Invariant metrics on unbounded strongly pseudoconvex domains with non-compact automorphism group

Abstract

We study invariant metrics on unbounded strongly pseudoconvex domains with non-compact automorphism group. The main result is that the corresponding Bergman and K$\ddot{\mathrm{a}}$hler–Einstein metrics are metrically equivalent. We also determine the comparisons among invariant metrics, including the Carathéodory and Kobayashi pseudo-metrics additionally.

Appendix: Carathéodory and Kobayashi pseudo-metrics and pseudo-distances

As is widely known, the Carathéodory and Kobayashi pseudo-metrics squeeze all pseudo-differential metrics on complex manifolds satisfying the Schwarz lemma with respect to holomorphic mappings and coinciding with the Poincaré metric on the unit disc. By contrast, the Bergman metric does not admit the Schwarz lemma. For this reason, one can ask whether the Bergman metric is compared with some invariant metrics. For a bounded or Kobayashi-hyperbolic domain $\Omega$ in $\mathbb {C}^n$, it is well known that

$$\begin{aligned} C_{\Omega }\le K_{\Omega }\quad \text {and}\quad C_{\Omega }\le B_{\Omega }, \end{aligned}$$

where $C_{\Omega }, K_{\Omega }$, and $B_{\Omega }$ are the Carathéodory pseudo-metric, the Kobayashi pseudo-metric, and the Bergman metric, respectively.

We shall focus our attention first on the behaviors of the Carathéodory and Kobayashi pseudo-distances along two different leaves on $D_{1,1}$. In addition, further comparisons among the Bergman distance and the above two pseudo-distances on $D_{1,1}$ will be given. Then we will explain the corresponding relation for the case of $D_{n,m}$ in higher dimensions.

Before moving to description of the behaviors of the Carathéodory and Kobayashi pseudo-metrics on $D_{1,1}$, let us briefly consider some basics on these two pseudo-metrics on an arbitrary domain $\Omega$ in $\mathbb {C}^n$. Let $\mathbb {D}$ denote the unit disc in $\mathbb {C}$. Then we define the Poincaré distance $d_{\mathbb {D}}$ on $\mathbb {D}$ by setting

$$\begin{aligned} d_{\mathbb {D}}(a, b)=\frac{1}{2}\log \frac{|1-a\bar{b}|+|a-b|}{|1-a\bar{b}|-|a-b|}=\tanh ^{-1}\left| \frac{a-b}{1-a\bar{b}}\right| \end{aligned}$$

for all $a,b\in \mathbb {D}.$ Given two complex spaces $\Omega _1$ and $\Omega _2$, let $\mathrm {Hol}(\Omega _1, \Omega _2)$ denote the set of all holomorphic mappings from $\Omega _1$ into $\Omega _2$. The Carathéodory pseudo-distance $d^{\mathrm {C}}_{\Omega }$ between two points p and q in a domain $\Omega \subset \mathbb {C}^n$ is defined by

$$\begin{aligned} d^{\mathrm {C}}_{\Omega }(p,q)=\sup \limits _{f}\left\{ d_{\mathbb {D}}(f(p),f(q)):\;f\in \mathrm {Hol}(\Omega ,\mathbb {D})\right\} . \end{aligned}$$

This Carathéodory pseudo-distance is closely related to the following pseudo-metric through the consideration of its integrated form: The infinitesimal Carathéodory pseudo-metric at a point $p\in \Omega$ and $\xi \in T_{p}\Omega$ is defined by

$$\begin{aligned} {C}_{\Omega }(p;\xi )=\sup \limits _{f}\left\{ |f_{*}(p)\xi |:\;f\in \mathrm {Hol}(\Omega ,\mathbb {D}),\;f(p)=0\right\} , \end{aligned}$$

where $f_{*}(p)$ denotes the $\mathbb {C}$-differential of f at p. Then, given two points $p, q\in \Omega$, the integrated form of the infinitesimal Carathéodory pseudo-metric is defined by

$$\begin{aligned} c_{\Omega }(p,q)=\inf \limits _{\gamma }\int _{0}^{1}C_{\Omega }(\gamma (t);\gamma '(t))\mathrm{d}t, \end{aligned}$$

where the infimum is taken over all piecewise $C^1$ curves $\gamma :[0,1] \rightarrow \Omega$ with $\gamma (0)=p$ and $\gamma (1)=q$. It has been known that $d^{\mathrm {C}}_{\Omega }\le c_{\Omega }$ holds for a domain $\Omega$ in $\mathbb {C}^n$.

