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

Hsu, Shu-Yu

J. Math. Kyoto Univ., Tome 49 (2009) no. 1, p. 503-571 / Harvested from Project Euclid

Résumé

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 $00$, 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

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Publié le : 2009-05-15
Classification:  58J35,  53C44,  58C99