Smooth Structures and Normalized Ricci Flows on Non-Simply Connected Four-Manifolds

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 [14, 26], we study the existence or non-existence of non-singular solutions of the normalized Ricci flow on 4−manifolds with non-trivial fundamental group and the relation with the smooth structures. For example, we prove that, for any finite cyclic group Zd, where d > 1, there exists a compact topological 4−manifold X with fundamental group Zd, which admits at least one smooth structure for which non-singular solutions of the normalized Ricci flow exist, but also admits infinitely many distinct smooth structures for which no non-singular solution of the normalized Ricci flow exists. Related non-existence results on non-singular solutions are also proved. Among others, we show that there are no non-singular Zd−equivariant solutions to the normalized Ricci flow on appropriate connected sums of CP s and CPs (d > 1).