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

Abstract

It is known that connected sums $X\# K 3 \# (\Sigma_g \times \Sigma_h) \# \ell_1 (S^1 \times S^3) \# \ell_2 \overline{\mathbb{C}P^2}$ satisfy the Gromov-Hitchin-Thorpe type inequality, but can not admit non-singular solutions of the normalized Ricci flow for any initial metric, where $\Sigma_g \times \Sigma_h$ is the product of two Riemann surfaces of odd genus, $\ell_1, \ell_2 \gt 0$ are sufficiently large positive integers, $g, h \gt 3$ are also sufficiently large positive odd integers, and $X$ is a certain irreducible symplectic 4-manifold. These examples are closely related with a conjecture of Fang, Zhang and Zhang. In the current article, we point out that there still exist 4-manifolds with the same property even if $\ell_1 = \ell_2 = 0$ and $g = h = 3$. The topology of these new examples are smaller than that of previously known examples.