Complete open Kahler manifolds with nonnegative bisectional curvature and non-maximal volume growth

TL;DR

This paper investigates conditions under which complete open Kähler manifolds with nonnegative bisectional curvature are Stein, focusing on volume growth restrictions to partially resolve an open problem.

Contribution

It introduces a weaker volume growth condition that extends previous results, advancing understanding of the structure of such Kähler manifolds.

Findings

Established partial criteria for Steinness under volume growth restrictions

Improved previous observations on volume growth conditions

Provided new insights into the geometry of nonnegative bisectional curvature manifolds

Abstract

It is still an open problem that a complete open Kahler manifold with positive bisectional curvature is Stein. This paper partially resolve the problem by putting a restriction to volume growth condition. The partial solution here improves the observation in ([8], page 341). The improvement is based on assuming a weaker volume growth condition that is not sufficiently maximal.

Equations

$$b_p(x)=\underset{r\to \infty }{\lim }\{r-d(x,\partial (B(p,r))\},$$

$$C(N)=\{q\in M|\text{there is a minimizing geodesic γ from p to q such that}{\gamma }^{\prime }(0)\in N\}$$

$$B_N(p,r)=B(p,r)\cap C(N)$$

$$B_{\Sigma }(p,r)=B(p,r)\cap C(\Sigma ).$$

$$\underset{r\to \infty }{\lim }{r}^{-n}Vol(B(p,r))={\alpha }_M,{\alpha }_M>0$$

$$\underset{r\to \infty }{\lim }{r}^{-n}Vol(B_{\Sigma }(p,r))={\alpha }_M$$

$$Vol(B_{\Sigma }(p,r))\sim Vol(B(p,r)),$$

$$\sim meansasymptotic$$

$$\underset{r\to \infty }{\lim }\text{inf}\frac{Vol(B_{\Sigma }(p,r))}{Vol(B(p_1,r))}$$

$$\underset{r\to \infty }{\lim }\text{inf}\frac{Vol(B_{\Sigma }(p,r))}{Vol(B(p_1,r))}\le \underset{r\to \infty }{\lim }\text{inf}\frac{Vol(B_{\Sigma }(q,r)}{Vol(B(p_1,r))}$$

$$\underset{r\to \infty }{\lim }\text{inf}[Vol(B_{\Sigma }(p,r))][Vol(B(p_1,r)){]}^{-1}=\alpha (n)$$

$$Vol(B_{\Sigma }(p,r))\sim Vol(B(p,r)),$$

$$\underset{r\to \infty }{\lim }[Vol(B_{\Sigma }(p,r))][Vol(B(p,r)){]}^{-1}=a$$

$$\underset{r\to \infty }{\lim }f(r)[g(r){]}^{-1}=a,a>0$$

$$\underset{r\to \infty }{\lim }\text{inf}[Vol(B_{\Sigma }(p,r))][Vol(B(p,r)){]}^{-1}=a,$$

$$Vol(B_{\Sigma }(p,r)){\sim }_{a}Vol(B(p,r)),$$

$$e_{p,q}=d(p,x)+d(x,q)-d(p,q).$$

$$e_{p,q}(x)\le 2h(x)$$

$$r_p(x)-\underset{t\to \infty }{\lim }\{t-d(x,\gamma (t))\}\le 2h_{\gamma }(x),$$

$$r_p(x)-b_p(x)\le r_p(x)-\underset{t\to \infty }{\lim }\{t-d(x,\gamma (t))\},$$

$$r_p(x)-b_p(x)\le 2h_{\gamma }(x)$$

$$r_p(x)-b_p(x)\le 2h_p(x)$$

$$C(\Sigma )\cap C({\Sigma }^c)=\varnothing .$$

$$B_{\Sigma }(p,r)\cap B_{{\Sigma }^c}(p,r)=\varnothing$$

$$\frac{Vol(B_{{\Sigma }^c}(p,r_m-h_m))}{Vol(B_{{\Sigma }^c}(p,r_m+h_m))}\ge {\left[\frac{r_m-h_m}{r_m+h_m}\right]}^n$$

