The Invariant Szeg\H{o} metric on strongly pseudoconvex domains

The Fefferman--Szeg\H{o} metric $g_{\operatorname{FS}}^\Omega$ 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: $\dim H^{p,q}_2(\Omega)=0$ if $p+q\ne n$, while $\dim H^{p,q}_2(\Omega)=\infty$ 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--Szeg\H{o} 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--Szeg\H{o} 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--Szeg\H{o} kernel vanishes to infinite order. Consequently, the boundary is locally spherical; if, in addition, $\Omega$ is simply connected, then $\Omega$ is biholomorphic to $\mathbb B^2$.