In complex dimension three, vanishing of the second-order coefficient in the boundary expansion of the normalized determinant of the Fefferman-Szegő metric is equivalent to local CR sphericity, as it equals a multiple of the squared Chern-Moser curvature.
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Explicit Fefferman-Szegő metric on egg domains D_{2m} is Kähler-Einstein and proportional to Bergman metric iff m=1.
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The Fefferman-Szeg\H{o} Sphericity Criterion in Complex Dimension Three
In complex dimension three, vanishing of the second-order coefficient in the boundary expansion of the normalized determinant of the Fefferman-Szegő metric is equivalent to local CR sphericity, as it equals a multiple of the squared Chern-Moser curvature.
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The invariant Szeg\H{o} metric on Egg domains
Explicit Fefferman-Szegő metric on egg domains D_{2m} is Kähler-Einstein and proportional to Bergman metric iff m=1.