Abstract: The study of conformal metrics with constant scalar curvature on compact manifolds, known as the Yamabe problem, has been completely solved by Neil Trudinger, Thierry Aubin and Richard Schoen. The nonlinear Yamabe problem, concerned with the Schouten tensor, defined on compact manifolds, are also formulated by elliptic approach or flow method. Another generalization to Yamabe problem is to consider non-compact manifolds. On the other hand, in the spirit of Lohkamp's theorem, which asserts that any smooth manifolds admit complete metrics with negative Ricci curvature. It seems an interesting quesiton whether there always exists a complete conformal metric with negative Ricci curvature. The answer is obviously negative for compact manifolds without boundary from the maximum principle. Guan has proved that the answer is yes on compact manifolds with boundary. Motivated by the formulation of Yamabe problem as well as Guan's existence theorem regarding Ricci curvature on compact manifolds with boundary, a natural question to ask is whether such complete conformal metrics, with prescribed symmetric functions of the eigenvalues of the Ricci tensor defined on negative cones, exist on certain non-compact manifolds? The affirmative answer is obtained for Euclidean spaces, on which we are also able to detect the nonexistence of such metrics under certain decay conditions.
