The structure of pseudo-holomorphic subvarieties for a degenerate almost complex structure and symplectic form on $S^1 \times B^3$

Abstract

A self-dual harmonic 2–form on a 4–dimensional Riemannian manifold is symplectic where it does not vanish. Furthermore, away from the form’s zero set, the metric and the 2–form give a compatible almost complex structure and thus pseudo-holomorphic subvarieties. Such a subvariety is said to have finite energy when the integral over the variety of the given self-dual 2–form is finite. This article proves a regularity theorem for such finite energy subvarieties when the metric is particularly simple near the form’s zero set. To be more precise, this article’s main result asserts the following: Assume that the zero set of the form is non-degenerate and that the metric near the zero set has a certain canonical form. Then, except possibly for a finite set of points on the zero set, each point on the zero set has a ball neighborhood which intersects the subvariety as a finite set of components, and the closure of each component is a real analytically embedded half disk whose boundary coincides with the zero set of the form.