This work presents some classical as well as some very recent
results and techniques concerning the spectral geometry
corresponding to the Laplace-Beltrami operator and the Hodge-de
Rham operators. It treats many topics that are not usually dealt
with in this field, such as the continuous dependence of the
eigenvalues with respect to the Riemannian metric in the
CINFINITY-topology, and some of their consequences, such as
Uhlenbeck's genericity theorem; examples of non-isometric flat tori
in all dimensions greater than or equal to four; Gordon's classical
technique for constructing isospectral closed Riemannian manifolds;
a detailed presentation of Sunada's technique and Pesce's approach
to isospectrality; Gordon and Webb's example of non-isometric
convex domains in Rn (n>=4) that are isospectral for both
Dirichlet and Neumann boundary conditions; the Chanillo-TrA]ves
estimate for the first positive eigenvalue of the Hodge-de Rham
operator, etc. Significant applications are developed, and many
open problems, references and suggestions for further reading are
given. Several themes for additional research are pointed out.
Audience: This volume is designed as an introductory text for
mathematicians and physicists interested in global analysis,
analysis on manifolds, differential geometry, linear and
multilinear algebra, and matrix theory. It is accessible to readers
whose background includes basic Riemannian geometry and functional
analysis. These mathematical prerequisites are covered in the first
two chapters, thus making the book largely self-contained.
General
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