On Einstein metrics, normalized Ricci flow and smooth structures on 3CP2#k(CP2)over-bar

In this paper, first we consider the existence and nonexistence of Einstein metrics on the topological 4-manifolds 3CP(2)#k (CP2) over bar, the connected sum of CP2 with both choices of orientation, by using the idea of Rasdeaconu Suvaina, 2009, and the constructions in Park-Park-Shin, 2013. Then, we study the existence or nonexistence of nonsingular solutions of the normalized Ricci flow on the exotic smooth structures of these topological manifolds by employing the obstruction developed in Ishida, 2008.