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

In this paper, first we consider the existence and non-existence of Einstein metrics 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}sdeaconu and \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 the existence or non-existence of non-singular solutions of the normalized Ricci flow on the exotic smooth structures of these topological manifolds by employing the obstruction developed in Ishida (2008).