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The covering spectrum of a compact length space Christina Sormani Guofang Wei y

Summary: The covering spectrum of a compact length space
Christina Sormani  Guofang Wei y
We de ne 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 speci cally, the covering spectrum is a set of real numbers > 0 which identify the distinct
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 Gromov-Hausdor convergence and study its gap phenomenon.
1 Introduction
One of the most important sub elds of Riemannian Geometry is the study of the Laplace spectrum of
a compact Riemannian manifold. Recall that the Laplace spectrum is de ned 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 de ned in an entirely di erent 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 in nite. A common alternative de nition of
the multiplicity of a given length is the number of free homotopy classes of geodesics that contain a


Source: Akhmedov, Azer - Department of Mathematics, University of California at Santa Barbara


Collections: Mathematics