K$\ddot{\operatorname{a}}$hlerity of invariant metrics on pseudoconvex domain of dimension two

TL;DR

This paper establishes uniformization theorems for two-dimensional pseudoconvex domains of finite type using Kähler metrics, characterizing when such domains are biholomorphic to the unit ball and analyzing curvature properties.

Contribution

It introduces new uniformization results for pseudoconvex domains of finite type in two dimensions using Kähler metrics with quasi-finite geometry, and characterizes the unit ball among Reinhardt domains.

Findings

Pseudoconvex Reinhardt domain of finite type is the unit ball iff Bergman metric is a scalar multiple of Kobayashi or Carathéodory metric.

Established a rigidity theorem for the holomorphic sectional curvature of the Bergman metric.

Proved uniformization theorems via Kähler-Kobayashi and Kähler-Carathéodory metrics with quasi-finite geometry.

Abstract

For a two dimensional bounded pseudoconvex domain of finite type, we prove uniformization theorems via K$\ddot{\operatorname{a}}$hler-Kobayashi metric or K$\ddot{\operatorname{a}}$hler-Carath$\acute{\operatorname{e}}$odory metric with quasi-finite geometry of order three. In particular, a pseudoconvex Reinhardt domain of finite type is the unit ball if and only if the Bergman metric is a scalar multiple of the Kobayashi metric or Carath$\acute{\operatorname{e}}$odory metric. Moreover, we establish a rigidity theorem concerning holomorphic sectional curvature of Bergman metric and Lu constant.

Equations

$$d_F(x,y):=\inf \{L_F(\gamma ):$$

$$\mathrm{curve}\mathrm{with}\gamma (0)=x\mathrm{and}\gamma (1)=y\}$$

$$K_{\Omega }(z,X):=\inf \left\{|\xi |:\exists f\in \mathrm{Hol}(\Delta ,\Omega ),\mathrm{with}f(0)=z,d(f{)}_0(\xi )=X\right\}$$

$$C_{\Omega }(z,X):=\sup \left\{|d{f}_z(X)|:f\in \mathrm{Hol}(\Omega ,\Delta ),f(z)=0\right\}$$

$${\kappa }_{\Omega }(z,z):=\sup \{|f(z){|}^2:f\in \mathrm{Hol}(\Omega ,\mathbb{C}),||f|{|}_{L^2(\Omega )}\le 1\}$$

$$B_{\Omega }(z,X):=\frac{\sup \{|X(f)(z)|:f\in \mathrm{Hol}(\Omega ,\mathbb{C}),f(z)=0,||f|{|}_{L^2(\Omega )}\le 1\}}{{\kappa }_{\Omega }^{1/2}(z,z)}.$$

$$L(\Omega ):=\underset{z\in \Omega ,0\ne v\in T_z(\Omega )}{\sup }\frac{C_{\Omega }(z,v)}{B_{\Omega }(z,v)}.$$

$$H(g)(X):=\frac{R(X,JX,X,JX)}{g(X,X{)}^2}.$$

$$K(d{s}^2):=-\frac{1}{2g}\Delta \log g.$$

$$H(F)(p,v):=\sup \left\{K({\phi }^{\ast }G(0))\right\}.$$

$$\frac{1}{C}{g}_0\le {\phi }^{\ast }g\le C{g}_0,$$

$$\underset{x\in U}{\sup }\left|\frac{{\partial }^{|\mu |+|\nu |}({\phi }^{\ast }g{)}_{i\bar{k}}}{\partial z^{\mu }\bar{\partial }{z}^{\nu }}(x)\right|\le A_{q}for\; all|\mu |+|\nu |\le q\mathrm{and}1\le i,k\le n,$$

$$\rho (z,w)=\mathrm{Re}w+H_m(z)+o\left(|z{|}^{m+1}+|z||w|\right)$$

$${\psi }_n^{-1}(z,w):=\left(p_n^{(1)}+z,p_n^{(2)}+{ϵ}_n+d_{n,0}w+\sum _{k=2}^m{d}_{n,k}{z}^k\right)$$

$$\rho \circ {\psi }_n^{-1}(z,w)=\mathrm{Re}w+P_n(z)+o\left(|z{|}^{m+1}+|z||w|\right).$$

$$||P_n({\tau }_n\cdot )||={ϵ}_n,$$

$${\delta }_n(z,w):=\left(\frac{z}{{\tau }_n},\frac{w}{{ϵ}_n}\right)$$

$$q_n:={\alpha }_n(p_n)=\left(0,-d_{n,0}^{-1}\right)\to (0,-1)=q_{\infty },$$

$$\underset{n\to \infty }{\lim }{K}_{{\Omega }_n}(z,v)=K_{{\Omega }_{\infty }}(z,v)$$

$$\underset{n\to \infty }{\mathrm{limsup}}{C}_{{\Omega }_n}(z,v)\le C_{{\Omega }_{\infty }}(z,v).$$

$$\underset{n\to \infty }{\lim }{C}_{{\Omega }_n}(z,v)=C_{{\Omega }_{\infty }}(z,v)$$

$$\left|C_{{\Omega }_j}(z_j,v_j)-C_{{\Omega }_{\infty }}(z_j,v_j)\right|>ϵ$$

$$\left|C_{{\Omega }_j}(z_j,v_j)-C_{{\Omega }_{\infty }}(z_0,v_0)\right|>ϵ/2.$$

$$\left|\frac{C_{{\Omega }_j}(z_j,v_j)}{C_{{\Omega }_{\infty }}(z_0,v_0)}-1\right|>ϵ/c.$$

