Gromov-Hausdorff limits of K¨ ahler manifolds and the finite generation conjecture

Gromov-Hausdorff limits of K¨ ahler manifolds and the finite generation conjecture

By Gang Liu

Abstract

We study the uniformization conjecture of Yau by using the Gromov- Hausdorff convergence. As a consequence, we confirm Yau’s finite genera- tion conjecture. More precisely, on a complete noncompact K¨ahler manifold with nonnegative bisectional curvature, the ring of polynomial growth holo- morphic functions is finitely generated. During the course of the proof, we prove ifMⁿ is a complete noncompact K¨ahler manifold with nonnegative bisectional curvature and maximal volume growth, thenM is biholomor- phic to an affine algebraic variety. We also confirm a conjecture of Ni on the existence of polynomial growth holomorphic functions on K¨ahler manifolds with nonnegative bisectional curvature.

1. Introduction

In [34], Yau proposed to study the uniformization of complete K¨ahler manifolds with nonnegative curvature. In particular, one wishes to determine whether or not a complete noncompact K¨ahler manifold with positive bisec- tional curvature is biholomorphic to a complex Euclidean space. For this sake, Yau further asked in [34] (see also page 117 in [33]) whether or not the ring of polynomial growth holomorphic functions is finitely generated and whether or not dimension of the spaces of holomorphic functions of polynomial growth is bounded from above by the dimension of the corresponding spaces of polyno- mials onCⁿ. Let us summarize Yau’s questions in the three conjectures below:

Conjecture 1. LetMⁿ be a complete noncompact K¨ahler manifold with positive bisectional curvature. Then M is biholomorphic toCⁿ.

Conjecture 2. LetMⁿ be a complete noncompact K¨ahler manifold with nonnegative bisectional curvature. Then the ring OP(M) is finitely generated.

The author was partially supported by NSF grant #DMS-1406593.

c 2016 Department of Mathematics, Princeton University.

Conjecture 3. LetMⁿ be a complete noncompact K¨ahler manifold with nonnegative bisectional curvature. Then given any d > 0, dim(Od(M)) ≤ dim(Od(Cⁿ)).

On a complete K¨ahler manifold M, we say a holomorphic functionf has polynomial growth, denoted by f ∈ Od(M) if there exists some C > 0 such that |f(x)| ≤C(1 +d(x, x₀))^d for all x ∈M. Here x₀ is a fixed point on M. Let OP(M) =∪d>0Od(M).

Conjecture 1 is open so far. However, there has been much important progress due to various authors. In earlier works, Mok-Siu-Yau [24] and Mok [23] considered embedding by using holomorphic functions of polynomial growth. Later, with the K¨ahler-Ricci flow, results were improved significantly.

See, for example, [30], [31], [9], [27], [1], [8], [17].

Conjecture 3 was confirmed by Ni [26] with the assumption that M has maximal volume growth. Later, by using Ni’s method, Chen-Fu-Le-Zhu [7]

removed the extra condition. See also [21] for a different proof. The key of Ni’s method is a monotonicity formula for heat flow on K¨ahler manifold with nonnegative bisectional curvature.

Despite great progress ofConjectures 1and3, not much was known about Conjecture 2. In [23], Mok proved the following:

Theorem 1.1(Mok). LetMⁿ be a complete noncompact K¨ahler manifold with positive bisectional curvature such that for some fixed point p∈M,

• scalar curvature ≤ _d(p,x)^{C0 2} for some C₀ >0;

• Vol(B(p, r))≥C₁r²ⁿ for some C₁>0.

Then Mⁿ is biholomorphic to an affine algebraic variety.

In Mok’s proof, the biholomorphism was given by holomorphic functions of polynomial growth. Therefore, OP(M) is finitely generated. In the general case, it was proved by Ni [26] that the transcendental dimension of OP(M) over C is at most n. However, this does not imply the finite generation of OP(M). The main result in this paper is the confirmation ofConjecture 2in the general case:

Theorem 1.2. Let Mⁿ be a complete noncompact K¨ahler manifold with nonnegative bisectional curvature. Then the ring O^P(M) is finitely generated.

During the course of the proof, we obtain a partial result forConjecture 1:

Theorem 1.3. Let Mⁿ be a complete noncompact K¨ahler manifold with nonnegative bisectional curvature. Assume M is of maximal volume growth.

Then M is biholomorphic to an affine algebraic variety.

Here maximal volume growth means Vol(B(p, r))≥Cr²ⁿfor someC >0.

This seems to be the first uniformization type result without assuming the curvature upper bound.

If one wishes to proveConjecture 1by consideringOP(M), it is important to know when OP(M) 6= C. In [26], Ni proposed the following interesting conjecture:

Conjecture 4. LetMⁿ be a complete noncompact K¨ahler manifold with nonnegative bisectional curvature. Assume M has positive bisectional curva- ture at one point p. Then the following three conditions are equivalent: (1) OP(M)6=C;

(2) M has maximal volume growth;

(3) there exists a constant C independent of r so that −R

B(p,r)S ≤ _r^C2. Here S is the scalar curvature and −R

means the average.

In the complex one-dimensional case, the conjecture is known to hold, e.g., [19]. For higher dimensions, Ni proved (1) implies (3) in [26]. The proof used the heat flow method. Then in [29], Ni and Tam proved that (3) also implies (1). Their proof employs the Poincar´e-Lelong equation and the heat flow method. Thus, it remains to prove (1) and (2) are equivalent. Under some extra conditions, Ni [27] and Ni-Tam [28] were able to prove the equivalence of (1) and (2). In [22], the author proved that (1) implies (2). In fact, the con- dition thatM has positive bisectional curvature at one point could be relaxed to that the universal cover of M is not a product of two K¨ahler manifolds.

In this paper, we prove that (2) also implies (1). Thus Conjecture 4 is solved in full generality. More precisely, we prove

Theorem 1.4. Let (Mⁿ, g) be a complete K¨ahler manifold with nonneg- ative bisectional curvature and maximal volume growth. Then there exists a nonconstant holomorphic function of polynomial growth on M.

