Reinhardt Models of Rank-Two Rotation Domains

Definitions and mathematical statement

For an integer $n\geq 1$, a Reinhardt domain is a nonempty connected open set $D\subseteq{\mathbb{C}}^n$ invariant under coordinatewise rotations: for every $z\in D$ and $(\theta_1,\ldots,\theta_n)\in{\mathbb{R}}^n$, \[ (e^{i\theta_1}z_1,\ldots,e^{i\theta_n}z_n)\in D. \] For a complex manifold $U$ and a biholomorphic map $f:U\to U$, a diagonal rotational linear model consists of such a domain $D$, a biholomorphism $h:U\to D$, and a map \[ L(z_1,\ldots,z_n)=(\lambda_1z_1,\ldots,\lambda_nz_n),\qquad |\lambda_j|=1, \] satisfying $h\circ f\circ h^{-1}=L|_D$. One possible rank convention assigns rank two when the identity component of \[ \overline{\{(\lambda_1^k,\ldots,\lambda_n^k):k\in{\mathbb{Z}}\}}\subseteq(S^1)^n, \qquad S^1=\{w\in{\mathbb{C}}:|w|=1\}, \] is a real two-dimensional torus. The supplied record does not specify whether this is its intended definition of rank or of a rotation domain; these definitions therefore describe a conditional interpretation only.

Classification problem, with unresolved scope. Determine which Reinhardt domains occur as linear models of rank-two rotation domains in the class of dynamical systems intended by the source. The supplied background explicitly includes the realization questions for the bidisk and the ball, \[ \Delta^2=\{(z_1,z_2)\in{\mathbb{C}}^2:|z_1|<1,\ |z_2|<1\},\qquad B^2=\{(z_1,z_2)\in{\mathbb{C}}^2:|z_1|^2+|z_2|^2<1\}. \] The record does not identify the ambient manifold, the allowed maps, whether rotation domains must be maximal dynamical components, or the equivalence relation for classifying models. It also does not explicitly fix the complex dimension, boundedness, or boundary regularity. Consequently, a fully quantified realization problem cannot be recovered from this record alone; no ambient class or additional hypotheses are imposed here.

Short English statement

Classify the Reinhardt domains that can serve as linear models of the intended rank-two rotation domains, including whether the bidisk and the ball occur. The source leaves essential dynamical hypotheses and the precise rank convention unspecified, so this is a statement-only draft requiring human mathematical review.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30003924$, title Reinhardt Models of Rank-Two Rotation Domains, exact source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. No individual proposer is supplied; the recorded proposed year is 2018.
• Supplied source citation: Geometric Methods of Complex Analysis (2019), Oberwolfach Reports, pp. 2253--2302, DOI 10.4171/owr/2018/37. Supplied source and literature link, labeled Oberwolfach Report 37/2018: https://doi.org/10.4171/owr/2018/37. The citation's 2019 designation and the report's 2018 designation are retained as supplied.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open, states that the report includes the bidisk and ball realization questions, and reports that no definitive classification was found. This is supplied dated curation, not an independently verified literature assessment; time-sensitive status and cited claims require independent verification.
• The registry marks the statement corrected_verified and explains that notation, typography, and sentence structure were repaired while mathematical content was preserved. This supplied editorial label does not establish mathematical correctness or completeness. The cited report has not been independently consulted for this draft.