An introduction to potential theory in calibrated geometry

By F. R. Harvey, H. Lawson. 2007

74 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/a66ddfa0fd9822255f8f39dc4d3e8feea2088d21

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Dirichlet duality and the nonlinear Dirichlet problem (2007, background): “The attendant notions of -plurisubharmonic functions and -convexity are discussed in detail in [12].”
Geometric plurisubharmonicity and convexity - an introduction (2011, methodology): “tly Gl -plurisubharmonic on a small enough collar of ∂Ω. For the proof of this, compute Hess(−log(−ρ)) and mimick the proof of Theorem 5.7 on the hypersurfaces {ρ= ǫ} (see the proof of Theorem 5.6 in [HL1]). Remark 5.10. (Signed Distance). Recall that a deﬁning function ρfor Ω satisﬁes |∇ρ| ≡ 1 in a”
Plurisubharmonicity in a General Geometric Context (2008, unknown): “Let ρ denote the signed distance function to ∂, i.e., ρ(x) = −dist(x, ∂) for x ∈ and ρ(x) = dist(x, ∂) for x / ∈ (so ρ is a defining function for ∂). One computes (see [ HL2 , §5]) that for points x ∈ ∂”