A remark on odd dimensional normalized Ricci flow

TL;DR

This paper studies the behavior of odd-dimensional compact manifolds evolving under normalized Ricci flow, showing convergence to shrinking Ricci solitons under certain curvature bounds, with special results in three dimensions.

Contribution

It establishes convergence of normalized Ricci flow to shrinking Ricci solitons in odd dimensions under bounded curvature conditions, removing the integral bound in three dimensions.

Findings

Flow converges to shrinking Ricci soliton under assumptions

In 3D, integral curvature bound can be omitted

Provides conditions for long-term geometric behavior

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.