Science Journal of Physics

May 2012, Volume 2012, ISSN:2276-6367

Author(s) 2012. This work is distributed under the Creative Commons Attribution 3.0 License.

Research Article


On The Critical Point Structure of Eigenfunctions Belonging to the First Nonzero Eigenvalue of A Genus Two Closed Hyperbolic Surface

Author: Carlos A. Cadavid1, Maria C. Osorno2, Oscar E. Ruiz3

CAD/CAM/CAE Laboratory, Universidad EAFIT, Medellin, Colombia

Accepted 10 May 2012; Available Online 30 May 2012

doi: 10.7237/sjp/128


We develop a method based on spectral graph theory to approximate the eigenvalues and eigenfunctions of the Laplace-Beltrami operator of a compact riemannian manifold. The method is applied to a closed hyperbolic surface of genus two. The results obtained agree with the ones obtained by other authors by different methods, and they serve as experimental evidence supporting the conjectured fact that the generic eigenfunctions belonging to the first nonzero eigenvalue of a closed hyperbolic surface of arbitrary genus are Morse functions having the least possible total number of critical points among all Morse functions admitted by such manifolds.

Keywords:Closed hyperbolic surface, Laplacian, Eigenvalue, Eigenfunction, Critical point, Spectral graph theory