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
Rigidity of complete K\"ahler-Einstein metrics under cscK perturbations
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abstract
In this paper, we study constant scalar curvature K\"ahler (cscK) metrics on complete non-compact K\"ahler--Einstein manifolds. We give sufficient conditions under which a cscK perturbation of a K\"ahler--Einstein metric must remain K\"ahler--Einstein. As a model case, we prove that the Bergman metric on a bounded strictly pseudoconvex domain is K\"ahler--Einstein whenever it has constant scalar curvature. In particular, combined with Huang--Xiao's resolution of Cheng's conjecture, this yields the ball characterization for smooth bounded strictly pseudoconvex domains.
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If the Bergman metric of a pseudoconvex domain in C^n (n≥3) has constant scalar curvature, then every strongly pseudoconvex boundary point is spherical.
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The Invariant Szeg\H{o} metric on strongly pseudoconvex domains
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
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A note on csc Bergman metric
If the Bergman metric of a pseudoconvex domain in C^n (n≥3) has constant scalar curvature, then every strongly pseudoconvex boundary point is spherical.