ t) and for all t > 0, where ∇ and the norm are taken with respect to g(t).
Proof. By the assumptions and Lemma 2.1, there exists a constant C 1 such that for all T > 0 and r > 0, (2.14) R(x, t) R(p, 3T ) for all x ∈ B T (p, r √ T ) and T ≤ t ≤ 2T . Now consider the metricŝ g(s) = 1 T g(T + sT ). Thenĝ satisifies the Kähler-Ricci flow equation with nonnegative holomorphic bisectional curvature. By the assumption that tR(p, t) is uniformly bounded and (2.14) and the fact that g(s) has nonnegative holomorphic bisectional curvature, we have where C 2 is constant independent of r and T where Rm, || · || are the curvature tensor and the norm with respect toĝ. Thus by the local derivatives estimates of Shi [26], see also [20], for any integer k ≥ 0, there is a constant C 3 which is independent of t such that On the other hand, by the Kähler-Ricci flow equation, we conclude that in B 0 (p, r) for some positive constant C 4 independent of r and T .
Hence we have B 1 (p, ρ) ⊂ B 0 (p, r 2 ) where ρ = C . By Lemma 2.3, the fact thatg i has positive Ricci curvature, and the fact that the eigenvalues of Ric are nondecreasing by Theorem 6.1 in [13], we can proceed exactly as in the proof of Theorem 2.1(i) to show that the injectivity radius ofg i (s) at p is uniformly bounded below away from zero. Rescaling back to g(t), we see that (i) is true. Note that here we use the fact that if we take a limit along a sequence s i as in the proof of Theorem 2.1(i), then in the limit metric, the scalar curvature at the origin is constant in time.
Using Lemma 2.3 and (i), the proof of Theorem 1.1 in [11] can be carried over to prove that M is biholomorphic to C n . 2 Sinceg i (s) is decreasing, for any ρ > 0 and s 0 ≥ 0, the curvature together with its derivatives are uniformly bounded on B s 0 (p, ρ) for all s ≥ s 0 . Combining with (i), one can apply the proof of the local version [19, Corollary 3.18] of compactness of solutions of Hamilton [22], to conclude that for any s i → ∞ we can find a subsequence s i k such that (M,g i (s + s i k ), p) converges to a complete eternal solution of (2.17) which is in fact an expanding gradient Kähler-Ricci soliton with positive Ricci curvature and nonnegative holomorphic bisectional curvature. The fact that the limit solution is an expanding gradient Kähler-Ricci soliton can be proved as in [12, §2]. The fact that the limiting manifold is biholomorphic to C n follows from [10,2].
Remark 2.1. Theorem 2.2 was proved by the authors in [12] under the stronger assumption that tR(p, t) is uniformly bounded for all t ≥ 0 independently of p. In particular, it was proved there that M is biholomorphic to C n . We point out that the injectivity radius bound in Theorem 2.2 (i) allows a direct application of the methods in [11], and thus a more direct proof of this result.
By the result of [4], if tR(p, t) is not bounded, then tR(p, t) → ∞ as t → ∞. We consider this case in the next theorem. Proof. To prove (i), since inf t R(p, t) = 0 and Ric > 0 for all t, we can find T i → ∞ such that a i = R(p, 3T i ) = inf t∈[0,3T i ] R(p, t). Then a i > 0 and lim i→∞ a i = 0. Now the metrics g i (t) = a i g( is the scalar curvature of g i . On the other hand, by Lemma 1.1, there is a constant C 1 independent of i such that . By the assumption that tR(p, t) → ∞ as t → ∞, we conclude that lim i→∞ a i T i = ∞ and We now consider the sequence of metrics h i (t) = g i (s i + t) which solve (2.1) for −a i T i − s i ≤ t < ∞. By Lemma 2.1 and the definition of a i , there is a constand C 2 independent of i such that is the geodesic ball with respect to h (i) .
Since the eigenvalues of the Ricci curvature of g(t) at p are uniformly pinched, the Ricci curvature of h i (p, 0) is uniformly bounded from below. Noting that a i T i → ∞ as i → ∞, by (2.18) and (2.19), we can proceed as in the proof of Theorem 2.1(i) to show that the injectivity radius of h i (0) is uniformly bounded from below. Rescaling back to g(t), using the definition of a i we see that (i) is true.
