双曲曲面理论中的高斯映射：统一视角
Gauss Maps in Hyperbolic Surface Theory:A Unified Perspective
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中文总结 AI 辅助
该研究针对双曲三维空间中浸入曲面的多种高斯型映射，构建统一框架梳理其目标空间、解析性质与相互关系，重点探讨广义DPW方法等内容，厘清了曲面承载的各类几何信息。
中文摘要 AI 辅助
双曲三维空间中的浸入曲面承载着多种具有不同几何作用的自然高斯型映射。双曲高斯映射记录有向法向测地线的理想端点；勒让德高斯提升保留位置-法向数据及其切触结构；调整高斯映射源自魏尔斯特拉斯-健本（Weierstrass--Kenmotsu）表示中的规范归一化与岩泽（Iwasawa）分裂；共形高斯映射则编码默比乌斯（Möbius）几何中的平均曲率球汇。我们在统一框架下呈现这些构造，重点阐述其目标空间、解析性质及相互关系。研究特别关注广义DPW方法、调整后秩一数据平坦性的必要性，以及威尔莫尔（Willmore）曲面的调和映射刻画。由此得到的视角区分了\\(\mathbb{H}^{3}(-1)\\)中浸入曲面所承载的渐近信息、切触信息、可积信息与共形信息。
英文摘要
Immersed surfaces in hyperbolic three-space carry several natural Gauss-type maps with distinct geometric roles. The hyperbolic Gauss maps record the ideal endpoints of oriented normal geodesics; the Legendre Gauss lift retains the position-normal data and its contact structure; adjusted Gauss maps arise from gauge normalization and Iwasawa splitting in Weierstrass--Kenmotsu representations; and the conformal Gauss map encodes the mean-curvature sphere congruence in Möbius geometry. We present these constructions in a common framework, emphasizing their target spaces, analytic properties, and mutual relations. Particular attention is given to the generalized DPW method, the necessity of flatness in adjusted rank-one data, and the harmonic-map characterization of Willmore surfaces. The resulting viewpoint distinguishes the asymptotic, contact, integrable, and conformal information carried by an immersed surface in \(\mathbb{H}^{3}(-1)\).