$$\underset{n\to \infty }{\mathrm{limsup}}\frac{C_{{\Omega }_n}(z,v)}{K_{{\Omega }_n}(z,v)}\le \frac{C_{{\Omega }_{\infty }}(z,v)}{K_{{\Omega }_{\infty }}(z,v)}\le 1$$

$$1-{ϵ}_1<\frac{C_{\Omega }(z,v)}{K_{\Omega }(z,v)}<1+{ϵ}_1$$

$$\frac{C_{{\Omega }_n}(z,v)}{K_{{\Omega }_n}(z,v)}=\frac{C_{\Omega }({\alpha }_n^{-1}(z),(d{\alpha }_n^{-1})v)}{K_{\Omega }({\alpha }_n^{-1}(z),(d{\alpha }_n^{-1})v)}>1-{ϵ}_1$$

$$\underset{j\to \infty }{\lim }\frac{C_{{\Omega }_j}(z_j,v_j)}{C_{{\Omega }_{\infty }}(z_0,v_0)}=\underset{j\to \infty }{\lim }\frac{C_{{\Omega }_j}(z_j,v_j)}{K_{{\Omega }_{\infty }}(z_0,v_0)}=\underset{j\to \infty }{\lim }\frac{C_{{\Omega }_j}(z_j,v_j)}{K_{{\Omega }_j}(z_j,v_j)}\frac{K_{{\Omega }_j}(z_j,v_j)}{K_{{\Omega }_{\infty }}(z_0,v_0)}=1,$$

$$\frac{1}{A}{K}_{{\Omega }_n}(z,v)\le \sqrt{g_n(z)(v,v)}\le A{K}_{{\Omega }_n}(z,v)$$

$$\underset{n\ge J}{\sup }\underset{x\in K}{\max }\left|\frac{{\partial }^{|\mu |+|\nu |}(g_n{)}_{i\bar{k}}}{\partial z^{\mu }\bar{\partial }{z}^{\nu }}(x)\right|\le C(K,\mu ,\nu ).$$

$$\left|\frac{{\partial }^{|\mu |+|\nu |}(g_{n_l}{)}_{i\bar{k}}}{\partial z^{\mu }\bar{\partial }{z}^{\nu }}(x_l)\right|\ge l.$$

Full text

1. Introduction

Let $\Omega$ be a bounded pseudoconvex domain in $\mathbb{C}^{n}$, there are a number of important invariant metrics on it: the Kobayashi metric, the Carath$\acute{\operatorname{e}}$odory metric, the Bergman metric and the K$\ddot{\operatorname{a}}$hler-Einstein metric. All of these metrics coincide, up to a constant, on the unit ball. The following question arises naturally.

Question. For which domain, the previous metrics are not coincide?

In 1979, Cheng conjectured that if the Bergman metric on a smoothly bounded strongly pseudoconvex domain is K$\ddot{\operatorname{a}}$hler-Einstein, then it is biholomorphically equivalent to the unit ball. Two dimensional case was proved by Fu-Wong[5] and Nemirovskiĭ-Shafikov[11]. And this conjecture was completely confirmed by Huang-Xiao. Recently, Savale-Xiao [13] proved the case of two dimensional pseudoconvex domain of finite type.

Theorem 1.1** ([7, Theorem 1.1]).**

Suppose $\Omega\subset\mathbb{C}^{n}$ is a smoothly bounded strongly pseudoconvex domain, then the Bergman metric is K$\ddot{a}$hler-Einstein if and only if it is biholomorphically equivalent to the unit ball.

As for invariant Finsler metrics, such as Kobayashi metric and Carath$\acute{\operatorname{e}}$odory metric, Gaussier-Zimmer firstly proved the following theorem.

Theorem 1.2** ([6, Theorem 1.5]).**

Let $\Omega\subset{\mathbb{C}^{n}}$ be a bounded strongly pseudoconvex domain with $C^{2}$ smooth boundary. Then the following are equivalent:

(1) $\Omega$ is biholomorphically equivalent to a ball quotient.

(2) the Kobayashi metric is a K$\ddot{a}$hler metric,

(3) the Kobayashi metric is a K$\ddot{a}$hler metric with constant holomorphic sectional curvature.

Furthermore, Theorem 1.2 means that for a bounded strongly pseudoconvex domain with $C^{2}$ smooth boundary, if the Bergman metric is a scalar multiple of the Kobayashi metric, then it is biholomorphic to the unit ball. After Theorem 1.2, Dong-Wang-Wong[3] proved a similar version of Carath$\acute{\operatorname{e}}$odory metric. Moreover, they also provided a rigidity theorem about holomorphic sectional curvature of Bergman metric with respect to Lu constant.

For a bounded domain $\Omega\subset\mathbb{C}^{n}$, we say that Kobayashi metric asymptotically equals Carath$\acute{\operatorname{e}}$odory metric at infinity, if $\lim\limits_{z\rightarrow\partial\Omega}\frac{C_{\Omega}(z,v)}{K_{\Omega}(z,v)}=1$ holds uniformly for $0\neq v\in\mathbb{C}^{n}$.

