Conformal Dirichlet-Neumann Maps and Poincaré-Einstein Manifolds

Abstract

A conformal description of Poincare-Einstein manifolds is developed: these structures are seen to be a special case of a natural weakening of the Einstein condition termed an almost Einstein structure. This is used for two purposes: to shed light on the relationship between the scattering construction of Graham-Zworski and the higher order conformal Dirichlet-Neumann maps of Branson and the author; to sketch a new construction of non-local (Dirichlet-to-Neumann type) conformal operators between tensor bundles.

This is a contribution to the Proceedings of the 2007 Midwest Geometry Conference in honor of Thomas P. Branson, published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA/

References in corpus (7)

• Conformally invariant powers of the Laplacian, Q-curvature, and tractor calculus
• Almost conformally Einstein manifolds and obstructions
• Special Kähler-Ricci potentials on compact Kähler manifolds
• Yang-Mills detour complexes and conformal geometry
• Standard Tractors and the Conformal Ambient Metric Construction
• Curved Casimir Operators and the BGG Machinery
• L^2 curvature and volume renormalization of AHE metrics on 4-manifolds

Cited by in corpus (10)

• Asymptotic Shear and the Intrinsic Conformal Geometry of Null-Infinity
• Holonomy reductions of Cartan geometries and curved orbit decompositions
• Tractor geometry of asymptotically flat spacetimes
• Conformal boundary operators, T-curvatures, and conformal fractional Laplacians of odd order
• Almost Einstein and Poincare-Einstein manifolds in Riemannian signature
• Higher fundamental forms of the conformal boundary of asymptotically de Sitter spacetimes
• Rigidity of conformal functionals on spheres
• Einstein gravity as a gauge theory for the conformal group
• Unit Invariance as a Unifying Principle of Physics
• Conformal Geometry of Embedded Manifolds with Boundary from Universal Holographic Formulae