Coordinate neighborhoods of arcs and the approximation of maps into (almost) complex manifolds

Abstract

We study the approximation of J-holomorphic maps continuous to the boundary from a domain in into an almost complex manifold by maps Jholomorphic to the boundary, giving partial results in the non-integrable case. For the integrable case, we study arcs in complex manifolds and establish the existence of neighborhoods biholomorphic to open sets in Euclidean space for several classes of arcs. As an application we obtain C k approximation of holomorphic maps continuous to the boundary into complex manifolds by maps holomorphic to the boundary, provided the boundary is nice enough.2000 Mathematics Subject Classification. 32Q60,32Q65,32H02,30E10

Sets of Approximation and Interpolation in ℂ for Manifold-Valued Maps

“…2 below, we prove a general result (Theorem 4) regarding approximation of sections of submersions over a set K which admits a decomposition K = K 1 ∪ K 2 into a "good pair" (K 1 , K 2 ) of compact sets, provided the image of the section has neighborhoods in C n . This is a direct generalization of the results in [2], Sect. 4 for maps into manifolds.…”

“…The proof of Theorem 1 is a direct application of Theorem 4, taking advantage of the one-dimensional character of the sets. For Theorem 3, we require a result (Theorem 5) which is found in [2], which asserts the existence of coordinate neighborhoods of arcs or a certain smoothness in complex manifolds. This in effect allows us to split up any set of class C 2 into a good pair, to which Theorem 4 can be applied.…”

“…For a proof, see [2], Lemma 4.4, where it is assumed that G = GL n (C), and each component of the intersection is star shaped, but the proof is valid for general G.…”

“…We will use the three lemmas above to give a proof of the following result (the main component of which is due to Rosay, see [13], and remarks in [12]) regarding the solution of a non-linear Cousin problem (see Lemma 4.5 of [2].) It is simply a translation of Theorem 4 to coordinates.…”

Approximation of holomorphic mappings on strongly pseudoconvex domains

“…We prove the analogous approximation result for sections of holomorphic submersions (Theorem 5.1). When r ≥ 2, a different proof can be obtained by using the fact that the graph G f = {(z, f (z) For approximation of holomorphic maps from planar Jordan domains to (almost) complex manifolds see D. Chakrabarti [3, Theorem 1.1.4], [4].…”