A GENERALIZATION OF CHENG'S THEOREM ∗
A GENERALIZATION OF CHENG'S THEOREM ∗
复制标题
程定理的推广 *
DOI：
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发表时间：
2008
期刊：
影响因子：
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通讯作者：
Jiaping Wang
中科院分区：
文献类型：
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作者：
Peter Li;Jiaping Wang
0。导言。本文证明了完备黎曼流形的L谱的底的上界的一个定理的推广。在文献[C]中，程证明了测地球的第一个Dirichlet特征值的比较定理。通过将球的半径取为无穷大，他得到了L光谱底部的估计。特别地，他证明了：如果M是n维完备黎曼流形，其Ricci曲率对于某个常数K>；0由−(n−1)K自下有界，则L谱的底λ1(M)由λ1(M)≤(n−1)K4有界。
0. Introduction. In this paper, we prove a generalization of a theorem of S.Y. Cheng on the upper bound of the bottom of the L spectrum for a complete Riemannian manifold. In [C], Cheng proved a comparison theorem for the first Dirichlet eigenvalue of a geodesic ball. By taking the radius of the ball to infinity, he obtained an estimate for the bottom of the L spectrum. In particular, he showed that if M is an n-dimensional complete Riemannian manifold whose Ricci curvature is bounded from below by −(n− 1)K for some constant K > 0, then the bottom of the L spectrum, λ1(M), is bounded by λ1(M) ≤ (n− 1)K 4 .