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

Abstract: We compute the Fefferman boundary measure and the associated Fefferman--Szegő kernel for the egg domains $$ E_{n,m}={(z,w)\in\mathbb{C}^{{n-1}\times\mathbb{C}:\} |z|^{2+|w|^{2m}<1}.} $$ 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--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.

• The paper computes the explicit form of the Fefferman–Szegő kernel on egg domains, detailing the boundary measure and monomial expansion techniques.
• It establishes analytical rigidity by linking kernel blow-up rates with the D’Angelo type, thereby quantifying finite-type behavior in weakly pseudoconvex settings.
• The study demonstrates that constant scalar curvature and Kähler–Einstein properties occur uniquely in the unit ball case, underscoring strict finite-type restrictions.

Fefferman–Szegő Kernels and Rigidity Phenomena on Egg Domains

Introduction and Mathematical Context

This paper undertakes the computation of the Fefferman–Szegő kernel and the associated invariant Kähler metric for a canonical class of weakly pseudoconvex domains, the so-called egg domains

$$E_{n,m} = \{ (z, w) \in \mathbb{C}^{n-1} \times \mathbb{C} : |z|^2 + |w|^{2m} < 1 \},$$

where $n \geq 2$ and $m \geq 1$ is an integer. For $m=1$, $E_{n,1}$ is the unit ball; for $m > 1$, $E_{n,m}$ retains a smooth boundary but exhibits degeneracy (finite-type weak pseudoconvexity) along $w=0$. These domains serve as minimal models where finite D'Angelo type features arise and yet retain enough symmetry to permit explicit calculations.

The core technical objective is the exact computation of the Fefferman–Szegő kernel for $E_{n,m}$. Consequent to this, the paper establishes strong rigidity phenomena: the associated Fefferman–Szegő metric on $E_{n,m}$ possesses constant scalar curvature, constant Ricci spectrum, and Kähler–Einstein and Bergman-proportionality properties only in the unit ball case $n \geq 2$0. Thus, the geometry and complex function theory of egg domains is shown to be rigidly controlled by the integer parameter $n \geq 2$1, which encodes the weak finite type.

Main Technical Results

Computation of Fefferman's Boundary Measure

The Fefferman boundary measure on $n \geq 2$2 is explicitly computed using the Monge–Ampère determinant associated to the defining function $n \geq 2$3. The calculation yields

$n \geq 2$4

on the boundary, and the explicit form of $n \geq 2$5 (up to normalization) in radial-angular coordinates. This precise description is central for subsequent Hilbert space constructions and the Szegő kernel analysis.

Hilbert Space and Monomial Basis

The Hardy-type space $n \geq 2$6 is identified as the closure in $n \geq 2$7 of holomorphic boundary functions, and an explicit orthogonal family of monomials is constructed. The norms of these monomials are computed as

$n \geq 2$8

where $n \geq 2$9 and $m \geq 1$0 is affine in $m \geq 1$1 and $m \geq 1$2. This representation is key to building the kernel in terms of a power series.

Explicit Series and Closed Form for the Kernel

The Fefferman–Szegő kernel $m \geq 1$3 is computed as an absolutely convergent double series expansion in monomials, which is reorganized using multinomial identities to a single series involving the auxiliary variable

$m \geq 1$4

Leveraging the integrality of $m \geq 1$5, this series is further recast as a finite closed-form quotient

$m \geq 1$6

where $m \geq 1$7 is a degree-$m \geq 1$8 polynomial determined by a finite generating function mechanism. Here, $m \geq 1$9 recovers the type parameter; the polynomial numerator structure enables an exact analysis of boundary singularities and weak-type exponents.

Weak Boundary Exponent and Type Detection

By restricting to the weak axis ($m=1$0), the paper shows that the blow-up rate of the diagonal kernel as $m=1$1 approaches the unit sphere is

$m=1$2

with

$m=1$3

This exponent is monotonic in $m=1$4 and uniquely determines $m=1$5, rigorously connecting the analytic data of the kernel to the boundary's D'Angelo type $m=1$6 at weak pseudoconvex points.

Inverse Rigidity and Boundary Regularity

A boundary-regular biholomorphism between $m=1$7 and $m=1$8 (one extending holomorphically across boundaries) is shown to exist if and only if $m=1$9 and $E_{n,1}$0. This result follows from detailed analysis of the spectrum of boundary types (strongly pseudoconvex points of type 2, weak points of type $E_{n,1}$1), which is detected by the D'Angelo invariant and preserved under biholomorphism.

Kähler Geometry and Scalar Curvature Rigidity

On the functional-analytic side, the diagonal form of the Fefferman–Szegő kernel induces an invariant Kähler metric. The scalar curvature of this metric is explicitly computed as a rational function of the finite-type variable $E_{n,1}$2.

The rigidity theorem demonstrates that constant scalar curvature, constant Ricci spectrum, Kähler–Einstein property, and proportionality between the Fefferman–Szegő and Bergman metrics can occur only if $E_{n,1}$3. The proof utilizes the structure of the kernel's numerator polynomial: any root beyond $E_{n,1}$4 leads to algebraic contradictions when enforcing the constancy of the curvature quantities.

Strong Numerical and Structural Findings

• Explicit closed-form for $E_{n,1}$5: The kernel's dependence on $E_{n,1}$6 and $E_{n,1}$7 is given in closed algebraic terms, with all singularity and blow-up behavior fully described.
• Rigidity of geometric invariants: The presence of constant scalar curvature, Kähler–Einstein structure, or Szegő–Bergman proportionality rigidly forces $E_{n,1}$8, i.e., the unit ball case — no other egg domain admits these invariant geometric structures.
• Boundary type and analytic invariants coincide: The weak-normal exponent in kernel blow-up directly matches the D'Angelo type, establishing an analytic test for geometric type.

Implications and Speculation on Future Developments

The explicit analysis of Fefferman–Szegő kernels in the egg domain context establishes a precise link between boundary type, kernel asymptotics, and invariant Kähler geometry. From a complex analysis perspective, it demonstrates that weak pseudoconvexity of finite type is analytically rigid, precluding the existence of extremal Kähler metrics other than in the symmetric (unit ball) case. These constraints have strong implications for classification in several complex variables and CR geometry.

Practically, these results suggest that attempts to construct canonical Kähler metrics (such as Kähler–Einstein or constant scalar curvature metrics) using kernel methods must fail in higher-type pseudoconvex domains unless the geometry reduces to a strongly pseudoconvex model. On the theoretical front, the methods provide a blueprint for extending kernel analyses to other families of weakly pseudoconvex and finite-type domains, though the coupling of terms in more general Reinhardt settings may require significant new ideas.

The paper also outlines natural extensions, including the study of multi-egg domains and more complicated domains of finite type without the high symmetry of the present model. The techniques introduced, especially the use of auxiliary-variable closed forms and the root structure of kernel numerator polynomials, are likely to inform future work in the explicit computation of invariant metrics and the study of their curvature properties.

Conclusion

This paper provides a comprehensive explicit computation of the Fefferman–Szegő kernel and the associated invariant metrics for egg domains of the form $E_{n,1}$9. The kernel analysis not only furnishes closed-form expressions and exposes the weak-type exponents but also yields rigidity results for boundary geometry and Kähler metric properties: constant scalar curvature, Kähler–Einstein structure, and Bergman–Szegő proportionality are confined strictly to the unit ball. The analytic and algebraic techniques developed here set a robust foundation for further exploration of kernel-based invariants on more general domains of finite type in several complex variables.