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

Clifford Henry Taubes

Geometry & Topology 2 (1998) 221–332

arXiv: math.SG/9901142

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.

Keywords
4–manifold invariants, symplectic geometry
Mathematical Subject Classification
Primary: 53C07
Secondary: 52C15
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Publication
Received: 2 February 1998
Revised: 20 November 1998
Accepted: 3 January 1999
Published: 6 January 1999
Proposed: Robion Kirby
Seconded: Gang Tian, Tomasz Mrowka
Authors
Clifford Henry Taubes
Department of Mathematics
Harvard University
Cambridge
Massachusetts 02138
USA