Hartogs domains and the Diederich-Fornæss index (Q2008345)

The authors study the Diederich-Fornæss index on bounded pseudoconvex Hartogs domains in \(\mathbb{C}^{2}\) with \(C^{2}\) smooth boundary when the weakly pseudoconvex points in the boundary have an annular structure. The main innovation is the determination of upper and lower bounds on the Diederich-Fornæss index based on a curvature term on the annulus. The method in particular computes the Diederich-Fornæss index of the standard worm domains, recovering a recent result of \textit{B. Liu} [Adv. Math. 353, 776--801 (2019; Zbl 1429.32046)]. Additionally, the size of the curvature term determines whether the domain admits a Stein neighborhood basis. In the special case of the worm domains, the authors recover the sharp cutoff on the allowable amount of winding that was recently obtained by \textit{J. Yum} [J. Geom. Anal. 29, No. 2, 1583--1607 (2019; Zbl 1437.32006)]. Moreover, under a reasonable hypothesis on the set of weakly pseudoconvex points, the Diederich-Fornæss index is equal to \(1\) if and only if there exists a family of good vector fields in the sense of \textit{H. P. Boas} and \textit{E. J. Straube} [Math. Sci. Res. Inst. Publ. 37, 79--111 (1999; Zbl 0967.32033)]. The article under review is adapted from the first author's 2018 PhD dissertation written under the direction of the second author.