## 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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### References:

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