  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 with 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.
