Are there any nontrivial compact Einstein fourmanifolds of negative or nonpositive sectional curvature? by nontrivial we mean not quotients of $\mathbb{H}^4$, $\mathbb{C}H^2$, $\mathbb{H}^2\times\mathbb{H}^2$, or $\mathbb{H}^2\times\mathbb{R}^2$. I did some search but couldn't find any.
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3$\begingroup$ I think this is an open problem, and things have not changed since Anderson's survey arxiv.org/abs/0810.4830. At some point Anderson claimed the existence of such examples, see arxiv.org/abs/math/0310041, but it was later retracted. $\endgroup$– Igor BelegradekOct 5, 2014 at 12:00

3$\begingroup$ you possibly want to add $H^2\times R^2$ to the list of trivial examples, since the product of a hyperbolic surface and the 2torus is KählerEinstein $\endgroup$– Vladimir S MatveevOct 5, 2014 at 13:48

4$\begingroup$ using Dehn filling on cusped hyperbolic manifolds (see the papers of Anderson and Bamler) you can construct plenty of manifolds that admit both Einstein and nonpositive sectional curvature metrics, but not simultaneously as far as I understand. $\endgroup$– Bruno MartelliNov 4, 2014 at 19:45

2$\begingroup$ An example has been announced by Joel Fine and Bruno Premoselli, see here scgp.stonybrook.edu/video_portal/video.php?id=3382 $\endgroup$– YangMillsNov 3, 2017 at 12:42

3$\begingroup$ And the paper just came out: arxiv.org/abs/1802.00608 $\endgroup$– YangMillsFeb 5, 2018 at 9:16

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1 Answer
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G. Mostow and Y.T. Siu, A compact Kahler surface of negative curvature not covered by the ball, Ann. Math. 112 (1980) 321360

2$\begingroup$ If memory serves, MostowSiu metric is not Einstein. I think there is a conjecture that any KahlerEinstein surface of negative sectional curvature is covered by $CH^2$. $\endgroup$ Oct 5, 2014 at 3:41

$\begingroup$ @IgorBelegradek I believe it is KE, as follows from the Calabi conjecture. $\endgroup$ Oct 5, 2014 at 3:56

14$\begingroup$ 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 KählerEinstein surfaces of nonpositive bisectional curvature. Invent. Math. 64 (1981), no. 3, 471–487] whose authors discuss MostowSie example in the introduction. $\endgroup$ Oct 5, 2014 at 4:01