Plurisubharmonicity in a General Geometric Context

TL;DR

This paper extends the concept of plurisubharmonicity to a broad geometric context, including calibrated manifolds and non-geometric situations, establishing fundamental properties, regularity, and solving related Monge-Ampère equations.

Contribution

It introduces a wide-ranging framework for plurisubharmonicity applicable to various geometric and non-geometric contexts, with classical analytic methods.

Findings

Properties of $P^+$-plurisubharmonic functions established

Solutions to Dirichlet problem for Monge-Ampère-type equations provided

Topological restrictions and constructions of $P^+$-convex domains analyzed

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.