Monopoles and foliations without holonomy-invariant transverse measure

TL;DR

This paper extends analysis of Seiberg-Witten equations on non-compact 4-manifolds with symplectic ends and constructs a new invariant for certain foliations, bypassing traditional perturbation methods.

Contribution

It provides a uniform exponential decay estimate for Seiberg-Witten equations on specific non-compact manifolds and introduces a novel invariant for foliations without holonomy-invariant transverse measure.

Findings

Established exponential decay estimates for Seiberg-Witten solutions.

Constructed a new invariant for smooth foliations without holonomy-invariant transverse measure.

Connected the invariant to boundary-stable monopole Floer homology.

Abstract

This article proves a uniform exponential decay estimate for Seiberg-Witten equations on non-compact 4-manifolds with exact symplectic ends of bounded geometry. This is an extension of the analysis for asymptotically flat almost K\"ahler (AFAK) structures by Kronheimer and Mrowka. As an application, we construct an invariant for smooth foliations without holonomy-invariant transverse measure, which takes value in the boundary-stable version of the monopole Floer homology group, without invoking the Eliashberg-Thurston perturbation.