Estimates of eigenvalues of a compact Riemannian manifold
Estimates of eigenvalues of a compact Riemannian manifold
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DOI：
10.1090/pspum/036/573435
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发表时间：
1980
期刊：
影响因子：
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通讯作者：
Peter Li;S. Yau
中科院分区：
文献类型：
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作者：
Peter Li;S. Yau
由于庞加莱不等式在分析中起着非常重要的作用，并且由于第一特征值的下界给出了庞加莱不等式中常数的上限，因此非常需要找到第一特征值的良好下限估计。对于欧几里得空间中的域，有Faber-Krahn、Polyá-Szegő、Payne、Weinberger等人的经典著作。这些作者的著作不仅优美且重要，而且对弯曲空间特征值的估计产生了深远的影响。对于许多几何问题，我们经常需要估计弯曲空间上域的庞加莱不等式。因此，在本文中，我们将注意力集中在这个案例上。这个方向的第一个主要成果归功于 Lichnerowicz [10] 和 Obata [11]。在他们出色的工作中，他们假设紧流形（无边界）的里奇曲率大于正常数，并根据该常数估计了下面的第一个特征值。值得注意的是，这个常数是尖锐的。 Reilly [14] 后来将 Lichnerowicz-Obata 的这种估计推广到他处理过的具有边界的流形
Since the Poincaré inequality plays a very important role in analysis and since a lower bound of the first eigenvalue gives an upper bound of the constant in the Poincaré inequality, it is very desirable to find a good lower estimate of the first eigenvalue. For domains in euclidean space, there are classical works of Faber-Krahn, Polyá-Szegő, Payne, Weinberger, etc. The works of these authors are not only beautiful and important, but also give a deep impact to estimate eigenvalues on curved spaces. For many geometric problems, we often need to estimate the Poincaré inequality for domains on a curved space. Thus in this paper, we concentrate our attention to this case. The first major result in this direction was due to Lichnerowicz [10] and Obata [11]. In their beautiful work, they assumed the Ricci curvature of the compact manifold (without boundary) is greater than a positive constant and they estimated the first eigenvalue from below in terms of this constant. It is remarkable that this constant is sharp. This estimate of Lichnerowicz-Obata was generalized later by Reilly [14] to manifolds with boundary where he treated