Non-Einstein Bergman metrics on bounded domains with compact quotients

This paper constructs a bounded contractible pseudoconvex domain in complex dimension two with a smooth projective surface as its compact quotient. The domain is not biholomorphic to a product of positive-dimensional complex manifolds, and its Bergman metric is not Kähler–Einstein. This gives a negative answer to the question in Problem 44 of whether the Bergman and complete Kähler–Einstein metrics necessarily coincide on such domains. Repository: https://github.com/opensource120/Some-Answers-to-Yau-s-120-open-problems-in-geometryPDF at the deposited version: https://github.com/opensource120/Some-Answers-to-Yau-s-120-open-problems-in-geometry/blob/b4f17a71d1712073bf82196a01c29ae04292e5c1/papers/problem_44.pdfTeX source at the deposited version: https://github.com/opensource120/Some-Answers-to-Yau-s-120-open-problems-in-geometry/blob/b4f17a71d1712073bf82196a01c29ae04292e5c1/papers/yau_problem_44.tex