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A basic problem in geometry is to ?nd canonical metrics on smooth
manifolds. Such metrics can be speci?ed, for instance, by curvature
conditions or extremality properties, and are expected to contain
basic information on the topology of the underlying manifold.
Constant curvature metrics on surfaces are such canonical metrics.
Their distinguished role is emphasized by classical uniformization
theory. Amorerecentcharacterizationofthesemetrics describes them
ascriticalpoints of the determinant functional for the
Laplacian.The key tool here is Polyakov'sva- ationalformula for the
determinant. In higher dimensions, however,it is necessary to
further restrict the problem, for instance, to the search for
canonical metrics in conformal classes. Here two metrics are
considered to belong to the same conf- mal class if they di?er by a
nowhere vanishing factor. A typical question in that direction is
the Yamabe problem ([165]), which asks for constant scalar
curvature metrics in conformal classes. In connection with the
problem of understanding the structure of Polyakov type formulas
for the determinants of conformally covariant di?erential operators
in higher dimensions, Branson ([31]) discovered a remarkable
curvature quantity which now is called Branson's Q-curvature. It is
one of the main objects in this book.
The periodic orbits of the geodesic flow of compact locally
symmetric spaces of negative curvature give rise to meromorphic
zeta functions (generalized Selberg zeta functions, Ruelle zeta
functions). The book treats various aspects of the idea to
understand the analytical properties of these zeta functions on the
basis of appropriate analogs of the Lefschetz fixed point formula
in which the periodic orbits of the flow take the place of the
fixed points. According to geometric quantization the Anosov
foliations of the sphere bundle provide a natural source for the
definition of the cohomological data in the Lefschetz formula. The
Lefschetz formula method can be considered as a link between the
automorphic approach (Selberg trace formula) and Ruelle's approach
(transfer operators). It yields a uniform cohomological
characterization of the zeros and poles of the zeta functions and a
new understanding of the functional equations from an index
theoretical point of view. The divisors of the Selberg zeta
functions also admit characterizations in terms of harmonic
currents on the sphere bundle which represent the cohomology
classes in the Lefschetz formulas in the sense of a Hodge theory.
The concept of harmonic currents to be used for that purpose is
introduced here for the first time. Harmonic currents for the
geodesic flow of a noncompact hyperbolic space with a compact
convex core generalize the Patterson-Sullivan measure on the limit
set and are responsible for the zeros and poles of the
corresponding zeta function. The book describes the present state
of the research in a new field on the cutting edge of global
analysis, harmonic analysis and dynamical systems. It should be
appealing notonly to the specialists on zeta functions which will
find their object of favorite interest connected in new ways with
index theory, geometric quantization methods, foliation theory and
representation theory. There are many unsolved problems and the
book hopefully promotes further progress along the lines indicated
here.
Dynamical zeta functions are associated to dynamical systems with a
countable set of periodic orbits. The dynamical zeta functions of
the geodesic flow of lo cally symmetric spaces of rank one are
known also as the generalized Selberg zeta functions. The present
book is concerned with these zeta functions from a cohomological
point of view. Originally, the Selberg zeta function appeared in
the spectral theory of automorphic forms and were suggested by an
analogy between Weil's explicit formula for the Riemann zeta
function and Selberg's trace formula ( 261]). The purpose of the
cohomological theory is to understand the analytical properties of
the zeta functions on the basis of suitable analogs of the
Lefschetz fixed point formula in which periodic orbits of the
geodesic flow take the place of fixed points. This approach is
parallel to Weil's idea to analyze the zeta functions of pro
jective algebraic varieties over finite fields on the basis of
suitable versions of the Lefschetz fixed point formula. The
Lefschetz formula formalism shows that the divisors of the rational
Hassc-Wcil zeta functions are determined by the spectra of
Frobenius operators on l-adic cohomology."
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