Geometric plurisubharmonicity and convexity - an introduction

By F. R. Harvey, H. Lawson. 2011

107 citations recorded by the source corpus.

This is an essay on potential theory for geometric plurisubharmonic functions. It begins with a given closed subset G of the Grassmann bundle $G(p,TX)$ of tangent $p$-planes to a riemannian manifold $X$. This determines a nonlinear partial differential equation which is convex but never uniformly elliptic (p < dim X). A surprising number of results in complex analysis carry over to this more general setting. The notions of: a G-submanifold, an upper semi-continuous G-plurisubharmonic function, a G-convex domain, a G-harmonic function, and a G-free submanifold, are defined. Results include a restriction theorem as well as the existence and uniqueness of solutions to the Dirichlet Problem for G-harmonic functions on G-convex domains.

Subfields: Applied Mathematics, Astronomy and Astrophysics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/8f8bec2c8821ea9542ed3fc52c94925c2a574019

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In Applied Mathematics

In Astronomy and Astrophysics

In Geometry and Topology

Citations in the corpus

p-convexity, p-plurisubharmonicity and the Levi problem (2011, background): “f K to be the set Kb of points x ∈ X such that u(x) ≤ sup K u for all smooth p-plurisubharmonic functions u on X. Then X is said to be p-convex if K ⊂⊂ X ⇒ Kb ⊂⊂ X. The following result was proven in [HL7]. A riemannian manifold X is p-convex if and only if X admits a smooth p-plurisubharmonic prope”
The Monge-Ampère equation for (n − 1)-plurisubharmonic functions on a compact Kähler manifold (2013, background): “Harvey-Lawson [27, 28] introduced a more general notion of k-plurisubharmonicity, where the complex lines in this second definition are replaced by complex k-planes.”