The Yamabe problem on manifolds with boundary: Existence and compactness results

Let (M, g) be an n-dimensional, compact, smooth, Riemannian manifold without boundary. For n = 2, we know from the uniformization theorem of Poincar that there exist metrics that are pointwise conformal to g and have constant Gauss curvature. For n 3, the well-known Yamabe conjecture states that there exist metrics that are pointwise conformal to g and have constant scalar curvature. The answer to the Yamabe conjecture is proved to be affirmative through the work of Yamabe See Lee and Parker See also Bahri and Brezis [3] and Bahri [2] for works on the Yamabe problem and related ones. For n 3, let g = u 4/(n-2) g for some positive function u > 0 on M; the scalar curvature R g of g can be calculated as