1.1. The k-Yamabe problem. The k-Yamabe problem is a higher order extension of the celebrated Yamabe problem for scalar curvature. It was initially proposed by Viaclovsky [72] and also arose in the study of Q-curvatures in [11]. Viaclovsky found that in a conformal metric, the resultant k-curvature equation can be expressed as an equation similar to the k-Hessian equation, which has been studied by many authors [77]. The k-Yamabe problem can be formulated as follows. Let (Mn, g) be a closed Riemannian manifold of dimension n ≥ 3. It is well-known that there is an orthonormal decomposition of the Riemannian curvature tensor