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The Ambient Metric (AM-178) by C. Graham, Charles Fefferman

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Chapter Four

 

Poincaré Metrics

In this chapter we consider the formal theory for Poincaré metrics associated to a conformal manifold (M, [g]). We will see that even Poincaré metrics are in one-to-one correspondence with straight ambient metrics, if both are in normal form. Thus the formal theory for Poincaré metrics is a consequence of the results of Chapter 3. The derivation of a Poincaré metric from an ambient metric was described in [FG], and the inverse construction of an ambient metric as the cone metric over a Poincaré metric was given in §5 of [GrL].

The definition of Poincaré metrics is motivated by the example of the hyperbolic metric 4(1 – |x|2)–2ge on the ball, where ge denotes the Euclidean metric. Let (M, [g]) be a smooth manifold ...

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