 
Summary: Contemporary Mathematics
Finite Part of Spectrum and Isospectrality
Xianzhe Dai and Guofang Wei
Abstract. We study geometric conditions under which finitely many eigen
values are sufficient to determine all spectral data. We also discuss briefly the
implication of this result to the local structure of moduli space of isopectral
metrics.
1. Introduction
For a compact (smooth) manifold Mn
, a Riemannian metric gives rise to a
canonical differential operator, namely, the Laplacian (acting on smooth func
tions). If M = , we put the Dirichlet boundary condition on the boundary. This
makes into a selfadjoint second order elliptic operator. Hence it has a discrete
spectrum all consisting of eigenvalues of finite multiplicity:
0 = 0 < 1 2 · · · .
Here the eigenvalues are repeated according to their multiplicity.
Two Riemannian manifolds (M, g) and (M , g ) are said to be isospectral if
their corresponding eigenvalues are identical:
i(M, g) = i(M , g ), i = 1, 2, · · · .
One of the main questions, popularized by Kac's question "can one hear the shape
