Ricci curvature and monopole classes on 3-manifolds

TL;DR

This paper establishes an L^2-estimate linking Ricci curvature and harmonic 1-forms on 3-manifolds with solutions to rescaled Seiberg-Witten equations, and provides conditions for monopole classes on connected sums.

Contribution

It introduces a new L^2-estimate involving Ricci curvature and harmonic forms, and characterizes monopole classes on certain connected sums of 3-manifolds.

Findings

L^2-estimate involving Ricci curvature and harmonic 1-forms

Necessary condition for monopole classes on connected sums

Application to solutions of rescaled Seiberg-Witten equations

Abstract

We prove an L^2-estimate involving Ricci curvature and a harmonic 1-form on a closed oriented Riemannian 3-manifold admitting a solution of any rescaled Seiberg-Witten equations. We also give a necessary condition to be a monopole class on some special connected sums.