Construction of a Closed Hyperbolic Surface of Arbitrarily Small Eigenvalue of Prescribed Serial Number

TL;DR

This paper presents a geometric construction method to produce closed hyperbolic surfaces with arbitrarily small eigenvalues at a prescribed position, extending previous theoretical results with a new proof approach.

Contribution

It introduces a geometric construction technique to generate hyperbolic surfaces with small eigenvalues at any desired serial number, building on and providing a new proof of earlier theoretical results.

Findings

Constructed hyperbolic surfaces with eigenvalues below any positive threshold

Demonstrated the eigenvalue control at a prescribed position

Provided a new geometric proof of the main spectral property

Abstract

In this paper we construct, for given any small positive number $\epsilon$ and given natural number $n$, and given any closed hyperbolic surface $M$, a closed hyperbolic covering surface $\widetilde{M}$, such that its $n$-th eigenvalue is less than $\epsilon$. An application of this result will also be discussed. The main result follows from the techniques used in B.Randol's paper in 1974 [Ran]. Here I give a new and geometric proof of the main result.