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In this paper, first we consider the existence and non-existence of Einsteinmetrics on the topological 4-manifolds $3\mathbb{CP}^2 # k \bar{\mathbb{CP}}^2$(for $k \in {11, 13, 14, 15, 16, 17, 18}$) by using the idea of R\u{a}sdeaconuand \c{S}uvaina (2009) and the constructions in Park, Park, and Shin(arXiv:0906.5195v2) and in Park, Park, and Shin (2009). Then, we study theexistence or non-existence of non-singular solutions of the normalized Ricciflow on the exotic smooth structures of these topological manifolds byemploying the obstruction developed in Ishida (2008).

On Einstein metrics, normalized Ricci flow and smooth structures on $3\mathbb{CP}^2 # k \bar{\mathbb{CP}}^2$