Generalized $\Cal{L}$-geodesic and monotonicity of the generalized reduced volume in the Ricci flow
| dc.creator | Hsu, Shu-Yu | |
| dc.date | 2006-08-08 | |
| dc.date | 2008-03-03 | |
| dc.date.accessioned | 2026-07-07T09:24:07Z | |
| dc.date.available | 2026-07-07T09:24:07Z | |
| dc.description | 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 $\widetilde{V}_p(τ)$, $τ=t_0-t$, 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 $\Cal{L}_p(q,\barτ)$-length between $p_0$ and $q$ for any $\barτ>0$. 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 $L_p(q,τ)$-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. | |
| dc.description | 64 pages | |
| dc.identifier | https://arxiv.org/abs/math/0608197 | |
| dc.identifier | http://arxiv.org/abs/math/0608197 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/155980 | |
| dc.subject | Differential Geometry | |
| dc.subject | 58J35 | |
| dc.title | Generalized $\Cal{L}$-geodesic and monotonicity of the generalized reduced volume in the Ricci flow | |
| dc.type | text |