Dirichlet duality and the nonlinear dirichlet problem on riemannian manifolds

In this paper we study the Dirichlet problem for fully nonlinear second-order equations on a riemannian manifold. As in our previous paper [HL₄], we define equations via closed subsets of the 2-jet bundle where each equation has a natural dual equation. Basic existence and uniqueness theorems are established in a wide variety of settings. However, the emphasis is on starting with a constant coefficient equation as a model, which then universally determines an equation on every riemannian manifold which is equipped with a topological reduction of the structure group to the invariance group of the model. For example, this covers all branches of the homogeneous complex Monge-Ampére equation on an almost complex hermitian manifold X. In general, for an equation F on a manifold X and a smooth domain Ω −−X, we give geometric conditions which imply that the Dirichlet problem on is uniquely solvable for all continuous boundary functions. We begin by introducing a weakened form of comparison which has the advantage that local implies global. We then introduce two fundamental concepts. The first is the notion of a monotonicity cone M for F. If X carries a global M- subharmonic function, then weak comparison implies full comparison. The second notion is that of boundary F-convexity, which is defined in terms of the asymptotics of F and is used to define barriers. In combining these notions the Dirichlet problem becomes uniquely solvable as claimed. This article also introduces the notion of local affine jet-equiva- lence for subequations. It is used in treating the cases above, but gives results for a much broader spectrum of equations on manifolds, including inhomogeneous equations and the Calabi-Yau equation on almost complex hermitian manifolds. A considerable portion of the paper is concerned with specific examples. They include a wide variety of equations which make sense on any riemannian manifold, and many which hold universally on almost complex or quaternionic hermitian manifolds, or topologically calibrated manifolds.