Generalized $\Cal{L}$-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 $\bar{g}_{ij}(x,t)$ that evolves by the Ricci flow $\partial_t \bar{g}_{ij}=-2\bar{R}_{ij}$ 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 $\Cal{L}_p$-length between $p_0$ and $q$, $\Cal{L}_p$-geodesic, the generalized reduced distance $l_p$ and the generalized reduced volume $\widetilde{V}_p(\tau)$, $\tau=t_0-t$, corresponding to the $\Cal{L}_p$-geodesic at the point $p_0$ at time $t_0$. Under the condition $\bar{R}_{ij}\ge -c_1\bar{g}_{ij}$ on $M\times (0,t_0)$ for some constant $c_1>0$, we will prove the existence of a $\Cal{L}_p$-geodesic which minimize the $\Cal{L}_p(q,\bar{\tau})$-length between $p_0$ and $q$ for any $\bar{\tau}>0$. This result for the case $p=1/2$ is conjectured and used many times but no proof of it was given in Perelman's papers on Ricci flow. My result is new and answers in affirmative the existence of such $\Cal{L}$-geodesic minimizer for the $L_p(q,\tau)$-length which is crucial to the proof of many results in Perelman's papers on Ricci flow. We also obtain many other properties of the generalized $\Cal{L}_p$-geodesic and generalized reduced volume.