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Duality of positive currents and plurisubharmonic functions in calibrated geometry

By F. Reese Harvey and H. Blaine Lawson


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

Year: 2009
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