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On Poletsky theory of discs in compact manifolds. (English) Zbl 1407.32014

Let \(M\) be a smooth, connected compact manifold, equipped with a regular almost complex structure \(J\) which has the doubly tangent property. Then for any \(p\in M\) and for any open subset \(U\subset M\) there exists a Poletsky disc, i.e., a \(J\)-holomorphic map \(u\) from a neighborhood of the closed unit disc into \(M\) with \(u(0)=p\) and such that \(|t\in[0,2\pi): u(e^{it})\notin U|<\epsilon\). This gives a partial answer to a question raised in [J.-P. Rosay, Indiana Univ. Math. J. 52, No. 1, 157–169 (2003; Zbl 1033.31006)].
The doubly tangent property was defined in [A. Gournay, Geom. Funct. Anal. 22, No. 2, 311–351 (2012; Zbl 1267.32007)], where examples are given and a Runge-type theorem is proved. The proof relies on this theorem and local arc approximation.

MSC:

32Q60 Almost complex manifolds
32Q65 Pseudoholomorphic curves
32E30 Holomorphic, polynomial and rational approximation, and interpolation in several complex variables; Runge pairs
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References:

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