$$B(x_m,h_m)\subset B_{{\Sigma }^c}(p,r_m+h_m)\backslash B_{{\Sigma }^c}(p,r_m-h_m)$$

$$Vol(B(x_m,h_m)$$

$$r_p(x)-b_p(x)\le 2h_p(x)$$

$$r_p(x)+h(x)\le 3h(x)+a,\text{when x∈bp−1(a)}$$

$$\underset{m\to \infty }{\lim }\text{sup}\frac{r_1(x_m)}{h_p(x_m)}$$

Full text

1. Introduction

One of the most useful tool in studying structures of nonnegatively curved complete open manifolds is the Busemann function. The spherical Busemann function is defined as

[TABLE]

In the process of proving soul theorem in [3], Cheeger and Gromoll proved that a complete open Riemaniann manifold with nonnegative sectional curvature admits a convex and exhaustion Busemann function, $b_{p}$.

It still remains unknown whether complete open manifolds with nonnegative Ricci curvature admit exhaustion Busemann function, except with the restriction to maximum volume growth as was proved by Shen in [7]. In this paper we drop the maximum volume growth condition and adapt a weaker condition.

Let $S_{p}M\subset T_{p}M$ be a unit tangent sphere in the tangent space $T_{p}M$ for a point $p\in M$. For any subset $N\subset S_{p}M$ , define

[TABLE]

to be the cone over $N$. The restriction of a geodesic ball of radius $r$ centered at $p$ to C(N) is denoted by

[TABLE]

.

Let $\Sigma=\{v\subset S_{p}M\ |\ \text{exp}_{p}(rv):[0,\infty)\to M\text{ is a ray}\}$. A cone of rays is defined by $C(\Sigma)$. Consequently,

[TABLE]

From lemma 4 in [5] we have

Lemma 1.1** ([5, Ordway-Stephens-Yang]).**

Let $M^{n}$ be a complete open manifold with $\text{Ric}_{M}\geq 0$. Suppose that $M$ has a maximum volume growth i.e

[TABLE]

then

[TABLE]

By limit properties, we obtain the following corollary

Corollary 1.2**.**

Let $M^{n}$ be a complete open manifold with $\text{Ric}_{M}\geq 0$. Suppose that $M$ has a maximum volume growth. Then

[TABLE]

[TABLE]

It is essential to note that nonnegative Ricci curvature ensures that the volume growth condition in corollary 1.2 above is independent of the base point: let $p,q\in M^{n}$ and $d=d(p,q)$. Then it is clear that $B(p,r)\subset B(q,r+d)$ and $B(q,r)\subset B(p,r+d)$. By Bishop-Gromov volume comparison theorem,

[TABLE]

Likewise

[TABLE]

Lemma 1.3**.**

Let $M^{n}$ be a complete open manifold with $Ric_{M}\geq 0$. For a fixed $p_{1}\in M$, the volume growth

[TABLE]

is independent of the base point $p\in M$.

The converse to corollary 1.2 above is not true. In other words, the volume growth condition

[TABLE]

does not necessarily imply maximum volume growth.

For example, the vertex 0 of a paraboloid $M\subset\mathbb{R}^{n+1}$ has an empty cut locus. Thus volume growth condition (1.1) holds at 0 and extends to other points by lemma 1.3 above. On the other hand, as a special case of lemma 4.1 in [6], the paraboloid $M$ in $\mathbb{R}^{n+1}$ defined by $M=\{(x_{1},x_{2},\cdot\cdot\cdot,x_{n},z):z=x_{1}^{2}+x_{2}^{2}+\cdot\cdot\cdot+x_{n}^{2}\}$ has a volume growth of at most $r^{\frac{n+1}{2}}$ which is not maximal. Furthermore, we can creat a non-empty cut locus of the point [math] at the same time maintaining positive curvature and manifesting volume growth conditions like that of (1.1).

