Notes on of Seiberg-Witten map on manifold with flat scalar curvature

TL;DR

This paper investigates the properties of the Seiberg-Witten moduli space on non-compact manifolds with flat scalar curvature, proving its compactness under specific topological and geometric conditions.

Contribution

It establishes the compactness of the Seiberg-Witten moduli space on certain non-compact manifolds with flat scalar curvature and specific cohomological restrictions.

Findings

Moduli space is compact under given conditions

Scalar curvature zero on the periodic end

Vanishing of certain cohomology groups

Abstract

In this paper, we focus on the moduli space of Seiberg-Witten equation on non-compact manifold with periodic end. Suppose that the scalar curvature on the periodic end is identically zero and the topological conditions: the first de-Rham cohomology and the self-dual cohomology restricting on the periodic end vanish. Then, we will show that the moduli space of the perturbed Seiberg-Witten equation is compact.