The invariant Szegő metric on Egg domains

We study the Fefferman–Szegő metric on egg domains $${\mathcal{D}}_{2m}=\{(z,w)\in {\mathbb{C}}^2:\mathrm{\mid }z{\mathrm{\mid }}^2+\mathrm{\mid }w{\mathrm{\mid }}^{2m}<1\},m\in {\mathbb{Z}}^{+}\mathrm{.}$$ Our first main result establishes the existence of the Fefferman–Szegő kernel on ${\mathcal{D}}_{2m}$ by verifying that the Fefferman weight lies in the Muckenhoupt class $A_2(\mathrm{\partial }{\mathcal{D}}_{2m})$. 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–Szegő metric and its Ricci curvature. As applications, we prove several rigidity results: the metric is Kähler–Einstein if and only if $m=1$; proportionality to the Bergman metric or to some complete Kähler 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–Szegő metric.