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Let $G\subseteq \mathbb{C}^n$ be a bounded domain. Consider the Carathéodory metric $C_G$ on $G$. If $G=\mathbb{D}^n$ (unit polydisc), then $C_G(a,z)=\max_{1\leq j\leq n}p(a_j,z_j)$, where $p$ denotes the Poincaré metric on $\mathbb{D}$.

My question: Is there a similar formula in case $D=\mathbb{B}^2\times \mathbb{D}\subseteq \mathbb{C}^3$, where $\mathbb{B}^2$ denotes the unit ball in $\mathbb{C}^2$? More generally, is there a formula (or some estimate) for the Carathéodory metric of the product domain, in terms of the Carathéodory metric on the components?

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For a general formula, you should check out Kobayashi's book Hyperbolic Complex Spaces. In Proposition 3.1.11 of that text, he proves the following:

Let $X$ and $Y$ be complex spaces. For $(x,y), (x',y') \in X\times Y$, we have $$ \text{max}\{ c_X(x,x'),c_Y(y,y')\} \leq c_{X\times Y}((x,y),(x',y')) \leq c_X(x,x') + c_Y(y,y'). $$

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  • $\begingroup$ Theorem 4.9.1 of this text says that for complex spaces $X,Y$, the left inequality becomes equality. I am not familiar with "complex spaces", can you refer some text dealing with "complex spaces" so that I have some examples of complex spaces. $\endgroup$
    – Jean
    Commented Dec 30, 2021 at 0:53
  • $\begingroup$ Is a ball (Euclidean) an example of a complex space? In general, what about a bounded domain? $\endgroup$
    – Jean
    Commented Dec 30, 2021 at 0:54
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    $\begingroup$ @Jean: a complex space is a generalization of a complex manifold and also a generalization of a complex analytic variety. In particular, every bounded domain is a complex space. $\endgroup$
    – Ben McKay
    Commented Dec 30, 2021 at 12:09
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    $\begingroup$ @Jean the below answer by Peter Pflug is the one you should look at. In particular, Theorem 4.9.1 of the text refers to the reference he mentions in his answer. $\endgroup$
    – Jackson Morrow
    Commented Dec 30, 2021 at 15:48
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In fact, there is the product property $\max\{c_X(x,x′),c_Y(y,y′)\}=c_{X×Y}((x,y),(x′,y′))$ for arbitrary domains $X, Y$. See chapter 18 in "Invariant distances and metrics in complex analysis" de Gruyter (2013).

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