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, φ). Analogues of totally real submanifolds are introduced and used to construct enormous families of strictly φ-convex spaces with every topological type allowed by Morse Theory. Specific calibrations are used as examples throughout.

In a sequel, the duality between φ-plurisubharmonic functions and φ-positive currents is investigated. This study involves boundaries, generalized Jensen measures, and other geometric objects on a calibrated manifold.

Recently the authors showed that there is a robust potential theory attached to any calibrated manifold $(X, ø). In particular, on X there exist ø -plurisubharmonic functions, ø-convex domains, ø-convex boundaries, etc., all inter-related and having a number of good properties. In this paper we show that, in a strong sense, the plurisubharmonic functions are the polar duals of the ø-submanifolds, or more generally, the ø currents studied in the original paper on calibrations. In particular, we establish an analogue of Duval-Sibony Duality which characterizes points in the ø-convex hull of a compact set K C X in terms of ø-positive Green's currents on X and Jensen measures on K. We also characterize boundaries of ø currents entirely in terms of ø-plurisubharmonic functions. Specific calibrations are used as examples throughout. Analogues of the Hodge Conjecture in calibrated geometry are considered.