On largeness and multiplicity of the first eigenvalue of hyperbolic surfaces

Abstract

We apply topological methods to study the smallest non-zero number ${\lambda }_1$ in the spectrum of the Laplacian on finite area hyperbolic surfaces. For closed hyperbolic surfaces of genus two we show that the set $\{S\in {\mathcal{M}}_2:{\lambda }_1(S)>1/4\}$ is unbounded and disconnects the moduli space ${\mathcal{M}}_2$.

16 pages, 2 figures

References in corpus (3)

• An Algorithm for the Computation of Eigenvalues, Spectral Zeta Functions and Zeta-Determinants on Hyperbolic Surfaces
• Pour toute surface hyperbolique de genre $g$, $λ_{2g-2}>\frac 14}$
• On topological upper-bounds on the number of small cuspidal eigenvalues

Cited by in corpus (4)

• Random hyperbolic surfaces of large genus have first eigenvalues greater than $\frac{3}{16}-\epsilon$
• Properties of Eigenvalues on Riemann Surfaces with Large Symmetry Groups
• Small eigenvalues of closed Riemann surfaces for large genus
• Degenerating hyperbolic surfaces and spectral gaps for large genus