Continuous solutions of nonlinear Cauchy-Riemann equations and pseudoholomorphic curves in normal coordinates

TL;DR

This paper proves elliptic regularity for nonlinear Cauchy-Riemann equations with minimal assumptions, provides explicit solutions in special cases, and offers local formulas for pseudoholomorphic curves under continuous almost complex structures.

Contribution

It establishes new regularity results for nonlinear inhomogeneous Cauchy-Riemann equations and derives explicit solutions and local formulas in specific scenarios.

Findings

Elliptic regularity holds under minimal assumptions.

Counterexample provided in a borderline case.

Explicit solutions when inhomogeneous terms are separable.

Abstract

We establish elliptic regularity for nonlinear inhomogeneous Cauchy-Riemann equations under minimal assumptions, and give a counterexample in a borderline case. In some cases where the inhomogeneous term has a separable factorization, the solution set can be explicitly calculated. The methods also give local parametric formulas for pseudoholomorphic curves with respect to some continuous almost complex structures.