Abstract:The Fefferman--Szegő 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--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 \(\mathbb B^2\).
| Comments: | 25 pages; This is a preliminary draft. Comments are welcome |
| Subjects: | Complex Variables (math.CV) |
| MSC classes: | Primary: 32F45, Secondary: 32A25 |
| Cite as: | arXiv:2605.25455 [math.CV] |
| (or arXiv:2605.25455v1 [math.CV] for this version) | |
| https://doi.org/10.48550/arXiv.2605.25455 arXiv-issued DOI via DataCite (pending registration) |
From: Anjali Bhatnagar [view email]
[v1]
Mon, 25 May 2026 06:06:30 UTC (29 KB)
此内容由惯性聚合(RSS阅读器)自动聚合整理，仅供阅读参考。 原文来自 — 版权归原作者所有。