Non-singular solutions to the normalized Ricci flow equation

Abstract

In this paper we study non-singular solutions of Ricci flow on a closed manifold of dimension at least 4. Amongst others we prove that, if M is a closed 4-manifold on which the normalized Ricci flow exists for all time t>0 with uniformly bounded sectional curvature, then the Euler characteristic $χ(M)\ge 0$. Moreover, the 4-manifold satisfies one of the following \noindent (i) M is a shrinking Ricci solition; \noindent (ii) M admits a positive rank F-structure; \noindent (iii) the Hitchin-Thorpe type inequality holds 2χ(M)\ge 3|τ(M)| where $χ(M)$ (resp. $τ(M)$) is the Euler characteristic (resp. signature) of M.