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Poincaré–Einstein holography for forms via conformal geometry in the bulk

About this Title

A. Rod Gover, Department of Mathematics, The University of Auckland, Private Bag 92019, Auckland 1, New Zealand – and – Mathematical Sciences Institute, Australian National University, ACT 0200, AustraliDepartment of Mathematics, The University of Auckland, Private Bag 92019, Auckland 1, New Zealand – and – Mathematical Sciences Institute, Australian National University, ACT 0200, Australia, Emanuele Latini, Institut für Mathematik, Universität Zürich-Irchel, Winterthurerstrasse 190, CH-8057 Zürich, Switzerland – and – INFN, Laboratori Nazionali di Frascati, CP 13, I-00044 Frascati, Italy and Andrew Waldron, Department of Mathematics, University of California, Davis, California 95616

Publication: Memoirs of the American Mathematical Society
Publication Year: 2015; Volume 235, Number 1106
ISBNs: 978-1-4704-1092-6 (print); 978-1-4704-2224-0 (online)
DOI: https://doi.org/10.1090/memo/1106
Published electronically: September 23, 2014
Keywords: Differential forms, conformally compact, Q-curvature, conformal harmonics, Poincaré–Einstein, AdS/CFT, holography, scattering, Poisson transform.
MSC: Primary 51P05, 70S99, 53Z05, 53B15, 81T20, 53A30, 58A10, 81R20

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Table of Contents

Chapters

  • 1. Introduction
  • 2. Bulk conformal geometry and extension problems
  • 3. Tractor exterior calculus
  • 4. The exterior calculus of scale
  • 5. Higher form Proca equations
  • 6. Obstructions, detours, gauge operators and $Q$-curvature
  • A. The ambient manifold
  • B. List of common symbols

Abstract

We study higher form Proca equations on Einstein manifolds with boundary data along conformal infinity. We solve these Laplace-type boundary problems formally, and to all orders, by constructing an operator which projects arbitrary forms to solutions. We also develop a product formula for solving these asymptotic problems in general. The central tools of our approach are (i) the conformal geometry of differential forms and the associated exterior tractor calculus, and (ii) a generalised notion of scale which encodes the connection between the underlying geometry and its boundary. The latter also controls the breaking of conformal invariance in a very strict way by coupling conformally invariant equations to the scale tractor associated with the generalised scale. From this, we obtain a map from existing solutions to new ones that exchanges Dirichlet and Neumann boundary conditions. Together, the scale tractor and exterior structure extend the solution generating algebra of Gover and Waldron to a conformally invariant, Poincaré–Einstein calculus on (tractor) differential forms. This calculus leads to explicit holographic formulæ for all the higher order conformal operators on weighted differential forms, differential complexes, and Q-operators of Branson and Gover (2005). This complements the results of Aubry and Guillarmou where associated conformal harmonic spaces parametrise smooth solutions.

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