Parabolic exhaustions for strictly convex domains
Given \(D\subset {\mathbb{C}}^ n\) a strictly convex domain and \(p\in D\) any point, the author defines a quasi-strictly parabolic exhaustion \(\tau\) of D with center p. The construction uses essentially the work of \textit{L. Lempert} on the Kobayashi metric [see Bull. Soc. Math. Fr. 109, 427-474 (1981; Zbl 0492.32025)]. Then to D and p are associated in a canonical way a complete circular domain \(G\subset {\mathbb{C}}^ n\) and a homeomorphism \(h: \bar G\to \bar D\) such that \(h(0)=p\), called the circular representation. The main result is: The circular representation is biholomorphic iff the Monge-Ampère foliation induced by the quasi-strictly parabolic exhaustion \(\tau\) of D centered at p is holomorphic. As a consequence a characterisation of the strictly convex domains D biholomorphic to a circular domain or to a ball in terms of the properties of the automorphism group of D is obtained.
scientific article; zbMATH DE number 3929427
scientific article