乘积空间中的双守恒与双调和超曲面

中文摘要 AI 辅助

本文研究了乘积空间 $L^m(\varepsilon)\times {\mathbb R}$ 中的双守恒与双调和超曲面，其中 $L^m(\varepsilon)$ 表示常截面曲率 $\varepsilon=1$ 或 $\varepsilon=-1$ 的空间形式 ${\mathbb S}^m$ 或 ${\mathbb H}^m$。我们获得了等参超曲面和常角超曲面的完整描述。此外，我们证明了对于平均曲率函数的梯度为主方向的那些双守恒与双调和超曲面的若干分类结果。在此背景下，我们在 ${\mathbb H}^m\times {\mathbb R}$ 中获得了完整的、非柱面的、非全脐的双守恒超曲面的新例子。然后我们证明了与这些例子相关的刚性定理。最后，我们证明了在 ${\mathbb H}^m \times {\mathbb R}$ 中，若平均曲率函数的平方有界且里奇曲率下有界，则不存在完整的真双调和超曲面的全局不存在性定理。

英文摘要

In this paper we investigate biconservative and biharmonic hypersurfaces in product spaces $L^m(\varepsilon)\times {\mathbb R}$, where $L^m(\varepsilon)$ denotes the space form ${\mathbb S}^m$ or ${\mathbb H}^m$ of constant sectional curvature $\varepsilon=1$ or $\varepsilon=-1$ respectively. We obtain their complete description in the case of isoparametric hypersurfaces and hypersurfaces of constant angle. Moreover, we prove some classification results for biconservative and biharmonic hypersurfaces for which the gradient of the mean curvature function is a principal direction. In this context we obtain new examples of complete, non-cylindrical, non-totally umbilical biconservative hypersurfaces in ${\mathbb H}^m\times {\mathbb R}$. Then we prove a rigidity theorem related to these examples. Finally, we prove a global non-existence theorem for complete proper biharmonic hypersurfaces in ${\mathbb H}^m \times {\mathbb R}$ such that the square of the mean curvature function is bounded and the Ricci curvature is bounded from below.

发表机构

• University of Rostock（罗斯托克大学）
• Università degli Studi di Cagliari（卡利亚里大学）

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