Monge-Ampère Equations on Riemannian Manifolds

where gij denotes the metric of M , g = det(gij) > 0 and φ ∈ C∞(∂Ω), ψ > 0 is C∞ with respect to (x, z, p) ∈ Ω̄× R× TxM , TxM denotes the tangent space at x ∈M . Monge-Ampère equations arise naturally from some problems in differential geometry. The Dirichlet problem in Euclidean space R has been widely investigated. In this case the solvability has been reduced to the existence of strictly convex subsolutions by Caffarelli, Nirenberg and Spruck [2] and independently by Krylov [8] for strictly convex domains Ω in R. More recently, Spruck and the first author [7] treated the problem for non-convex domains in R as well as on S in connection with the geometric problem of finding hypersurfaces in R of constant Gauss curvature with prescribed boundary. In this paper we extend some of the known results in R to arbitrary Riemannian Manifolds. Our main result is the following analogue of Theorem 0.3 of [7].