指導教授：李瑩英臺灣大學：數學研究所趙凱衞Zhao, Kai-WeiKai-WeiZhao2014-11-302018-06-282014-11-302018-06-282014http://ntur.lib.ntu.edu.tw//handle/246246/264022本論文將統整一些在球面、投影實平面以及輪胎面上求得以面積表示的拉普拉斯算子第一特徵值最優上界之方法。In this thesis, we will summarize some approaches to obtain sharp upper bounds of the first nonzero eigenvalues of the Laplacian operators on closed surfaces, including sphere S2, real projective plane RP2 and torus T2, in terms of their areas.口試委員審定書i 致謝ii 中文摘要iii Abstract iv 1 Introduction 1 2 Sharp upper bound on S2 7 2.1 Conformal automorphisms on sphere 7 2.2 Hersch’s theorem 16 3 Sharp upper bound on RP2 19 3.1 Minimal immersions into sphere 19 3.2 Conformal area 24 3.3 λ1-minimal surfaces 31 4 Sharp upper bound on T2 37 4.1 Variational method on eigenvalue problem 37 4.2 The structure of extremal metrics of eigenvalue functionals 42 4.3 Sharp upper bound of the first eigenvalue on torus 47 5 Open problems 49 References 50714680 bytesapplication/pdf論文公開時間：2014/08/21論文使用權限：同意無償授權拉普拉斯算子特徵值問題譜幾何最優上界最優不等式thesishttp://ntur.lib.ntu.edu.tw/bitstream/246246/264022/1/ntu-103-R01221005-1.pdf