We study bounded pseudoconvex domains in complex Euclidean space. We define an index associated to the boundary and show this new index is equivalent to the Diederich-Fornaess index defined in 1977. This connects the Diederich-Fornaess index to boundary conditions and refines the Levi pseudoconvexity. We also prove the β-worm domain is of index π/(2β). It is the first time that a precise non-trivial Diederich-Fornaess index in Euclidean spaces is obtained. This finding also indicates that the Diederich-Fornaess index is a continuum in (0, 1], not a discrete set. The ideas of proof involve a new complex geometric analytic technique on the boundary and detailed estimates on differential equations.global regularity for many other pseudoconvex domains (with smooth boundary) is still lacking a complete answer (see ). In particular, the∂-Neumann operator may or may not preserve the sobolev space W s (p,q) (Ω) of (p, q)-forms for all s > 0 (see Barrett [3], Kiselman [19] and Boas-Straube [5]) even under the category of smoothly bounded pseudoconvexity. This demands a refinement of pseudoconvexity to further classify the (weakly) pseudoconvex domains.In 1977, Diederich-Fornaess proved a celebrated theorem in [10]. The idea of the theorem is to replace the classical log composition in the notion of Hartogs pseudoconvexity with a power function composition. This classifies the notion of pseudoconvexity in the detail.Definition 0.2. Let Ω be a bounded pseudoconvex domain with C 2 boundary in C n . The function ρ is called a defining function if following conditions are satisfied:1. the function ρ is C 2 on a neighborhood of Ω, 2. the domain Ω = {z ∈ C n : ρ(z) < 0}, and 3. the gradient ∇ρ = 0 on ∂Ω.where the supremum is taken over all defining functions of Ω, is called the Diederich-Fornaess index of the domain Ω (see ).The index does a fundamental job to refine the notion of Hartogs pseudoconvexity in terms of the Sobolev regularity for the∂-Neumann operator and Bergman projection. For example, Berndtsson-Charpentier showed in [4], that the∂-Neumann operator and the Bergman projection preserves W k (Ω) for k < η 0 /2 when the Diederich-Fornaess index is η 0 . However, the Diederich-Fornaess index has not been understood thoroughly because the verifications and computations are rather difficult. In this paper, we understand the index by refining the notion of Levi pseudoconvexity. In other words, we define a natural index of the boundary and show this new index is equal to the Diederich-Fornaess index. This is given by Theorem 2.9. This theorem can be thought of as a refinement of the equivalence of Hartogs pseudoconvexity and Levi pseudoconvexity.To prove the theorem, we first simplify some common quantities and establish several useful identities. Many of the simplifications are motivated by the geometric analysis for Riemannian and Kähler manifolds. We also pass the plurisubharmonicity to the study of quadratic equations. Using the useful identities and the discriminant for quadratic equations, we are able to obtain the ...

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.