Some Analytical Results on the Seiberg-Witten Equations and the Bogomolny Equations

Abstract

This thesis contains some analytical results on the Bogomolny equations and the Seiberg-Witten equations. Chapter one studies the Bogomolny equations. In this chapter, first I briefly introduce Taubes' analytical approach to studying the moduli space of the Bogomolny equations with certain asymptotic conditions on R^3. Then I illustrate how to use an adaption of Taubes' method to study the Bogomolny equations on R^3 with a knot singularity. In chapter two, the first two sections consist of a brief introduction to the Seiberg-Witten theory and the ECH (embedded contact homology) theory on a contact manifold. In particular, Hutchings' ECH spectral invariants are introduced here. The third section studies the asymptotic behaviors of the ECH spectral invariants. This study is based on some analysis in the Seiberg-Witten theory. Appendix A includes some analytical details which tie up the loose ends in chapter one. Appendix B gives a rigorous definition of the min-max action in Seiberg-Witten theory that is used in chapter two.