Compactness for Yamabe metrics in low dimensions

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Compactness of solutions to the Yamabe problem (2004, unknown): “The case n = 5 was proved independently by Druet [ 45 ].”
A Compactness Theorem for the Yamabe Problem (2009, background): “[9] O. Druet, Compactness for Yamabe metrics in low dimensions, Int.”
Blow-up phenomena for the Yamabe equation II (2009, background): “Druet [6] proved the conjecture in dimensions 4 and 5.”
A Neumann problem with critical exponent in nonconvex domains and Lin-Ni’s conjecture (2010, unknown): “See [18], [33], [35], [36] and [39].”