 
Summary: The covering spectrum of a compact length space
Christina Sormani Guofang Wei y
Abstract
We dene a new spectrum for compact length spaces and Riemannian manifolds called the
\covering spectrum" which roughly measures the size of the one dimensional holes in the space.
More specically, the covering spectrum is a set of real numbers Æ > 0 which identify the distinct
Æ covers of the space. We investigate the relationship between this covering spectrum, the length
spectrum, the marked length spectrum and the Laplace spectrum. We analyze the behavior of
the covering spectrum under GromovHausdor convergence and study its gap phenomenon.
1 Introduction
One of the most important subelds of Riemannian Geometry is the study of the Laplace spectrum of
a compact Riemannian manifold. Recall that the Laplace spectrum is dened as the set of eigenvalues
of the Laplace operator. The elements of the Laplace spectrum are assigned a multiplicity equal to
the dimension of the corresponding eigenspace.
Another spectrum dened in an entirely dierent manner is the length spectrum of a manifold:
the set of lengths of smoothly closed geodesics. There are various methods used to assign a mul
tiplicity to each element of the length spectrum. The simplest notion is to count all geodesics of
a given length. This becomes uninteresting when one has continua of geodesics of the same length
as in a torus, so that all or some multiplicities become innite. A common alternative denition of
the multiplicity of a given length is the number of free homotopy classes of geodesics that contain a
