Potential theory on almost complex manifolds

On almost complex manifolds the pseudo-holomorphic curves have been much more intensely studied than their "dual" objects, the plurisubharmonic functions. These functions are standardly defined by requiring that the restriction to each pseudo-holomorphic curve be subharmonic. In this paper plurisubharmonic functions are defined by applying the viscosity approach to a version of the complex hessian which exists intrinsically on any almost complex manifold. Three theorems are proven. The first is a restriction theorem which establishes the equivalence of our definition with the "standard" definition. In the second theorem, using our "viscosity" definitions, the Dirichlet problem is solved for the complex Monge-Ampère equation in both the homogeneous and inhomogeneous forms. Finally, it is shown that the plurisubharmonic functions considered here agree in a precise way with the plurisubharmonic distributions. In particular, this proves a conjecture of Nefton Pali.