Type: Article
Publication Date: 1999-09-15
Citations: 149
DOI: https://doi.org/10.1215/s0012-7094-99-09916-7
Let (M, g) be an n-dimensional, compact, smooth, Riemannian manifold without boundary.For n = 2, we know from the uniformization theorem of Poincaré 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 [39], Trudinger [38], Aubin [1], and Schoen [31].See Lee and Parker [23] for a survey.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