Yamabe problem 的相關研究

Survey on Yamabe problem

張瑞恩(Jui-En Chang)

指導教授 ： 李瑩英

國立臺灣大學/理學院/數學研究所/碩士(2009年)

https://doi.org/10.6342/NTU.2009.01164

摘要

Yamabe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。

關鍵字

保角變換 ； 純量曲率 ； Yamabe 問題

並列摘要

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.

並列關鍵字

conformal diffeomorphism ； scalar curvature ； Yamabe problem

參考文獻

[1] R. M. Schoen and S.-T. Yau, Lectures on Differential Geometry, 1994, International

[2] Simon Brendle, On the conformal scalar curvature equation and related problems,

arXiv: 0802.0295v1

[3] R. M. Schoen, Cyclic Coverings, Calabi-Yau Manifolds, and Complex multiplication,

Ch. Vatiational Theory for the Total Scalar Curvature Functional for Riemannian Metrics