Celso M. Doria | Searcharxiv

arXiv2002

A Necessary Condition for the Existence of SW-Monopoles

Originally, the SW-equations discovered by Seiberg-Witten are 1st-order PDE, which solutions (A,ϕ), with ϕ\ne 0, are known as SW-monopoles. It is known that the solutions of these 1st-order eq correspond to the minimum of SW-functional. However, it is not true, that for all spin^{c} class α, the minimum is always attained by this sort of solution. In fact, there are only a finite number of αsuch that the minimum is a SW-monopole. We show a necessary condition to be satisfied by the class αin order to the minimum be a SW-monopole.

math-ph↗

arXiv2001

On the Existence of Critical Points to the Seiberg-Witten Functional

Considering that the Seiberg-Witten functional satisfies the Palais-Smale Condition, up to gauge equivalence, the Minimax Principle can be applied on the moduli space to prove the existence of critical points, which correspond to solutions of the second-order SW-equations, up to gauge equivalence.

math.DG↗