发表机构
Chern Institute of Mathematics, Nankai University（南开大学陈省身数学研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究利用Chodosh-Gianocca的思路，得到了$\text{R}^N$中嵌入双侧极小超曲面的指标下界，证明了$\text{Spin}(7)$下指标为1且总曲率有限的完全双侧极小浸入是高维悬链面，相关估计对高维悬链面尖锐。
AI 中文摘要
我们利用Chodosh-Gianocca近期工作的思路，证明了对于$\boldsymbol{\text{R}}^N$中的嵌入双侧极小超曲面，存在以第一贝蒂数和端点数表示的指标下界，该估计对高维悬链面是尖锐的。我们还得到了Chodosh-Gianocca定理10.1的$\boldsymbol{\text{Spin}(7)}$类似结果：指标为1且总曲率有限的完全双侧极小浸入$\boldsymbol{M}^7\to\boldsymbol{\text{R}}^8$是高维悬链面。
英文摘要
We show that for an embedded two-sided minimal hypersurface in $\mathbb{R}^N$, there is a lower bound for the index in terms of the first Betti number and the number of ends, using ideas from the recent work of Chodosh--Gianocca. This estimate is sharp for the higher-dimensional catenoid. We also obtain a $\operatorname{Spin}(7)$ analogue of Theorem 10.1 of Chodosh--Gianocca: a complete two-sided minimal immersion $M^7\to\mathbb{R}^8$ of index one and finite total curvature is a higher-dimensional catenoid.
Comments14 pages. Comments are very welcome!
