-
An introduction to potential theory in calibrated geometry
- American Journal of Mathematics
- Johns Hopkins University Press
- Volume 131, Number 4, August 2009
- pp. 893-944
- 10.1353/ajm.0.0067
- Article
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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 $(X,\phi)$.
Analogues of totally real submanifolds are 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.
In a sequel, the duality between $\phi$-pluri\-sub\-harmonic functions and
$\phi$-positive
currents is investigated.
This study involves boundaries,
generalized Jensen measures, and other geometric objects on a calibrated manifold.