Plurisubharmonicity in a General Geometric Context

Abstract

Recently the authors have explored new concepts of plurisubharmonicity and pseudoconvexity, with much of the attendant analysis, in the context of calibrated manifolds. Here a much broader extension is made. This development covers a wide variety of geometric situations, including, for example, Lagrangian plurisubhamonicity and convexity. It also applies in a number of non-geometric situations. Results include: fundamental properties of $P^{+}$-plurisubharmonic functions, plurisubharmonic distributions and regularity, $P^{+}$-convex domains and $P^{+}$-convex boundaries, topological restrictions on and construction of such domains, continuity of upper envelopes, and solutions of the Dirichlet problem for related Monge-Ampere-type equations. Many results in this paper have been generalized in recent work of the authors. However, this article covers many cases of geometric interest, and certain convexity assumptions here allow the use of classical analytic methods, making the exposition more accessible.

Cited by in corpus (9)

• Dirichlet Duality and the Nonlinear Dirichlet Problem on Riemannian Manifolds
• Removable singularities for nonlinear subequations
• Hyperbolic domains in real Euclidean spaces
• Lagrangian potential theory and a Lagrangian equation of Monge-Ampère type
• Parallel calibrations and minimal submanifolds
• Every smoothly bounded p-convex domain in R^n admits a p-plurisubharmonic defining function
• The Dirichlet Problem with Prescribed Asymptotic Singularities
• The equivalence of viscosity and distributional subsolutions for convex subequations - a strong Bellman principle
• The quaternionic Monge-Ampère operator and plurisubharmonic functions on the Heisenberg group