Local properties of J-complex curves in Lipschitz-continuous structures

Abstract

We prove the existence of primitive curves and positivity of intersections of J-complex curves for Lipschitz-continuous almost complex structures. These results are deduced from the Comparison Theorem for J-holomorphic maps in Lipschitz structures, previously known for J of class \({\mathcal{C}^{1, Lip}}\). We also give the optimal regularity of curves in Lipschitz structures. It occurs to be \({\mathcal{C}^{1,LnLip}}\), i.e. the first derivatives of a J-complex curve for Lipschitz J are Log-Lipschitz-continuous. A simple example that nothing better can be achieved is given. Further we prove the Genus Formula for J-complex curves and determine their principal Puiseux exponents (all this for Lipschitz-continuous J-s).