Local Geometry of Decoupled Pseudoconvex Domains

Abstract

For any smoothly bounded, strongly pseudoconvex domain Ω in ℂⁿ, the boundary of a suitable ball in ℂⁿ locally approximates b Ω near p ∈ b Ω, i.e., their defining functions agree up to 3-jets near p. This approximation is often the first step taken when analyzing various domain dependent functions and operators on Ω (e.g., the Bergman and Szegö kernels, the ∂̄-Neumann operator, Poisson kernels), as it allows certain results on the behavior of the operators (or functions) associated to the ball and various tools of classical analysis to be brought to bear on questions about the behavior of the operators on Ω. See, for instance, [F],[F-S],[P]. When the domain Ω is only weakly pseudoconvex, there is no single model domain which locally approximates b Ω, in any reasonable sense, near some z ⁰ ∈ b Ω. Moreover, the description of an appropriate approximating domain in terms of analytic information on the defining function is not known in general, even when the Levi form degenerates to finite order. Additionally, even when an approximating domain for b Ω near z ⁰ ∈ b Ω can be found, its shape may vary greatly from the shape of an approximating domain for b Ω near another z ¹ ∈ bΩ, even when z ¹ is close to z ⁰.