Geometric Plurisubharmonicity and Convexity - an Introduction

This is an essay on potential theory for geometric plurisubharmonic functions. It begins with a given closed subset Gl 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 < dimX). A surprising number of results in complex analysis carry over to this more general setting. The notions of: a Gl -submanifold, an upper semi-continuous Gl -plurisubharmonic function, a Gl -convex domain, a Gl -harmonic function, and aGl -free submanifold, are defined. Results include a restriction theorem as well as the existence and uniqueness of solutions to the Dirichlet Problem for Gl -harmonic functions on Gl -convex domains.