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

TL;DR

This paper establishes fundamental properties of J-complex curves in Lipschitz-continuous structures, including existence, intersection positivity, optimal regularity, and genus formulas, extending prior results to less regular settings.

Contribution

It proves the existence and intersection positivity of J-complex curves in Lipschitz structures, and determines their optimal regularity and genus formulas, extending classical results to Lipschitz continuous almost complex structures.

Findings

Existence of primitive J-complex curves in Lipschitz structures

Positivity of intersections for J-complex curves

Optimal regularity of curves as C^{1,LnLip}

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 $C^{1, Lip}$. We also give the optimal regularity of curves in Lipschitz structures. It occurs to be $C^{1,LnLip}$, 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 Puisieux exponents (all this for Lipschitz-continuous $J$-s).