Eigenvalues of the Laplacian on forms
Eigenvalues of the Laplacian on forms
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形式上拉普拉斯算子的特征值
DOI：
10.1090/s0002-9939-1982-0656119-2
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发表时间：
1982
影响因子：
1.7
通讯作者：
J. Dodziuk
中科院分区：
文献类型：
--
作者：
J. Dodziuk
给出了作用在紧致黎曼流形上形式上的拉普拉斯算子的特征值的一些界。在流形无边界的情况下，我们给出的曲率，其协变导数和内射半径的上界。对于小测地线球，给出了特征值的上、下界。在（2）中，Cheng证明了黎曼流形中测地球上函数的Laplacian Ao的第一特征值的一个漂亮的比较定理，并由此导出了紧致流形上函数的Laplacian Ao的高特征值的上界。这些上界是通过取小球的Dirichlet边界条件的第一本征函数，将它们由零扩展到整个流形，并估计适当的线性组合的Rayleigh-Ritz商而得出的。相同的过程不能应用于正次数的形式，因为满足绝对或相对边界条件的球的本征函数在通过零扩展到整个流形时将不在索伯列夫空间771中。本文对Eichhorn（4）的方法作了一个改进，证明了对于球上在边界上为零的某些形式，可以用几何量来估计Rayleigh-Ritz商。当扩展为零时，这些形式处于H1中，并且应用于它们的Cheng的论点给出了封闭流形上特征值的上界。我们的结果远不如Cheng的定理优美。首先，我们必须要求测地线球包含在切割轨迹内。因此，所有高本征值的估计依赖于内射半径。其次，Cheng用几何量的形式得到了Laplacian Ao的特征值的显式估计（cf.（2，推论2.2，2.3，定理2.4）），而我们只说哪些几何量决定特征值的界，但没有给出实际界的估计。从我们的方法可以得到显式的估计，但常数是如此复杂，我们无法从中获得任何有用的信息。这是一个有趣的问题，是否所有出现在我们估计中的几何量（截面曲率、内射半径、克恩张量R^的边界）都是真的必要的。我们不知道闭流形的答案。然而，在§3中，我们表明，根据J. Cheeger的建议，对于半径小于内射性半径的测地线球，拉普拉斯算子Ap的本征值可以根据截面曲率的界限从上下估计。我很高兴地感谢J. Cheeger提出的建议，这些建议导致了本文的改进。
Some bounds for eigenvalues of the Laplace operator acting on forms on a compact Riemannian manifold are derived. In case of manifolds without boundary we give upper bounds in terms of the curvature, its covariant derivative and the injectivity radius. For a small geodesic ball upper and lower bounds of eigenvalues in terms of bounds of sectional curvature are given. In (2) Cheng proves a beautiful comparison theorem for the first eigenvalue of the Laplacian Ao on functions for a geodesic ball in a Riemannian manifold, and derives as a consequence, upper bounds for higher eigenvalues of the Laplacian on functions for a compact manifold. These upper bounds are derived by taking the first eigenfunctions with Dirichlet boundary conditions for small balls, extending them by zero to the whole manifold, and estimating the Rayleigh-Ritz quotient of an appropriate linear combination. The same procedure cannot be applied to forms of positive degree since an eigenfunction for a ball satisfying either absolute or relative boundary conditions will not be in the Sobolev space 771 when extended by zero to the whole manifold. In this note a modification of the method of Eichhorn (4) is used to prove that, for certain forms on a ball which vanish on the boundary, it is possible to estimate the Rayleigh-Ritz quotient in terms of geometric quantities. These forms are in H1 when extended by zero and Cheng's argument applied to them gives upper bounds for eigenvalues on a closed manifold. Our result is much less elegant than Cheng's theorem. In the first place we have to require that the geodesic ball is contained within the cut locus. Hence, all estimates of higher eigenvalues depend on the injectivity radius. Secondly, Cheng obtained explicit estimates of eigenvalues of the Laplacian Ao in terms of geometric quantities (cf. (2, Corollaries 2.2, 2.3, Theorem 2.4)), whereas we only say which geometric quantities determine the bounds of eigenvalues but give no estimates of the actual bounds. Explicit estimates could be derived from our method, but the constants are so complicated that we were unable to obtain any useful information from them. It is an interesting question, whether all of the geometric quantities which appear in our estimates (sectional curvature, injectivity radius, the bounds for the Kern tensor R^) are really necessary. We do not know the answer for closed manifolds. However, in §3 we show, following a suggestion of J. Cheeger, that for a geodesic ball of radius smaller than the radius of injectivity, eigenvalues of the Laplacian Ap can be estimated from above and below in terms of bounds of sectional curvature. It is a pleasure to thank J. Cheeger for suggestions which led to improvement of this paper.