arXivDaily arXiv每日学术速递 周一至周五更新
arXiv 1802.03059math.DG
收藏

关于有限强全曲率的$\mathbb{H}^n\times \mathbb{R}$超曲面的结构

On the structure of hypersurfaces in $\mathbb{H}^n\times \mathbb{R}$ with finite strong total curvature

Maria Fernanda Elbert, Barbara Nelli

阅读 PDF
arXiv 页面 ↗HTML 原文 ↗

摘要：

证明了若$X:M^n\to\mathbb{H}^n\times \mathbb{R}$，$n\geq 3$是一个具有有限强全曲率的可定向完全浸入，则$X$是真的，且$M$同构于一个紧流形$\bar M$减去有限个点$q_1, \dots q_k$ .添加一些额外的假设，包括$H_r=0,$，其中$H_r$是高阶平均曲率，我们获得更多的信息，每个穿刺的邻居的几何形状。读者还将在本文中找到满足$H_r=0$且双曲平移不变的$\mathbb{H}^n\times \mathbb{R}$的超曲面的分类结果以及这些超曲面的半空间中的极大值原理。

英文摘要：

We prove that if $X:M^n\to\mathbb{H}^n\times \mathbb{R}$, $n\geq 3$, is a an orientable, complete immersion with finite strong total curvature, then $X$ is proper and $M$ is diffeomorphic to a compact manifold $\bar M$ minus a finite number of points $q_1, \dots q_k$. Adding some extra hypothesis, including $H_r=0,$ where $H_r$ is a higher order mean curvature, we obtain more information about the geometry of a neighbourhood of each puncture. The reader will also find in this paper a classification result for the hypersurfaces of $\mathbb{H}^n\times \mathbb{R}$ which satisfy $H_r=0$ and are invariant by hyperbolic translations and a maximum principle in a half space for these hypersurfaces.

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。

↑