Suppose $\Omega\subset\mathbb{C}^{n}$ is a strongly pseudoconvex domain with $C^{2}$ smooth boundary, then Kobayashi metric asymptotically equals Carath$\acute{\operatorname{e}}$odory metric at infinity. Moreover, if $\Omega\subset\mathbb{C}^{2}$ is a smoothly bounded pseudoconvex domain of finite type, which is biholomorphic to a ball quotient, then [9, Proposition 1.3] implies that Kobayashi metric asymptotically equals Carath$\acute{\operatorname{e}}$odory metric at infinity. Since the covering map is a local isometry from the unit ball to $\Omega$ with respect to Kobayashi metric and K$\ddot{\operatorname{a}}$hler-Einstein metric, then Kobayashi metric on $\Omega$ is a K$\ddot{\operatorname{a}}$hler metric with quasi-finite geometry of order infinity (see section 2). In this paper, we firstly prove the converse result as follows.

Theorem 1.3**.**

Let $\Omega$ be a smoothly bounded pseudoconvex domain of finite type in $\mathbb{C}^{2}$. Then the following are equivalent:

(1) $\Omega$ is biholomorphically equivalent to a ball quotient,

(2) the Kobayashi metric is a K$\ddot{a}$hler metric with quasi-finite geometry of order three and Kobayashi metric asymptotically equals Carath$\acute{\operatorname{e}}$odory metric at infinity,

(3) the Kobayashi metric is a K$\ddot{a}$hler metric with constant holomorphic sectional curvature.

Since [9, Proposition 3.11], then the Bergman metric on $\Omega$ has quasi-finite geometry of order infinity, where $\Omega\subset\mathbb{C}^{2}$ is a smoothly bounded pseudoconvex domain of finite type. As a corollary, the following holds.

Corollary 1.4**.**

Suppose $\Omega\subset\mathbb{C}^{2}$ is a smoothly bounded domain of finite type, then the following are equivalent:

(1) $\Omega$ is biholomorphic to the unit ball,

(2) the Bergman metric is a scalar multiple of the Kobayashi metric and Kobayashi metric asymptotically equals Carath$\acute{e}$odory metric at infinity.

Moreover, the following uniformazition theorem with respect to K$\ddot{\operatorname{a}}$hler-Einstien metric holds.

Corollary 1.5**.**

Suppose $\Omega\subset\mathbb{C}^{2}$ is a smoothly bounded domain of finite type, then the following are equivalent:

(1) $\Omega$ is biholomorphic to a ball quotient,

(2) the K$\ddot{a}$hler-Einstein metric is a scalar multiple of the Kobayashi metric and Kobayashi metric asymptotically equals Carath$\acute{e}$odory metric at infinity.

As for Carath$\acute{\operatorname{e}}$odory metric, we obtain the following theorem by using a similar proof of Theorem 1.3.

Theorem 1.6**.**

Let $\Omega$ be a smoothly bounded pseudoconvex domain of finite type in $\mathbb{C}^{2}$. Then the following are equivalent:

(1) $\Omega$ is biholomorphically equivalent to the unit ball,

(2) the Carath$\acute{e}$odory metric is a K$\ddot{a}$hler metric with quasi-finite geometry of order three and Kobayashi metric asymptotically equals Carath$\acute{e}$odory metric at infinity,

(3) the Carath$\acute{e}$odory metric on $\Omega$ is a K$\ddot{a}$hler metric with constant holomorphic sectional curvature,

(4) the Bergman metric or K$\ddot{a}$hler-Einstein metric is a scalar multiple of the Carath$\acute{e}$odory metric and Kobayashi metric asymptotically equals Carath$\acute{e}$odory metric at infinity.

Suppose $\Omega\subset\mathbb{C}^{n}$ is a bounded domain, and $B_{\Omega}(z,v)$ is Bergman metric on $\Omega$. Then we have $B_{\Omega}(z,v)\geq C_{\Omega}(z,v)$ for each $(z,v)\in\Omega\times\mathbb{C}^{n}$, where $C_{\Omega}$ is the Carath$\acute{\operatorname{e}}$odory metric on $\Omega$. For the optimal control of these metrics, it is natural to consider the Lu constant $L(\Omega)$ (see section 2.2). Combining with the holomorphic sectional curvature of Bergman metric and Lu constant, we obtain the following result.

Theorem 1.7**.**

Let $\Omega$ be a smoothly bounded domain of finite type in $\mathbb{C}^{2}$, then the following are equivalent:

(1) $\Omega$ is biholomorphic to the unit ball,

(2) the holomorphic sectional curvature of $B_{\Omega}(z,v)$ is bounded above by $-4(L(\Omega))^{2}$ and Kobayashi metric asymptotically equals Carath$\acute{e}$odory metric at infinity.

Let $\Omega$ be a bounded domain in $\mathbb{C}^{n}$ and $z\in\partial\Omega$ be a boundary point, we say that $\Omega$ is locally convexifiable at $z$, if there exists a neighbourhood $U$ of $z$ and a biholomorphic mapping $\Phi$ from $U$ into $\mathbb{C}^{n}$ such that $\Phi(U\cap\Omega)$ is convex.

For a smoothly bounded pseudoconvex Reinhardt domain of finite type in $\mathbb{C}^{2}$, Theorem 3.2 in [4] implies that it is locally convexifiable at every boundary point. Combining with the proofs of [12, Theorem 1.2] and [8, Theorem 1], we know that for such domain, Kobayashi metric asymptotically equals Carath$\acute{\operatorname{e}}$odory metric at infinity. Thus we obtain the following result.

Theorem 1.8**.**

Let $\Omega$ be a smoothly bounded pseudoconvex Reinhardt domain of finite type in $\mathbb{C}^{2}$, then the following are equivalent:

(1) $\Omega$ is biholomorphically equivalent to a ball quotient,

(2) the Kobayashi metric is a K$\ddot{a}$hler metric with quasi-finite geometry of order three,

(3) the K$\ddot{a}$hler-Einstein metric is a scalar multiple of Kobayashi metric,

(4) the Kobayashi metric is a K$\ddot{a}$hler metric with constant holomorphic sectional curvature.

