Bergman coordinates

Steven R. Bell¹

¹ Mathematics Department Purdue University West Lafayette, IN 47907, U.S.A.

Studia Mathematica, Tome 176 (2006) no. 1, pp. 69-83

Résumé

Various incarnations of Stefan Bergman's notion of representative coordinates will be given that are useful in a variety of contexts. Bergman wanted his coordinates to map to canonical regions, but they fail to do this for multiply connected regions. We show, however, that it is possible to define generalized Bergman coordinates that map multiply connected domains to quadrature domains which satisfy a long list of desirable properties, making them excellent candidates to be called Bergman representative domains. We also construct a kind of Bergman coordinate that maps a domain to an algebraic variety in $\mathbb C^2$ in a natural way, and thereby show that Bergman-style coordinates can be used to convert problems in conformal mapping to problems in algebraic geometry. Many of these results generalize routinely to finite Riemann surfaces.

Détail

DOI : 10.4064/sm176-1-5

Keywords: various incarnations stefan bergmans notion representative coordinates given useful variety contexts bergman wanted his coordinates map canonical regions fail multiply connected regions however possible define generalized bergman coordinates map multiply connected domains quadrature domains which satisfy long list desirable properties making excellent candidates called bergman representative domains construct kind bergman coordinate maps domain algebraic variety mathbb natural thereby bergman style coordinates convert problems conformal mapping problems algebraic geometry many these results generalize routinely finite riemann surfaces