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The Ambient Metric (AM-178)$
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Charles Fefferman and C. Robin Graham

Print publication date: 2011

Print ISBN-13: 9780691153131

Published to Princeton Scholarship Online: October 2017

DOI: 10.23943/princeton/9780691153131.001.0001

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Conformally Flat and Conformally Einstein Spaces

Conformally Flat and Conformally Einstein Spaces

Chapter Seven Conformally Flat and Conformally Einstein Spaces
The Ambient Metric (AM-178)

Charles Fefferman

C. Robin Graham

Princeton University Press

This chapter analyzes the ambient and Poincaré metrics for locally conformally flat manifolds and for conformal classes containing an Einstein metric. The obstruction tensor vanishes for even dimensional conformal structures of these types. It shows that for these special conformal classes, there is a way to uniquely specify the formally undetermined term at order n/2 in an invariant way and thereby obtain a unique ambient metric up to terms vanishing to infinite order and up to diffeomorphism, just like in odd dimensions. It derives a formula of Skenderis and Solodukhin [SS] for the ambient or Poincaré metric in the locally conformally flat case which is in normal form relative to an arbitrary metric in the conformal class, and proves an elated unique continuation result for hyperbolic metrics in terms of data at conformal infinity. The case n = 2 is special for all of these considerations. The chapter also derives the form of the GJMS operators for an Einstein metric.

Keywords:   conformal geometry, ambient metric, Poincaré metrics, flat manifolds, Eistein metric

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