A remark on odd dimensional normalized Ricci flow

Abstract

Let $(M^n,g_0)$ ($n$ odd) be a compact Riemannian manifold with $\lambda(g_0)>0$, where $\lambda(g_0)$ is the first eigenvalue of the operator $-4\Delta_{g_0}+R(g_0)$, and $R(g_0)$ is the scalar curvature of $(M^n,g_0)$. Assume the maximal solution $g(t)$ to the normalized Ricci flow with initial data $(M^n,g_0)$ satisfies $|R(g(t))| \leq C$ and $\int_M |Rm(g(t))|^{n/2}d\mu_t \leq C$ uniformly for a constant $C$. Then we show that the solution sub-converges to a shrinking Ricci soliton. Moreover,when $n=3$, the condition $\int_M |Rm(g(t))|^{n/2}d\mu_t \leq C$ can be removed.