Part (ii) follows from part (i) as in the proof of Theorem 2.1. By the above facts, in particular that (2.18) holds on a sequence of balls with radii going to ∞, one can proceed as in the proof of Theorem 2.2 (iii) to conclude that (M, h i (0), p) converges a complete steady gradient Kähler-Ricci soliton with positive Ricci curvature and nonnegative holomorphic bisectional curvature. Note that the gradient of the function defining the gradient Kähler-Ricci soliton has a unique zero by (2.18) and the fact that the Ricci curvature is positive. Hence the gradient Kähler-Ricci soliton is biholomorphic to C n by the results in [10,2].
The methods of proof in the theorems above allow us to study the holomorphic isometries of (M, g(0)).
Theorem 2.4. Let (M, g(t)) be as in the Basic Assumption 1. Suppose in addition that tR(p, t) → ∞ and the eigenvalues of the Rc(p, t) are uniformly pinched. Let Γ be a finite subset of the holomorphic isometry group of (M, g(0)). Then there exists q ∈ M such that q is fixed by any γ ∈ Γ.
Proof. By Theorems 2.1 and 2.3, we can find t i → ∞ and a bounded sequence a i > 0 such that (M, a i g(t i ), p) converge to a steady gradient Kähler-Ricci soltion with positive Ricci curvature. By the proof of Theorem 2.1, we conclude that there exists a > 0 such that ∇ i R (i) has a unique zero at q i ∈ B i (p, a) which lies inside B i (p, a 3 ). Here B i is the geodesic ball, ∇ i is the covariant derivative and R (i) is the scalar curvature with respect to a i g(t i ).
We claim that for any holomorphic isometry of (M, g(0)), there is i 0 such that γ(q i ) = q i for all i ≥ i 0 . It is easy to see that the theorem follows from this claim as Γ is finite.
By the uniqueness of Kähler-Ricci flow [18] (see also [21]), γ is also an isometry of g(t) and hence of a i g(t i ) for all i. So γ(q i ) is also a zero of ∇ i R (i) . Since q i is the unique zero of ∇ i R (i) in B i (p, a) it is easy to see that the claim is true if γ fixes p.
Suppose γ(p) = p. Since for any ǫ > 0, B t i (p, ǫ) exhaust M, see [13], and since a i is uniformly bounded, we conclude that there In case tR(p, t) is uniformly bounded in t or if g(t) is an eternal solution, then we have the following much stronger result.
Theorem 2.5. Let (M, g(t)) be as in the Basic Assumption 1. Suppose in addition that either (i) for some p ∈ M, tR(p, t) is uniformly bounded in t, or (ii) g(t) is an eternal solution. Then there exists q ∈ M such that q is fixed by any holomorphic isometry of (M, g(0)).
Proof. First consider the case that g(t) is an eternal solution. Then by the monotonicity of the eigenvalues of the Ricci curvature in t [13] (Theorem 6.1), the Ricci curvature at p is uniformly bounded below away from zero for all t. On the other hand, by the LYH type differential inequality by Cao [6], we know that d dt R(p, t) ≥ 0. Hence as in [12], for any t i → ∞, (M, g(t i ), p) will subconverge to a gradient Kähler-Ricci soliton with positive Ricci curvature and nonnegative holomorphic bisectional curvature. Hence as in the proof of Theorem 2.4, we can prove that there exist T > 0 and some a > 0 such that for t ≥ T there exists q(t) ∈ B t (p, a) which is the unique point satisfying ∇ t R(q(t), t) = 0 in B t (p, 3a).
Also as in the proof of Theorem 2.4, we can prove that for every holomorphic isometry γ, there exists T γ such that γ(q(t)) = q(t) for all t ≥ T γ .
We want to prove that every isometry γ fixes q(T ). Suppose γ(q(T )) = q(T ). Since d T (p, q(T )) < a by uniqueness of q(T ), we must have d T (γ(q(T )), q(T )) ≥ 2a. On the other hand, we have d Tγ (γ(q(T γ )), q(T γ )) = 0 and T < T γ by the choice of T γ . Now by uniqueness of q(t) and the fact that g(t) is a solution of the Kähler-Ricci flow equation, it is not hard to see that q(t) describes a continuous path on M with respect to t, and thus the same is true for γ(q(t)). It follows then, that for some T < T 0 < T γ we must have d T 0 (γ(q(T 0 )), q(T 0 )) = a. But this would violate the fact that q(T 0 ) is the unique zero of ∇ T 0 R in B T 0 (p, 3a). This proves the case when g(t) is an eternal solution.
The case that t(R(p, t) is uniformly bounded in t can be proved similarly using Theorem 2.2.
