J-Holomorphic Curves in Rough Almost Complex Structures

Abstract

A pseudoholomorphic curve, or J-holomorphic curve, is a differentiable map from a Riemann surface to a manifold with almost complex structure J, that satisfies an analogue of the Cauchy-Riemann equations. When J is smooth, pseudoholomorphic curves have well-known regularity and uniqueness properties. I will survey what can happen when J is only continuous or satisfies a Hölder condition