Small four-manifolds without non-singular solutions of normalized Ricci flows

It is known [6] that connected sums X#K3#(σ_g × σ_h)#l₁(S¹×S³)#l₂CP² satisfy the Gromov-Hitchin-Thorpe type inequality, but can not admit non-singular solutions of the normal- ized Ricci flow for any initial metric, where σ_g × σ_h is the product of two Riemann surfaces of odd genus, l₁, l₂ > 0 are sufficiently large positive integers, g, h > 3 are also sufficiently large positive odd integers, and X is a certain irreducible symplectic 4-manifold. These exmples are closely related with a conjecture of Fang, Zhang and Zhang [10]. In the current article, we point out that there still exist 4-manifolds with the same property even if l₁ = l₂ = 0 and g = h = 3. The topology of these new examples are smaller than that of previously known examples.