Description of All Complex Geodesics in the Symmetrized Bidisc

Abstract

Effective formulae are found for the complex geodesics in the symmetrized bidisc.

Geometry of the symmetrized polydisc

“…[2,3,4], [7], [16]). Description of automorphisms in G 2 was given in [12], description of proper mappings in G 2 was given in [9].…”

Schwarz lemma for the tetrablock

“…Note that the symmetrized bidisc is (1, 2)-circular. The uniqueness result on geodesics and their complete description in G 2 are given in [16] and [5]. Here, we show how one can use our results to give a simpler proof of the description of all geodesics in G 2 passing through the origin.…”

“…The symmetrized bidisc and its higher dimensional analogue, the symmetrized polydisc G n , has been recently extensively studied (see e.g. [3], [4], [7], [16], [11], [5] and others). It was shown that G 2 cannot be exhausted by domains biholomorphic to convex ones (see [7], [10]).…”

“…It is simple to see that this condition is not only necessary but also sufficient (see e. g. [16] and [5]). …”

On automorphisms of the symmetrized bidisc

“…It has been recently studied by many authors, e.g. [1], [2], [4], [8]. The domain G 2 is the first known bounded pseudoconvex domain for which the Carathéodory and Lempert functions coincide, but which cannot be exhausted by domains biholomorphic to convex domains; cf.…”