Moreover, the following are also equivalent:

(4) $\Omega$ is biholomorphically equivalent to the unit ball,

(5) the Carath$\acute{e}$odory metric is a K$\ddot{a}$hler metric with quasi-finite geometry of order three,

(6) the Carath$\acute{e}$odory metric is a K$\ddot{a}$hler metric with constant holomorphic sectional curvature,

(7) the Bergman metric or K$\ddot{a}$hler-Einstein metric is a scalar multiple of the Kobayashi metric or Carath$\acute{e}$odory metric,

(8) the holomorphic sectional curvature of Bergman metric is bounded above by $-4(L(\Omega))^{2}$.

This paper is orgnized as follows. We give the preliminaries in section 2. Section 3 is denoted to prove main theorems.

2. Preliminaries

2.1. Notations

(1) Let $|\cdot|$ be the standard Euclidean norm in $\mathbb{C}^{n}$.

(2) $\Delta$ will be denoted as the unit disc in $\mathbb{C}$, and $\Delta_{r}:=\left\{z\in\mathbb{C}:|z|<r\right\}$. The unit ball in $\mathbb{C}^{n}$ for $n\geq 2$ is denoted by $\mathbb{B}^{n}$, and $B_{r}(z_{0}):=\left\{z\in\mathbb{C}^{n}:|z-z_{0}|<r\right\}$.

(3) For any $n\geq 1$, we let $(\mathbb{C}^{n})^{*}:=\mathbb{C}^{n}\setminus\{0\}$.

(4) For $\Omega\varsubsetneq\mathbb{C}^{n}$, we denote $\delta_{\Omega}(x):=\inf\left\{|z-x|:z\in\partial\Omega\right\}$.

2.2. Invariant metric

Definition 2.1**.**

Suppose $\Omega$ is a domain in $\mathbb{C}^{n}$. For an upper semicontinuous function $F:\Omega\times\mathbb{C}^{n}\rightarrow[0,\infty)$, we say that $F$ is a $Finsler\ metric$ on $\Omega$, if the equality $F(z,tX)=|t|F(z,X)$ holds for any $z\in\Omega,t\in\mathbb{C},X\in\mathbb{C}^{n}$.

Let $F$ be a Finsler metric on $\Omega$, it can induce a distance

[TABLE]

for any $x,y\in\Omega$. Here $L_{F}(\gamma)$ is defined as $L_{F}(\gamma):=\int_{0}^{1}F(\gamma(t),\dot{\gamma}(t))dt$.

For a domain $\Omega\subset\mathbb{C}^{n}$, the Kobayashi metric on $\Omega$ is defined by

[TABLE]

for any $z\in\Omega$ and $0\neq X\in\mathbb{C}^{n}$. And the Carath$\acute{\operatorname{e}}$odory metric is denoted as

[TABLE]

for $z\in\Omega,0\neq X\in\mathbb{C}^{n}$.

Suppose

[TABLE]

is the Bergman kernel in the diagonal of $\Omega\times\Omega$. Then the Bergman metric on $\Omega$ is defined by

[TABLE]

Definition 2.2**.**

Let $\Omega$ be a bounded domain in $\mathbb{C}^{n}$, then the $Lu\ constant\ L(\Omega)$ is defined as

[TABLE]

Note that $L(\Omega)\leq 1$ by the definition of Lu constant. Moreover, $L(\Omega)=\frac{1}{\sqrt{n+1}}$ for the unit ball $\mathbb{B}^{n}$, and $L(\Omega)=\sqrt{2n}$ for the unit polydisk $\Delta^{n}$. When $\Omega$ is a bounded strongly pseudoconvex domain with $C^{2}$ smooth boundary in $\mathbb{C}^{n}$, since $\lim\limits_{z\rightarrow\partial\Omega}\frac{C_{\Omega}(z,v)}{B_{\Omega}(z,v)}=\frac{1}{\sqrt{n+1}}$ holds uniformly for $0\neq v\in\mathbb{C}^{n}$, then $L(\Omega)\geq\frac{1}{\sqrt{n+1}}$.

2.3. Holomorphic sectional curvature

Let $(M,J,g)$ be a K$\ddot{\operatorname{a}}$hler manifold, and $R(g)$ be the curvature tensor of $(M,g)$. For any $z\in M$ and non-zero tangent vector $X\in T_{z}M$, the holomorphic sectional curvature is defined by

[TABLE]

The above definition relies on the smoothness of metric $g$. For a Finsler metric, we have a similar definition. We firstly recall the Gaussian curvature for pseudo-metric on $\Delta$.

Definition 2.3**.**

Suppose $ds^{2}=gdz\otimes d\bar{z}$ is a pseudo-metric on $\Delta$, where $g$ is an upper semicontinuous function. Then the Gaussian curvature of $ds^{2}$ on $\Delta\setminus\{g=0\}$ is defined by

[TABLE]

Here $\Delta u(\zeta):=4\liminf\limits_{r\rightarrow 0}\frac{1}{r^{2}}\left\{\frac{1}{2\pi}\int_{0}^{2\pi}u(\zeta+re^{i\theta})d\theta-u(\zeta)\right\}$ for an upper semicontinuous function $u$.

According to [1], we can define the holomorphic sectional curvature of a Finsler metric as follows.

