On Einstein metrics, normalized Ricci flow and smooth structures on 3ℂℙ^2#kℂℙ^2

In this paper, first we consider the existence and nonexistence of Einstein metrics on the topological 4-manifolds 3ℂℙ^2#kℂℙ^2, the connected sum of ℂℙ^2 with both choices of orientation, by using the idea of Răsdeaconu-Şuvaina, 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