The Diederich–Fornæss index I: For domains of non-trivial index

Abstract

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–Fornæss index defined in 1977. This connects the Diederich–Fornæss 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–Fornæss index in Euclidean spaces is obtained. This finding also indicates that the Diederich–Fornæss 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.