Convergence of the Yamabe flow in dimension 6 and higher

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A Positive Mass Theorem in Three Dimensional Cauchy–Riemann Geometry (2013, background): “The positive mass theorem played a key role in other problems in conformal geometry, for example in the compactness of the Yamabe equation or the Yamabe flow, see [7], [8], [30].”
An existence theorem of conformal scalar-flat metrics on manifolds with boundary (2009, methodology): “[8] S. Brendle, ”Convergence of the Yamabe flow in dimension 6 and higher”, Invent.”
The classification of locally conformally flat Yamabe solitons (2011, background): “In [2] and [3] it has been showed that if 3 ≤ n ≤ 5 or if n ≥ 6 (in the latter case Brendle imposes that the metric is either locally conformally flat or he assumes a certain condition on the rate of vanishing of Weyl tensor at the points at which it”