Geometry & Topology

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

Clifford Henry Taubes

Full-text: Open access

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.

Article information

Source
Geom. Topol., Volume 2, Number 1 (1998), 221-332.

Dates
Received: 2 February 1998
Revised: 20 November 1998
Accepted: 3 January 1999
First available in Project Euclid: 21 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.gt/1513882908

Digital Object Identifier
doi:10.2140/gt.1998.2.221

Mathematical Reviews number (MathSciNet)
MR1658028

Zentralblatt MATH identifier
0908.53013

Subjects
Primary: 53C07: Special connections and metrics on vector bundles (Hermite-Einstein- Yang-Mills) [See also 32Q20]
Secondary: 52C15: Packing and covering in $2$ dimensions [See also 05B40, 11H31]

Keywords
4–manifold invariants symplectic geometry

Citation

Taubes, Clifford Henry. The structure of pseudo-holomorphic subvarieties for a degenerate almost complex structure and symplectic form on $S^1 \times B^3$. Geom. Topol. 2 (1998), no. 1, 221--332. doi:10.2140/gt.1998.2.221. https://projecteuclid.org/euclid.gt/1513882908


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