The strategy of the proofs in this paper is very different from earlier works.

Here we make use of several different techniques:

• the Gromov-Hausdorff convergence theory developed by Cheeger-Colding [2], [3], [4], [5], Cheeger-Colding-Tian [6];

• the heat flow method by Ni [26] and Ni-Tam [28], [29];

• the H¨ormander L²-estimate of ∂ [16], [11];

• the three circle theorem [21].

We point out that recently, the Gromov-Hausdorff convergence theory was shown to be a very powerful tool to study K¨ahler manifolds; see, e.g., [12], [32].

The first key point is to proveTheorem 1.4. By H¨ormander’sL²-technique, to produce holomorphic functions of polynomial growth, it suffices to construct strictly plurisubharmonic function of logarithmic growth. However, it is not obvious how to construct such functions by only assuming the maximal volume growth condition. In [24], [23], Mok-Siu-Yau and Mok considered the Poincar´e- Lelong equation√

−1∂∂u= Ric. When the curvature has pointwise quadratic

decay, they were able to prove the existence of a solution with logarithmic growth. Later, Ni and Tam [28], [29] were able to relax the condition to that the curvature has average quadratic decay. Then it suffices to prove that maximal volume growth implies the average curvature decay.

We prove Theorem 1.4 by a different strategy. We first blow down the manifold. Then by using the Cheeger-Colding theory, heat flow technique and H¨ormanderL²-theory, we construct holomorphic functions with controlled growth in a sequence of exhaustion domains onM. Then the three circle theo- rem ensures that we can take subsequence to obtain a nonconstant holomorphic function with polynomial growth.

Once Theorem 1.4is proved, H¨ormander’s L²-technique produces a lot of holomorphic functions of polynomial growth. It turns out O^P(M) separates points and tangent spaces onM. However, since the manifold is not compact, it does not follow directly that M is affine algebraic. To overcome this difficulty, we prove inTheorem 6.1that the map given by Od(M) is proper. We will use induction on the dimension of the splitting factor of a tangent cone.

Once we have proved the properness of the holomorphic map, it is straight- forward to prove M is affine algebraic by using techniques from complex an- alytic geometry. Here the argument resembles some part in [12]. Then we conclude Conjecture 2 when the manifold has maximal volume growth. For the general case, we apply the main result in [22]. It suffices to handle the case when the universal cover of the manifold splits. Then we need to consider group actions. The final result follows from an algebraic result of Nagata [25].

This paper is organized as follows. InSection 2, we collect some prelimi- nary results necessary for this paper. InSection 3, we prove a result which con- trols the size of a holomorphic chart when the manifold is Gromov-Hausdorff close to a Euclidean ball. As the first application, inSection 4we prove a gap theorem for the complex structure of Cⁿ. Section 5 deals with the proof of Theorem 1.4. The proof ofTheorem 6.1is contained inSection 6. Finally, the proof of Theorem 1.2is givenSection 7.

There are two appendices. InAppendix A, we present a result of Ni-Tam in [28] which was not stated explicitly. (Here we are not claiming any credit.) InAppendix B, we introduce some results of Nagata [25] to conclude the proof of the main theorem.

Acknowledgments. The author would like to express his deep gratitude to Professors John Lott, Lei Ni, and Jiaping Wang for many valuable discussions during the work. He thanks Professor Xinyi Yuan for some discussions on al- gebra. He also thanks Professors Richard Bamler, Peter Li, Luen-Fai Tam and Shing-Tung Yau for their interest in this work. Finally, he thanks the anony- mous referee for pointing out some inaccuracies and improving the exposition of this paper.

2. Preliminary results

First recall some convergence results for manifolds with Ricci curvature lower bound. Let (M_iⁿ, yi, ρi) be a sequence of pointed complete Riemannian manifolds, whereyi ∈M_iⁿandρi is the metric onM_iⁿ. By Gromov’s compact- ness theorem if (M_iⁿ, yi, ρi) have a uniform lower bound of the Ricci curvature, then a subsequence converges to some (M_∞, y_∞, ρ_∞) in the Gromov-Hausdorff topology. See [14] for the definition and basic properties of Gromov-Hausdorff convergence.

Definition 2.1. Let Ki ⊂ M_iⁿ → K_∞ ⊂ M_∞ in the Gromov-Hausdorff topology. Assume that {fi}^∞i=1 are functions on M_iⁿ and f_∞ is a function on M_∞. Assume that Φiareεi-Gromov-Hausdorff approximations and lim

i→∞εi= 0.

Iffi◦Φi converges tof∞uniformly, we sayfi →f∞uniformly overKi→K∞. In many applications, fi are equicontinuous. The Arzela-Ascoli theorem applies to the case when the spaces are different. When

(M_iⁿ, yi, ρi)→(M_∞, y_∞, ρ_∞)

in the Gromov-Hausdorff topology, any bounded, equicontinuous sequence of functions fi has a subsequence converging uniformly to some f_∞on M_∞.

Let the complete pointed metric space (M_∞^m, y) be the Gromov-Hausdorff limit of a sequence of connected pointed Riemannian manifolds, {(M_iⁿ, pi)}, with Ric(Mi) ≥ 0. Here M_∞^m has Hausdorff dimension m with m ≤ n.

A tangent cone at y ∈ M_∞^m is a complete pointed Gromov-Hausdorff limit ((M_∞)_y, d_∞, y_∞) of{(M_∞, r_i⁻¹d, y)}, whered, d_∞are the metrics ofM_∞,(M_∞)_y respectively, and{r_i} is a positive sequence converging to 0.

Definition 2.2. A point y ∈M_∞ is called regular, if there exists some k so that every tangent cone at y is isometric toR^k. A point is called singular, if it is not regular.

In [3], the following theorem was proved:

Theorem 2.1. Regular points are dense in the Gromov-Hausdorff limits of manifolds with Ricci curvature lower bound.

