Three Structure Theorems in Several Complex Variables by Reese Harvey

p-convexity, p-plurisubharmonicity and the Levi Problem

Three results in p-convex geometry are established. First is the analogue of the Levi problem in several complex variables: namely, local p-convexity implies global p-convexity. The second asserts that the support of a minimal p-dimensional current is contained in the union of the p-hull of the boundary with the “core” of the space. Lastly, the extreme rays in the convex cone of p-positive matr...

Potential Theory on Almost Complex Manifolds

Pseudo-holomorphic curves on almost complex manifolds have been much more intensely studied than their “dual” objects, the plurisubharmonic functions. These functions are standardly defined by requiring that the restriction to each pseudo-holomorphic curve be subharmonic. In this paper subharmonic functions are defined by applying the viscosity approach to a version of the complex hessian which...

Existence, Uniqueness and Removable Singularities for Nonlinear Partial Differential Equations in Geometry

This paper surveys some recent results on existence, uniqueness and removable singularities for fully nonlinear differential equations on manifolds. The discussion also treats restriction theorems and the strong Bellman principle.

Tangents of plurisubharmonic functions

Tangents of plurisubharmonic functions Local properties of plurisubharmonic functions are studied by means of the notion of tangent which describes the behavior of the function near a given point. We show that there are plurisubharmonic functions with several tangents, disproving a conjecture of Reese Harvey.

Holomorphic Approximation on Totally Real Sub- Manifolds of a Complex Manifold