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

The Normalized Ricci Flow on Four-Manifolds and Exotic Smooth Structures