Below is a result of Ni-Tam [28] on the heat flow on K¨ahler manifolds:

Theorem 2.2. Let Mⁿ be a complete noncompact K¨ahler manifold with nonnegative bisectional curvature. Let u be a smooth function on M with compact support. Let

v(x, t) = Z

M

H(x, y, t)u(y)dy.

Here H(x, y, t) is the heat kernel of M. Let η(x, t)_αβ = v_αβ and λ(x) be the minimum eigenvalue for η(x,0). Let

λ(x, t) = Z

M

H(x, y, t)λ(y)dy.

Thenη(x, t)−λ(x, t)g_αβ is a nonnegative(1,1)tensor fort∈[0, T]forT >0.

A detailed proof of this theorem is presented inAppendix A.

Recall the H¨ormander L²-theory:

Theorem 2.3. Let (X, ω) be a connected but not necessarily complete K¨ahler manifold withRic≥0. Assume X is Stein. Let ϕbe aC^∞ function on X with √

−1∂∂ϕ≥c ω for some positive function c on X. Let g be a smooth (0,1) form satisfying ∂g = 0 and ^R_X ^|g|_c²e^−ϕωⁿ < +∞. Then there exists a smooth function f onX with ∂f=g and ^R_X|f|²e^−ϕωⁿ≤^RX

|g|²

c e^−ϕωⁿ. The proof can be found in [11, pp. 38–39]. Also compare Lemma 4.4.1 in [16]. Note that the theorem also applies to singular metrics with positive curvature in the current sense.

Recall the three circle theorem in [21]:

Theorem 2.4. Let M be a complete noncompact K¨ahler manifold with nonnegative holomorphic sectional curvature, p ∈M. Let f be a holomorphic function on M. Let M(r) = sup

B(p,r)|f(x)|. Then logM(r) is a convex function of logr. Therefore, given any k >1, ^M_M(r)^(kr) is monotonic increasing.

Theorem 2.4 has the following consequences:

Corollary 2.1. Given the same condition as in Theorem 2.4, if f ∈ Od(M), then ^M(r)_rd is nonincreasing.

Corollary 2.2.Given the same condition as inTheorem2.4,iff(p) = 0, then ^M_r^(r) is nondecreasing.

Remark2.1. The three circle theorem is still true for holomorphic sections on nonpositive bundles. See page 17 of [21] for a proof.

Finally, we need the multiplicity estimate by by Ni [26] (see also [7]):

Theorem 2.5. Let Mⁿ be a complete noncompact K¨ahler manifold with nonnegative bisectional curvature. Then dim(Od(M))≤dim(Od(Cⁿ)).

Note that this result also follows from Corollary 2.1.

In this paper, we will denote by Φ(u₁, . . . , u_k|. . .) any nonnegative func- tions depending onu₁, . . . , u_k and some additional parameters such that when these parameters are fixed,

ulimk→0· · · lim

u1→0Φ(u₁, . . . , u_k|. . .) = 0.

LetC(n), C(n, v) be large positive constants depending only onnorn, v, and let c(n), c(n, v) be small positive constants depending only on n or n, v. The values might change from line to line.

3. Construct holomorphic charts with uniform size

In this section, we introduce the following proposition, which is crucial for the construction of holomorphic functions.

Proposition 3.1. Let Mⁿ be a complete K¨ahler manifold with nonnega- tive bisectional curvature, x∈M. There exist ε(n)>0, δ=δ(n) >0 so that the following holds: For ε < ε(n), if dGH(B(x,¹_εr), BCⁿ(0,¹_εr)) < εr, there exists a holomorphic chart (w₁, . . . , w_n) containing B(x, δr) so that

• ws(x) = 0(1≤s≤n);

• Pⁿ

s=1

ws|²(y)−r²(y)| ≤Φ(ε|n)r² in B(x, δr) — here r(y) =d(x, y);

• |dws(y)| ≤C(n) in B(x, δr).

Proof. By scaling, we may assume r 1, to be determined. SetR= ₁₀₀^r 1. According to the assumptions and the Cheeger-Colding theory [2] (see also equation (1.23) in [4]), there exist real harmonic functions b₁, . . . , b_2n on B(x,4r) so that

(3.1) −

Z

B(x,2r)

X

j

|∇(∇bj)|²+^X

j,l

|h∇bj,∇b_li −δ_jl|²≤Φ(ε|n, r) and

(3.2) b_j(x) = 0(1≤j ≤2n), |∇b_j| ≤C(n)

in B(x,2r). Furthermore, the map F(y) = (b₁(y), . . . , b_2n(y)) is a Φ(ε|n)r Gromov-Hausdorff approximation from B(x,2r) toBR²ⁿ(0,2r). According to the argument above Lemma 9.14 in [6] (see also (20) in [22]), after a suitable orthogonal transformation, we may assume

(3.3) −

Z

B(x,r)|J∇b_2s−1− ∇b_2s|² ≤Φ(ε|n, r) for 1≤s≤n. Setw_s⁰ =b2s−1+√

−1b2s. Then

(3.4) −

Z

B(x,r)|∂w⁰_s|²≤Φ(ε|n, r).

The idea is to perturb w⁰_s so that they become a holomorphic chart. We would like to apply the H¨ormanderL²-estimate. First, we construct the weight

function. Consider the function h(y) =

X2n j=1

b²_j(y).

Then in B(x, r),

(3.5) |h(y)−r²(y)| ≤Φ(ε|n)r². By (3.2),

(3.6) |∇h(y)| ≤C(n)r(y)

inB(x, r). The real Hessian ofh satisfies (3.7)

Z

B(x,5R)

X

u,v

|h_uv(y)−2g_uv|²≤Φ(ε|n, R).

