Communications in Analysis and Geometry

Volume 28 (2020)

Number 8

The Second of Two Special Issues in Honor of Karen Uhlenbeck’s 75th Birthday

Special-Issue Editors: Georgios Daskalopoulos (Brown University), Kefeng Liu, Chuu-Lian Terng (U. of Cal. Irvine), and Shing-Tung Yau

The Richberg technique for subsolutions

Pages: 1787 – 1806

DOI: https://dx.doi.org/10.4310/CAG.2020.v28.n8.a2

Authors

F. Reese Harvey (Department of Mathematics, Rice University, Houston, Texas, U.S.A.)

H. Blaine Lawson, Jr. (Department of Mathematics, Stony Brook University, Stony Brook, New York, U.S.A.)

Szymon Pliś (Institute of Mathematics, Cracow University of Technology, Kraków, Poland)

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 $ \in C(X)$ there exists a strictly $F$-subharmonic function $w \in C^\infty (X)$ with $u \lt w \lt u + h$. Applications include all convex constant coefficient subequations on $\mathbb{R}^n$, various nonlinear subequations on complex and almost complex manifolds, and many more.

Received 29 October 2017

Accepted 4 August 2019

Published 8 January 2021