发表机构
Universidad de Alcalá（阿尔卡拉大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文通过圈群方法，在反德西特三维空间中构造常平均曲率和常高斯曲率的类时曲面，证明二者仅差一个归一化，并给出显式公式与闭式例子。
AI 中文摘要
已知在平坦洛伦兹三维空间中，具有洛伦兹调和的高斯映射的常平均曲率类时曲面，以及在三维球面中具有常高斯曲率的曲面，其圈群描述已经建立。我们针对反德西特三维空间 $\mathbb{H}^3_1\cong SL(2,\mathbb{R})$ 中的类时曲面进行了相应的构造：一类是常平均曲率曲面，其高斯映射关于第一基本形式的共形结构是调和的；另一类是常高斯曲率 $K>-1$ 且 $K\neq0$ 的曲面，其高斯映射此时为浸入，并且关于第二基本形式是调和的。对于这两类曲面，我们证明了调和性刻画，通过 Sym-Bobenko 型公式从扩展标架恢复曲面，并利用广义 DPW 方法求解几何柯西问题，其中势函数直接用数据显式写出，且无需对高斯映射进行归一化。随后我们证明这两种构造仅相差一个归一化：在替换 $\lambda=\zeta^2$ 并施加一个常数规范变换后，常平均曲率扩展标架 $\hat F$ 是其高斯映射的扩展标架，并且映射 $\hat F|_{\lambda=e^{4\theta}}\exp(\theta e_3)\hat F|_{\lambda=1}^{-1}$ 和 $e_3\hat F|_{\lambda=e^{4\theta}}e_3\exp(\theta e_3)\hat F|_{\lambda=1}^{-1}$（其中 $e_3=\mathrm{diag}(-1,1)$）在常平均曲率曲面非平坦处分别是常高斯曲率为 $1/\sinh^2\theta$ 和 $-1/\cosh^2\theta$ 的类时曲面。前者是经典的平行曲面，后者是其极曲面。我们给出了显式族，包括闭式形式的扩展标架和曲面，以说明这些构造。
英文摘要
The loop group description of surfaces whose Gauss map is Lorentz harmonic is known for timelike surfaces of constant mean curvature in the flat Lorentzian 3-space, and for surfaces of constant Gaussian curvature in the 3-sphere. We carry out the corresponding construction for timelike surfaces in anti-de Sitter 3-space $\mathbb{H}^3_1\cong SL(2,\mathbb{R})$: those of constant mean curvature, whose Gauss map is harmonic for the conformal structure of the first fundamental form, and those of constant Gaussian curvature $K>-1$, $K\neq0$, whose Gauss map is then an immersion and is harmonic for the second fundamental form. For both classes we prove the harmonicity characterization, recover the surfaces from extended frames by Sym-Bobenko-type formulas, and solve the geometric Cauchy problem by the generalized DPW method, with potentials written explicitly in terms of the data and no normalization of the Gauss map. We then show that the two constructions differ only by a normalization: after the substitution $λ=ζ^2$ and a constant gauge, a constant mean curvature extended frame $\hat F$ is an extended frame of its Gauss map, and the maps $\hat F|_{λ=e^{4θ}}\exp(θe_3)\hat F|_{λ=1}^{-1}$ and $e_3\hat F|_{λ=e^{4θ}}e_3\exp(θe_3)\hat F|_{λ=1}^{-1}$, with $e_3=\mathrm{diag}(-1,1)$, are timelike surfaces of constant Gaussian curvature $1/\sinh^2θ$ and $-1/\cosh^2θ$ wherever the constant mean curvature surface is not flat. The first is the classical parallel surface, the second its polar surface. Explicit families, with extended frames and surfaces in closed form, illustrate the constructions.
Comments39 pages
