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Local properties of \(J\)-complex curves in Lipschitz-continuous structures. (English) Zbl 1233.32015

Let \(M\) be a smooth manifold and let \(J\) be a Lipschitz continuous section of \(\mathrm{End} \, TM\) such that \(J^2=-\mathrm{id}_{TM}\). The authors consider \(J\)-holomorphic curves in \(M\), i.e., \(C^1\) maps \(u: \Sigma \rightarrow M\) from a Riemann surface \((\Sigma, j)\) such that \(du \circ j = J \circ du\).
The authors extend to this case properties that are known in the case of an integrable complex structure, and, more generally, when the almost complex structure \(J\) is sufficiently smooth (say \(C^2\)).
In particular, they prove the existence of primitive parametrizations (Theorem A), the positivity of intersections, and the genus formula in the case when \(M\) is complex two-dimensional (Theorems B and D), as well as a comparison theorem for \(J\)-holomorphic curves that are tangent to each other at some point. They also prove that the differential \(du\) of a \(J\)-holomorphic curve is log-Lipschitz continuous (Theorem C) and establish an analogue of the Puiseux series expansion (Theorem E).

MSC:

32Q65 Pseudoholomorphic curves
32Q60 Almost complex manifolds
14H50 Plane and space curves
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