Existence of a good direction for Green’s mappings

Establish the existence of a direction ψ ∈ [0,2] such that the radial variation integral ∫_0^1 |T''(r e^{iπψ})| dr is finite for any Green’s mapping T: H → 𝛀̃ associated with a non‑polar planar domain Ω, where H denotes the Green’s fundamental domain of Ω. Determine whether this property holds at least in the special case where the boundary ∂Ω is a uniformly perfect subset of the real line.

Background

The paper studies boundary behavior of analytic maps related to Green’s functions and compares them to known radial variation estimates for conformal and universal covering maps. For conformal maps and for universal coverings onto uniformly perfect sets, sharp radial variation bounds are known, including the existence of directions along which second derivatives have integrable radial variation.

In March 1999, Binder posed an analogous problem for Green’s mappings T:H→𝛀̃ of general domains Ω: whether there exists a “good direction” ψ along which ∫_01 |T''(r e{iπψ})| dr < ∞. The authors note that this problem remains open even when ∂Ω is uniformly perfect and lies on the real line.

The present paper confirms the existence of such directions for a specific class: Denjoy domains with boundary equal to a real Julia set of an expanding quadratic polynomial P(z)=z2−λ with λ>2+√2, establishing integrable radial variation along a large set of directions. However, the general uniformly perfect case and the full generality of Ω still remain unresolved.

References

In March 1999, the first named author (Binder) posed the problem of showing that a "good direction" ψ∈[0,2] exists, for any Green's mapping T:H→\tilde \Omega, i.e., \begin{equation}\label{binder} \int\limits_0\limits{1} |T''(re{i\pi\psi})|dr <\infty, \quad\text{ for at least one } \quad \psi\in [0,2], \end{equation} and delineated a path connecting binder to ucova-int1. Presently this problem is open even in the special case where $\partial \Omega $ is a uniformly perfect subset of the real line.

binder:

∫01∣T′′(reiπψ)∣dr<∞, for at least one ψ∈[0,2],\int\limits_0\limits^{1} |T''(re^{i\pi\psi})|dr <\infty, \quad\text{ for at least one } \quad \psi\in [0,2],

ucova-int1:

∫01∣PE′′(reiπψ)∣dr<∞.\int _{0}^{1}|P_E''(re^{i\pi\psi})|dr<\infty.

— Greenś Mapping and Julia Sets (2508.04207 - Binder et al., 6 Aug 2025) in Subsection “Green’s mapping” (Introduction), after equation (binder)