狄克技巧

Abstract

摘要: 這份報告中，主要是專注於瑞奇流的局部解以及給定瑞奇曲率，求解黎曼度量的局部解。這兩個主題之證明，主要是判定他們是否分別為拋物型與橢圓型方程，進而使用偏微分方程的已知理論求解。不過就是由於他們並非能恰巧滿足已知理論條件,所以我們必須去修改方程使其能滿足條件。 這篇報告之所有內容都是D.M. Deturck所得到，故詳細內容可參閱其所著相關論文。

In this review, we would concentrate on two main results 'Short-time existence of Ricci flow' and 'Local existence of metrics with prescribed Ricci curvature'. All of these materials could be found in the original papers by D.M. Deturck.([2],[3]) Both proofs depended on whether they're strictly parabolic and elliptic, resp. Because θgθt= -2 Ricc(g) and Ricc'(g)h are not strictly parabolic and elliptic, we must modify the equations (i.e. adding some terms) to make them to satisfy the requirements. Without complete proofs we would just point out the key steps in Chapter 3 after giving some preliminaries.