发表机构
Hunan First Normal University; Shenzhen Polytechnic University（湖南第一师范学院; 深圳职业技术大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对调和拟共形映射，引入仿射外接半径与仿射直径作为仿射不变量，刻画最优仿射平衡下的最小畸变，并建立锐利双边界及二阶估计，统一极值畸变问题。
AI 中文摘要
对于保向平面调和映射，实仿射映射的后复合在复伸张空间上诱导自同构。受此事实以及经典畸变理论中缺乏自然仿射不变几何量的启发，我们引入与复伸张像相关联的仿射外接半径和仿射直径。指数化的仿射外接半径给出了仿射归一化下最小拟共形畸变的刻画。最优仿射平衡在相似变换意义下唯一，并产生具有中心对称伸张的典范调和映射。利用三点支撑原理和锐利的双曲Jung定理，我们建立了这些不变量的普适锐利双边界，并给出其伪双曲形式的重述。对于典范平衡映射，我们证明了预Schwarz导数的锐利二阶估计，表明仿射归一化抵消了前导阶共形差异。我们为调和拟共形映射族发展了仿射稳定的分层分类，包括正规性和不变序。该框架统一了极值畸变问题，并将经典解析函数理论与调和拟共形映射的畸变理论联系起来。
英文摘要
For orientation-preserving planar harmonic mappings, postcomposition by real-affine mappings induces automorphisms on the space of complex dilatations. Motivated by this fact and the lack of natural affine-invariant geometric quantities in classical distortion theory, we introduce the affine circumradius and affine diameter associated with the image of the complex dilatation. The exponentiated affine circumradius gives a characterization of minimal quasiconformal distortion under affine normalization. Optimal affine balancing is unique up to similarity and yields canonical harmonic mappings with centrally symmetric dilatation. Using a three-point support principle and a sharp hyperbolic Jung theorem, we establish universal sharp two-sided bounds for these invariants and provide their pseudohyperbolic reformulations. For canonically balanced mappings, we prove sharp second-order estimates for pre-Schwarzian derivatives, showing that affine normalization cancels leading-order conformal discrepancies. We develop an affine-stable hierarchical classification for families of harmonic quasiconformal mappings, including normality and an invariant ordering. This framework unifies extremal distortion problems and connects classical analytic function theory with the distortion theory of harmonic quasiconformal mappings.
Comments39 pages, 4 figures, it was submitted to a juornal in August 2026
