Of central interest in conformal
differential geometry are conformal invariants, for example,
conformally covariant differential operators, conformal curvature
tensors, conformal holonomy groups or groups of conformal
diffeomorphisms. Conformally covariant operators arise often in
physics. For example, the classical Maxwell equation on 4-dimensional
Minkowski space is conformally covariant. Further conformally covariant
operators are the Dirac operator, the Yamabe operator, the Paneitz
operator and the twistor operator. In recent years the
AdS/CFTcorrespondence in quantum gravity motivated new studies in
conformal differerential geometry. The aim of the seminar is to present
some of these ideas and developments.
The seminar is organized like a summer school and adressed to graduate
students and post doc's. There will be 2 lecture series, one on
Q-curvature, its origin and relevance in geometry, spectral theory and
physics and one on holonomy theory of conformal manifolds and its
relation to Einstein metrics and to conformally invariant twistor
equations. In the two courses we intend to cover the following subjects:
Andreas Juhl: Q-curvature
Helga Baum: Holonomy theory of conformal structures
- The flat model of conformal geometry. Conformal group. Actions
- The Fefferman-Graham construction and conformal invariants.
Poincare-Einstein metrics. Conformally covariant powers of the
Laplacian (GJMS-operators). GJMS-operators and scattering theory of
asymptotically hyperbolic spaces.
- Origins and various routes to Q-curvature. Q-curvature in
spectral theory and geometry. Q-curvature in dimension 4,
Paneitz-operator and Paneitz-curvature.
- Holography and Q-curvature. The holographic formula and its
- Families of conformally covariant differential operators. Residue
families and Q-polynomials. Recursive structures.
- Eastwoods' curved translation principle: from Verma modules to
tractor calculus. Applications to families of conformally covariant
operators. Tractor construction of Q-curvature. Extrinsic Q-curvature.
- Cartan connections, curvature and holonomy groups. Tractor
bundles and tractor connections.
- Conformal structures and the normal conformal Cartan connection.
The conformal tractor connection and its curvature. Holonomy groups of
- Conformal holonomy groups and Einstein metrics. Splitting theorem
in conformal holonomy theory. Cone and ambient metric constructions and
relation to metric holonomy groups. Classification results for
Riemannian and Lorentzian conformal holonomy groups.
- Twistor equation on spinors, conformal Killing spinors and
related geometries. Link to conformal holonomy.
- CR-geometry and Fefferman spaces.
- Conformal structures with unitary and special unitary holonomy
groups. Results for other holonomy groups.