2 A fully nonlinear version of the Yamabe problem and a Harnack type inequality

We present some results in [9], a continuation of our earlier works [7] and [8]. One result is the existence and compactness of solutions to a fully nonlinear version of the Yamabe problem on locally conformally flat Riemannian manifolds, and the other is a Harnack type inequality for general conformally invariant fully nonlinear second order elliptic equations. Let (M, g) be an n−dimensional, compact, smooth Riemannian manifold without boundary, n ≥ 3, consider the Weyl-Schouten tensor Ag = 1 n−2 ( Ricg − Rg 2(n−1)g ) , where Ricg and Rg denote respectively the Ricci tensor and the scalar curvature associated with g. We use λ(Ag) to denote the eigenvalues of Ag with respect to g. Let ĝ = u 4 n−2 g be a conformal change of metrics, then (see, e.g., [17]),