Example 1.4**.**

Consider $\mathcal{M}=\{(x_{1},x_{2},\cdot\cdot\cdot,x_{n},z):z=x_{1}^{2}+x_{2}^{2}+\cdot\cdot\cdot+x_{n}^{2}\}\subset\mathbb{R}^{n+1}$. $(\mathcal{M},ds_{\mathcal{M}}^{2})$ is a complete open manifold with positive Ricci curvature ($n>2$). Here, $ds_{M}^{2}$ is an induced Euclidean metric. For $0\neq q\in\mathcal{M}$, let $D_{l}(q)$ be a geodesic ball of radius $l$ centered at $q$. Consider a smooth function $f:\overline{D_{l}(q)}\to\mathbb{R}$. For a small neiborhood $U$ of $\overline{D_{l}(q)}$, there exists a smooth function $h:\mathcal{M}\to\mathbb{R}$ such that $h|_{\overline{D_{l}(q)}}=f$ and $\text{supp}\ h\subset U$. For $\varepsilon>0$, denote $\mathcal{M_{\varepsilon}}=(\mathcal{M},ds_{\mathcal{M}}^{2}+\varepsilon hds_{\mathcal{M}}^{2})$. We can choose $\varepsilon$ small enough such that the Ricci curvature remains positive throughout $\mathcal{M}$ and an extension $\gamma:[0,\infty)\to\mathcal{M_{\varepsilon}}$ of a minimizing geodesic from [math] to $q$ leaves $D_{l}(q)$ and intersect a ray at a point. It follows that the cut locus of the point [math] is no longer empty. Since only rays intersecting and neighboring $D_{l}(q)$ are affected in this new manifold, for a fixed $a\approx 1$, $a\leq 1$, we can choose $l>0$ and $\varepsilon>0$ small enough such that

[TABLE]

.

Given two real valued functions $f,g:\mathbb{R}\to\mathbb{R}$. Denote the limit

[TABLE]

if it exists by $f\sim_{a}g$.

Now we extend the result by Shen in [7] by replacing the maximum volume growth condition with a weaker volume growth condition.

Lemma 1.5**.**

Let $M$ be a complete open manifold with $\text{Ric}_{M}\geq 0$. Let $[9^{n}-1]9^{-n}<a\leq 1$ where $n=dim_{\mathbb{R}}M$. If

[TABLE]

then for any $t\in\mathbb{R}$, $b_{p}^{-1}(t)$ is compact .

The following theorem is the main result in this paper:

Theorem 1.6**.**

Let $M$ be a complete open K$\ddot{a}$hler manifold with nonnegative bisectional curvature. Then $M$ is a Stein manifold if the followings holds

• (a)

The bisectional curvature is positive outside a compact set

• (b)

[TABLE]

where $[9^{2n}-1]9^{-2n}<a\leq 1$ and $n=dim_{\mathbb{C}}M$

2. Proofs

We will prove Lemma 1.5 first then Theorem 1.6.

Proof of Lemma 1.5.

Proving by contradiction, we assume that $b^{-1}(t)$ is non-compact and then show that the assumed volume growth condition doesn’t hold.

Define the excess function for two points $p,q$ as

[TABLE]

By the triangle inequality, we have that

[TABLE]

Denote $r_{p}(x)=d(p,x)$. Assume that the minimizing geodesic between $p$ and $q$ is part of a ray emanating from $p$. Now, taking the limit of inequality (2.1) as $q$ goes to infinity, we end up with the following inequality

[TABLE]

where $h_{\gamma}(x)$ is a distance from $x$ to a ray $\gamma$ emenating from $p$. Since

[TABLE]

for each ray $\gamma$ emanating from $p$, inequality (2.2) implies that

[TABLE]

Let $h_{p}(x)=d(x,Rp)$, where $R_{p}$ is a union of rays emanating from $p$. Since inequality (2.3) holds for any ray $\gamma$, we have that

[TABLE]

Next, note that

[TABLE]

Therefore, for any $r>0$ and $p\in M$, we have that

[TABLE]

Observe that $B(x,h_{p}(x))\subset C(\Sigma^{c})$. It follows that $B(x,h_{p}(x))\subset B_{\Sigma^{c}}(p,r_{p}(x)+h_{p}(x))$.

