For every pair of positive integers (m, n) with n >= 2 there exists an R-parameter family of mutually Bergman-inequivalent Reinhardt domains in C^n whose Bergman metrics are locally isometric to m times the Fubini-Study metric.
Yamamori,Origin rigidity induced by monomial structure of Bergman spaces on Reinhardt domains, J
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Abundance of Bergman metrics with constant positive holomorphic sectional curvature
For every pair of positive integers (m, n) with n >= 2 there exists an R-parameter family of mutually Bergman-inequivalent Reinhardt domains in C^n whose Bergman metrics are locally isometric to m times the Fubini-Study metric.