CR Paneitz operator on non-embeddable 3D tori has infinitely many negative eigenvalues under mild assumptions.
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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.
The Fefferman-Szegő metric on C^∞-smooth bounded strongly pseudoconvex domains in C^n has vanishing L2-Dolbeault cohomology outside middle degree, C^∞ bounded geometry, and yields rigidity results implying the domain is biholomorphic to the ball under gradient Kahler-Ricci soliton or constant scalar
Explicit Fefferman-Szegő metric on egg domains D_{2m} is Kähler-Einstein and proportional to Bergman metric iff m=1.
Establishes lower bound for Kähler hyperbolicity modulus on complete Kähler manifolds via boundary gradient length of plurisubharmonic functions, with applications to symmetric and strongly pseudoconvex domains.
Survey of known results on the bottom of the spectrum of the Hodge Laplacian on complete noncompact Kähler manifolds, including upper bounds under curvature assumptions and rigidity theorems.
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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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Bottom of the spectrum of complete noncompact K\"{a}hler manifolds
Survey of known results on the bottom of the spectrum of the Hodge Laplacian on complete noncompact Kähler manifolds, including upper bounds under curvature assumptions and rigidity theorems.