Fractals and the Base Eigenvalue of the Laplacian on Certain Noncompact Surfaces

Abstract

In this paper, we find bounds on the smallest or base eigenvalue for the Laplacian on the noncompact surfaces within a certain parameterized class. We attempt to fit a curve to the resulting data. These eigenvalues are related to the Hausdorff dimension of certain fractals and to some problems in number theory.

Keywords:

• Primary 35P15, 37D05, 37F35
• Minimal eigenvalue
• base eigenvalue
• Laplacian
• Hausdorff dimension
• Hecke group
• Schottky group