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 Teichm├╝ller 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.


32F45 Invariant metrics and pseudodistances in several complex variables
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