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

Abstract

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.