Bergman isometries between convex domains in \(\mathbb{C}^ 2\) which are polyhedral (Q1816545)

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 can then ask if all the isometries are indeed complex analytic up to an obvious complex conjugation. There are several affirmative answers to this question which dates back to the time of Oka. In the present work, we study the case of convex polyhedral domains in \(\mathbb{C}^2\). The main result is: Theorem. Let \(\Omega \subset \mathbb{C}^2\) be a convex polyhedral domain with piecewise Levi flat boundary. Then \(\Aut (\Omega)\) has index at most 4 as subgroup of \(\text{Iso} (\Omega)\).