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 \(\mathbb B^2\).