## 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

#### 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
Revised: 20 November 1998
Accepted: 3 January 1999
First available in Project Euclid: 21 December 2017

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

#### 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

#### References

• N Aronszajn, A unique continuation theorem for elliptic differential equations or inequalities of the second order, J. Math. Pures Appl. 36 (1957) 235–249
• S K Donaldson, An application of gauge theory to the topology of $4$–manifolds, J. Diff. Geom. 18 (1983) 279–315
• S K Donaldson, Connections, cohomology and the intersection forms of $4$–manifolds, J. Diff. Geom. 24 (1986) 275–341
• M H Freedman, The topology of four dimensional manifolds, J. Diff. Geom. 17 (1982) 357–453
• M Gromov, Pseudo-holomorphic curves in symplectic manifolds, Invent. Math. 82 (1985) 307–347
• H Hofer, K Wysocki, E Zehnder, Properties of pseudo-holomorphic curves in symplectisations, I: Asymptotics, Analyse Nonlinear 13 (1996) 337–379
• H Hofer, K Wysocki, E Zehnder, Properties of pseudo-holomorphic curves in symplectisations, IV: Asymptotics with degeneracies, from: “Contact and Symplectic Geometry”, C Thomas, editor, Cambridge University Press (1996)
• H Hofer, M Kreiner, Holomorphic curves in contact dynamics, to appear
• K Honda, Harmonic forms for generic metrics, preprint (1997)
• K Honda, Local properties of self-dual harmonic $2$–forms on a $4$–manifold, preprint (1997)
• C LeBrun, Yamabe constants and the perturbed Seiberg–Witten equations, preprint (1997)
• C Luttinger, unpublished
• D McDuff, Singularities and positivity of intersections of $J$–holomorphic curves, with Appendix by Gang Liu, from: “Proceedings of CIMPA Summer School of Symplectic Topology”, Nice 1992, Birkhauser (1994)
• D McDuff, D Salamon, $J$–Holomorphic curves and quantum cohomology, American Mathematical Society, Providence (1996)
• C B Morrey, Multiple Integrals in the Calculus of Variations, Springer, New York (1996)
• P Pansu, Pseudo-holomorphic curves in symplectic manifolds, from: “Holomorphic Curves in Symplectic Geometry”, M. Audin and F. Lafontaine, editors, Progress in Math 117, Birkhauser (1994) 233–250
• T Parker, J Wolfson, Pseudo-holomorphic maps and bubble trees, Journ. Geom. Anal. 3 (1993) 63–98
• C H Taubes, The geometry of the Seiberg–Witten invariants, from: “Surveys in Differential Geometry”, 1996, S T Yau, editor, International Press, to appear
• C H Taubes, Self-dual connections on $4$–manifolds with indefinite intersection matrix, J. Diff. Geom. 19 (1984) 517–560
• C H Taubes, $SW \Rightarrow Gr$: From the Seiberg–Witten equations to pseudo-holomorhic curves, Jour. Amer. Math. Soc. 9 (1996) 845–918
• C H Taubes, $SW \Rightarrow Gr$: From the Seiberg–Witten equations to pseudo-holomorphic curves, from: “$SW = Gr$: The Equivalence of the Seiberg–Witten and Gromov Invariants”, R Stern, editor, International Press, to appear
• C H Taubes, Seiberg-Witten invariants and pseudoholomorphic subvarieties for self-dual, harmonic 2–forms, preprint (1998)
• C H Taubes, The geometry of the Seiberg-Witten invariants, from: “Proceedings of the International Congress of Mathematicians, Berlin, 1998”, Vol II, Documenta Mathematica (1998) 493-506
• R Ye, Gromov's compactness theorem for pseudo-holomorphic curves, Trans. Amer. Math. Soc. 342 (1994) 671–694