Global Existence for the Seiberg-Witten Flow

TL;DR

This paper introduces a gradient flow for the Seiberg-Witten functional on 4-manifolds, proving global existence and convergence to critical points, advancing understanding of the flow's long-term behavior.

Contribution

It establishes the global existence and convergence of the Seiberg-Witten flow on compact 4-manifolds, a novel result in gauge theory dynamics.

Findings

Proved global existence of the flow.

Showed convergence to critical points.

Established uniqueness of the limit.

Abstract

We introduce the gradient flow of the Seiberg-Witten functional on a compact, orientable Riemannian 4-manifold and show the global existence of a unique smooth solution to the flow. The flow converges uniquely in $C^\infty$ up to gauge to a critical point of the Seiberg-Witten functional.