在這篇文章裡我們描述了兩種由Perelman提出建立沿著瑞奇 流的κ-noncollapsing定理的方法。第一種方法是使用Perelman entropy。第二種方法是利用Perelman’s reduced volume的單調性來建立。Reduced volume是對non-collapsing定理更局部的看法，因此我們學習Perelman的証明中關於龐加萊猜想裡ancient κ-noncollapsing的解時(這種解不必是緊緻因此不被總體的量所控制)，第二個方法是重要的。我們的論述主要是依據Cao-Zhu [6]，關於Perelman’s Wfunctional我們參考O. Rothaus [3]給予更詳細的說明。 In this paper we report on the two methods pioneered by G. Perelman[1] to establish his κ-noncollapsing thm of the Ricci flow. The first method uses the Perelman entropy. The second proof uses the monotonicity of the Perelman’s reduced volume. The second proof is important, because the reduced volume is a more localized quantity in its definition and so one can in fact establish local versions of the non-collapsing theorem which turn out to be important when we study ancient κ-noncollapsing solutions in Perelman’s proof of the Poincar´e conjecture. Such solutions need not be compact and so cannot be controlled by global quantities (such as the Perelman entropy). Our treatment follows closely the cuticle by Cao-Zhu [6], with some more details on Perelman’s W functional by O. Rothaus [3].
