李瑩英臺灣大學:數學研究所張瑞恩Chang, Jui-EnJui-EnChang2010-05-052018-06-282010-05-052018-06-282009U0001-1407200913495700http://ntur.lib.ntu.edu.tw//handle/246246/180621Yamabe問題內容如下：給定任何一個三維以上的黎曼緊緻流形，找一個保角的度量使得在此度量之下的純量曲率為一定值。此問題已經被Richard Schoen 在 1984 年完全解決。本論文研究此問題的證明。並且對於此問題的拋物偏微分方程版本， Yamabe 流，在本論文中也收錄了一些相關的結果。The Yamabe problem is as following. Given a compact Riemannian manifold of dimension n≥3, find a conformal metric with constant scalar curvature. The problem is completely solved by Richard Schoen in 1984. This thesis studies the proof of the Yamabe problem. It also includes some results of the Yamabe flow, which is the parabolic counterpart of the Yamabe problem.Contents謝 ii文摘要 iii文摘要 iii Introduction 1 The P.D.E and the scalar curvature functional 4.1 Derive the P.D.E. for the Yamabe problem 4.2 The scalar curvarure functional 8 The proof of Yamabe problem 14.1 The best constant of Sobolev inequality 15.2 Relation between lambda(M) and the solution of the Yamabe problem 17.3 Construction of the conformal normal coordinate 22.4 The asymptotic expansion of the Green function of the conformal Laplacianperator 28.5 Construction of the test function on the asymptotically flat manifold 35.6 Positive mass theorem and the complete solution of the Yamabe problem 42 Use Yamabe flow to solve Yamabe problem 44.1 The long time existence 45.2 The case that lambda(M) is non-positive 46.3 The case when 3 ≤ dim(M) ≤ 5 or M is locally conformally flat 47.4 The other cases 49eferences 50application/pdf669349 bytesapplication/pdfen-US保角變換純量曲率Yamabe 問題conformal diffeomorphismscalar curvatureYamabe problemthesishttp://ntur.lib.ntu.edu.tw/bitstream/246246/180621/1/ntu-98-R96221002-1.pdf