Smooth structures, normalized Ricci flows, and finite cyclic groups

Abstract

A solution to the normalized Ricci flow is called non-singular if it exists for all time with uniformly bounded sectional curvature. By using the techniques developed by the present authors [Ishida, The normalized Ricci flow on four-manifolds and exotic smooth structures; Şuvaina, Einstein metrics and smooth structures on non-simply connected 4-manifolds] we prove that for any finite cyclic group \({\mathbb{Z}_{d}}\) , where d > 1, there exist infinitely many compact topological 4-manifolds, with fundamental group \({\mathbb{Z}_{d}}\) , which admit at least one smooth structure for which non-singular solutions of the normalized Ricci flow exist, but also admit infinitely many distinct smooth structures for which no non-singular solution of the normalized Ricci flow exists. We show that there are no non-singular \({\mathbb{Z}_{d}}\) -equivariant, d > 1, solutions to the normalized Ricci flow on appropriate connected sums of \({\mathbb{CP}^{2}\rm{s}}\) and \({\overline{\mathbb {CP}}^{2}\rm{s}}\).