Are there any nontrivial compact Einstein four-manifolds 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 BelegradekCommented Oct 5, 2014 at 12:00
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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 2-torus is Kähler-Einstein $\endgroup$– Vladimir S MatveevCommented Oct 5, 2014 at 13:48
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5$\begingroup$ 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 non-positive sectional curvature metrics, but not simultaneously as far as I understand. $\endgroup$– Bruno MartelliCommented Nov 4, 2014 at 19:45
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5$\begingroup$ An example has been announced by Joel Fine and Bruno Premoselli, see here scgp.stonybrook.edu/video_portal/video.php?id=3382 $\endgroup$– YangMillsCommented Nov 3, 2017 at 12:42
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5$\begingroup$ And the paper just came out: arxiv.org/abs/1802.00608 $\endgroup$– YangMillsCommented Feb 5, 2018 at 9:16
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2 Answers
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This is answered in this paper, based on a construction of Gromov-Thurston (as commented by @YangMills):
Fine, Joel; Premoselli, Bruno, Examples of compact Einstein four-manifolds with negative curvature, J. Am. Math. Soc. 33, No. 4, 991-1038 (2020). MR4155218 ZBL1467.53055.
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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) 321-360
answered Oct 5, 2014 at 3:30
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2$\begingroup$ 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 $CH^2$. $\endgroup$ Commented Oct 5, 2014 at 3:41
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$\begingroup$ @IgorBelegradek I believe it is K-E, as follows from the Calabi conjecture. $\endgroup$ Commented Oct 5, 2014 at 3:56
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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 Kähler-Einstein surfaces of nonpositive bisectional curvature. Invent. Math. 64 (1981), no. 3, 471–487] whose authors discuss Mostow-Sie example in the introduction. $\endgroup$ Commented Oct 5, 2014 at 4:01