AI 中文总结
本文通过双曲正交环图案的统一化变量研究 sinh-Gordon 方程的解的近似，证明其在 $C^\infty$ 范数下以 $\varepsilon^2$ 误差收敛于原解，并推导出其收敛于双曲平面的调和映射。
AI 中文摘要
我们考虑了在 arXiv:2409.06573 中引入的双曲正交环图案，并聚焦于通过在环中心进行统一化变量来对其进行表征。给定 sinh-Gordon 方程 $Δu -\sinh(2u)=0$ 的光滑解，我们限制在其定义域的一个紧致子集上，并通过边长为 $\varepsilon$ 的正方形格点对其进行离散化。将 $u$ 的值作为狄利克雷边界条件，我们证明了对应的双曲环图案的统一化变量 $u^\varepsilon$ 在 $C^\infty$ 范数下以阶 $\varepsilon^2$ 的误差收敛于 $u$，给定环对合适地收敛于圆。作为结果，我们推导出双曲正交环图案收敛于到双曲平面的调和映射。
英文摘要
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 $Δu -\sinh(2u)=0$, we restrict to a compact subset of its domain and discretize it by square grid lattices with edge length $\varepsilon$. Taking the values of $u$ as Dirichlet boundary conditions, we prove that the corresponding uniformizing variables $u^\varepsilon$ of the hyperbolic ring patterns converge to $u$ in $C^\infty$ with error of order $\varepsilon^2$, 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.
Comments12 pages, 2 figures
