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.

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Clifford Henry Taubes
Department of Mathematics
Harvard University
Cambridge, MA 02138, USA

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