The Nitsche-threshold conjecture for minimizing harmonic diffeomorphisms of annuli

Let $\mathbb{A}(r,1)$ and $\mathbb{A}(R,1)$ be concentric annuli, and let $\rho$ be a metric on $\mathbb{A}(R,1)$ with bounded Gaussian curvature and finite area. Define $r_\diamond$ to be the infimum of those $r$ for which there exists a minimizing $\rho$-harmonic diffeomorphism from $\mathbb{A}(r,1)$ onto $\mathbb{A}(R,1)$. For a doubly connected bounded plane domain $\mathbb{Y}$, let $\Psi:\mathbb{A}(R,1)\to\mathbb{Y}$ be conformal, where $R=R(\mathbb{Y})\in(0,1)$.

Nitsche-threshold conjecture. One has $r_\diamond<R$, and for every $r>r_\diamond$ there exists a minimizing $\rho$-harmonic diffeomorphism from $\mathbb{A}(r,1)$ onto $\mathbb{A}(R,1)$ that, together with its inverse, is $\mathscr{C}^{1,\alpha}$ up to the boundary. In particular, there exists $r_\diamond<R(\mathbb{Y})$ such that for every $r>r_\diamond$ there is a Euclidean harmonic energy-minimizing diffeomorphism $h:\mathbb{A}(r,1)\to\mathbb{Y}$ that, together with its inverse, is $\mathscr{C}^{1,\alpha}$ up to the boundary whenever $\partial\mathbb{Y}\in\mathscr{C}^{1,\alpha}$.

The result would extend the existence and boundary regularity theory for energy-minimizing harmonic diffeomorphisms beyond the critical Nitsche threshold. The paper notes that Proposition 4 gives only $r_\diamond\leq R$; the strict inequality and the asserted regularity remain conjectural.