Definition 2.4**.**

Let $F$ be a Finsler metric on domain $\Omega\subset\mathbb{C}^{n}$ and $G=F^{2}$. For $p\in\Omega$ and $0\neq v\in T^{1,0}_{z}\Omega$, the holomorphic sectional curvature is defined by

[TABLE]

Here the supremum ranges through all holomorphic mappings $\varphi:\Delta\rightarrow\Omega$, satisfying $\varphi(0)=p$ and $\varphi^{\prime}(0)=\lambda v$ with some $\lambda\in\mathbb{C^{*}}$, and $K(\varphi^{*}G)$ is the Gaussian curvature of pseudo-metric $\varphi^{*}G$ on $\Delta$.

Remark 2.5*.*

If the smooth Finsler metric $F$ is a Hermitian metric, then two definitions above of holomorphic sectional curvature coincide. It is worth noting that for any $(p,v)$, if $F$ is the Kobayashi metric or Carath$\acute{\operatorname{e}}$odory metric on a Kobayashi hyperbolic or Carath$\acute{\operatorname{e}}$odory hyperbolic domain $\Omega$, then $H(F)(p,v)\geq-4$ or $H(F)(p,v)\leq-4$ (see [14, 16] for more details). Moreover, if $\Omega$ is a bounded convex domain, then Kobayashi metric coincides with Carath$\acute{\operatorname{e}}$odory metric, which implies that $H(F)(p,v)\equiv-4$.

2.4. Quasi-finite geometry

Now we recall the notion of quasi-finite geometry appeared in [15], which is an important tool in K$\ddot{\operatorname{a}}$hler geometry.

Definition 2.6**.**

Suppose $(M^{n},g)$ is a K$\ddot{\operatorname{a}}$hler manifold. We say that $(M^{n},g)$ has $quasi$-$finite\ geometry\ of\ order\ \alpha\geq 0$, if there exist constants $r_{2}>r_{1}>0$ such that: for any point $z\in M^{n}$ there exists a neighbourhood $U\subset\mathbb{C}^{n}$ and a nonsingular holomorphic mapping $\varphi:U\rightarrow M^{n}$ such that

(1) $\varphi(0)=z$,

(2) $B_{r_{1}}(0)\subset U\subset B_{r_{2}}(0)$,

(3) there exists a constant $C\geq 1$ determined only by $r_{1},r_{2},n$ such that

[TABLE]

where $g_{0}$ is the standard Euclidean metric on $\mathbb{C}^{n}$,

(4) for any integer $\alpha\geq q\geq 0$, there exists a constant $A_{q}$ determined by $q,r_{1},r_{2},n$ such that

[TABLE]

where $(\varphi^{*}g)_{i\bar{k}}$ is the component of $\varphi^{*}g$ on $U$ in terms of the natural coordinates $z=(z_{1},\cdots,z_{n})$ and $\mu,\nu$ are the multiple indices with $|\mu|=\mu_{1}+\cdots+\mu_{n}$.

3. Proofs of main theorems

In this section, we prove our main theorems. For a smoothly bounded pseudoconvex domain of finite type $\Omega\subset\mathbb{C}^{2}$, we will construct the scaling sequence with respect to a boundary point of $\Omega$. We firstly recall the notion of normal convergence.

Definition 3.1**.**

Suppose $\left\{\Omega_{n}\right\}\subset\mathbb{C}^{n}$ is a sequence of domains, we say that $\Omega_{n}$ $converges\ normally$ to a domain $\Omega_{\infty}\subset\mathbb{C}^{n}$ if the following hold:

(1) for any compact set $K$, if $K$ is contained in the interior of $\bigcap_{i>k}\Omega_{i}$ for some constant $k$, then $K\subset\Omega_{\infty}$,

(2) for any compact set $K_{1}\subset\Omega_{\infty}$, there exists a constant $k_{1}$ such that $K_{1}\subset\bigcap_{i>k_{1}}\Omega_{i}$.

Suppose $\Omega\subset\mathbb{C}^{2}$ is a smoothly bounded pseudoconvex domain of finite type, and $p\in\partial\Omega$ is a boundary point of $m$-type. Now we recall the scaling sequence with respect to boundary point $p$. Without loss of generality, we may assume that $\Omega=\left\{(z,w)\in\mathbb{C}^{2}:\rho(z,w)<0\right\}$, where $\rho(z,w)$ is a smooth function and $p=(0,0)$. Then after changing of coordinates, we may assume that

[TABLE]

in a neighbourhood of $p$, where $H_{m}(z)$ is a homogeneous polynomial of degree $m$ from $\mathbb{C}$ to $\mathbb{R}$, subharmonic and without harmonic terms.

Suppose $\{p_{n}\}_{n\in\mathbb{N}}\subset\Omega$ is a sequence that converges to $p$, and for each $n$ we consider constant $\epsilon_{n}>0$ such that $\left(p_{n}^{(1)},p_{n}^{(2)}+\epsilon_{n}\right)\in\partial\Omega$ with $p_{n}=\left(p_{n}^{(1)},p_{n}^{(2)}\right)$. Let

[TABLE]

be an automorphism of $\mathbb{C}^{2}$, where $d_{n,k}$ are constants such that

[TABLE]

Here $P_{n}(z):\mathbb{C}\rightarrow\mathbb{R}$ is a subharmonic polynomial without harmonic terms and $P_{n}(0)=0$ with degree $m$. Furthermore, we select constant $\tau_{n}>0$ such that

[TABLE]

where $||\cdot||$ denote the norm in the space of polynomials with degree at most $m$.

