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Oct 5, 2014 at 4:01 comment added Igor Belegradek I think the metric coming from the solution of Calabi conjecture need not have negative sectional curvature. The conjecture I mention in the first comment is taken from [Siu, Yum Tong; Yang, Paul Compact Kähler-Einstein surfaces of nonpositive bisectional curvature. Invent. Math. 64 (1981), no. 3, 471–487] whose authors discuss Mostow-Sie example in the introduction.
Oct 5, 2014 at 3:56 comment added Igor Rivin @IgorBelegradek I believe it is K-E, as follows from the Calabi conjecture.
Oct 5, 2014 at 3:41 comment added Igor Belegradek If memory serves, Mostow-Siu metric is not Einstein. I think there is a conjecture that any Kahler-Einstein surface of negative sectional curvature is covered by $CH^2$.
Oct 5, 2014 at 3:30 history answered Igor Rivin CC BY-SA 3.0