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In Besse's "EInstein manifolds", p. 354, the question is posed if the volume of Einstein metrics on a given compact manifold (normalized such that $Ric=\pm(n-1)g$) take only finitely many values.

For negative values, they say that no example of more than one volume on a fixed manifold is known. Are there still no such examples?

I'd like to know if there is any recent progress which is related to this question.

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