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Differential inequalities of continuous functions and removing singularities of Radó type for \(J\)-holomorphic maps. (English) Zbl 1159.32008

A classical theorem by Radó asserts that if a continuous complex-valued function \(f\) defined on an open subset \(\Omega \subset \mathbb{C}\) is holomorphic on \(\Omega \setminus f^{-1}(0)\), then it is holomorphic on the whole of \(\Omega\).
The paper under review deals with generalizations of this theorem in two different contexts: (1) continuous maps \(f: \Omega \subset\mathbb{C}^n\rightarrow \mathbb{C}^m\) and (2) continuous maps from \(\Omega \subset \mathbb{C}\) to an almost complex manifold \((M,J)\).
The two cases are studied independently.
Part 1 of the paper is concerned with case (1). Among the results proved we quote the following (Prop. A): If \(m=1\) and \( |\overline{\partial}f| \leq K |f| \) on \(\Omega \setminus f^{-1}(0)\) for some \(K>0\), then \(f^{-1}(0)\) is an analytic subset. The precise meaning of the differential inequality \( |\overline{\partial}f| \leq K |f| \) is the following: there is \(A\in L^\infty_{(0,1)} (\Omega \setminus f^{-1}(0))\) such that \( \overline{\partial}f = A f\) on \(\Omega \setminus f^{-1}(0)\) in the sense of distributions. This proposition depends on a more technical result (Thm. A) which asserts that \(e^{-u}f\) is holomorphic for some locally defined function \(u\). Another result (Prop. C) deals with the case \(n=1\) and \(m>1\). The authors remark that the analogue of Prop.1 in the general case (\(n>1, m>1\)) is still open. Related results are contained in [N. Pali, Math. Ann. 336, No. 3, 571–615 (2006; Zbl 1110.32003)].
Let us now describe some of the results in case (2). A closed subset \(C\) in an almost complex manifold \((M,J)\) is called a Radó subset if any continuous map from a region \(\Omega \subset \mathbb{C}\) to \(M\) which is \(J\)-holomorphic on \(\Omega \setminus u^{-1} (C)\) is automatically \(J\)-holomorphic on the whole of \(\Omega\) (unless \(u(\Omega) \subset C\)). Then (Cor. to Thm. C) the image of a proper \(J\)-holomorphic curve is Radó. In particular, discrete sets are Radó.

MSC:

32H02 Holomorphic mappings, (holomorphic) embeddings and related questions in several complex variables
32A60 Zero sets of holomorphic functions of several complex variables
32Q65 Pseudoholomorphic curves

Citations:

Zbl 1110.32003
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