ρ–Nitsche conjecture for ρ-harmonic homeomorphisms between annuli

Prove the ρ–Nitsche conjecture: Given annuli A1 = {z : 1 ≤ |z| ≤ r} and A2 = {w : 1 ≤ |w| ≤ R} in the complex plane and a Riemannian metric ρ on A2, demonstrate that if there exists a ρ–harmonic homeomorphism h : A1 → A2 (i.e., h satisfies hzz + (log ρ2) ◦ h · hz hz = 0), then r ≤ exp(∫1R ρ(s) / (s ρ(s) − α) ds), where α = inf1≤s≤R (ρ(s)) s.

Background

The classical Nitsche conjecture characterizes when a harmonic homeomorphism exists between two concentric annuli; it was resolved by Iwaniec, Kovalev, and Onninen. Building on this, Kalaj proposed a generalization—the ρ–Nitsche conjecture—replacing the Euclidean setting with a Riemannian metric ρ, leading to an inequality involving ρ that constrains the moduli of the annuli.

Partial progress toward the ρ–Nitsche conjecture has been obtained, and special cases (e.g., ρ(w) = |w|−2) are known to satisfy the corresponding inequality. The conjecture remains an explicit open formulation in general, and this paper recalls it as context for the extremal problems studied for weighted combined energy and distortion.

References

On the basis of Nitsche conjecture, Kalaj [13] proposed the so-called ρ−Nitsche conjecture as follows ρ−Nitsche conjecture If there exists a ρ−harmonic homeomorphism of the annuli A1 into A2, then r ≤ exp ∫1R ρ(s) / (s ρ (s) − α) ds , where α = inf (ρ (s))s .

— The extremal problem for weighted combined energy and the generalization of Nitsche inequality  (2401.09948 - Feng et al., 2024) in Section 1 (Introduction), equation (1.10)

A compactness approach for normalized quasiconformal homeomorphisms suggests a possible existence theorem for extremals, but a complete proof would have to control the boundary correspondence, rule out collapse of boundary components, and preserve the degree in the limit. We therefore treat attainment and uniqueness of extremals as part of the open extremal problem stated in Section~\ref{sec:conclusion}.

— On an Iwaniec--Kovalev--Onninen Conjecture for harmonic quasiconformal annulus mappings  (2609.29795 - Kalaj et al., 24 Sep 2026) in Remark 3.5, Section 3; Problem 6.1, Section 6