Non-singular solutions to the normalized Ricci flow equation

Abstract

In this paper, we study non-singular solutions to Ricci flow on a closed manifold of dimension at least 4. Amongst other things 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 \(\chi (M)\ge 0\) . Moreover, the 4-manifold satisfies one of the followings (i) M is a shrinking Ricci soliton; (ii) M admits a positive rank F-structure; (iii) the Hitchin–Thorpe type inequality holds $$2\chi (M)\ge 3|\tau(M)|$$where \(\chi (M)\) (resp. \(\tau(M)\)) is the Euler characteristic (resp. signature) of M.