Mathematics – Differential Geometry
Scientific paper
2005-05-26
Mathematics
Differential Geometry
26 pages; changed content; added reference; typos added; changed content; Proposition 3.1 is made better;correct mistake, easy
Scientific paper
The concern of this paper is to clarify a relationship between the curvatures at infinity and the spectral structure of the Laplacian. In particular, this paper discusses the question of whether there is an eigenvalue of the Laplacian embedded in the essential spectrum or not. The borderline-behavior of the radial curvatures for this problem will be determined: we will assume that the radial curvature $K_{\rm rad.}$ of an end converges to a constant -1 at infinity with the decay order $K_{\rm rad.} + 1 = o(r^{-1})$ and prove the absence of eigenvalues embedded in the essential spectrum. Furthermore, in order to show that this decay order $K_{\rm rad.} + 1 = o(r^{-1})$ is sharp, we will construct a manifold with the radial curvature decay $K_{\rm rad.} + 1 = O(r^{-1})$ and with an eigenvalue $\frac{(n-1)^2}{4} + 1$ embedded in the essential spectrum $[ \frac{(n-1)^2}{4}, \infty)$ of the Laplacian.
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