On the spectrum of a finite-volume negatively-curved manifold
On the spectrum of a finite-volume negatively-curved manifold
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有限体积负曲流形的谱
DOI：
10.1353/ajm.2001.0012
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发表时间：
1999
影响因子：
1.7
通讯作者：
J. Lott
中科院分区：
文献类型：
--
作者：
J. Lott
我们证明了具有有界截面曲率的非紧流形，其端点足够接近于射线的Gromov-Hausdorff，具有有限维的平方可积调和形式空间。在有限体积流形的特殊情况下，捏负截面曲率，我们表明，本质谱的p-irons拉普拉斯算子是一个收集的常微分算子的本质谱的联合的目的。我们给出的例子，这样的流形曲率捏任意接近？1，并且在函数Laplacian的谱中具有无限数量的间隙。1.导论.本文考虑有限体积的黎曼流形和收缩负截面曲率。给出了微分形式拉普拉斯算子的核及其本质谱的一些结果。我们的第一个结果是有限维的平方可积调和形式的空间为更一般的一类黎曼流形，这可以大致表征为那些有界的截面曲率和端部足够Gromov-Hausdorff接近射线。设M是以m为基点的完备连通r ~ c维黎曼流形。设Br（m）表示围绕m的距离球，并且设Sr（m）= dBr（m）是围绕m的距离球。放
We show that a noncompact manifold with bounded sectional curvature, whose ends are sufficiently Gromov-Hausdorff close to rays, has a finite dimensional space of square-integrable harmonic forms. In the special case of a finite-volume manifold with pinched negative sectional curvature, we show that the essential spectrum of the p-iorm Laplacian is the union of the essential spectra of a collection of ordinary differential operators associated to the ends. We give examples of such manifolds with curvature pinched arbitrarily close to ?1 and with an infinite number of gaps in the spectrum of the function Laplacian. 1. Introduction. In this paper we consider Riemannian manifolds of finite volume and pinched negative sectional curvature. We give results about the kernel of the differential form Laplacian and about its essential spectrum. Our first result is the finite dimensionality of the space of square-integrable harmonic forms for a more general class of Riemannian manifolds, which can be roughly characterized as those with bounded sectional curvature and with ends that are sufficiently Gromov-Hausdorff close to rays. Let M be a complete connected rc-dimensional Riemannian manifold with a basepoint m. Let Br(m) denote the distance ball around m and let Sr(m) = dBr(m) be the distance sphere around m. Put