On Normalized Ricci Flow and Smooth Structures on Four-Manifolds with $b^+=1$

TL;DR

This paper establishes an obstruction to non-singular solutions of the normalized Ricci flow on certain four-manifolds with $b^+=1$, linking geometric flow behavior to exotic smooth structures on specific topological 4-manifolds.

Contribution

It introduces a new obstruction criterion for the normalized Ricci flow on four-manifolds with $b^+=1$, connecting geometric analysis with smooth structure classification.

Findings

Obstruction to non-singular Ricci flow solutions on $b^+=1$ four-manifolds.

Relationship between Ricci flow solutions and exotic smooth structures.

Application to topological manifolds ${f C}P^2 atural l ar{f C}P^2$ for $5 \,\leq\, l \leq 8$.

Abstract

We find an obstruction to the existence of non-singular solutions to the normalized Ricci flow on four-manifolds with $b^+=1$. By using this obstruction, we study the relationship between the existence or non-existence of non-singular solutions of the normalized Ricci flow and exotic smooth structures on the topological 4-manifolds ${\mathbb C}{P}^2 # l \overline{{\mathbb C}{P}^2}$, where $5 \leq l \leq 8$.