The Diederich–Fornæss index II: For domains of trivial index

Abstract

We study bounded pseudoconvex domains in complex Euclidean spaces. We find analytical necessary conditions and geometric sufficient conditions for a domain being of trivial Diederich-Fornaess index (i.e. the index equals to 1). We also connect a differential equation to the index. This reveals how a topological condition affects the solution of the associated differential equation and consequently obstructs the index being trivial. The proofs relies on a new method of study of the complex geometry of the boundary. The method was motivated by geometric analysis of Riemannian manifolds. We also generalize our main theorems under the context of de Rham cohomology.

Plurisubharmonic Defining Functions in $\mathbb C^2$

“…A similar problem was considered by Liu [10], [11]. Recall that a Diederich-Fornaess exponent of Ω ⊂⊂ C n is a number η ∈ (0, 1] for which there exists a smooth defining function ρ such that −(−ρ) η is strictly plurisubharmonic.…”

The Diederich--Fornæss index and the regularities on the $\bar{\partial}$-Neumann problem

“…The β-worm domains are of nontrivial index (see Diederich-Fornaess [24] and Liu [35]) and the strongly pseudoconvex domains are of trivial index. The Diederich-Fornaess index has been intensely studied by Adachi [2], Adachi-Brinkschulte [3], Harrington [27], Abdulsahib-Harrington [1], Krantz-Liu-Peloso [34], and Liu [37], [35], [36].…”

“…Looking at both global regularities and the Diederich-Fornaess index, there are many pieces of evidence indicating the global regularities and trivial index happens simultaneously (compare [12] [9] with [35] [36]), it is of interest to find a direct relation between the Diederich-Fornaess index and the regularities. However, it turns out this problem is rather difficult.…”

“…Assume Σ is a real curve. It is known the Diederich-Fornaess index is equal to 1 for the case of C 2 (see Theorem 4.5 of Liu [36]). Combining this fact with Theorem 0.2, one gets the following corollary.…”

Local Plurisubharmonic Defining Functions on the Boundary

“…Kohn [16] showed that if the Diederich-Fornaess index is 1 then the ∂-Neumann operator is globally regular. For more general necessary condition needed for domains to have index 1 see [18]. In general, plurisubharmonic defining functions are of importance in the study of the ∂-Neumann operator as it is a sufficient condition for global regularity of ∂-Neumann operator and the Bergman projection [2], [3], [4], [5].…”