Since $b_{p}$ is exhaustion whenever $h_{p}$ is bounded, we assume that $h_{p}$ is unbounded. Due to noncompactness of $b_{p}^{-1}(t)$, we can construct a diverging sequence $\{x_{m}\}\subset b_{p}^{-1}(t)$. Consequently, $\{h_{p}(x_{m})\}$ is a divergence sequence.

Denote $h_{m}=h_{p}(x_{m})$ and $r_{m}=r(x_{m})$. By Bishop-Gromov volume comparison theorem,

[TABLE]

It is easy to verify that

[TABLE]

and that

[TABLE]

Inequality 2.6 is due to the fact that $h\leq r$ and

[TABLE]

In particular

[TABLE]

Now, denote $r_{1}(x)=d(x_{1},x)$. By triangle inequality and (2.4),

[TABLE]

Also note that

[TABLE]

By volume comparison theorem we obtain

[TABLE]

Denote f_{p}(r)=Vol(B(p,r))\text{for a fixed p\in M}. From (2.7), (2.8), and (2.9), we have

[TABLE]

The last inequality is due to the fact that the volume growth

[TABLE]

is independent of the base point $x_{1}$.

From inequalities (2.6), (2.10), and the volume comparison theorem, we have

[TABLE]

Which leads to the inequality

[TABLE]

Since

[TABLE]

the volume growth condition assumption implies that

[TABLE]

Evidently, inequality (2.12) contradicts inequality (2.13). Hence $b_{p}^{-1}(t)$ must be compact.

∎

Proof of Theorem 1.6.

The $Ricci$ curvature is nonnegative and positive outside a compact set because the bisectional curvature is assumed. The Busemann function $b_{p}$ is a continuous plurisubharmonic exhaustion by lemma 1.5 and a result by H.Wu in [9]. In the same paper (Theorem C [9]), it follows that there exist a strictly plurisubharmonic exhaustion function. This completes the proof.

∎

3. Applications

Let $H_{k}(M,\mathbb{Z})$ denote the k-th singular homology group of $M$ with integer coefficients. It is well known that if $M$ is a complete proper Riemannian n-dimensional manifold with $Ric_{M}\geq 0$, then using Morse theorem, $M$ has the homotopy type of a CW complex with cells each of dimension $\leq n-2$ and $H_{i}(M,\mathbb{Z})=0$, $i\geq n-1$. ([6], [4])

As an application of lemma 1.5, we have the following result.

Corollary 3.1**.**

Let $(M,g)$ be a complete open manifold with $Ric_{M}\geq 0$ . If

[TABLE]

where $[9^{n}-1]9^{-n}<a\leq 1$, then $M$ has the homotopy type of a $CW$ complex with cells each of dimension $\leq n-2$. In particular, $H_{i}(M,\mathbb{Z})=0$, $i\geq n-1$

It is also known that if $M$ is a Stein manifold of n-complex dimension, then the homology groups $H_{k}(M,\mathbb{Z})$ are zero if $k>n$ and $H_{n}(M,\mathbb{Z})$ is torsion free (theorem 1 [1]), [2]. As an application of theorem 1.6, we have the following result.

Corollary 3.2**.**

Let $M$ be a complete open K$\ddot{a}$hler manifold with nonnegative bisectional curvature. If the followings holds

• (a)

The bisectional curvature is positive outside a compact set

• (b)

[TABLE]

where $[9^{2n}-1]9^{-2n}<a\leq 1$ and $n=dim_{\mathbb{C}}M$

then

[TABLE]

and $H_{n}(M,\mathbb{Z})$ is torsion free

Acknowledgement: Thanks to Professor Bun Wong for his advice.