Variational Aspects of the Seiberg-Witten Functional

TL;DR

This paper investigates the variational properties of the Seiberg-Witten functional, establishing regularity of solutions and the Palais-Smale condition, which are crucial for understanding the solutions' structure in four-dimensional geometry.

Contribution

It proves regularity of weak solutions and verifies the Palais-Smale condition for the Seiberg-Witten functional, advancing the mathematical understanding of these equations.

Findings

Proved regularity of weak solutions.

Established Palais-Smale condition for the functional.

Enhanced understanding of the variational structure in four-dimensional geometry.

Abstract

The Seiberg-Witten equations that have recently found important applications for four-dimensional geometry are the Euler-Lagrange equations for a functional involving a connection $A$ on a line bundle $L$ and a section $\phi$ of another bundle $W^+$ constructed from $L$ and a spinor bundle on a given four-dimensional Riemannian manifold. We show the regularity of weak solutions and the Palais-Smale condition for this functional.