For each $n\geq 1$, we define the following mapping

[TABLE]

and $\alpha_{n}:=\delta_{n}\circ\psi_{n}$, which is an automorphism of $\mathbb{C}^{2}$. Note that if we let $\Omega_{n}=\alpha_{n}(\Omega)$ for each $n=1,2,\cdots$, then $\Omega_{n}$ converges to $\Omega_{\infty}:=\left\{(z,w)\in\mathbb{C}^{2}:\operatorname{Re}w+P_{\infty}(z)<0\right\}$ in the sense of Definition 3.1. Here $P_{\infty}(z):\mathbb{C}\rightarrow\mathbb{R}$ is a real-valued subharmonic polynomial without harmonic terms and its degree is $m$. Moreover, we have

[TABLE]

and the domain $\Omega_{\infty}$ is called a $model\ domain$.

Under the scaling sequence above, we obtain the following result about stability of Kobayashi metrics.

Lemma 3.2 [10, Lemma 5.2].

For $(z,v)\in\Omega_{\infty}\times\mathbb{C}^{2}$, we have

[TABLE]

and the convergence is uniform on compact subsets of $\Omega_{\infty}\times\mathbb{C}^{2}$.

By considering the extremal mappings of Carath$\acute{\operatorname{e}}$odory metric and using Montel’s theorem, the stability of the family $\left\{C_{\Omega_{n}}\right\}$ holds as follows.

Lemma 3.3**.**

For $(z,v)\in\Omega_{\infty}\times\mathbb{C}^{2}$, then

[TABLE]

Although we can not obtain the stability of $\left\{C_{\Omega_{n}}\right\}$ like $\left\{K_{\Omega_{n}}\right\}$, by considering boundary behaviour of Kobayashi metric and Carath$\acute{\operatorname{e}}$odory metric, we can get the following lemma.

Lemma 3.4.

If $\lim\limits_{z\rightarrow p}\frac{C_{\Omega}(z,v)}{K_{\Omega}(z,v)}=1$ holds uniformly for $0\neq v\in\mathbb{C}^{2}$. Then

[TABLE]

holds uniformly on compact sets of $\Omega_{\infty}\times\mathbb{C}^{2}$.

Proof.

After taking a subsequence, we may assume that there exists a constant $\epsilon>0$ such that

[TABLE]

holds for some $(z_{j},v_{j})$. Here each $(z_{j},v_{j})$ lies in a compact set $K\subset\Omega_{\infty}\times\mathbb{C}^{2}$. And we may assume that $z_{j}\rightarrow z_{0},v_{j}\rightarrow v_{0}$. Without loss of generality, we suppose that

[TABLE]

Hence, there exists a constant $c>0$, independently of $j$, such that

[TABLE]

Since $\lim\limits_{z\rightarrow p}\frac{C_{\Omega}(z,v)}{K_{\Omega}(z,v)}=1$ holds uniformly for $0\neq v\in\mathbb{C}^{2}$, then for any $(z,v)\in\Omega_{\infty}\times\mathbb{C}^{2}$, we have $C_{\Omega_{\infty}}(z,v)=K_{\Omega_{\infty}}(z,v)$. To see this, we may assume $0\neq v$ and observe that

[TABLE]

from Lemma 3.2 and Lemma 3.3. Note that $\lim\limits_{z\rightarrow\partial\Omega}\frac{C_{\Omega}(z,v)}{K_{\Omega}(z,v)}=1$ uniformly for $0\neq v$ implies that for any $\epsilon_{1}>0$, there exists a constant $\delta>0$ such that

[TABLE]

for any $0\neq v$ and $z\in\Omega$ with $\delta_{\Omega}(z)<\delta$. Since $\alpha_{n}^{-1}(z)$ converges to $p\in\partial\Omega$, hence for large $n$ we obtain $\delta_{\Omega}(\alpha_{n}^{-1}(z))<\delta$. Then

[TABLE]

for large $n$. Taking limit for $n$ implies that $C_{\Omega_{\infty}}(z,v)=K_{\Omega_{\infty}}(z,v)$. Therefore, we can deduce that

[TABLE]

which is a contracdiction. ∎

Remark 3.5*.*

If $\lim\limits_{z\rightarrow p}\frac{C_{\Omega}(z,v)}{K_{\Omega}(z,v)}=1$ holds uniformly for $0\neq v\in\mathbb{C}^{2}$, then from the proof of Lemma 3.4 and Remark 2.5, we know that the holomorphic sectional curvature of $K_{\Omega_{\infty}}$ equals -4.

According to the notion of quasi-finite geometry, we obtain the following result, which concerns the convergence of K$\ddot{\operatorname{a}}$hler metrics under the scaling sequence. The proof is similar with [2, Proposition 6.1], we provide it for completeness.

Proposition 3.6**.**

Suppose $g_{n}$ is a complete K$\ddot{a}$hler metric on $\Omega_{n}$ for each $n$. If

(1) there exists a constant $A\geq 1$, independently on $n$, such that

[TABLE]

for all $z\in\Omega_{n}$ and $v\in\mathbb{C}^{2}$,

(2) each $(\Omega_{n},g_{n})$ has quasi-finite geometry of order $\alpha\geq 1$, where the constants appeared in Definition 2.6 are independent on $n$.

Then after passing to a subsequence, there exists a complete $C^{\alpha-1}$ smooth metric $g_{\infty}$ on $\Omega_{\infty}$ such that $g_{n}$ converges to $g_{\infty}$ locally uniformly in $C^{\alpha-1}$ topology.

