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Is the space of metrics of negative sectional curvature over a closed 3-manifold connected? If so, in what paper is this result stated?

Note: as the Ricci flow hyperbolizes negatively curved metrics, the question is whether this or some similar flow remains within the space of negatively curved metrics.

Thanks in advance!

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    $\begingroup$ I think this might be open. The space of hyperbolic metrics is contractible by a result of Gabai. So if one can deform any negatively curved metric to a hyperbolic metric, then those would also be connected. Chow-Hamilton studied the cross curvature flow for negatively curved metrics, showing short time existence. They expect it to converge to the hyperbolic metric, but this still appears to be open. journals.tubitak.gov.tr/math/vol28/iss1/1 $\endgroup$
    – Ian Agol
    Commented yesterday

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