The Upper Bound for the Eigenvalue of the Laplace-Beltrami Operator on 2-d Sphere

摘要

This paper is aimed to summary some previous conclusions about the upper bound for the eigenvalues of the Laplace-Beltrami operator on sphere and to give more details about their proof.In addition, we propose some of our own thinking about some open problem about the eigenvalues of Laplace operator.This paper is aimed to summary some previous conclusions about the upper bound for the eigenvalues of the Laplace-Beltrami operator on sphere and to give more details about their proof.In addition, we propose some of our own thinking about some open problem about the eigenvalues of Laplace operator.This paper is aimed to summary some previous conclusions about the upper bound for the eigenvalues of the Laplace-Beltrami operator on sphere and to give more details about their proof.In addition, we propose some of our own thinking about some open problem about the eigenvalues of Laplace operator.This paper is aimed to summary some previous conclusions about the upper bound for the eigenvalues of the Laplace-Beltrami operator on sphere and to give more details about their proof.In addition, we propose some of our own thinking about some open problem about the eigenvalues of Laplace operator.

目录

声明

ABSTRACT

第1章 Introduction

1．2 Some Fundamental Conclusions

第2章 The proof of Hersch’s theorem

2．0．1 The proof of J．Hersch theorem

第3章 The proof of A2(S2)=8π

3．1 An important example

3．2 Main Theorem

3．3 The proof of the N．Nadirashvili‘s theorem

第4章 Conformal Spectrum and Harmonic Maps

4．1 The main theorem

4．2 The convergence of the metric and its conformal factor

4．3 The Boundness of the eigenfunction

4．4 The proof of Theorem 4．1

第5章 Higher order eigenvalue

5．1 The convergence of the conformal factor

5．1．1 The eigenvalue of the singular part

5．2 The mass of each singular point

第6章 The third eigenvalue of the Laplace operator on the sphere

6．1 There is at least one singular point

6．2 There is no singular point

第7章 Further thinking

参考文献

附录

Acknowledgements