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Does a compact four-dimensional self-dual Einstein manifold with negative scalar curvature have negative sectional curvature? This would be true if we believe the folklore conjecture that a compact negative-scalar-curvature SD Einstein 4-manifold is either a real-hyperbolic 4-manifold or a complex-hyperbolic 4-manifold.

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A mathematical comment: there are examples of complete (but not compact) SD Einstein metrics with negative scalar curvature which have non-trivial pi_2 and so don't admit any negatively curved metric. (See, e.g., Calderbank-Singer "Einstein metrics and complex singularities".) –  Joel Fine Nov 13 '13 at 12:52
    
I think that this is an open question. The fact that the folklore conjecture you state is still open shows that the only known examples of compact negative scalar curvature SD Einstein metrics are hyperbolic and complex-hyperbolic. In fact, its interesting that you call this a "folklore conjecture". I've asked several people whether they believe there are any other compact, negative scalar-curvature SD Einstein manifolds. Some say they hope so, some say they have no idea, but no-one has been brave enough to conjecture either way! –  Joel Fine Nov 13 '13 at 12:55

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