Smooth approximation of plurisubharmonic functions on almost complex manifolds

Abstract

This note establishes smooth approximation from above for Jplurisubharmonic 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 ∈ C ∞ (X) of smooth strictly Jplurisubharmonic functions point-wise decreasing down to u.In 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.

The Monge–Ampère equation for non-integrable almost complex structures

“…functions ϕ for which √ −1∂∂ϕ > 0 (see e.g. [11,25,27,28,33,36,37]), and our main theorem fits well into this picture.…”

On the regularization of J-plurisubharmonic functions

“…Harvey and Lawson, after viewing a preliminary version of this paper, informed the author that using viscosity methods it is possible to prove it in any dimension. It will be explained in [5].…”

The Geometry of m-Hyperconvex Domains

“…If the approximating sequence is assumed to be only continuous on , then the corresponding result was proved by Cegrell [18, Theorem 2.1] in the case , and Lu [46, Theorem 1.7.1] for general m . In connection with Theorem 5.2, we would like to make a remark on Theorem 6.1 in a recent paper by Harvey et al [34]. There they prove a similar approximation theorem, but there is an essential difference.…”