Approximation of solutions of the sinh-Gordon equation $\mathrm{\Delta }u-\sinh (2u)=0$ by hyperbolic orthogonal ring patterns

We consider hyperbolic orthogonal ring patterns as introduced in arXiv:2409.06573 and focus on their characterization by uniformizing variables at the centers of the rings. Given a smooth solution of the sinh-Gordon equation $\mathrm{\Delta }u-\sinh (2u)=0$, we restrict to a compact subset of its domain and discretize it by square grid lattices with edge length $\epsilon$. Taking the values of $u$ as Dirichlet boundary conditions, we prove that the corresponding uniformizing variables $u^{\epsilon }$ of the hyperbolic ring patterns converge to $u$ in $C^{\mathrm{\infty }}$ with error of order ${\epsilon }^2$, given that the pairs of rings suitably converge to circles. As a consequence we deduce that the hyperbolic orthogonal ring patterns converge to a harmonic map to the hyperbolic plane.