Yamabe Invariants and the $\mathrm{Pin}^-(2)$-monopole Equations

TL;DR

This paper introduces a novel method using $ ext{Pin}^-(2)$-monopole equations to compute Yamabe invariants for certain 4-manifolds, providing new insights into geometric structures and obstructions to Einstein metrics.

Contribution

It develops a twisted Seiberg-Witten approach with $ ext{Pin}^-(2)$-monopole equations to calculate invariants and identify obstructions to Einstein metrics on 4-manifolds.

Findings

Computed Yamabe invariants for new classes of 4-manifolds

Established obstructions to Einstein metrics and Ricci flow solutions

Demonstrated the effectiveness of twisted monopole equations in geometric analysis

Abstract

We compute the Yamabe invariants for a new infinite class of closed $4$-dimensional manifolds by using a "twisted" version of the Seiberg-Witten equations, the $\mathrm{Pin}^-(2)$-monopole equations. The same technique also provides a new obstruction to the existence of Einstein metrics or long-time solutions of the normalised Ricci flow with uniformly bounded scalar curvature.