标题： 宽度和Ricci流的有限灭绝时间 显示英文标题

标题： Width and finite extinction time of Ricci flow

摘要： 这是一篇阐述性文章，包含完整的证明，旨在面向一般的非专业读者。 结果分为两部分。 首先，我们讨论一种几何不变量，我们称之为宽度，它是一个流形的属性，并展示如何将其视为最小2球面面积的和。 例如，当$M$是一个同伦3-球时，宽度大致来说是将$M$“翻转”所需的最小2球的面积。 其次，我们利用这一点得出结论，在任何同伦3-球上，哈密顿的里奇流会在有限时间内消失。 我们选择撰写本文是因为这里给出的结果和想法非常有用，并似乎对广大读者有兴趣。 显示英文摘要

摘要： This is an expository article with complete proofs intended for a general non-specialist audience. The results are two-fold. First, we discuss a geometric invariant, that we call the width, of a manifold and show how it can be realized as the sum of areas of minimal 2-spheres. For instance, when $M$ is a homotopy 3-sphere, the width is loosely speaking the area of the smallest 2-sphere needed to ``pull over'' $M$. Second, we use this to conclude that Hamilton's Ricci flow becomes extinct in finite time on any homotopy 3-sphere. We have chosen to write this since the results and ideas given here are quite useful and seem to be of interest to a wide audience.