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The Kobayashi metric of a complex ellipsoid in $$\mathbb{C}^ 2$$. (English) Zbl 0783.32012
In this paper the invariant Kobayashi metric on the complex ellipsoid in $$\mathbb{C}^ 2$$: $E_ m=\bigl\{(z_ 1,z_ 2)\in\mathbb{C}^ 2:| z_ 1|^ 2+| z_ 2|^{2m}<1\bigr\}$ for real $$m\geq{1\over 2}$$ is calculated, an explicit formula is given, in which an involved parameter is determined through solving a transcendental equation. Thus, complex ellipsoid becomes the third kind of domains for which the invariant metric is found explicitly, the other two are symmetric domains and Teichmüller space. It is also proved the obtained metric is $$C^ 1$$ on the tangent bundle away from the zero section. Besides, using the Monte-Carlo method, it is developed an algorithm implemented by software to calculate the infinitesimal Kobayashi metric on domains of the general form: $\Omega_ \rho=\bigl\{(z_ 1,z_ 2)\in\mathbb{C}^ 2:\rho(z_ 1,z_ 2)<0\bigr\}$ where $$\rho$$ is a real-valued polynomial. A comparison of the results obtained by computer calculation and from the explicit formula is presented.

##### MSC:
 32F45 Invariant metrics and pseudodistances in several complex variables
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