CR Paneitz operator on non-embeddable 3D tori has infinitely many negative eigenvalues under mild assumptions.
Acta Math.148, 159–192 (1982)
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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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Non-embeddable torus and CR Paneitz operator
CR Paneitz operator on non-embeddable 3D tori has infinitely many negative eigenvalues under mild assumptions.
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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.