The normalized Ricci flow on four-manifolds and exotic smooth structures

Abstract

In this article, we shall investigate the relationship between the existence or non-existence of non-singular solutions to the normalized Ricci flow and smooth structures on closed 4-manifolds, where non-singular solutions to the normalized Ricci flow are solutions which exist for all time $t \in [0, \infty)$ with uniformly bounded sectional curvature. In dimension 4, there exist many compact topological manifolds admitting distinct smooth structures, i.e., exotic smooth structures. Interestingly, in this article, the difference between existence and non-existence of non-singular solutions to the normalized Ricci flow on 4-manifolds turns out to strictly depend on the choice of smooth structure. In fact, we shall prove that, for every natural number $\ell$, there exists a compact topological 4-manifold $X_{\ell}$ which admits smooth structures for which non-singular solutions of the normalized Ricci flow exist, but also admits smooth structures for which no non-singular solution of the normalized Ricci flow exists. Hence, in dimension 4, smooth structures become definite obstructions to the existence of non-singular solutions to the normalized Ricci flow.