Negative curvature and embedded eigenvalues
Negative curvature and embedded eigenvalues
复制标题
负曲率和嵌入特征值
DOI：
10.1007/bf02570738
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发表时间：
1990
影响因子：
0.8
通讯作者：
H. Donnelly
中科院分区：
文献类型：
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作者：
H. Donnelly
设M~是截面曲率K< 0的完备单连通黎曼流形.若K~-1在无穷远处，则M的Laplacian A有由半直线[（n-1）2/4，oo）组成的本质谱[4].常负曲率的模型空间没有特征值。然而，很容易在一个大的紧集上构造-89 < K< 0的例子，而在一个更大的紧集之外构造K=-1的例子。在这种情况下，极大极小原理表明，任意多个特征值~。<（n-1）2/4的情形，本文证明了不存在嵌入特征值2>（n-1）2/4。前一段时间，作者[3]给出了K+ 1及其前两个协变导数的衰减条件，这些条件保证不出现嵌入特征值。不幸的是，曲率的协变导数的衰减条件对大多数微分几何学家来说是相当没有吸引力的。目前的文件的目的是删除所有假设的曲率导数。我们的方法需要K+ 1本身的更强的衰变条件。主要结果是定理3.1。
Let M~ be a complete simply connected Riemannian manifold with sectional curvature K< 0. If K~-1 at infinity then the Laplacian A of M has essential spectrum [4] consisting of the half line [(n-1) 2/4, oo). The model space of constant negative curvature has no eigenvalues. However, it is easy to construct examples with-89< K< 0 on a large compact set and K=-1 outside a bigger compact set. In such cases, the minimax principle shows that arbitrarily many eigenvalues~.<(n-1) 2/4 may occur.This paper is concerned with proving the absence of embedded eigenvalues 2>(n-1) 2/4. Some time ago, the author [3] gave decay conditions on K+ 1 and its first two covariant derivatives which guarantee that no embedded eigenvalues occur. Unfortunately, the decay conditions on covariant derivatives of curvature are rather unappealing to most differential geometers. The purpose of the current paper is to remove all hypotheses on derivatives of curvature. Our method requires somewhat stronger decay conditions on K+ 1 itself. The main result is Theorem 3.1.