Small Eigenvalues of Hyperbolic Polygons

Abstract

The hyperbolic geometric structure is a type of non-Euclidean geometry. We first examine the geodesics in hyperbolic space using the properties of Mobius transformations in the upper half-plane. We derive a distance formula and use it to determine the hyperbolic versions of the Pythagorean theorem, the Law of Sines, the Law of Cosines, Ceva's Theorem, and Menelaus's Theorem. We then examine the spectral properties of hyperbolic triangles. We determine a differential equation for a family of triangles with constant first eigenvalue of the hyperbolic Laplacian with Dirichlet boundary conditions