Prescribed k-Curvature Problems on Complete Noncompact Riemannian Manifolds

Abstract To study the prescribed $k$-curvature problem of 2nd-order symmetric curvature tensors on complete noncompact Riemannian manifolds, we consider a class of fully nonlinear elliptic partial differential equations. It is proved that on a Riemannian manifold with negative sectional curvature and Ricci curvature bounded from below, the equation is solvable provided that all the eigenvalues of the tensor are negative. The result is applicable to the prescribed $k$-curvature problems of modified Schouten tensor and Bakry–Émery Ricci tensor.