To establish the dual concept of the Carathéodory pseudo-distance, we first define the Lempert function $\delta ^{\mathrm {K}}_{\Omega }$ for $\Omega$ by setting

$$\begin{aligned} \delta ^{\mathrm {K}}_{\Omega }(p,q)=\inf \limits _{h}\left\{ d_{\mathbb {D}}(a,b):\;h\in \mathrm {Hol}(\mathbb {D},\Omega ), h(a)=p, h(b)=q\right\} ,\quad \text {for}\;p,q\in \Omega . \end{aligned}$$

Moreover, it is well known that the Lempert function is not a pseudo-distance because it does not satisfy the triangle inequality. As the largest pseudo-distance bounded by $\delta ^{\mathrm {K}}_{\Omega }$, we now define the Kobayashi pseudo-distance $d^{\mathrm {K}}_{\Omega }$ by setting

$$\begin{aligned} d^{\mathrm {K}}_{\Omega }(p,q)=\mathrm {inf}\sum _{j=1}^{N}\delta ^{\mathrm {K}}_{\Omega }(p_{j-1},p_{j}),\quad \text {for}\;p,q\in \Omega , \end{aligned}$$

(75)

where the infimum is taken over all the possible chains of holomorphic discs from p to q. Then it follows obviously from the definitions above that

$$\begin{aligned} d^{\mathrm {K}}_{\Omega }(p,q)\le \delta ^{\mathrm {K}}_{\Omega }(p,q)\le d_{\mathbb {D}}(a,b), \end{aligned}$$

(76)

where $h(a)=p$ and $h(b)=q$ for $h\in \mathrm {Hol}(\mathbb {D},\Omega )$.

Let us now consider the concept of the infinitesimal Kobayashi pseudo-metric due to H. L. Royden in 1971: For a domain $\Omega$ in $\mathbb {C}^n$, the infinitesimal Kobayashi pseudo-metric at a point $p\in \Omega$ and $\xi \in T_{p}\Omega$ is defined by

$$\begin{aligned} K_{\Omega }(p;\xi )=\inf \limits _{h}\left\{ |\lambda |:\;h\in \mathrm {Hol}(\mathbb {D},\Omega ), h(0)=p, h_{*}(0)\lambda =\xi \right\} . \end{aligned}$$

Analogously to that of the Carathéodory pseudo-distance, given two points $p,q\in \Omega$, we define the integrated form of the infinitesimal Kobayashi pseudo-metric by setting

$$\begin{aligned} k_{\Omega }(p,q)=\inf \limits _{\gamma }\int _{0}^{1}K_{\Omega }(\gamma (t);\gamma '(t))\mathrm{d}t, \end{aligned}$$

where the infimum is taken over all piecewise $C^1$ curves $\gamma :[0,1] \rightarrow \Omega$ with $\gamma (0)=p$ and $\gamma (1)=q.$ This concept is indeed identical to the Kobayashi pseudo-distance defined in (75). Altogether, one can reach the following comparison:

$$\begin{aligned} d^{\mathrm {C}}_{\Omega }(p,q)\le c_{\Omega }(p,q)\le k_{\Omega }(p,q)=d^{\mathrm {K}}_{\Omega }(p,q),\quad \text {for}\;p,q\in \Omega . \end{aligned}$$

Now we turn into the investigation of the behaviors of the Carathéodory and Kobayashi pseudo-distances on the domain $D_{1,1}$ in $\mathbb {C}^2$. Since both of the Carathéodory and Kobayashi pseudo-distances are invariant under holomorphic automorphisms, we shall utilize the explicit form of $\mathrm {Aut}(D_{1,1})$ in determining the behaviors of these pseudo-metrics on $D_{1,1}$. As described in Sect. 2, $\mathrm {Aut}(D_{n,m})$ is generated by the following mappings:

$$\begin{aligned} r_{U}&: D_{n,m} \rightarrow D_{n,m}, \quad (z, w) \mapsto ( Uz, w),\\ r_{U'}&: D_{n,m} \rightarrow D_{n,m}, \quad (z, w) \mapsto ( z, U'w),\\ \tau _v&: D_{n,m} \rightarrow D_{n,m}, \quad (z, w) \mapsto ( z + v, e^{-\langle z, v\rangle -\frac{1}{2}{\Vert v\Vert }^2 } w), \end{aligned}$$

where $U\in U(n)$, $U'\in U(m)$, and $v \in \mathbb C^n$. Considering $\gamma _{U'}$ and $\tau _{-z_0}$ for the case of $D_{1,1}$, we have

$$\begin{aligned} \gamma _{U'}\circ \tau _{-z_0}(z,w)=(z-z_0,U'e^{z\bar{z}_{0}-\frac{1}{2}{|z_{0}|}^2}w),\quad \text {for}\;(z,w)\in D_{1,1}. \end{aligned}$$