Now consider a smooth function ϕ: R⁺ → R⁺ with ϕ(t) = t for 0 ≤ t ≤ 1, ϕ(t) = 0 for t≥2, and|ϕ|,|ϕ⁰|,|ϕ⁰⁰| ≤C(n). Let H(x, y, t) be the heat kernel on M, and set

(3.8) u(y) = 5R²ϕ Çh(y)

5R² å

, ut(z) = Z

M

H(z, y, t)u(y)dy.

Claim 3.1. u₁(z) satisfies that (u₁)_αβ(z)≥c(n)g_αβ >0 in B^Äx,₁₀^Rä. Proof. Let λ(y) be the lowest eigenvalue of the complex Hessian u_αβ. By (3.7),

− Z

B(x,5R)|h_αβ−2g_αβ|²≤Φ(ε|n, R).

Then there exists E ⊂B(x,5R) with

(3.9) vol(B(x,5R)\E)≤Φ(ε|n, R), h_αβ ≥ 1 2g_αβ

on E. By (3.5), we may assume h(y) ≤ 5R² in B(x,2R). Then u = h in B(x,2R). We have

(3.10) ÇZ

B(x,2R)\E|λ²(y)|dy å¹

2

≤ ÑZ

B(x,4R)\E

X

α,β

|h_αβ|² é¹₂

≤4 ÑZ

B(x,4R)\E

X

α,β

|h_αβ−2g_αβ|²dy é¹

2

+ 4 ÑZ

B(x,4R)\E

X

α,β

|2g_αβ|²dy é¹₂

≤Φ(ε|n, R)

and

(3.11)

|λ| ≤ |u_αβ|

=

ϕ⁰h_αβ+ ϕ⁰⁰ 5R²hαh_β

≤ |ϕ⁰(h_αβ−2g_αβ)|+

2ϕ⁰g_αβ+ ϕ⁰⁰ 5R²hαh_β

≤C(n)(|h_αβ−2g_αβ|+ 1).

Therefore, (3.12)

Z

B(x,5R)|λ(y)|dy≤C(n)R²ⁿ.

Let λ(z,1) =^R H(z, y,1)λ(y)dy. Note that by definition (3.8),u is supported inB(x,4R). By (3.9), λ≥ ¹₂ in E. Forz∈B^Äx,₁₀^Rä,

(3.13)

Z

H(z, y,1)λ(y)dy= Z

B(x,4R)

H(z, y,1)λ(y)dy

≥ Z

B(x,2R)\EH(z, y,1)λ(y)dy +

Z

B(x,4R)\B(x,2R)H(z, y,1)λ(y)dy +

Z

E∩B(z,1)H(z, y,1)λ(y)dy.

By heat kernel estimates of Li-Yau [20], H(z, y,1)≥c(n)>0 for y∈B(z,1).

Also, with volume comparison, we findH(z, y,1)≤C(n) fory, z ∈B(x,4R).As a consequence,

(3.14) Z

B(x,2R)\E|H(z, y,1)λ(y)|dy

≤C(n) Z

B(x,2R)\E|λ(y)|dy

≤C(n) ÇZ

B(x,2R)\E|λ²(y)|dy å¹

2

(vol(B(x,2R)\E))¹²

≤Φ(ε|n, R), (3.15)

Z

E∩B(z,1)H(z, y,1)λ(y)dy≥ 1 2

Z

E∩B(z,1)H(z, y,1)dy≥c(n)>0.

Note that d(y, z) ≥ R for y ∈ B(x,4R)\B(x,2R). The heat kernel esti- mate says H(y, z,1)≤C(n)e^−R

2

5 . Therefore, by (3.12), (3.16)

Z

B(x,4R)\B(x,2R)|H(z, y,1)λ(y)|dy≤C(n)e^−R

2

5 R²ⁿ<Φ Å1

R ã

.

Putting (3.14), (3.15), (3.16) in (3.13), we find (3.17) λ(z,1) =

Z

H(z, y,1)λ(y)dy≥c(n)−Φ Å1

R|n)−Φ(ε|n, R ã

for z ∈B^Äx,₁₀^Rä. We first let R be large and then let ε be very small. Then λ(z,1)> c(n). We conclude the proof of the claim from Theorem 2.2.

Recall thatutis defined in (3.8). We claim that there existsε₀ =ε₀(n)>0 so that for large R,

(3.18) min

y∈∂B(x,20^R)

u₁(y)>4 sup

y∈B(x,ε⁰20^R)

u₁(y).

This is a simple exercise by using the heat kernel estimate. One can also apply Proposition A.1to conclude the proof. From now on, we freeze the value ofR.

That is to say, R=R(n)>0 satisfies Claim 3.1, (3.18) and ₂₀^Rε₀ >100.

Let Ω be the connected component of



y∈B Å

x, R 20

ãu₁(y)<2 sup

y∈B(x,ε0R 20)

u₁(y)





containing B(x, ε₀₂₀^R). Then Ω is relatively compact in B(x,₂₀^R) and Ω is a Stein manifold byClaim 3.1.

Now we applyTheorem 2.3to Ω, with the K¨ahler metric induced fromM. Take smooth (0,1) formsgs=∂w⁰_s defined in (3.4), and take the weight func- tion ψ=u₁. We find smooth functionsfs in Ω with∂fs=gs and

(3.19)

Z

Ω|fs|²e^−ψωⁿ≤ Z

Ω

|gs|²

c e^−ψωⁿ≤ R

Ω|∂w_s⁰|²ωⁿ

c(n) ≤Φ(ε|n).

Here we used the fact that r= 100R= 100R(n). By Proposition A.1, we find ψ=u₁≤C(R, n) =C(n) in B(x, R). Therefore,

(3.20)

Z

B(x,10)|fs|²ωⁿ≤ Z

Ω|fs|²ωⁿ≤Φ(ε|n).

Note thatw_s=w⁰_s−f_s is holomorphic, as∂w_s=∂w_s⁰ −g_s= 0. Sincew_s⁰ is harmonic (complex),fs is also harmonic. By the mean value inequality [18]

and Cheng-Yau’s gradient estimate [10], we find that in B(x,5), (3.21) |fs| ≤Φ(ε|n), |∇fs| ≤Φ(ε|n).

