An introduction to potential theory in calibrated geometry

In this paper we introduce and study the notion of plurisubharmonic functions in calibrated geometry. These functions generalize the classical plurisubharmonic functions from complex geometry and enjoy their important properties. Moreover, they exist in abundance whereas the corresponding pluriharmonics are generally quite scarce. A number of the results established in complex analysis via plurisubharmonic functions are extended to calibrated manifolds. This paper introduces and investigates questions of pseudo-convexity in the context of a general calibrated manifold (X, &phis;). Analogues of totally real submanifolds are introduced and used to construct enormous families of strictly &phis;-convex spaces with every topological type allowed by Morse Theory. Specific calibrations are used as examples throughout. In a sequel, the duality between &phis;-plurisubharmonic functions and &phis;-positive currents is investigated. This study involves boundaries, generalized Jensen measures, and other geometric objects on a calibrated manifold.