Then, for each $(z_{0},w_{0})\in D_{1,1}$, we obtain

$$\begin{aligned} \gamma _{U'}\circ \tau _{-z_0}(0,0)=(-z_0,0)\quad \text {and}\quad \gamma _{U'}\circ \tau _{-z_0}(z_{0},w_{0})=(0,U'e^{\frac{1}{2}{|z_0|}^2}w_{0}). \end{aligned}$$

(77)

Combining (77) with the invariance of the Carathéodory pseudo-distance under biholomorphic mappings, one can deduce that

$$\begin{aligned} d^{\mathrm {C}}_{D_{1,1}}((0,0),(z_{0},w_{0}))=d^{\mathrm {C}}_{D_{1,1}}((-z_0,0),(0,w^{*}_{0})), \end{aligned}$$

where ${|w^{*}_{0}|}^{2}=e^{{|z_0|}^2}{|w_{0}|}^{2}$. Since $d^{\mathrm {C}}_{D_{1,1}}$ vanishes identically along a complex line $L:=\{(z,0):\;z\in \mathbb {C}\}\subset D_{1,1}$, it follows from the triangle inequality that

$$\begin{aligned} d^{\mathrm {C}}_{D_{1,1}}((0,0),(z_0,w_0))&=d^{\mathrm {C}}_{D_{1,1}}((z_0,0),(z_0,w_0)) \nonumber \\&=d^{\mathrm {C}}_{D_{1,1}}((0,0),(0,w^{*}_{0})). \end{aligned}$$

(78)

The latter equality comes from the invariance of the Carathéodory pseudo-distance under the composition of the above automorphism $\gamma _{U'}\circ \tau _{-z_0}$ and a unitary transformation in the w-coordinate. The relation (78) tells us that the Carathéodory pseudo-distance between the leaf

$$\begin{aligned} \{(z,w)\in D_{1,1}:\;e^{{|z|}^2}{|w|}^2=c\;\text {for a fixed}\;c\in [0,1)\} \end{aligned}$$

and the complex line L is constant with respect to the constant c.

Now we shall show that

$$\begin{aligned} d^{\mathrm {C}}_{D_{1,1}}((0,0),(0,w^{*}_{0}))=d_{\mathbb {D}}(0,|w^{*}_{0}|)=d^{\mathrm {K}}_{D_{1,1}}((0,0),(0,w^{*}_{0})). \end{aligned}$$

(79)

For the proof, one starts with a holomorphic mapping $f: D_{1,1}\rightarrow \mathbb {D}$ defined by

$$\begin{aligned} f(z,w)=w. \end{aligned}$$

Then it follows from the definition that

$$\begin{aligned} f(0,w^{*}_{0})=w^{*}_{0}\quad \text {and}\quad f(0,0)=0. \end{aligned}$$

This, in conjunction with the definition of $d^{\mathrm {C}}_{D_{1,1}}$, yields

$$\begin{aligned} d^{\mathrm {C}}_{D_{1,1}}((0,0),(0,w^{*}_{0}))&\ge d_{\mathbb {D}}(f(0,0),f(0,w^{*}_{0})) \nonumber \\&=d_{\mathbb {D}}(0,w^{*}_{0})\nonumber \\&=d_{\mathbb {D}}(0,|w^{*}_{0}|). \end{aligned}$$

(80)

Let us now consider a holomorphic disc $h:\mathbb {D}\rightarrow D_{1,1}$ defined by

$$\begin{aligned} h(w)=(0,w). \end{aligned}$$

Since $d^{\mathrm {C}}_{D_{1,1}}(p,q)\le d^{\mathrm {K}}_{D_{1,1}}(p,q)$ holds for all $p,q\in D_{1,1}$, (76) and (80) imply that

$$\begin{aligned} d^{\mathrm {C}}_{D_{1,1}}((0,0),(0,w^{*}_{0}))&\le d^{\mathrm {K}}_{D_{1,1}}((0,0),(0,w^{*}_{0}))\\&\le \delta ^{\mathrm {K}}_{D_{1,1}}((0,0),(0,w^{*}_{0}))\\&=\delta ^{\mathrm {K}}_{D_{1,1}}(h(0),h(w^{*}_{0}))\\&\le d_{\mathbb {D}}(0,w^{*}_{0})\\&=d_{\mathbb {D}}(0,|w^{*}_{0}|)\\&\le d^{\mathrm {C}}_{D_{1,1}}((0,0),(0,w^{*}_{0})). \end{aligned}$$

This finishes the proof of Eq. (79).

