Approximation of non-holomorphic maps -

Abstract

We will study the approximation of nonholomorphic maps from the unit disc to a complex manifold. This starts by generalizations of some theorems from one complex variable to several complex variables like the generalizations of Mittag-lefler and Weirstrass factorization theorem to the famous Cousin problems. Through these generalizations we will face local to global problems, like the d-bar problem which can be solved by some cohomology conditions. The work is based on a paper by Jean-Pierre Rosay which deals with approximation of nonholomorphic maps and applications to the Poletsky theory of discs. The main question to be answered is whether we can approximate a map with a small d-bar from the unit disc to a complex manifold by a holomorphic map. Lempert gives an example that negatively answers this question by taking any smooth map from the unit disc to any compact Riemann surface of genus greater than or equal to two. However, by taking a condition on the map to be restricted we will prove that the answer is positive.