Therefore, equation (3.1) implies (3.22)

Z

B(x,4)|(ws)i(wt)jg^ij −2δst| ≤Φ(ε|n).

Claim 3.2. ws(s= 1, . . . , n) is a holomorphic chart in B(x,1).

Proof. Recall that (b₁, . . . , b_2n) is a Φ(ε|n) Gromov-Hausdorff approxima- tion to the image in R²ⁿ. According to (3.21), on B(x,1), w = (w1, . . . , wn) is also a Φ(ε|n) Gromov Hausdorff approximation to BCⁿ(0,1). Therefore, w⁻¹(BCⁿ(0,1)) is compact inB(x,1 + Φ(ε|n)). First we prove that the degree dof the map w is 1. By (3.22) and the fact that holomorphic maps preserves the orientation, d≥1. We also have

(3.23)

d·vol(BCⁿ(0,1)) = 1 (−2√

−1)ⁿ Z

w⁻¹(B(0,1))

dw₁∧dw₁∧ · · · ∧dw_n∧dw_n

≤(1 + Φ(ε|n)) Vol(B(x,(1 + Φ(ε|n))) + Φ(ε|n)

by (3.21) and (3.22). This means that if ε is sufficiently small, then d = 1.

That is to say, (w₁, . . . , wn) is generically one to one in B(x,1). Moreover, (w₁, . . . , wn) must be a finite map: the preimage of a point must be a subvariety which is compact in the Stein manifold Ω, thus finitely many points. According to Proposition 14.7 on page 87 of [13], this is an isomorphism.

We can make a small perturbation so that ws(x) = 0 for 1≤s≤n. This

completes the proof ofProposition 3.1.

4. A gap theorem for the complex structure of Cⁿ

As the first application ofProposition 3.1, we prove a gap theorem for the complex structure of Cⁿ. The conditions are rather restrictive. However, we shall expand some of the ideas in later sections.

Theorem 4.1. Let Mⁿ be a complete noncompact K¨ahler manifold with nonnegative bisectional curvature and p∈M. There exists ε(n)>0 so that if ε < ε(n) and

(4.1) vol(B(p, r))

r²ⁿ ≥ω_2n−ε

for all r >0, then M is biholomorphic to Cⁿ. Here ω_2n is the volume of the unit ball in Cⁿ. Furthermore,the ringOP(M) is finitely generated. In fact, it is generated by nfunctions which form a coordinate in Cⁿ.

Proof. Consider the blow-down sequence (Mi, pi, gi) = ^ÄM, p,_s¹²

ig^ä for s_i→ ∞. According toProposition 3.1and the Cheeger-Colding theory [2], ifε is sufficiently small, there exists a holomorphic chart (w₁ⁱ, . . . , w_nⁱ) onB(pi,1).

Moreover, the map (wⁱ₁, . . . , wⁱ_n) is a Φ(ε|n) Gromov-Hausdorff approximation toBCⁿ(0,1). We may assume

(4.2) wⁱ_s(pi) = 0

for s= 1, . . . , n. We can also regard w_sⁱ as holomorphic functions on B(p, si)

⊂M. For eachi, we can find a new basisv_sⁱ for span{wⁱ_s} so that (4.3)

Z

B(p,1)

v_sⁱvⁱ_t=δ_st. Set

(4.4) M_sⁱ(r) = sup

x∈B(p,r)|v_sⁱ(x)|. Claim 4.1. _M^M_i^si(si)

s(¹2si) ≤2 + Φ(ε|n) for 1≤s≤n.

Proof. It suffices to prove this fors= 1. Letv₁ⁱ = ^Pn

j=1cⁱ_jw_jⁱ. Without loss of generality, assume|cⁱ₁|= max

1≤j≤n|cⁱ_j|>0. Then ^v_cⁱ¹i 1

=wⁱ₁+^Pn

j=2

α_ijwⁱ_j and|α_ij| ≤1.

Since on Mi, (wⁱ₁, . . . , wⁱ_n) is a Φ(ε|n) Gromov-Hausdorff approximation to BCⁿ(0,1), we find

(4.5)

M_sⁱ(si) M_sⁱ(¹₂si) =

sup

x∈B(p,si)

wⁱ₁(x) + ^Pn

j=2αijwⁱ_j(x) sup

x∈B(p,^si₂)

w₁ⁱ(x) + ^Pn

j=2

α_ijw_jⁱ(x)

= sup

x∈B(pi,1)

wⁱ₁(x) + ^Pn

j=2αijwⁱ_j(x) sup

x∈B(pi,¹₂)

w₁ⁱ(x) + ^Pn

j=2αijw_jⁱ(x)

≤2 + Φ(ε|n).

This concludes the proof.

According to the three circleTheorem 2.4, ^M_M^sii^(2r)

s(r) is monotonic increasing for 0< r < ¹₂si. ThenClaim 4.1 implies

(4.6) M_sⁱ(2r)

M_sⁱ(r) ≤2 + Φ(ε|n)

for 0< r < ¹₂si. From (4.3), we findM_sⁱ(¹₂)≤C(n). Equation (4.6) implies

(4.7) M_sⁱ(r)≤C(n)(r^α+ 1)

forα= 1 + Φ(ε|n). Ass_i → ∞, by taking subsequence, we can assumev_sⁱ →v_s uniformly on each compact set of M. Set

(4.8) M_s(r) = sup

x∈B(p,r)|v_s(x)|.

Then

(4.9) Ms(r)≤C(n)(r^α+ 1)

forα= 1 + Φ(ε|n) andr≥0. We may assumevs ∈ O³₂(M). Note thatvs also satisfies

(4.10) vs(p) = 0(1≤s≤n), Z

B(p,1)vsvt=δst.

Our goal is to prove that (v₁, . . . , vn) is a biholomorphism fromM toCⁿ. Claim 4.2. Let ε in (4.1) be sufficiently small (depending only on n).

