# Examples of Einstein four-manifolds of negative sectional curvature

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.

• 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. Oct 5, 2014 at 12:00
• 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 Oct 5, 2014 at 13:48
• 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. Nov 4, 2014 at 19:45
• An example has been announced by Joel Fine and Bruno Premoselli, see here scgp.stonybrook.edu/video_portal/video.php?id=3382 Nov 3, 2017 at 12:42
• And the paper just came out: arxiv.org/abs/1802.00608 Feb 5, 2018 at 9:16

• 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 3:41