Coordinate Neighborhoods of Arcs and the Approximation of Maps into (Almost) Complex Manifolds

摘要

This paper is divided into three sections, which, though mostly independent of each other, are devoted to the study of the following question. Let Ω ∈ C, and let (X, J) be an almost complex manifold. Suppose that f is a continuous map from the closure Ω to X that is J-holomorphic on Ω . Can we approximate f by maps J-holomorphic on (shrinking) neighborhoods of Ω ?rnIn Section 2 we give some conditions under which such a map f can be approximated by J-holomorphic maps in a neighborhood of Ω Unfortunately, this involves smoothness assumptions on the boundary partial deriv and on f as well (see Theorem 1).