If we rescale each v_s so that sup

B(p,1)|v_s| = 1, then in B(p,1), (v₁, . . . , v_n) is a

1

100n-Gromov-Hausdorff approximation to BCⁿ(0,1).

Proof. We argue by contradiction. Assume there exists a positive sequence εa→0(a∈N), and let (M_a⁰, qa) be a sequence of n-dimensional complete non- compact K¨ahler manifolds with nonnegative bisectional curvature and

(4.11) vol(B(qa, r))

r²ⁿ ≥ω_2n−εa

for all r > 0. Assume there exist holomorphic functions u^a_s(s = 1, . . . , n) on M_a⁰ so that

(4.12) u^a_s(qa) = 0, u^a_s ∈ O³

2(M_a⁰), Z

B(qa,1)

u^a_su^a_t =c^a_stδst, sup

B(qa,1)|u^a_s|= 1.

Here c^a_st are constants. Assume in B(qa,1) that (u^a₁, . . . , u^a_n) is not a _100n¹ - Gromov-Hausdorff approximation toBCⁿ(0,1). According to Cheeger-Colding theory [2] and (4.11), (M_a⁰, qa) converges to (R²ⁿ,0) in the pointed Gromov- Hausdorff sense. By the three circle theorem and (4.12), we have a uniform bound for u^a_s in B(qa, r) for any r >0. Let a → ∞. Then there is a subse- quence so that u^a_s →us uniformly on each compact set. Moreover, by Remark 9.3 of [6] (see also (21) in [22]), there is a natural linear complex structure on R²ⁿ. Thus we can identify the limit space with Cⁿ. By Lemma 4 in [22], the limits of holomorphic functions are still holomorphic. Moreover, {u_s} satisfy (4.12), according to the three circle theorem. Thusus are all linear functions which form a standard complex coordinate in Cⁿ. Therefore, (u₁, . . . , un) is an isometry fromBCⁿ(0,1) toBCⁿ(0,1). This contradicts the assumption that (u^a₁, . . . , u^a_n) is not a _100n¹ -Gromov-Hausdorff approximation toBCⁿ(0,1).

According to Claim 4.2, (v1, . . . , vn)(∂B(p,1))∩BCⁿ(0,¹₂) =∅. Let U be the connected component of (v₁, . . . , vn)⁻¹(BCⁿ(0,¹₂)) containing p. Then U is relatively compact in B(p,1). We claim that (v1, . . . , vn)(U) has complex dimensionn. Otherwise, for a generic pointq in (v1, . . . , vn)(U), let the preim- age in B(p,1) be Σq. Then Σq has complex dimension at least one. But Σq

is a compact analytic set in a Stein manifold. Thus it contains only finitely many points. This is a contradiction.

Therefore, dv1 ∧ · · · ∧dvn is not identically zero. By (4.9) and Cheng- Yau’s gradient estimate, |dvi| ≤ C(n)(r^Φ(ε|n)+ 1). Thus |dv1 ∧ · · · ∧dvn| ≤ C(n)(r^Φ(ε|n)+ 1). The canonical line bundle K_M has nonpositive curvature.

Note that by the remark followingCorollary 2.2, the three circle theorem also holds for holomorphic sections of nonpositive bundles. Therefore, if the holo- morphicn-formdv₁∧···∧dvnvanishes at some point inM, then|dv₁∧···∧dvn| must be of at least linear growth, byCorollary 2.2. Therefore,dv₁∧ · · · ∧dvn

is vanishing identically on M. This is a contradiction.

Next we prove that the map (v₁, . . . , vn): M → Cⁿ is proper. Given any R > 1, we can define a norm | · |^R for the span of v₁, . . . , vn induced by R

B(p,R)vsvt. There exists a basisv₁^R, . . . , v^R_n for the span ofv₁, . . . , vn so that (4.13)

Z

B(p,1)v^R_sv^R_t =δst, Z

B(p,R)v_s^Rv_t^R=c(R)stδst.

That is, we diagonalize the two norms| · |1and | · |^Rsimultaneously. Obviously we have

(4.14)

Xn s=1

|vs(x)|² = Xn s=1

|v^R_s(x)|²

for anyx∈M. To prove (v₁, . . . , vn) is proper, it suffices to prove ^Pn

s=1|v_s^R(x)|² is large for x∈∂B(p, R) and large R. Define

(4.15) w_s^R(x) = v^R_s(x)

c^R_s , where c^R_s are positive constants so that

(4.16) sup

x∈B(p,R)

w^R_s(x) = 1

for s = 1, . . . , n. Note that ^R_B(p,1)|v_s^R|² = 1 and v^R_s(p) = 0. According to Corollary 2.2 and (4.13),

(4.17) c^R_s ≥cR,

where c =c(n) >0, R > 1. We can applyClaim 4.2 to v^R_s inB(p, R). Here we have to rescale the radius to 1. Then we obtain that (Rw₁^R, . . . , Rw^R_n) is a

R

100n-Gromov-Hausdorff approximation fromB(p, R) to BCⁿ(0, R). In partic- ular, for anyx∈∂B(p, R), there exists somes₀ with|w_s^R₀(x)| ≥ _2n¹ . Then

|v^R_s0(x)|=c^R_s0|w_s^R₀(x)| ≥ 1 2ncR, Xn

s=1

|vs(x)|² = Xn s=1

|v_s^R(x)|² ≥c(n)R². (4.18)

The properness is proved.

Asdv₁∧ · · · ∧dvn is not vanishing at any point on M and (v₁, . . . , vn) is a proper map to Cⁿ, we conclude that (v1, . . . , vn) is a biholomorphism from M toCⁿ.

