Geometric plurisubharmonicity and convexity - an introduction

TL;DR

This paper introduces potential theory for geometric plurisubharmonic functions on Riemannian manifolds, extending complex analysis concepts to a broader geometric setting with nonlinear PDEs.

Contribution

It develops a framework for G-plurisubharmonic functions and related notions, establishing existence, uniqueness, and restriction results for G-harmonic functions.

Findings

Extension of complex analysis concepts to Riemannian geometry

Existence and uniqueness of solutions to the Dirichlet Problem for G-harmonic functions

Development of a nonlinear PDE framework for geometric plurisubharmonicity

Abstract

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.