Prof. Dr. Andreas Juhl
Research Monographs
1.
Cohomological Theory of Dynamical
Zeta Functions,
by Andreas Juhl
Series: Progress in Mathematics, vol. 194, Birkhäuser,
2001, 709pp.
Birkhäuser-Springer
The dynamical zeta functions in the title of this
research monograph are generalizations of
Selberg's zeta function.
These
functions are defined by the closed orbits (and their monodromies)
of the geodesic flows of rank one locally symmetric
spaces They can be studied by methods of
automorphic forms
(Selberg's trace formula) and
hyperbolic dynamics (Ruelle transfer operators).
The
monograph takes
an intermediate point of view by systematically
deriving the properties
of
the zeta functions from
so-called dynamical Lefschetz formulas. These constitute analogs of
Lefschetz fixed point formulas in which closed orbits take the role of
fixed points. The relevant
cohomology for that purpose is tangential cohomology of the
stable foliation (Anosov structure)
of the geodesic flow. The
fact that the stable leaves of the geodesic flows define a Legendrian
foliation of the sphere bundle puts the whole approach into the
framework of
geometric quantization.
A central result is a
natural description of
the divisors of the Selberg zeta functions of compact locally
symmetric
spaces of rank one in
terms of indices of canonical complexes. From that point of view,
the
functional equations of the
zeta functions appear as duality theorems
in index theory. Although
the
underlying dynamics largely motives
the approach, the language of the book is dominated by
representation
theory and differential geometry.
2. Families of Conformally
Covariant Differential
Operators,
Q-Curvature and Holography,
by Andreas Juhl
Series: Progress in Mathematics, vol. 275, Birkhäuser, 2009, 488pp.
Birkhäuser-Springer
The central
object of this monograph is Branson's Q-curvature. This important and subtle
scalar
Riemannian curvature quantity in even dimensions was
introduced by Thomas Branson about 15 year
ago in connection with
the study of variational formulas for determinants of conformally
covariant
differential operators.
In the book we develop a
new approach which rests on a theory of families of conformally
covariant
differential operators which are associated to hypersurfaces
in conformal manifolds. The new
approach
is at the cutting edge of many central developments in conformal
differential geometry in the last two
decades
(Fefferman-Graham ambient metric, spectral theory on
Poincare-Einstein spaces, tractor
calculus, Verma modules and
Cartan geometry). In addition, the theory of conformally covariant
families
is
inspired by the idea of holography in the AdS/CFT-duality. Among
other things, it naturally leads to a
holographic description of Q-curvature.
3. Conformal Differential Geometry:
Q-curvature and Conformal Holonomy,
by Helga
Baum and Andreas Juhl.
Series: Oberwolfach
Seminars, vol. 40, Birkhäuser, 2010, 165pp.
Birkhäuser-Springer
Conformal
invariants (conformally invariant tensors, conformally covariant
differential operators,
conformal holonomy groups etc.) are of central
significance in differential geometry and physics.
Well-known examples
of conformally covariant operators are the Yamabe, the Paneitz,
the
Dirac
and the twistor operator. These operators are intimely connected
with the notion of Branson’s
Q-curvature.
The aim of these lectures is
to present the basic ideas and some of the recent developments
around Q
-curvature and conformal holonomy. The part on Q -curvature starts with
a
discussion
of its origins and
its relevance in geometry and spectral theory. The following lectures
describe
the fundamental relation between Q -curvature and scattering
theory on asymptotically hyperbolic
manifolds. Building on this, they
introduce the recent concept of Q -curvature polynomials and
use these
to reveal the recursive structure of Q -curvatures. The part on
conformal holonomy starts
with an introduction to Cartan
connections and its holonomy groups. Then we define holonomy groups
of
conformal manifolds, discuss its relation to Einstein metrics and
recent classification results in
Riemannian and Lorentzian signature.
In particular, we explain the connection between conformal
holonomy and
conformal Killing forms and spinors, and describe Fefferman metrics in
CR geometry
as Lorentzian manifold with conformal holonomy SU(1,m).
Some recent papers
R. Graham and A. Juhl, Holographic
formula for Q-curvature, Adv. in Math. 216 (2), 841-853, 2007.
arXiv
C. Falk and A. Juhl, Universal recursive
formulas for Q-curvature, J. Reine Angew.Math. (to appear).
arXiv
A. Juhl, On conformally covariant powers of the Laplacian, (submitted).
arXiv
C. Krattenthaler and A. Juhl, Summation formulas for GJMS-operators and
Q-curvatures on the
Moebius sphere, (submitted).
arXiv
A. Juhl, On Branson's Q-curvature of order eight, (submitted).
arXiv