Duality of Positive Currents and Plurisubharmonic Functions in Calibrated Geometry

TL;DR

This paper explores the duality between urrents and unctions in calibrated geometry, establishing a duality principle and characterizations that extend potential theory concepts to this setting.

Contribution

It demonstrates that unctions are polar duals of urrents in calibrated manifolds, extending duality principles and characterizations in potential theory.

Findings

Established an analogue of Duval-Sibony Duality in calibrated geometry.

Characterized urrents' boundaries using unctions.

Linked urrents and unctions through duality principles.

Abstract

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