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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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Formal Theory

Formal Theory

(p.17) Chapter Three Formal Theory
The Ambient Metric (AM-178)

Charles Fefferman

C. Robin Graham

Princeton University Press

This chapter presents proof of Theorem 2.9 for n > 2. It further notes that similar arguments using the form of the perturbation formulae (3.32) for the Ricci curvature show that the metrics constructed in Theorems 3.7, 3.9 and 3.10 are the only formal expansions of metrics for ρ‎ > 0 or ρ‎ < 0 involving positive powers of ¦ ρ‎ r ρ‎ and log ¦ ρ‎ r ρ‎, which are homogeneous of degree 2, Ricci-flat to infinite order, and in normal form. Convergence of formal series determined by Fuchsian problems such as these in the case of real-analytic data has been considered by several authors. In particular, results of [BaoG] can be applied to establish the convergence of the series occurring in Theorems 3.7 and 3.9 (and also in Theorem 3.10 if the obstruction tensor vanishes) if g and h are real-analytic. Convergence results including also the case when log terms occur in Theorem 3.10 are contained in [K].

Keywords:   conformal geometry, ambient metric, theorem, Fuchsian problems, Ricci curvature

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