A remark on odd dimensional normalized Ricci flow

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 $-4{\mathrm{\Delta }}_{g_0}+R(g_0)$, 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 $\mathrm{\mid }R(g(t))\mathrm{\mid }\le C$ and $\int_M |Rm(g(t))|^{n/2}d\mu_t \leq C$uniformlyforaconstant$C$. Then we show that the solution sub-converges to a shrinking Ricci soliton. Moreover,when $n=3$, the condition ${\int }_M\mathrm{\mid }Rm(g(t)){\mathrm{\mid }}^{n\mathrm{/}2}d{\mu }_t\le C$ can be removed.