$$G$$ -monopole classes, Ricci flow, and Yamabe invariants of 4-manifolds

Abstract

On a smooth closed oriented 4-manifold \(M\) with a smooth action by a finite group \(G\), we show that a \(G\)-monopole class gives the \(L^2\)-estimate of the Ricci curvature of a \(G\)-invariant Riemannian metric, and derive a topological obstruction to the existence of a \(G\)-invariant nonsingular solution to the normalized Ricci flow on \(M\). In particular, for certain \(m\) and \(n, m\mathbb C P_2 \# n\overline{\mathbb{C P}}_2\) admits an infinite family of topologically equivalent but smoothly distinct non-free actions of \(\mathbb Z _d\) such that it admits no nonsingular solution to the normalized Ricci flow for any initial metric invariant under such an action, where \(d>1\) is a non-prime integer. We also compute the \(G\)-Yamabe invariants of some 4-manifolds with \(G\)-monopole classes and the orbifold Yamabe invariants of some 4-orbifolds.