Generalized L-geodesic and monotonicity of the generalized reduced volume in the Ricci flow

Abstract

Suppose $M$ is a complete n-dimensional manifold, $n\ge 2$, with a metric $\overline{g}_{ij}(x,t)$ that evolves by the Ricci flow $\partial_t \overline{g}_{ij}=-2\overline{R}_{ij}$ in $M\times (0,T)$. For any $0<p<1$, $(p_0,t_0)\in M\times (0,T)$, $q\in M$, we define the $\mathcal{L}_p$-length between $p_0$ and $q$, $\mathcal{L}_p$-geodesic, the generalized reduced distance $l_p$ and the generalized reduced volume $\widetilde{V}_p(\tau)$, $\tau=t_0-t$, corresponding to the $\mathcal{L}_p$-geodesic at the point $p_0$ at time $t_0$. Under the condition $\overline{R}_{ij} \ge -c_1\overline{g}_{ij}$ on $M\times (0,t_0)$ for some constant $c_1>0$, we will prove the existence of a $\mathcal{L}_p$-geodesic which minimize the $\mathcal{L}_p(q,\overline{\tau})$-length between $p_0$ and $q$ for any $\overline{\tau}>0$. This result for the case $p=1/2$ was mentioned and used many times by G. Perelman but no proof of it was given in Perelman's papers on Ricci flow. Let $g(\tau)=\overline{g}(t_0-\tau)$ and let $\widetilde{V}_p^{\overline{\tau}}(\tau)$ be the rescaled generalized reduced volume. Suppose $M$ also has nonnegative curvature operator with respect to the metric $\overline{g}(t)$ for any $t\in (0,T)$ and when $1/2<p<1$, $M$ has uniformly bounded scalar curvature on $(0,T)$. Let $0<c<1$ and let $\tau_0=\min ((2(1-p))^{-1/(2p-1)},t_0)$. For any $1/2\le p<1$ we prove that there exists a constant $A_0\ge 0$ with $A_0=0$ for $p=1/2$ such that $e^{-A_0\tau}\widetilde{V}_p(\tau)$ is a monotone decreasing function in $(0,\overline{\tau}_1)$ where $\overline{\tau}_1=(1-c)\tau_0$ if $1/2<p<1$ and $\overline{\tau}_1=t_0$ if $p=1/2$. When $(M,\overline{g})$ is an ancient $\kappa$-solution of the Ricci flow, we will prove a monotonicity property of the rescaled generalized volume $\widetilde{V}_p^{\overline{\tau}}(\tau)$ with respect to $\overline{\tau}$ for any $1/2\le p<1$. When $p=1/2$, the $\mathcal{L}_p$-length, $\mathcal{L}_p$-geodesic, the $l_p$ function and $\widetilde{V}_p(\tau)$ are equal to the $\mathcal{L}$-length, $\mathcal{L}$-geodesic, the reduced distance $l$ and the reduced volume $\widetilde{V}(\tau)$ introduced by Perelman in his papers on Ricci flow. We will also prove a result on the reduced distance $l$ and the reduced volume $\widetilde{V}$ which was used by Perelman without proof in [18].