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 $|R(g(t))|\le C$ and ${\int }_M|Rm(g(t)){|}^{n/2}d{\mu }_t\le C$ uniformly for a constant $C$. Then we show that the solution sub-converges to a shrinking Ricci soliton. Moreover,when $n=3$, the condition ${\int }_M|Rm(g(t)){|}^{n/2}d{\mu }_t\le C$ can be removed.

2 pages, some minor corrections and improvements

References in corpus (3)

• Ricci flow with surgery on three-manifolds
• The logarithmic Sobolev inequality along the Ricci flow
• Bounding sectional curvature along a Kähler-Ricci flow