A Compactness Theorem for the Yamabe Problem

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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].”
Bifurcation and local rigidity of homogeneous solutions to the Yamabe problem on spheres (2011, background): “[13] obtained remarkable results on compactness and non-compactness of this space of solutions.”
Blow-up solutions for asymptotically critical elliptic equations on Riemannian manifolds (2009, background): “His conjecture was very recently proved to be true in case n ≤ 24 by Khuri–Marques–Schoen [13].”
A Neumann problem with critical exponent in nonconvex domains and Lin-Ni’s conjecture (2010, unknown): “See [18], [33], [35], [36] and [39].”
A compactness theorem for a fully nonlinear Yamabe problem under a lower Ricci curvature bound (2012, background): “For 8 ≤ n ≤ 24, it was proved that the answer to Problem 1.2 is positive provided that the positive mass theorem holds in these dimensions; see Li and Zhang [32, 33] for 8 ≤ n ≤ 11, and Khuri, Marques and Schoen [24] for 12 ≤ n ≤ 24.”
An existence theorem of conformal scalar-flat metrics on manifolds with boundary (2009, methodology): “[26] R. Schoen, ”Conformal deformation of a Riemannian metric to constant scalar curvature”, J. Differential Geom.”