The modified Yamabe problem and geometry of modified scalar curvatures

Study of modified scalar curvatures and the modified Yamabe problem is given. An existence theorem of type of Trudinger and Aubin is obtained. Interpreting Isotropic curvature as a certain modified scalar curvature we can extend the four-dimensional sphere theorem of 23]. Another modified scalar curvature is used in solving open problems concerning the conjecture of self-dual Einstein 4-manifold and the Goldberg conjecture, both being under negative Ricci condition.