Proof.

In this proof, the distance induced by Kobayashi metric on $\Omega$ is denoted by $d_{\Omega}$. It is enough to show that: for any compact set $K\subset\Omega_{\infty}$ and multi-indices $\mu,\nu$ with $|\mu|+|\nu|\leq\alpha$, there exist constants $J\geq 0,C(K,\mu,\nu)\geq 1$ depending only on $K,\mu,\nu$, such that

[TABLE]

We suppose that there exists a compact set $K\subset\Omega_{\infty}$ and $\mu,\nu$ with $|\mu|+|\nu|\leq\alpha$ such that: for each integer $l\geq 0$, there exists a constant $n_{l}\geq l$ and $x_{l}\in K$ such that

[TABLE]

Without loss of generality, we assume that $x_{l}\rightarrow x_{\infty}\in\Omega_{\infty}$ and $K\subset\Omega_{n_{l}}$ for all $l\geq 1$. From Difinition 2.6, we know that for each $l$, there is a domain $U_{l}\subset\mathbb{C}^{2}$ and a nonsingular mapping $\varphi_{l}:U_{l}\rightarrow\Omega_{n_{l}}$ with $\varphi_{l}(0)=x_{l}$ satisfying conditions in Definition 2.6, where the constants in Definition 2.6 are independent on $l$.

Suppose $r<r_{1}$ is a fixed constant, then

[TABLE]

If $w\in\varphi_{l}(B_{r}(0))$, then we have that

[TABLE]

for some constant $M>0$, independently of $l$. Hence from [10, Lemma 5.6] each $\varphi_{l}(B_{r}(0))$ is compactly contained in $\Omega_{\infty}$ for all $l$ large. And it implies that $\varphi_{l}(B_{r}(0))$ is uniformly bounded.

From condition (3) in Definition 2.6, we obtain that

[TABLE]

for each $l$ and $v\in\mathbb{C}^{2}$. It implies that

[TABLE]

and

[TABLE]

Then Montel’s theorem implies that $\varphi:=\lim\limits_{l\rightarrow\infty}\varphi_{l}:B_{r_{1}}(0)\rightarrow\Omega_{\infty}$ is holomorphic and nonsingular at 0.

Note that $\varphi|_{U_{1}}$ is invertible in some neighbourhood $U_{1}$, and $\varphi_{l}$ converges $\varphi$ in $C^{\infty}$ topology. Then $\varphi_{l}|_{U_{2}}$ is invertible for large $l$ in some neighbourhood $U_{2}\subset U_{1}$. Moreover, we know that $\left(\varphi_{l}|_{U_{2}}\right)^{-1}$ converges locally uniformly to $\left(\varphi|_{U_{2}}\right)^{-1}$.

After choosing a neighbourhood $V$ of $x_{\infty}\in\Omega_{\infty}$ such that $V\subset\varphi(U_{2})$, we may assume that $V\subset\varphi_{l}\left(U_{2}\right)$ with $x_{l}\in V$ for large $l$ and

[TABLE]

whenever $|a|+|b|\leq|\mu|+|\nu|$. Combining with condition (4) in Definition 2.6, we obtain that

[TABLE]

which is a contracdiction.

Note that after passing to a subsequence, we may assume that $g_{n}$ converges locally uniformly to a $C^{\alpha-1}$ smooth pseudo-metric $g_{\infty}$ on $\Omega_{\infty}$ in $C^{\alpha-1}$ topology. From Lemma 3.2, we have

[TABLE]

and

[TABLE]

then $g_{\infty}$ is a complete $C^{\alpha-1}$ smooth metric on $\Omega_{\infty}$. And it completes the proof. ∎

$Proof\ of\ Theorem\ \ref{kob}$: We just need to prove $(2)\Rightarrow(1)$ and $(3)\Rightarrow(1)$.

$(3)\Rightarrow(1)$: Suppose $\Omega$ satisfies condition (2) in Theorem 1.3, then it is covered by $\mathbb{B}^{2}$ or $\mathbb{C}^{2}$ or $\mathbb{CP}^{2}$. Since covering map is a local isometry for Kobayashi metrics and $\Omega$ is Kobayashi hyperbolic, then $\Omega$ is covered by $\mathbb{B}^{2}$.

$(2)\Rightarrow(1)$: Suppose $(\Omega,K_{\Omega})$ has quasi-finite geometry of order three, then each $(\Omega_{n},K_{\Omega_{n}})$ has quasi-finite geometry of order three, where the constants in Definition 2.6 are independently on $n$. In particular, the holomorphic sectional curvature of $K_{\Omega_{n}}$ is uniformly bounded.

Claim:$\lim\limits_{n\rightarrow\infty}H(K_{\Omega_{n}})(z,v)=H(K_{\Omega_{\infty}})(z,v)=-4$ holds uniformly on compact sets of $\Omega_{\infty}\times(\mathbb{C}^{2})^{*}$.

$Proof\ of\ claim$: Since $H(K_{\Omega_{n}})(z,v)\geq-4$ for each $n$ and $(z,v)\in\Omega_{n}\times(\mathbb{C}^{2})^{*}$, we need to show that

[TABLE]

holds uniformly on compact sets of $\Omega_{\infty}\times(\mathbb{C}^{2})^{*}$. For a contradiction, we assume there exist $(z_{j},v_{j})$ lying in a compact set of $\Omega_{\infty}\times(\mathbb{C}^{2})^{*}$ with $|v_{j}|=1$, such that

[TABLE]

We assume that $z_{j}\rightarrow z_{0}\in\Omega_{\infty}$ and $v_{j}\rightarrow v_{0}$ with $|v_{0}|=1$. From Proposition 3.6 and Lemma 3.2, we know that after passing to a subsequence, $K_{\Omega_{j}}$ converges to $K_{\Omega_{\infty}}$ locally uniformly in $C^{2}$ topology. And it means that $H({K_{\Omega_{j}}})(z_{j},v_{j})\rightarrow H({K_{\Omega_{\infty}}})(z_{0},v_{0})=-4$, and this is a contradiction.

