Jiliang Fan | Searcharxiv

arXiv2026

The invariant Szeg\H{o} metric on Egg domains

We study the Fefferman--Szeg\H{o} metric on egg domains \[ \mathcal D_{2m}=\{(z,w)\in\mathbb C^2: |z|^2+|w|^{2m}<1\},\qquad\qquad\qquad m\in\mathbb Z^+. \] Our first main result establishes the existence of the Fefferman--Szeg\H{o} kernel on $\mathcal{D}_{2m}$ by verifying that the Fefferman weight lies in the Muckenhoupt class $A_2(\partial\mathcal{D}_{2m})$. We then derive an explicit closed-form expression for this kernel, demonstrate that its blowup occurs precisely on the boundary diagonal, and determine its boundary asymptotic behaviour. Using this kernel, we compute the associated Fefferman--Szeg\H{o} metric and its Ricci curvature. As applications, we prove several rigidity results: the metric is K\"ahler--Einstein if and only if $m=1$; proportionality to the Bergman metric or to some complete K\"ahler metric $g_m^{\mathcal D_{2m}}$ is also equivalent to $m=1$. Finally, we establish the vanishing of the $L^2$-cohomology outside the middle dimension for the Fefferman--Szeg\H{o} metric.

math.CV↗

arXiv2026

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 $\mathbb B^2$.

math.CV↗