Fig. 2

Pseudo-distances between two leaves

Concerning the associated Carathéodory and Kobayashi pseudo-distances between the leaves $C_1$ and $C_2$ as shown in Fig. 2, one can get

$$\begin{aligned} d^{\mathrm {C}}_{D_{1,1}}(C_{1},C_{2})=d_{\mathbb {D}}(A',B')=d^{\mathrm {K}}_{D_{1,1}}(C_{1},C_{2}). \end{aligned}$$

(81)

Here is a proof of (81). Let us denote by O the origin (0, 0) in $\mathbb {C}^2$. For any point B in the leaf $C_{2}$ given in Figure 2, (79) and the triangle inequality imply that

$$\begin{aligned} d^{\mathrm {C}}_{D_{1,1}}(O,C_{1})+d^{\mathrm {C}}_{D_{1,1}}(C_{1},B)\ge & {} d^{\mathrm {C}}_{D_{1,1}}(O,B) \nonumber \\= & {} d^{\mathrm {C}}_{D_{1,1}}(O,B') \nonumber \\= & {} d_{\mathbb {D}}(O,B') \nonumber \\= & {} d_{\mathbb {D}}(O,A')+d_{\mathbb {D}}(A',B'), \end{aligned}$$

(82)

where $OB'$ is the geodesic with respect to the Poincaré metric. This, in conjunction with the fact that $d^{\mathrm {C}}_{D_{1,1}}(O,C_{1})=d_{\mathbb {D}}(O,A')$ inherited from (78) and (79), yields

$$\begin{aligned} d^{\mathrm {C}}_{D_{1,1}}(C_{1},B)\ge d_{\mathbb {D}}(A',B'). \end{aligned}$$

(83)

Through observation of orbits of $\mathrm {Aut}(D_{1,1})$, we can assure the existence of a point A in the leaf $C_{1}$ with the same first component as the point B such that

$$\begin{aligned} d^{\mathrm {C}}_{D_{1,1}}(A,B)=d_{\mathbb {D}}(A',B'). \end{aligned}$$

(84)

Then it follows from (84) that

$$\begin{aligned} d^{\mathrm {C}}_{D_{1,1}}(C_1,B)\le d^{\mathrm {C}}_{D_{1,1}}(A,B)=d_{\mathbb {D}}(A',B'). \end{aligned}$$

(85)

Since B was arbitrary, (83) and (85) conclude the verification of (81). Since we do not know the explicit form of geodesic in general, it leads to another highly intricate stage to compute the pseudo-distances between any two points in the leaves $C_1$ and $C_{2}$. In the present paper, we would not take such a kind of consideration.

We shall now investigate the relation among the Bergman distance, the Carathéodory and Kobayashi pseudo-distances on $D_{1,1}.$ In determining the associated comparison theorem, we utilize the invariance of the Bergman metric under biholomorphic mappings. Note that, for each piecewise $C^1$ curve $\gamma (t):=(u(t),v(t)): [0,1]\rightarrow D_{1,1}$, there exists a piecewise $C^1$ curve $\tilde{\gamma }(t):=(0,\tilde{v}(t)):[0,1]\rightarrow D_{1,1}$ such that

$$\begin{aligned} \begin{pmatrix} u'(t)&v'(t) \end{pmatrix} \begin{pmatrix} g_{i\bar{j}}(u,v) \end{pmatrix} \begin{pmatrix} \overline{u'(t)}\\ \overline{v'(t)} \end{pmatrix}= \begin{pmatrix} 0&\tilde{v}'(t) \end{pmatrix} \begin{pmatrix} S({|\tilde{v}(t)|}^2)&{}0\\ 0&{}S'({|\tilde{v}(t)|}^2) \end{pmatrix} \begin{pmatrix} 0\\ \overline{\tilde{v}'(t)} \end{pmatrix} \end{aligned}$$

