arXivDaily arXiv每日学术速递 周一至周五更新
arXiv 2008.09805math.DG
收藏

$M\times\mathbb{R}$中高阶中曲率为常数的超曲面

Hypersurfaces of Constant Higher Order Mean Curvature in $M\times\mathbb{R}$

R. F. de Lima, F. Manfio, J. P. dos Santos

阅读 PDF
arXiv 页面 ↗HTML 原文 ↗

摘要：

本文考虑具有常数$r$次中曲率$H_r\ge 0$的积$M\times\mathbb R$的超曲面（称为$H_r$ -超曲面），其中$M$是任意的黎曼$n$ -流形.我们开发了一个一般的方法来构建它们，并使用它来产生许多例子的各种流形$M,$，包括所有的单连通空间形式和双曲空间$\mathbb{H}_{\mathbb F}^m$（秩$1$对称空间的非紧型）。构造了$\mathbb{H}_{\mathbb F}^m\times\mathbb R$和$\mathbb S^n\times\mathbb R$中的完备旋转$H_r(\ge 0)$ -超曲面，并对其进行了分类.它们包括球，Delaunay型annuli，并在$\mathbb{H}_{\mathbb F}^m\times\mathbb R,$整个图的情况下。我们还构造并分类了完整的$H_r(\ge 0)$-$\mathbb{H}_{\mathbb F}^m\times\mathbb R$的超曲面，这些超曲面对于抛物等距或双曲平移都是不变的。通过证明$\mathbb H^n\times\mathbb R$或$\mathbb S^n\times\mathbb R$ $(n\ge 3)$的紧、连通、严格凸的$H_r$ -超曲面是旋转嵌入球面，建立了一个Jelvan-Liebmann型定理.得到了这些周围空间的完备$H_r$ -超曲面的其它唯一性结果。

英文摘要：

We consider hypersurfaces of products $M\times\mathbb R$ with constant $r$-th mean curvature $H_r\ge 0$ (to be called $H_r$-hypersurfaces), where $M$ is an arbitrary Riemannian $n$-manifold. We develop a general method for constructing them, and employ it to produce many examples for a variety of manifolds $M,$ including all simply connected space forms and the hyperbolic spaces $\mathbb{H}_{\mathbb F}^m$ (rank $1$ symmetric spaces of noncompact type). We construct and classify complete rotational $H_r(\ge 0)$-hypersurfaces in $\mathbb{H}_{\mathbb F}^m\times\mathbb R$ and in $\mathbb S^n\times\mathbb R$ as well. They include spheres, Delaunay-type annuli and, in the case of $\mathbb{H}_{\mathbb F}^m\times\mathbb R,$ entire graphs. We also construct and classify complete $H_r(\ge 0)$-hypersurfaces of $\mathbb{H}_{\mathbb F}^m\times\mathbb R$ which are invariant by either parabolic isometries or hyperbolic translations. We establish a Jellett-Liebmann-type theorem by showing that a compact, connected and strictly convex $H_r$-hypersurface of $\mathbb H^n\times\mathbb R$ or $\mathbb S^n\times\mathbb R$ $(n\ge 3)$ is a rotational embedded sphere. Other uniqueness results for complete $H_r$-hypersurfaces of these ambient spaces are obtained.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

↑