Smooth Structures and Normalized Ricci Flows on Non-Simply Connected Four-Manifolds

TL;DR

This paper investigates the existence of non-singular solutions to the normalized Ricci flow on 4-manifolds with non-trivial fundamental groups, revealing how smooth structures influence these solutions and establishing new non-existence results.

Contribution

It demonstrates the dependence of Ricci flow solutions on smooth structures in 4-manifolds with finite cyclic fundamental groups, providing new existence and non-existence results.

Findings

Existence of smooth structures with non-singular Ricci flow solutions.

Existence of infinitely many smooth structures with no such solutions.

Non-existence of non-singular solutions in certain equivariant cases.

Abstract

A solution to the normalized Ricci flow is called non-singular if it exists for all time with uniformly bounded sectional curvature. By using the techniques developed by the present authors, we study the existence or non-existence of non-singular solutions of the normalized Ricci flow on 4-manifolds with non-trivial fundamental group and the relation with the smooth structures. For example, we prove that, for any finite cyclic group ${\mathbb Z}_{d}$, where $d>1$, there exists a compact topological 4-manifold $X$ with fundamental group ${\mathbb Z}_{d}$, which admits at least one smooth structure for which non-singular solutions of the normalized Ricci flow exist, but also admits infinitely many distinct smooth structures for which {\it no} non-singular solution of the normalized Ricci flow exists. Related non-existence results on non-singular solutions are also proved. Among others, we show that there are no non-singular $\ZZ_d-$equivariant solutions to the normalized Ricci flow on appropriate connected sums of $\bcp ^2$s and $\cpb $s ($d>1$).