Next we prove that OP(M) is generated by (v1, . . . , vn). We can regard (v₁, . . . , v_n) as a global holomorphic coordinate system on M. If f ∈ Od(M), we can thinkf =f(v₁, . . . , v_n). It suffices to prove the right-hand side is a poly- nomial. Indeed,|f(x)| ≤C(1+d(x, p)^d). Note that by (4.18),|f(v₁, . . . , vn)| ≤ C^{ÄÄ Pn}

s=1|vs|^2ä

d

2 + 1^ä. This proves f is a polynomial ofv₁, . . . , vn. 5. Proof of Theorem 1.4

Proof. We only consider the case for n ≥ 2. Otherwise, the result is known. Pick p∈M. Let

(5.1) α= lim

r→∞

vol(B(p, r)) r²ⁿ >0.

Consider the blow-down sequence (Mi, pi, gi) = ^ÄM, p,_s¹2 i

g^ä for si → ∞. By the Cheeger-Colding theory [2], a subsequence converges to a metric cone (X, p∞, d∞). Define

(5.2) r(x) =d_∞(x, p_∞), x∈X, r_i(x) =d_gi(x, p_i), x∈M_i. Now pick two regular points y₀, z₀∈X with

(5.3) r(y₀) =r(z₀) = 1, d_∞(y₀, z₀)≥c(n, α)>0.

Note that the latter inequality is guaranteed by Theorem 2.1. There exists δ₀ >0 satisfying

(5.4) B(y0,2δ0)∩B(z0,2δ0) =∅, δ0 < 1 10 and

d_GH Å

B Å

y₀,1 εδ₀

ã , BR²ⁿ

Å 0,1

εδ₀ ãã

≤ 1 2εδ₀, dGH

Å B

Å z0,1

εδ0

ã , BR²ⁿ

Å 0,1

εδ0

ãã

≤ 1 2εδ0. (5.5)

Here ε = ¹₂ε(n), where ε(n) is given by Proposition 3.1. Therefore, if i is sufficiently large, we can find points yi, zi∈Mi withri(yi) =ri(zi) = 1 and

(5.6)

dGH

Å B

Å yi,1

εδ₀ ã

, BR²ⁿ

Å 0,1

εδ₀ ãã

≤εδ₀, d_GH

Å B

Å z_i,1

εδ₀ ã

, BR²ⁿ

Å 0,1

εδ₀ ãã

≤εδ₀.

Let w_sⁱ and v_sⁱ be the local holomorphic charts around yi and zi constructed in Proposition 3.1. Note that they have uniform size (independent of i). By

changing the value of δ₀, we may assume w_sⁱ, v_sⁱ are holomorphic charts in B(yi, δ0) andB(zi, δ0). Moreover,

|dw_sⁱ|,|dvⁱ_s| ≤C(n), wⁱ_s(yi) = 0, v_sⁱ(zi) = 0, (5.7)

Xn s=1

|wⁱ_s(y)|²−dgi(y, yi)²

≤Φ(ε|n)δ₀², (5.8)

Xn s=1

|v_sⁱ(z)|²−dgi(z, zi)²

≤Φ(ε|n)δ₀² (5.9)

for y ∈ B(yi, δ₀), z ∈ B(zi, δ₀). We need to construct a weight function on B(pi, R) for some large R to be determined later. The construction is similar toProposition 3.1. Set

(5.10) Ai =B(pi,5R)\B Å

pi, 1 5R

ã .

By the Cheeger-Colding theory [2, (4.43) and (4.82)], there exists a smooth functionρi on Mi so that

Z

Ai

|∇ρi− ∇1

2r²_i|²+|∇²ρi−gi|² <Φ Å1

i|R ã

, (5.11)

|ρi−r_i² 2 |<Φ

Å1 i|R

ã (5.12)

inA_i. According to (4.20)–(4.23) in [2], (5.13) ρi = 1

2(Gi)

2

2−2n, ∆Gi(x) = 0, x∈B(pi,10R)\B Å

pi, 1 10R

ã , and

(5.14) Gi =r_i²⁻²ⁿ

on ∂(B(pi,10R)\B(pi,_10R¹ )).Now

(5.15) |∇ρi(y)|=C(n)|Gi|¹⁻ⁿⁿ|∇Gi(y)|. By (5.12)–(5.14) and the Cheng-Yau’s gradient estimate, (5.16) |∇ρi(y)| ≤C(n)ri(y)

fory∈A_iand sufficiently largei. Now consider a smooth functionϕ: R⁺→R⁺ given by ϕ(t) = t fort≥2, ϕ(t) = 0 for 0≤t≤1, and |ϕ|,|ϕ⁰|,|ϕ⁰⁰| ≤ C(n).

Let

(5.17) u_i(x) = 1

R²ϕ(R²ρ_i(x)).

We set ui(x) = 0 for x∈B(pi,_5R¹ ). Then ui is smooth inB(pi,4R).

Claim 5.1. For sufficiently largei,^R_B(p

i,4R)|∇ui−∇¹₂r²_i|²+|∇²ui−gi|²<

Φ(_R¹), ui−^r₂²ⁱ<Φ^Ä_R^1ä, and|∇ui| ≤C(n)ri in B(pi,4R).

Proof. We have

∇u_i(x) =ϕ⁰(R²ρ_i(x))∇ρ_i(x), (5.18)

∇²ui(x) =R²ϕ⁰⁰(R²ρi(x))∇ρi⊗ ∇ρi+ϕ⁰(R²ρi(x))∇²ρi. (5.19)

The proof follows from a routine calculation, by (5.12), (5.13), and (5.16).

As in Proposition 3.1, consider a smooth function ϕ: R⁺ → R⁺ with ϕ(t) =t for 0≤t≤1, ϕ(t) = 0 for t≥2, and|ϕ|,|ϕ⁰|,|ϕ⁰⁰| ≤C(n). Set (5.20) vi(z) = 3R²ϕ

Çu_i(z) 3R²

å

, vi,t(z) = Z

M

Hi(z, y, t)vi(y)dy.

Here Hi(x, y, t) is the heat kernel on Mi. Then vi is supported in B(pi,4R).

