An Introduction to Potential Theory in Calibrated Geometry

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 pluriharmo...

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Bibliographic Details
Published in:American journal of mathematics Vol. 131; no. 4; pp. 893 - 944
Main Authors: Harvey, F. Reese, Lawson, H. Blaine
Format: Journal Article
Language:English
Published: Baltimore, MD Johns Hopkins University Press 01.08.2009
Subjects:
Calibration
Exact sciences and technology
Exhaustion
General logic
General mathematics
General, history and biography
Geometry
Lagrangian function
Laplacians
Logic and foundations
Mathematical analysis
Mathematical functions
Mathematical logic, foundations, set theory
Mathematical manifolds
Mathematics
Potential theory
Riemann manifold
Sciences and techniques of general use
Several complex variables and analytic spaces
Subharmonics
Theory
Topological manifolds
Complex function
Morse theory
Type theory
Submanifold
Convex space
Potential theory
Duality
Mathematics
Calibration
Géométric measure
Geometry
Manifold
Model matching
Topological space
Plurisubharmonic function
Convexity
ISSN:0002-9327, 1080-6377, 1080-6377
Online Access:Get full text
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Summary: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, φ). Analogues of totally real submanifolds are introduced and used to construct enormous families of strictly , φ-convex spaces with every topological type allowed by Morse Theory. Specific calibrations are used as examples throughout. In a sequel, the duality between , φ-plurisubharmonic functions and , φ-positive currents is investigated. This study involves boundaries, generalized Jensen measures, and other geometric objects on a calibrated manifold.
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ISSN:0002-9327
1080-6377
1080-6377
DOI:10.1353/ajm.0.0067