An Introduction to Potential Theory in Calibrated Geometry

TL;DR

This paper introduces and explores plurisubharmonic functions within calibrated geometry, extending complex analysis concepts to a broader geometric setting and establishing properties like pseudo-convexity and constructing diverse convex spaces.

Contribution

It generalizes classical plurisubharmonic functions to calibrated manifolds, introduces pseudo-convexity in this context, and constructs extensive families of convex spaces with varied topologies.

Findings

Plurisubharmonic functions exist abundantly in calibrated geometry.

Pseudo-convexity concepts are extended to calibrated manifolds.

Constructed diverse strictly -convex spaces with various topologies.

Abstract

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. In particular, the notion of pseudo-convexity for a calibrated manifold (X,\phi) is introduced and studied. Analogues of totally real submanifolds are also introduced and 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.