By similar arguments as in Claim 3.1, we arrive at the following:

Proposition 5.1. v_i,1(z) satisfies that (v₁)_αβ(z) ≥ c(n, α)g_αβ > 0 for z∈B(pi,₁₀^R). Here α >0 is given by (5.1).

Now define (5.21)

q_i(x) = 4n Ç

log Ç _n

X

s=1

|w_sⁱ|² å

λ Ç

4 Pn s=1|w_sⁱ|²

δ₀² å

+ log Ç _n

X

s=1

|v_sⁱ|² å

λ Ç

4 Pn s=1|v_sⁱ|²

δ²₀ åå

. Hereλis a standard cut-off functionR⁺→R⁺withλ(t) = 1 for 0≤t≤1 and λ(t) = 0 for t≥2. Note that by (5.8) and (5.9),qi(x) has compact support in B(yi, δ₀)∪B(zi, δ₀)⊂B(pi,2).

Lemma 5.1. √

−1∂∂q_i ≥ −C(n, δ₀)ω_i. Moreover, e^−qi(x) is not locally integrable at y_i and z_i.

Proof. We have

(5.22) |√

−1∂∂|w_sⁱ|²|=|∂wⁱ_s∧∂w_sⁱ| ≤ |dwⁱ_s|² ≤C(n) inB(yi, δ₀). When λ⁰

Ç 4

Pn s=1

|wⁱ_s|² δ²₀

å

6

= 0,

(5.23) δ²₀ ≥

Xn s=1

|w_sⁱ|²≥ 1 4δ²₀. Also note that

(5.24) √

−1∂∂log Xn s=1

|wⁱ_s|²

!

≥0

in the current sense. Then the proof of the first part follows from routine calculation.

For the second part, whenx∈B(yi,₁₀^δ0), e^−qi(x) = 1

Å _n P

s=1|wⁱ_s|² ã_4n.

As wⁱ_s(yi) = 0 for all s, a simple calculation shows e^−qi(x) is not locally inte- grable atyi. The same argument works for zi. PuttingProposition 5.1 and Lemma 5.1 together, we findC(n, α, δ0)>0 so that

(5.25) √

−1∂∂(qi(x) +C(n, α, δ₀)v_i,1(x))≥ωi

inB^Äpi,₁₅^Rä. Set

(5.26) ψi(x) =qi(x) +C(n, α, δ₀)v_i,1(x).

By the same argument as inProposition 3.1, we find ε0 =ε0(α, n)>0 so that for sufficiently large R,

(5.27) min

y∈∂B(pi,₂₀^R)

v_i,1(y)>4 sup

y∈B

Ä

pi,ε0R 20

äv_i,1(y).

Of course, we can assume

(5.28) ε0R

20 >4.

From now on, we freeze the value of R. That is,

(5.29) R=R(n, α)>0

satisfies the all the conditions above. Let Ωi be the connected component of



y∈B Å

pi, R 20

ãv_i,1(y)<2 sup

y∈B(^pi,ε0R 20)

v_i,1(y)



 containingB^Äpi, ε0R

20

ä. Then Ωi is relatively compact inB^Äpi,₂₀^Räand Ωi is a Stein manifold, by Proposition 5.1. Also B(pi,3)⊂Ωi.

Now consider a function fi(x) = 1 for x ∈ B^Äyi,^δ₄^0ä; fi has compact support in B(yi, δ₀) ⊂B(pi,2) and|∇fi| ≤C(n, α, δ₀). Then fi(zi) = 0. We solve the equation∂hi =∂fi in Ωi with

(5.30)

Z

Ωi

|hi|²e^−ψi ≤ Z

Ωi

|∂fi|²e^−ψi ≤C(n, α, δ₀).

ByLemma 5.1, we have thate^−qi is not locally integrable atyi andzi,hi(yi) = hi(zi) = 0. Define the holomorphic function µi = fi−hi. Recall that by the construction, fi(yi) = 1, fi(zi) = 0. Then

(5.31) µi(yi) =fi(yi)−hi(yi) = 1, µi(zi) =fi(zi)−hi(zi) = 0.

Therefore, µi is not constant on Ωi. It is easy to see thatψi(x)≤C(n, α, δ₀) inB(pi,3). Then

(5.32) 1

C(n, α, δ₀) Z

B(pi,3)|hi|² ≤ Z

Ωi

|hi|²e^−ψi ≤C(n, α, δ₀).

Thus (5.33)

Z

B(pi,3)|µi|² ≤2 Z

B(pi,3)(|hi|²+|fi|²)≤C(n, α, δ0).

Mean value inequality implies that

(5.34) |µi(x)| ≤C(n, α, δ₀) forx∈B(pi,2). Therefore, the holomorphic function (5.35) ν_i^∗(x) =µi(x)−µi(pi) is uniformly bounded in B(pi,2). Set

(5.36) M_i⁰(r) = sup

x∈B(pi,r)|ν_i^∗(x)|. Then

(5.37) M_i⁰(2)≤C(n, α, δ₀).

On the other hand, by (5.31), we find

(5.38) M_i⁰(1)≥ 1

2. Therefore,

(5.39) M_i⁰(2)

M_i⁰(1) ≤C(n, α, δ₀).

Now we are ready to apply the three circle theorem. More precisely, we consider the rescale functions ν^∗_i = βiν_i^∗ in B(p,2si) ⊂ M. Here βi are constants so that

(5.40)

Z

B(p,2)|ν^∗_i|² = 1.

This implies

(5.41) |ν^∗_i| ≤C(n, α)

in B(p,1). Set Mi(r) = sup

x∈B(p,r)|ν^∗_i|. The three circle theorem says ^M_M^i(2r)

i(r) is monotonic increasing for 0 < r ≤ si. By (5.39) and similar arguments as in (4.9), we obtain that

(5.42) Mi(r)≤C(n, α, δ0)^Är^C(n,α,δ0)+ 1^ä

for all iand si ≥r. Let i→ ∞. A subsequence ofν^∗_i converges uniformly on each compact set to a holomorphic functionv of polynomial growth. v cannot