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

Susovan Pal

Abstract

In this paper we construct, for given any small positive number $ε$ 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 $ε$. 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.

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

Abstract

In this paper we construct, for given any small positive number and given natural number , and given any closed hyperbolic surface , a closed hyperbolic covering surface , such that its -th eigenvalue is less than . 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.
Paper Structure (4 sections, 10 equations, 1 figure)

This paper contains 4 sections, 10 equations, 1 figure.

Table of Contents

  1. Introduction and Preliminaries
  2. Organization of the paper
  3. Statement and Proof of the main thorem
  4. Application of the main theorem

Figures (1)

  • Figure :