The Richberg technique for subsolutions

Abstract

This note adapts the sophisticated Richberg technique for approximation in pluripotential theory to the $F$-potential theory associated to a general nonlinear convex subequation $F\subset J^2(X)$ on a manifold $X$. The main theorem is the following "local to global" result. Suppose $u$ is a continuous strictly $F$-subharmonic function such that each point $x\in X$ has a fundamental neighborhood system consisting of domains for which a "quasi" form of $C^{\infty }$ approximation holds. Then for any positive $h\in C(X)$ there exists a strictly $F$-subharmonic function $w\in C^{\infty }(X)$ with $u<w<u+h$. Applications include all convex constant coefficient subequations on ${\mathbf{R}}^n$, various nonlinear subequations on complex and almost complex manifolds, and many more.