  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).

Differential Geometry;Algebraic Geometry