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Korean Mathematical Society JKMS

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J. Korean Math. Soc. 2018; 55(4): 897-921

Online first article March 21, 2018      Printed July 1, 2018

https://doi.org/10.4134/JKMS.j170515

Copyright © The Korean Mathematical Society.

Geometric analysis on the Diederich--Forn\ae ss index

Steven George Krantz, Bingyuan Liu, Marco Maria Peloso

Washington University, University of California, Universit? degli Studi di Milano

Abstract

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Given bounded pseudoconvex domains in 2-dimensional complex Euclidean space, we derive analytical and geometric conditions which guarantee the Diederich-Forn\ae ss index is 1. The analytical condition is independent of strongly pseudoconvex points and extends Forn\ae ss--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--Forn\ae ss index is 1. The index of this domain can not be verified by formerly known theorems.

Keywords: Diederich-Forn\ae ss index, pseudoconvex, domain, plurisub\-harmonic

MSC numbers: Primary 32U05; Secondary 53C21

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