Plurisubharmonic Functions in Calibrated Geometries

TL;DR

This paper introduces and explores plurisubharmonic functions in calibrated geometry, extending complex analysis concepts to a broader geometric setting and establishing foundational properties and examples.

Contribution

It generalizes classical plurisubharmonic functions to calibrated manifolds, providing new tools and results in calibrated geometry.

Findings

Existence of abundant plurisubharmonic functions in calibrated manifolds

Extension of complex analysis results to calibrated geometries

Construction of strictly phi-convex spaces with diverse topologies

Abstract

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 many of 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 investigates, in depth, questions of: pseudo-convexity and cores, positive phi-currents, Duval-Sibony Duality, and boundaries of phi-submanifolds, all in the context of a general calibrated manifold (X,phi). Analogues of totally real submanifolds are used to construct enormous families of strictly phi-convex spaces with every topological type allowed by Morse Theory. Specific calibrations are used as examples throughout. Analogues of the Hodge Conjecture in calibrated geometry are considered.