$L^2$ harmonic forms and the Seiberg-Witten map on non compact four manifolds

TL;DR

This paper investigates a new phenomenon in non-compact four-manifolds where the d^+ image of one-forms cannot densely cover L^2 self-dual forms, leading to a novel functional analytic framework for the Seiberg-Witten map.

Contribution

It introduces a new functional analytic framework for the Seiberg-Witten map based on a phenomenon related to L^2 harmonic forms on non-compact four-manifolds.

Findings

d^+ image of one-forms does not densely cover L^2 self-dual forms

Existence of a certain L^2 self-dual harmonic form affects the structure

New framework for analyzing the Seiberg-Witten map on non-compact manifolds

Abstract

We explain a new phenomenon on non compact complete Riemannian four manifolds, where d^+ image of one forms can not exhaust densely on L^2 self dual forms on each compact subset, if a certain L^2 self dual harmonic form exists. This leads to construct a new functional analytic framework on the Seiberg-Witten map.