Smooth Approximation of Plurisubharmonic Functions on Almost Complex Manifolds
Abstract
This note establishes smooth approximation from above for J-plurisubharmonic functions on an almost complex manifold (X,J). The following theorem is proved. Suppose X is J-pseudoconvex, i.e., X admits a smooth strictly J-plurisubharmonic exhaustion function. Let u be an (upper semi-continuous) J-plurisubharmonic function on X. Then there exists a sequence {u_j} of smooth, strictly J-plurisubharmonic functions point-wise decreasing down to u. On any almost complex manifold (X,J) each point has a fundamental neighborhood system of J-pseudoconvex domains, and so the theorem above establishes local smooth approximation on X. This result was proved in complex dimension 2 by the third author, who also showed that the result would hold in general dimensions if a parallel result for continuous approximation were known. This paper establishes the required step by solving the obstacle problem.
Cite
@article{arxiv.1411.7137,
title = {Smooth Approximation of Plurisubharmonic Functions on Almost Complex Manifolds},
author = {F. Reese Harvey and H. Blaine Lawson, and Szymon Pliś},
journal= {arXiv preprint arXiv:1411.7137},
year = {2017}
}