Conformal deformation of a Riemannian metric to constant scalar curvature. (English) Zbl 0576.53028

Schoen, Richard

J. Differ. Geom. 20, 479-495 (1984).

The author announces the existence of metrics of constant scalar curvature in any conformal class on compact manifolds which are conformally flat near a point or which are 3-, 4- or 5-dimensional. This together with previous arguments due to Thierry Aubin settles the proof of the Yamabe conjecture.
Two new ideas are involved. Assuming first that the metric is conformally flat near a point, the author constructs a function on which the Yamabe functional takes on a value below the critical one. This function solves an exterior boundary value problem near that point. Its existence relies on an extension to any dimension of the theorem due to S. T. Yau and the author asserting that the total mass is positive for asymptotically Euclidean metrics with nonnegative scalar curvature. (This part of the proof is not yet in print.) Secondly, a delicate perturbation argument working in dimensions 3, 4 and 5 allows the author to get rid of the assumption of conformal flatness at a point.

Reviewer: J.P.Bourguignon

MSC:

53C20	Global Riemannian geometry, including pinching
58E11	Critical metrics

Keywords:

conformal class; conformally flat near a point; Yamabe conjecture; total mass