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 $\beta$-worm domain is of index $\pi /(2\beta )$. 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.

Corrections on names of authors in references

References in corpus (1)

• A global estimate for the Diederich--Fornaess index of weakly pseudoconvex domains

Cited by in corpus (1)

• The Diederich--Fornæss index II: for domains of trivial index