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

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\Bbb CP_2 # n\bar{\Bbb CP}_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 oribifold Yamabe invariants of some 4-orbifolds.