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Discreteness conditions for the Laplacian on complete, non-compact Riemannian manifolds. (English) Zbl 0622.53024

This paper is a shortened version of the author’s thesis (Duisburg 1985; Zbl 0591.53040) and it contains a necessary and sufficient criterion for the discreteness of the spectrum of the Laplacian for manifolds with a finite number of ends. Topologically these ends are \({\mathbb{R}}^ +\times N\) (N is compact) with metric tensor \(g_{ij}(r,x)=\left( \begin{matrix} 1\\ 0\end{matrix} \begin{matrix} 0\\ \omega (r,x)\end{matrix} \right),\) (r,x)\(\in {\mathbb{R}}^ +\times N\), and \(\omega\) representing the metric of the hypersurfaces \(\{\) \(r\}\) \(\times N\). The result extends in a certain sense Baider’s reduction principle for warped products \({\mathbb{R}}^ +\times N\) [cf. A. Baider, J. Differ. Geom. 14, 41-57 (1979; Zbl 0411.58022)]. Links between geometry and discreteness are investigated.

MSC:

53C20 Global Riemannian geometry, including pinching
58J50 Spectral problems; spectral geometry; scattering theory on manifolds
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References:

[1] Baider, A.: Noncompact Riemannian Manifolds with Discrete Spectra. J. Differ. Geom.14, 41-57 (1979) · Zbl 0411.58022
[2] Bishop, R., Crittenden, R.: Geometry of manifolds. New York: Academic Press 1964 · Zbl 0132.16003
[3] Brooks, R.: A relation between Growth and the Spectrum of the Laplacian. Math. Z.178, 501-508 (1981) · Zbl 0468.58019 · doi:10.1007/BF01174771
[4] Brooks, R.: On the spectrum of noncompact manifolds with finite volume. Math. Z.187, 425-432 (1984) · Zbl 0537.58040 · doi:10.1007/BF01161957
[5] Chernoff, P.R.: Essential self-adjointness of powers of generators of hyperbolic equations. J. Funct. Anal.12, 401-414 (1973) · Zbl 0263.35066 · doi:10.1016/0022-1236(73)90003-7
[6] Do Carmo, M.P.: Differential geometry of curves and surfaces, Englewood Cliffs: Prentice Hall 1976 · Zbl 0326.53001
[7] Donnelly, H.: On the essential spectrum of a complete Riemannian manifold. Topology20, 1-14 (1981) · Zbl 0463.53027 · doi:10.1016/0040-9383(81)90012-4
[8] Donnelly, H., Li, P.: Pure point spectrum and negative curvature for noncompact manifolds. Duke Math. J. 46, 497-593 (1979) · Zbl 0416.58025 · doi:10.1215/S0012-7094-79-04624-6
[9] Eichhorn, J.: Riemannsche Mannigfaltigkeiten mit einer zylinderähnlichen Endenmetrik. Math. Nachr.114, 23-51 (1983) · Zbl 0549.53038 · doi:10.1002/mana.19831140103
[10] Gaffney, P.: The harmonic operator for exterior differential forms. Proc. Nat. Acad. Sci. USA37, 48-50 (1951) · Zbl 0042.10205 · doi:10.1073/pnas.37.1.48
[11] Gage, M.E.: Upper bounds for the first eigenvalue of the Laplace-Beltrami operator. Indiana Univ. Math. J.29, 897-912 (1980) · Zbl 0465.53031 · doi:10.1512/iumj.1980.29.29061
[12] Glazman, I.M.: Direct methods of qualitative spectral analysis of singular differential operators. Jerusalem (1965) · Zbl 0143.36505
[13] Gromoll, D., Klingenberg, W., Meyer, W.: Riemannsche Geometrie im Großen. Lecture Notes in Math.55. Berlin Heidelberg New York: Springer 1975 · Zbl 0293.53001
[14] Hartman, P.: Ordinary differential equations. New York: John Wiley and Sons Inc. 1984 · Zbl 0125.32102
[15] Heintze, E., Karcher, H.: A general comparison theorem with applications to volume estimates for submanifolds. Ann. Scient. Ec. Norm. Sup. 4o serie, t 11, 451-470 (1970) · Zbl 0416.53027
[16] Kufner, A., John, O., Fu?ik, S.: Function spaces. Leyden: Nordhoff 1977
[17] Maslov, V.P.: A criterion for discreteness of the spectrum of a Strurm-Liouville equation with an operator coefficient. Funct. Anal. Appl.2, 153-157 (1968) · Zbl 0188.46301 · doi:10.1007/BF01075949
[18] Müller-Pfeiffer, E.: Spektraleigenschaften singulärer gewöhnlicher Differentialoperatoren. Leipzig: Teubner 1977 · Zbl 0373.47022
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