Automorphisms of the Worm Domain
Fani Xerakia
TL;DR
The paper addresses the automorphism problem for the Diederich-Fornæss worm domain, a smooth pseudoconvex domain without a Stein neighborhood basis. It develops a CR-geometric approach, combining Segre-variety techniques and infinitesimal CR automorphisms with local extensions under Local Condition $R$, first for the unbounded worm boundary and then for the bounded worm. The main finding is that the automorphism group of the worm domain consists precisely of rotations in the $z$-variable, $F(z,w)=(e^{i\theta}z,w)$, with the boundary being locally spherical away from the exceptional locus and caps, and an explicit local biholomorphism to the sphere on the core. These results clarify how boundary geometry constrains holomorphic symmetries in non-Stein pseudoconvex domains and provide an explicit, geometrically meaningful description of the automorphism structure.
Abstract
The Diederich-Fornæss worm domain, an important example of a smoothly bounded pseudoconvex domain without a Stein neighborhood basis, provides key counterexamples in the theory of Several Complex Variables. In this paper, we examine its automorphism group and observe that its boundary is locally spherical everywhere except at the exceptional locus and the caps.