The Intrinsic Geometry on Bounded Pseudoconvex Domains

We study bounded pseudoconvex domains in 2-dimensional complex Euclidean spaces. We are interested in developing sufficient and necessary conditions for the Diederich-Forn'ss index to be 1. It was known that an obstructive complex Hessian prevents the index from 1. We derive two estimates on this obstructive complex Hessian and its multiplicative inverse, respectively. A consequence of the first estimate gives almost an equivalent condition of the index to be 1. Our method is a new way to resolve problems with the Diederich-Forn'ss index. This method contains a localization to the boundary and a geometric analysis on its Levi-flat sets.