(86)

using Eq. (8) in Example 2.1. Then, (86) forces the Bergman metric to satisfy

$$\begin{aligned} \int _{0}^1\sqrt{\begin{pmatrix} u'(t)&v'(t) \end{pmatrix} \begin{pmatrix} g_{i\bar{j}}(u,v) \end{pmatrix} \begin{pmatrix} \overline{u'(t)}\\ \overline{v'(t)} \end{pmatrix} } \mathrm{d}t= & {} \int _{0}^{1}\sqrt{S'(x){|\tilde{v}'(t)|}^2}\mathrm{d}t\\\ge & {} \int _{0}^{1}\frac{2|\tilde{v}'(t)|}{(1+x)(1-x)}\mathrm{d}t \nonumber \\\ge & {} \int _{0}^{1}\frac{|\tilde{v}'(t)|}{1-{|\tilde{v}(t)|}^2}\mathrm{d}t \nonumber \\= & {} d_{\mathbb {D}}(\tilde{v}(0),\tilde{v}(1)),\nonumber \end{aligned}$$

(87)

where $x={|\tilde{v}(t)|}^2$. Taking the infimum of the left-hand side of the inequality (87) over all the possible piecewise curves, one can deduce that the Bergman distance between two leaves in $D_{1,1}$ is always greater than or equal to the corresponding Carathéodory and Kobayashi pseudo-distances, that is,

$$\begin{aligned} B_{D_{1,1}}\ge C_{D_{1,1}}=K_{D_{1,1}}. \end{aligned}$$

Moreover, if we take a curve $\gamma (t)=(u(t),v(t))$ along the complex line L, then the associated Bergman distance is nothing but

$$\begin{aligned} \int _{0}^{1}|\tilde{v}'(t)|\mathrm{d}t \end{aligned}$$

which is exactly the Euclidean length of $\gamma (t)$ in L.

Remark 5

Similar arguments as above can be made for the cases of $D_{n,m}$ in higher dimensions. To compute the Carathéodory pseudo-distance, we first consider the following mappings:

$$\begin{aligned} \tau _{-z} :&D_{n,m} \rightarrow D_{n,m},\quad (z, w) \mapsto (0,w^{*}),\\ \psi _{w^{*}} :&D_{n,m} \rightarrow D_{n,m},\quad (z, w) \mapsto (z,\underbrace{0,\ldots ,0}_{m-1},\tilde{w}^{*}_{m}),\\ P_{n,m}&: D_{n,m} \rightarrow \mathbb {D},\quad (z,w) \mapsto w_{m},\\ \mu _{b} :&\mathbb {D} \rightarrow \mathbb {D},\quad a \mapsto \frac{a-b}{1-{\bar{b}}a}, \end{aligned}$$

where $\psi _{w^{*}}\in \mathrm {Id}_{n}\times \mathcal {U}(m)\subset \mathrm {Aut}(D_{n,m})$. Let us define a holomorphic mapping $f: D_{n,m}\rightarrow \mathbb {D}$ by setting

$$\begin{aligned} f(p,q)=\mu _{\tilde{w}^{*}_{m}}\circ P_{n,m}\circ \psi _{w^{*}}\circ \tau _{-z}(p,q) \end{aligned}$$

(88)

for a fixed point $(z,w)\in D_{n,m}$. Then (88) clearly forces f to satisfy $f(z,w)=0$.

To get the associated Kobayashi pseudo-distance, we next consider a M$\ddot{\mathrm{o}}$bius transformation $\tilde{\mu }_{c}: \mathbb {D}\rightarrow \mathbb {D}$ defined by $\tilde{\mu }_{c}=-\mu _{c}$. Adopting a unitary action on w-coordinate in $D_{n,m}$, we may choose a function $\varphi _{w^{*}}: \mathbb {D}\rightarrow D_{n,m}$ such that $\varphi _{w^{*}}(\tilde{w}^{*}_{m})=(0,w^{*})$ with $\Vert w^{*}\Vert =|\tilde{w}^{*}_{m}|$. Then we define a holomorphic mapping $h:\mathbb {D}\rightarrow D_{n,m}$ by setting

$$\begin{aligned} h(a)=\varphi _{w^{*}}\circ \tilde{\mu }_{\tilde{w}^{*}_{m}}(a). \end{aligned}$$

(89)

In particular, h satisfies $h(0)=(0,w^{*})$. These certain mappings given in (88) and (89) make it possible that the Carathéodory and Kobayashi pseudo-distances between two leaves in Fig. 2 are identical. Furthermore, the result of comparisons among the Bergman metric and the above two pseudo-metrics is essentially same as that of $D_{1,1}$.