By the results of Shi [28] (see also [12]), if (M, g) has bounded nonnegative holomorphic bisectional curvature with average quadratic curvature decay and positive Ricci somewhere on M, then there exists a long time solution to Kähler-Ricci flow such that tR(p, t) is uniformly bounded independently of p. We thus have the following immediate Corollary of Theorem 2.2 and 2.5: Corollary 2.1. Let (M, g) be a complete non-compact Kähler manifold with non-negative holomorphic bisectional curvature. Suppose k(x, r) ≤ C/(1 + r 2 ) for some C and all r and x, and that Rc(p) > 0 for some p ∈ M. Then M is biholomorphic to C n , and there exists q ∈ M such that q is fixed by any holomorphic isometry of (M, g).

Fiber bundle structures
In this section we study structure of complete noncompact Kähler manifold with nonnegative holomorphic bisectional curvature using the Kähler-Ricci flow. To motivate our discussion, let us consider an example. Take a complete noncompact Kähler metric h on C n with positive holomorphic bisectional curvature which is U(n) invariant, see [5,6,31]. LetM = C l × C n (l ≥ 1 with metric g = g e × h where g e is the standard metric on C l . Let k be a nonzero real number. Define γ :M →M as γ(ζ, z) = (ζ 1 + k, ζ 2 , . . . , ζ l , e 2πk √ −1 z) where ζ ∈ C l and z ∈ C n . Then γ is a holomorphic isometry of (M, g) and γ m has no fixed point unless m = 0. Let Γ be the cyclic group generated by γ. Then (M, g) = (M, g)/Γ is a complete noncompact manifold with nonnegative holomorphic bisectional curvature, which is a vector bundle over C l /Γ ′ where Γ ′ is the group generated by γ ′ which sends (ζ 1 , ζ 2 , . . . , ζ l ) to (ζ 1 + k, ζ 2 , . . . , ζ l ). Now let (M, g(t)) be long time solution of the Kähler-Ricci flow (2.1) on complete noncompact Kähler manifold with bounded nonnegative holomorphic bisectional curvature yet without any assumption on the positivity of Ricci curvature. If (M ,g i (t)) is the universal cover of M with corresponding liftg i (t), theng i (t) is also such a solution to Kähler-Ricci flow . As mentioned in the introduction, (M ,g i (t)) splits for each t > 0 as (C k × N, g e × h(t)) where g e is the standard Euclidean metric on C n and h(t) solves (1.2) and has positive Ricci curvature for t > 0. We will use our results from §2, for the factor (N, h(t)), together with the arguments in [25, §5] to provide a fiber bundle description of M. In Corollary 3.1 we treat the particular case of an initial complete non-compact Kähler manifold having bounded nonnegative holomorphic bisectional curvature which decays quadratically in the average sense.
We begin with the following Basic assumption 2: Let M n be a noncompact complex manifold and let (M n , g(t)) be a complete solution of the Kähler-Ricci flow Theorem 3.1. With the Basic Assumption 2, there is k ≥ 0 such that for all t, (M, g(t)) is a holomorphic Riemannian fiber bundle over C k /Γ where Γ is a discrete subgroup of holomorphic isometries of C k , with fiber being C n−k in Case 1 and a pseudoconvex domain in C n−k in Case 2. If in Case 2, g(t) is in fact an eternal solution, then the fiber is also C n−k .
Proof. LetM be the universal cover of M. By the splitting result of Cao [7] we haveM = C k ×Ñ and g(t) = g ǫ (t) ×h(t) where g ǫ (t) is flat andh(t) satisfies the Basic Assumption 1 of §1 onÑ . In particular, in Case 1Ñ is biholomorphic to C n−k by Theorem 2.2, and in Case 2Ñ is homeomorphic to the Euclidean space and biholomorphic to a pseudoconvex domain in C n−k by Theorem 2.1 and 2.3. Moreover, if in Case 2, g(t) and henceh(t) is in fact an eternal solution, then by [12] N is biholomorphic to C n .
Let F ∈ Γ, the first fundamental group of M with respect to the metric g(0). By the uniqueness of Kähler-Ricci flow (see [18,21]), F is an holomorphic isometry with respect to g(t) for all t. As a mapping onM , The following argument is basically as in [25, §5], using the fact that N is biholomorphic either to C n−k or a pseudoconvex domain in C n−k which is homeomorphic to R 2(n−k) .