Hence we obtain that

[TABLE]

uniformly for $0\neq v\in\mathbb{C}^{2}$. Theorem 1.2 in [9] means that $\Omega$ is strongly pseudoconvex, and we complete the proof since Theorem 1.2.

$Proof\ of\ Theorem\ \ref{kem}$: Suppose $\Omega$ is covered by the unit ball. Since holomorphic covering map is a local imsometry of Kobayashi metrics and K$\ddot{\operatorname{a}}$hler-Einstein metrics, then K$\ddot{\operatorname{a}}$hler-Einstein metric is a scalar multiple of Kobayashi metric on $\Omega$.

On the other hand, from [9, Proposition 3.11] we know that for any $x\in\Omega$, there exists a biholomorphism $\varphi_{x}$ from $B_{r}(0)$ to $\varphi_{x}(B_{r}(0))\subset\Omega$ satisfying

(1) $\varphi_{x}(0)=x,$

(2) $\frac{1}{A}g_{0}\leq\varphi_{x}^{*}B_{\Omega}\leq Ag_{0}$ with some constant $A\geq 1$.

Here $r,A$ are independent on $x$ and $g_{0}$ is the standard Euclidean metric on $\mathbb{C}^{2}$.

Since $B_{\Omega}$ is equivalent to the K$\ddot{\operatorname{a}}$hler-Einstein metric $g_{KE}$ on $\Omega$, after a similar proof of [17, Lemma 3] in coordinate $\left(B_{r}(0),\varphi_{x}\right)$, then $(\Omega,g_{KE})$ has quasi-finite geometry of order infinity. Thus if $(\Omega,g_{KE})$ satisfies condition (2) in Corollary 1.5, then Theorem 1.3 implies that $\Omega$ is biholomorphic to a ball quotient.

$Proof\ of\ Theorem\ \ref{car}$: It’s enough to prove $(2)\Rightarrow(1)$ and $(3)\Rightarrow(1)$.

$(3)\Rightarrow(1)$: For a bounded domain $\Omega$, we know that $K_{C_{\Omega}}(z,v)\leq-4$ for any $(z,v)\in\Omega\times(\mathbb{C}^{2})^{*}$. If $C_{\Omega}$ satisfies condition (3) in Theorem 1.6, then it has constant negative holomorphic sectional curvature. And [9, Theorem 1.2] implies that $\Omega$ is strongly pseudoconvex. We know that $\Omega$ is biholomorphically equivalent to a ball from Theorem 1.9 in [3].

$(2)\Rightarrow(1)$: If $C_{\Omega}$ satisfies condition (2) in Theorem 1.6, then Lemma 3.4 and Proposition 3.6 impliy that

[TABLE]

uniformly for $0\neq v\in\mathbb{C}^{2}$ after a similar argument of proof of Theorem 1.3. Hence $\Omega$ is strongly pseudoconvex from [9, Theorem 1.2], and this completes the proof of Theorem 1.6.

Now we prove Theorem 1.7, and it can be confirmed by the following proposition and Theorem 1.10 in [3].

Proposition 3.7**.**

Let $\Omega\subset\mathbb{C}^{2}$ be a smoothly bounded pseudoconvex domain of finite type, and $p\in\partial\Omega$ be a boundary point. If

[TABLE]

and

[TABLE]

uniformly for $0\neq v\in\mathbb{C}^{2}$, then $\Omega$ is strongly pseudoconvex at $p$.

Proof.

Let $\left\{\Omega_{n}\right\}$ be the scaling sequence with respect to $p$, and $\Omega_{\infty}$ be the corresponding model domain. Suppose $B_{\Omega_{n}}$ is the Bergman metric on $\Omega_{n}$, then Proposition 3.10 in [9] implies that there exists a complete K$\ddot{\operatorname{a}}$hler metric $B_{\infty}$ on $\Omega_{\infty}$ such that $B_{n}$ converges to $B_{\infty}$ on $\Omega_{\infty}$ locally uniformly in $C^{\infty}$ topology after taking a subsequence. Note that $H(B_{\infty})\leq-4L^{2}(\Omega)$ holds in $\Omega_{\infty}$. Since $C_{\Omega}$ and $B_{\Omega}$ are biholomorphically invariant and $C_{\Omega}\leq L(\Omega)B_{\Omega}$, then

[TABLE]

holds for any $(z,v)\in\Omega_{\infty}\times\mathbb{C}^{2}$ from Lemma 3.4. Ahlfors-Schwarz lemma implies that

[TABLE]

for any $(z,v)\in\Omega_{\infty}\times\mathbb{C}^{2}$. Since $K_{\Omega_{\infty}}=C_{\Omega_{\infty}}$, then $K_{\Omega_{\infty}}=L(\Omega)B_{\infty}=C_{\Omega_{\infty}}$, and it implies that $\Omega_{\infty}$ is biholomorphic to the unit ball. Then we obtain that $\Omega$ is strongly pseudoconvex at $p$. ∎