The Invariant Szegő metric on strongly pseudoconvex domains

The Fefferman–Szegő metric $g_{\mathrm{FS}}^{\Omega }$ on a $C^{\infty }$-smooth bounded strongly pseudoconvex domain $\Omega \subset {\mathbb{C}}^n$ is an invariant metric defined via the Fefferman surface measure. For this metric, we first establish the vanishing of its $L^2$-Dolbeault cohomology outside the middle degree: $\dim H_2^{p,q}(\Omega )=0$ if $p+q\ne n$, while $\dim H_2^{p,q}(\Omega )=\infty$ if $p+q=n$. We also prove that the metric has $C^{\infty }$-bounded geometry. Using this analytic property, we obtain several rigidity results. In particular, if the Fefferman–Szegő metric is a gradient Kahler–Ricci soliton, then $\Omega$ is biholomorphic to the unit ball ${\mathbb{B}}^n$. Moreover, if the metric has constant scalar curvature, then it is Einstein, and again $\Omega$ is biholomorphic to ${\mathbb{B}}^n$. We also give a Ramadanov-type criterion in terms of the Fefferman–Szegő invariant function. Finally, in dimension $n=2$, assuming the existence of a Kahler immersion into a finite-dimensional ball that maps boundary to boundary transversally, we show that the logarithmic term of the Fefferman–Szegő kernel vanishes to infinite order. Consequently, the boundary is locally spherical; if, in addition, $\Omega$ is simply connected, then $\Omega$ is biholomorphic to ${\mathbb{B}}^2$.