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

We compute the Fefferman boundary measure and the associated Fefferman--Szeg\H{o} kernel for the egg domains $$E_{n,m}=\{(z,w)\in\mathbb{C}^{n-1}\times\mathbb{C}:\ |z|^2+|w|^{2m}<1\}.$$ 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 $n\ge2$, the associated Fefferman--Szeg\H{o} metric has constant scalar curvature only in the ball case $m=1$, and the K\"ahler--Einstein, constant Ricci-spectrum, and Bergman-proportionality statements follow as corollaries of the same calculation.