Bergman isometries between convex domains in ℂ2which are polyhedral

Abstract

This article deals with the problem of analyticity of Bergman isometries. One of the most important properties of the Bergman metric of a bounded domain is that it is invariant under the action of the group of biholomorphic maps. One then can ask if all the isometries are indeed complex analytic up to an obvious complex conjugation. There are several affermative answers to this question. In the present work, we study the case of convex polyhedral domains in C and we prove that any Bergman isometry of such a domain is analytic up to a complex conjugation.