双曲空间中的极小平移超曲面

中文摘要 AI 辅助

在双曲空间 $\mathbb{H}^{n+1}$ 的上半空间模型中，I 型平移超曲面是形如 $x_{n+1}=f_1(x_1)+\cdots+f_n(x_n)$ 的图，而 II 型平移超曲面是形如 $x_n=f_1(x_1)+\cdots+f_{n-1}(x_{n-1})+g(x_{n+1})$ 的图。我们对每个 $n\geq 2$ 分类了两种类型的极小平移超曲面。我们将这些结果作为 $\mathbb{R}^{n+1}$ 中任意非零指数 $\alpha\neq 0$ 的奇异极小平移超曲面分类的特殊情形获得，该分类涉及与超曲面作为图所对应的超平面正交（I 型）或平行（II 型）的方向。I 型极小平移超曲面恰好是单参数拱族上的抛物柱面，这与文献中的一个不存在性定理相矛盾。II 型极小平移超曲面是抛物柱面、垂直全测地超平面，以及当 $n\geq 3$ 时，是 Scherk 曲面上的垂直柱面。

英文摘要

In the upper half-space model of the hyperbolic space $\mathbb{H}^{n+1}$, a translation hypersurface of type I is a graph $x_{n+1}=f_1(x_1)+\cdots+f_n(x_n)$ and a translation hypersurface of type II is a graph $x_n=f_1(x_1)+\cdots+f_{n-1}(x_{n-1})+g(x_{n+1})$. We classify the minimal translation hypersurfaces of both types for every $n\geq 2$. We obtain these results as particular cases of a classification of the singular minimal translation hypersurfaces of $\mathbb{R}^{n+1}$ for an arbitrary exponent $α\neq 0$, with respect to a direction orthogonal (type I) or parallel (type II) to the hyperplane over which the hypersurface is a graph. The minimal translation hypersurfaces of type I are exactly the parabolic cylinders over a one-parameter family of arches, which contradicts a nonexistence theorem in the literature. The minimal translation hypersurfaces of type II are the parabolic cylinders, the vertical totally geodesic hyperplanes and, when $n\geq 3$, the vertical cylinders over the Scherk surface.

发表机构

• Université Cheikh Anta Diop（谢赫·安达·迪奥普大学）
• FASTEF

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