We first prove f 2 is independent ofx. Suppose there isx 1 =x 2 andỹ such that f 2 (x 1 ,ỹ) = f 2 (x 2 ,ỹ). Let α be a line passing through x 1 andx 2 in C k . Then α(s) = (γ(s),ỹ) is a line inM . That is to say, α is a geodesic which is minimizing between any two points in α. Hence F •α(s) is a line ofM too. So its projection ontoÑ is minimizing between any two end points as a curve inÑ . But the projection is f 2 •α, which is not a constant curve because f 2 (x 1 ,ỹ) = f 2 (x 2 ,ỹ). Hence it must be a line inÑ . But this implies a factor C is splitted fromÑ by [15]. This is impossible becauseÑ has positive Ricci curvature. Thus the claim is established.
Next we want to prove that f 1 is independent ofỹ. First note that On the other hand, since F is an isometry, f 2 cannot increase length. For if γ(s) is a curve inÑ, then the length for the curve α(s) = (x, γ(s)) and the length of F • α(s) = (f 1 (x, γ(s)), f 2 (γ(s)) are equal. But the length of α is the same as the length of γ. So the length f 2 (γ(s)) cannot be larger than the length of γ. But this is also true for f −1 2 . Hence f 2 is an isometry. If this is the case, then f 1 (x, γ(s)) must be a point. Hence f 1 is independent toỹ. Let Λ 1 = {f 1 |(f 1 , f 2 ) ∈ Γ for some f 2 }. Then Λ 1 is a subgroup of the holomorphic isometries of (C k , g ǫ (t)) for all t. We claim that Λ 1 is fixed point free. Suppose there is Then G is a subgroup of the group of the holomorphic isometries of N. G acts onÑ freely because Γ acts onM freely. Nowh descends tõ N/G while still satisfying the Basic Assumption 1 in §1, and thus must be simply connected by Theorems 2.1,2.2 and 2.3 which leads to a contradiction if G is not trivial. Thus f ′ 1 must be trivial and Λ 1 is fixed point free.
Let β be the homomorphism from Γ to Λ 1 defined by β(F ) = f 1 , ∈ Γ and f ′ 1 is the identity map. Then by similar argument, we can prove that the group: is trivial. Hence β is an isomorphism. Then Λ 1 acts onM isometrically holomorphically and freely by f (x,ỹ) = (f (x), ρ 2 • β −1 (f )(ỹ)) where ρ 2 is the homomorphism given by mapping Hence as shown in [25, §5], M is a Riemannian and holomorphic fiber bundle over C k /Λ 1 with fiberÑ . By the comments in the first paragraph of the proof, we see that this completes the proof of the Theorem.
Theorem 3.2. If n − k = 1 in the previous Theorem, then (M, g(t)) is a holomorphic Riemannian line bundle over C k /Γ Proof. Let Γ andÑ as in the proof of previous theorem. ThenÑ is bihlomorphic to C and has a global holomorphic coordinate z. Thus for any F ∈ Γ with F = (f 1 , f 2 ), f 2 (z) = az + b for some a = 0 and b ∈ C. Now by Theorem 2.4, any f 2 has a fixed point. It follows that any nontrivial f 2 with F = (f 1 , f 2 ) ∈ Γ, has a unique fixed point. By Theorem 2.4 again, we conclude that there is z 0 ∈Ñ = C such that f 2 (z 0 ) = z 0 for any holomorphic isometry f 2 with F = (f 1 , f 2 ) ∈ Γ. We see that we can choose a global holomorphic coordinatez onÑ = C such that in this coordinate, every such f 2 has the form f 2 (z) = az for some complex constant a. Thus in this case, the fiber bundle structure constructed in the proof of Theorem 3.1 is in fact a line bundle structure. This completes the proof of the Theorem.
By the remarks preceding Corollary 2.1, we may have the following immediate corollary to Theorems 3.1 and 3.2: Corollary 3.1. Let (M, g) be a complete non-compact Kähler manifold with non-negative holomorphic bisectional curvature. Suppose k(x, r) ≤ C/(1 + r 2 ) for some C and all r and x. Then there is k ≥ 0 such that (M, g) is a holomorphic Riemannian fiber bundle over C k /Γ where Γ is a discrete subgroup of holomorphic isometries of C k , with fiber being C n−k .
Moreover, if n − k = 1 then the above fiber bundle is in fact a line bundle.

Gradient Kähler-Ricci Solitons
In this section we refine the results in the previous section in the case of a steady gradient Kähler-Ricci soliton with bounded and nonnegative holomorphic bisectional curvature. We remark that in the case of an expanding gradient Kähler-Ricci soliton satisfying the same curvature conditions, we have M biholomorphic to C n by [10], see also [2].
