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

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Chapter Five

 

Self-dual Poincaré Metrics

In [LeB], LeBrun showed using twistor methods that if g is a real-analytic metric on an oriented real-analytic 3-manifold M, then [g] is the conformal infinity of a real-analytic self-dual Einstein metric on a deleted collar neighborhood of M × {0} in M × [0, ∞), uniquely determined up to real-analytic diffeomorphism. As mentioned in [FG], LeBrun’s result can be proved as an application of our formal theory of Poincaré metrics. In this chapter we show that the corresponding formal power series statement is a consequence of Theorem 4.8. The self-duality condition can be viewed as providing a conformally invariant specification of the formally undetermined term .

Let M be an oriented 3-manifold. Give ...

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