Index and topology of minimal hypersurfaces in R^n
Index and topology of minimal hypersurfaces in R^n
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R^n 中最小超曲面的索引和拓扑
DOI：
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发表时间：
2016
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通讯作者：
Chao Li
中科院分区：
文献类型：
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作者：
Chao Li
在本文中，我们考虑 $\mathbb{R}^n$ 中具有有限总曲率的浸入式两侧最小超曲面。我们证明莫尔斯指数和雅可比算子的无效性之和自下以超曲面的端点数和第一个贝蒂数的线性函数为界。当$n=4$时，我们可以通过仔细研究刚性情况来删除无效项。我们的结果是 Li-Wang 的首次有效推广。利用我们的索引估计和 Chodosh-Ketover-Maximo 最近工作的想法，我们证明了具有有限索引的 $\mathbb{R}^4$ 中最小超曲面的紧凑性和有限性结果。
In this paper, we consider immersed two-sided minimal hypersurfaces in $\mathbb{R}^n$ with finite total curvature. We prove that the sum of the Morse index and the nullity of the Jacobi operator is bounded from below by a linear function of the number of ends and the first Betti number of the hypersurface. When $n=4$, we are able to drop the nullity term by a careful study for the rigidity case. Our result is the first effective generalization of Li-Wang. Using our index estimates and ideas from the recent work of Chodosh-Ketover-Maximo, we prove compactness and finiteness results of minimal hypersurfaces in $\mathbb{R}^4$ with finite index.