Geometric analysis on the diederich–Fornæss index

Given bounded pseudoconvex domains in 2-dimensional complex Euclidean space, we derive analytical and geometric conditions which guarantee the Diederich-Fornæss index is 1. The analytical condition is independent of strongly pseudoconvex points and extends Fornæss– Herbig’s theorem in 2007. The geometric condition reveals the index reflects topological properties of boundary. The proof uses an idea including differential equations and geometric analysis to find the optimal defining function. We also give a precise domain of which the Diederich– Fornæss index is 1. The index of this domain can not be verified by formerly known theorems.