Extremal holomorphic maps in special classes of domains

Abstract

In the paper we discuss three different notions of extremal holomorphic mappings: weak $m$-extremals, $m$-extremals and $m$-complex geodesics. We discuss relations between them in general case and in the special cases of unit ball, classical Cartan domains, symmetrised bidisc and tetrablock. In particular we show that weak $3$-extremal maps in the symmetrised bidisc are rational thus giving the (partial) answer to a problem posed in a recent paper by J. Agler, Z. Lykova and N. J. Young.

arXiv comment26 pages, revised version in the form that is accepted for publication in Annali della Scuola Normale Superiore di Pisa, Classe di Scienze