論瑞奇流的正則性

Abstract

本論文包含四章以及附錄。 在第一章中，我們從方程和幾何兩方面來探討瑞奇流的特性，並介紹瑞奇流的演變與基本思想。 我們試圖說明本論文的成果與瑞奇流的正則性之關聯。 我們在第二章利用某類旋轉對稱流形來構造一個奇異點發生在無窮遠處的解。 此外還利用佩雷曼的結果來討論遠古解的無塌陷性質。 在第三章中，我們證明在滿足比昂奇不等式的流形上，瑞奇張量滿足施皖雄型態的一階微分估計。 第四章將用來討論瑞奇孤立子的性質以及它們的分類問題。 我們證明當擴張孤立子的曲率遞減階數超過二次時，其無窮遠切錐為歐氏空間。

This thesis consists of four chapters and an appendix. The first chapter is dedicated to the fundamental ideas of the theory of Ricci flow, which shows how our works are connected to the whole story. In the second chapter, we construct a solution of Ricci flow on a rotationally symmetric manifold such that it remains a complete manifold at the maximal time. We also derive a noncollapsing property for certain ancient solutions near their maximal times. Both of these two results are related to the regularity of limits of solutions. In the third chapter, we show that a first order Shi-type estimate holds for Ricci tensor on manifolds which satisfy the weak Bianchi inequality. The last chapter is concerned with expanding gradient Ricci solitons. There we discuss the classification problem and show that every tangent cone at infinity of an expanding soliton with fast-than-quadratic-decay curvature must be $mathbb{R}^n$.