First let us fix some notations. Let (M n , g) be a complete steady gradient Kähler-Ricci Ricci soliton with bounded and nonnegative holomorphic bisectional curvature. Namely, there is a real valued function f such that Let (M , g) be the universal cover of (M n , g). By [7],M = C k ×Ñ and g = g e ×h, where g e is the standard metric and (Ñ ,h) is a steady gradient Kähler-Ricci soliton with bounded nonnegative holomorphic bisectional curvature and with positive Ricci curvature. Hence there exists some smooth real valued function f satisfying (4.1) onÑ. Now let us also assume that the scalar curvature of M attains its maximum at some point. Hence the same is true forÑ . Then by [8] ∇f (the real gradient of f relative toh) has a unique zero at somep. Let λ 1 ≥ λ 2 ≥ · · · ≥ λ l > 0 be the eigenvalues of the Ricci curvature atp, where l = n − k. Recall thatÑ is biholomorpic to C l by [10], [2]. Let Γ be the fundamental group of M. Then by the proof of Theorem 3.1, each F in Γ is of the form (γ 1 , γ 2 ) where γ 1 is an holomorphic isometry of C k and γ 2 is a holomorphic isometry of (Ñ,h). Let We want to describe the holomorphic isometries in Λ 2 . In fact, we want to describe any holomorphic isometry ofÑ in general. It was shown by Bryant [2] that there is a global holomorphic coordinates z i ofÑ, ranging over C l , which linearize the holomorphic vector field Z = 1 2 ∇f − √ −1J(∇f ) . That is to say, Such coordinates are called Poincarè coordinates.
Remark 4.1. We remark that in case of expanding gradient Kähler-Ricci soliton with nonnegative Ricci curvature, Poincaré coordinates also exist. This is because in this case, the potential function f has a unique critical point and J(∇f ) is also a Killing vector field. Now let φ : (Ñ ,h) → (Ñ ,h) be a holomorphic isometry. Thus φ * (h) =h.
Lemma 4.1. The following are true: Proof.
(2) follows from the definition of Z, the fact that φ * (h) =h and from (1). Proof. This is basically a result in [2]. For the sake of completeness, we sketch the proof here. Since φ * (Z) = Z and we are using Poincaré coordinates in the domain and target, for each i, we have where α is a multi-index. The fact that there is no constant term in (4.3) follows from the the fact that by their definition, both z and w must vanish atp ∈Ñ . Now by substituting the power series in (4.3) and its derivative with respect to z for both sides of (4.2), equating coefficients and using the fact that the λ i 's are all positive, the result follows.
In [32], Yang proves the following: Let M be a noncompact Kähler-Ricci soliton with nonnegative Ricci curvature. Suppose its scalar curvature attains a positive maximum at a compact complex submanifold K with codimension 1 and the Ricci curvature is positive away from K. Then M is a holomorphic line bundle over K.
In case the holomorphic bisectional curvature is bounded and nonnegative, from the lemma, the proof of Theorem 3.1 and the discussion before Lemma 4.1, we have the following: Theorem 4.1. Let (M, g) be a steady gradient Kähler-Ricci soliton with bounded non-negative holomorphic bisectional curvature such that the scalar curvature is maximal at some p ∈ M. Then assuming the above notation, there exists global complex coordinates onÑ such that M is a Riemannian and holomorphic fiber bundle over C k /Λ 1 with fiber C n−k such that each γ 2 ∈ Λ 2 is of upper triangular form. If in addition, for all positve λ j we have l i=1 m i λ i = λ j for all nonnegative integers m i such that l i=1 m i ≥ 2. Then M is a holomorphic vector bundle over C k /Λ 1 .
We end this section by proving a general volume growth estimate for expanding gradient solitons. The motivation is from the work of Ni [24] where he proved that a complete noncompact Kḧaler manifold with bounded and nonnegative holomorphic bisectional curvature and with maximum volume growth must have quadratic curvature decay as in (1) in the Introduction. Proof. Let f be such that f ij = R ij + g ij . By the proof in [23, §20] (see also [19]), R + |∇f | 2 − 2f is a constant, where R is the scalar curature. Modifying f by adding a constant to f , we may assume that R + |∇f | 2 − 2f = 0. Since R ≥ 0, we have f ≥ 0 and |∇f | 2 ≤ 2f.  for R ≥ 1.