Finiteness of index and total scalar curvature for minimal hypersurfaces
Finiteness of index and total scalar curvature for minimal hypersurfaces
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DOI：
10.1090/s0002-9939-1989-0946639-1
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发表时间：
1989-02
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影响因子：
--
通讯作者：
J. Tysk
中科院分区：
文献类型：
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作者：
J. Tysk
设Mn，n > 3，是欧氏空间中的定向极小浸入完备超曲面.我们表明，对于n = 3，4，5，或6，Mn的指数是有限的当且仅当Mn的总标量曲率是有限的，只要Mn的体积增长是有界的常数乘以rn，其中r是欧几里得距离函数。我们还注意到，当n > 8时，这个结果不成立。此外，我们证明了Mn的指数是有界的总标量曲率的常数倍，所有n > 3，没有任何假设的体积增长的Mn。
Let Mn, n > 3, be an oriented minimally immersed complete hypersurface in Euclidean space. We show that for n = 3, 4, 5, or 6, the index of Mn is finite if and only if the total scalar curvature of Mn is finite, provided that the volume growth of Mn is bounded by a constant times rn, where r is the Euclidean distance function. We also note that this result does not hold for n > 8. Moreover, we show that the index of Mn is bounded by a constant multiple of the total scalar curvature for all n > 3, without any assumptions on the volume growth of Mn.