On the first eigenvalue of the Laplacian for compact submanifolds of Euclidean space
On the first eigenvalue of the Laplacian for compact submanifolds of Euclidean space
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DOI：
10.1007/bf02567385
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发表时间：
1977-12
影响因子：
0.9
通讯作者：
R. Reilly
中科院分区：
文献类型：
--
作者：
R. Reilly
这篇论文的灵感来自于D. Bleecker和J. Weiner b[3]最近的工作。下列结果是本文所得结果的典型例子：定理。在欧几里德空间中等距浸没的紧n流形的拉普拉斯函数的第一个特征值以平均曲率矢量范数的平方的平均值n倍为界。此外，如果特征值达到这个界，则子流形实际上是欧几里德空间中某个超球的最小子流形。（参见定理a中r= 1的情况）
This paper was inspired by recent work of D. Bleecker and J. Weiner [3]. The following results are typical examples of those we obtain in this paper.THEOREM. The first eigenvalue of the Laplacian for a compact n-manifold isometrically immersed in Euclidean space is bounded above by n times the average value of the square of the norm of the mean curvature vector. Moreover, if the eigenvalue achieves this bound, then the submanifold is actually a minimal submanifold of some hypersphere in the Euclidean space.(See the case r= 1 in Theorem A.)