Potential Theory on Almost Complex Manifolds

Abstract

Pseudo-holomorphic curves on almost complex manifolds have been much more intensely studied than their "dual" objects, the plurisubharmonic functions. These functions are defined classically by requiring that the restriction to each pseudo-holomorphic curve is subharmonic. In this paper subharmonic 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 proved. The first is a restriction theorem which establishes the equivalence of our definition with the "classical" definition. In the second theorem, using our "viscosity" definitions, the Dirichlet problem is solved for the complex Monge-Amp\`ere equation in both the homogeneous and inhomogeneous forms. These two results are based on theorems found in two recent papers of the authors. Finally, it is shown that the plurisubharmonic functions considered here agree with the plurisubharmonic distributions. In particular, this proves a conjecture of Nefton Pali.