#### Geometry & Topology, Vol. 2 (1998)
Paper no. 10, pages 221--332.

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

### Clifford Henry Taubes

**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 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.
**Keywords**.
Four-manifold invariants, symplectic geometry

**AMS subject classification**.
Primary: 53C07.
Secondary: 52C15.

**DOI:** 10.2140/gt.1998.2.221

**E-print:** `arXiv:math.SG/9901142`

Submitted to GT on 2 February 1998.
(Revised 20 November 1998.)
Paper accepted 3 January 1999.
Paper published 6 January 1999.

Notes on file formats
Clifford Henry Taubes

Department of Mathematics

Harvard University

Cambridge, MA 02138, USA

Email: chtaubes@abel.math.harvard.edu

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