A study on eigenfunctions and eigenvalues on surfaces

Abstract

In this thesis, we summarize some of the characterization theorems of the sphere, the content of Engman’s paper [6]. We investigate the method used by Cheng [4] and we can find out that there is another method to characterize the sphere using eigenfunctions. We discover that if f is an eigenfunction of the Laplacian, with f satisfying the relation f2 + c1g(∇f,∇f) = c2 for some constant c1 and c2, then the manifold must be isometric to the standard sphere or the flat torus, depending on its topology. Next we investigate Engman’s method to surface of revolution and try to extend it into higher dimensions. We try to extend the metric on Sn for n ≥ 3, with metric g = ds ⊗ ds + [r(s)]2canSn-1 on Sn \ {n, s}. If the method of classical separation of variables is valid, then we can find out that the multiplicities of sphere must be the highest, compared with other kinds of “spherical revolutions”.

Date of Award	2008
Original language	English
Awarding Institution	The Hong Kong University of Science and Technology