In this article we prove upper bounds for the Laplace eigenvalues ${\lambda }_k$ below the essential spectrum for strictly negatively curved Cartan–Hadamard manifolds. Our bound is given in terms of k² and specific geometric data of the manifold. This applies also to the particular case of non-compact manifolds whose sectional curvature tends to $-\infty$, where no essential spectrum is present due to a theorem of Donnelly/Li. The result stands in clear contrast to Laplacians on graphs where such a bound fails to be true in general.

中文翻译：

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关于非紧流形的特征值界的注解

在本文中，我们证明了拉普拉斯特征值的上限 ${\lambda }_{克}$低于严格负弯曲 Cartan-Hadamard 流形的基本谱。我们的界限是根据k ²和流形的特定几何数据给出的。这也适用于非紧凑流形的特殊情况，其截面曲率趋于$-\infty$，由于 Donnelly/Li 定理，不存在本质谱。结果与图上的拉普拉斯算子形成鲜明对比，后者的边界一般不成立。