The structure of pseudo-holomorphic subvarieties for a degenerate almost complex structure and symplectic form on S1 × B3

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

References

Forward citations

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

Volume 2, issue 1 (1998)

Clifford Henry Taubes