Differential geometric study of conformal and CR invariant theory via the ambient metric construction due to Fefferman and Graham

• Applicant: Privatdozent Dr. Felipe Leitner
• Subject Area: Mathematics
• Term: from 2007 to 2010
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 42840418

Conformal and CR geometries are rigid structures, which are distinguishable by curvature and integral invariants. The invariant theory can be studied via parabolic Cartan geometry/tractor calculus. An alternative approach developed by Fefferman/Graham is the ambient metric construction, which is basically equivalent to the Poincare-Einstein model. Via these constructions invariants such as the GJMS-operators and Q-curvature can be defined for the purpose of purely mathematical studies in conformal and CR geometry. Moreover, "holographic relations" of objects on the (conformal) boundary and the interior "bulk" of the Poincare model can be established. This is of interest in physics, where the holographic principle (of quantum gravity) finds a concrete manifestation in connection with the AdS/CFT-correspondence, which aims to relate string theory/sup er gravity with sup er symmetric conformal field theories. This project considers as its main goal (geometric) realisations of Fefferman-Graham ambient and Poincare-Einstein models in curved situations. For such models explicit expressions for conformal and CR invariants (which are otherwise formally defined) shall be calculated. We aim to establish "holographic" relations between objects on the boundary and on the interior of the Poincare-Einstein model. In particular, we ask for Taylor expansions of Poincare spaces and explicit formulae for GJMS-operators and Q-curvature. A further question concerns geometric Poisson transformations for harmonic solutions of boundary problems on curved Poincare models in terms of integral formulae. It is also a task to relate symmetries such as solutions of overdetermined invariant differential equations (e.g. twistor forms/spinors) on the boundary to geometric objects on the Poincare "bulk".

• DFG Programme: Priority Programmes
• Subproject of: SPP 1154: Global Differential Geometry