Fefferman-Szegő kernels and finite-type rigidity on egg domains

We compute the Fefferman boundary measure and the associated Fefferman–Szegő kernel for the egg domains $$E_{n,m}=\{(z,w)\in {\mathbb{C}}^{n-1}\times \mathbb{C}:\mathrm{\mid }z{\mathrm{\mid }}^2+\mathrm{\mid }w{\mathrm{\mid }}^{2m}<1\}\mathrm{.}$$ The kernel is given both by an orthogonal monomial expansion and by a closed form in a natural auxiliary finite-type variable; its diagonal weak-boundary exponent recovers the integer $m$. For $n\ge 2$, the associated Fefferman–Szegő metric has constant scalar curvature only in the ball case $m=1$, and the Kähler–Einstein, constant Ricci-spectrum, and Bergman-proportionality statements follow as corollaries of the same calculation.