×

Riemannian submersions with discrete spectrum. (English) Zbl 1257.58018

Summary: We prove some estimates on the spectrum of the Laplacian on the total space of a Riemannian submersion in terms of the spectrum of the Laplacian on the base and the geometry of the fibers. When the fibers of the submersions are compact and minimal, we prove that the spectrum of the Laplacian on the total space is discrete if and only if the spectrum of the Laplacian on the base is discrete. When the fibers are not minimal, we prove a discreteness criterion for the total space in terms of the relative growth of the mean curvature of the fibers and the mean curvature of the geodesic spheres in the base. We discuss in particular the case of warped products.

MSC:

58J50 Spectral problems; spectral geometry; scattering theory on manifolds
53C20 Global Riemannian geometry, including pinching
58D15 Manifolds of mappings
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Baider, A.: Noncompact Riemannian manifolds with discrete spectra. J. Diff. Geom. 14, 41–57 (1979) · Zbl 0411.58022
[2] Bessa, G.P., Montenegro, J.F.: Eigenvalue estimates for submanifolds with locally bounded mean curvature. Ann. Global Anal. Geom. 24, 279–290 (2003) · Zbl 1060.53063 · doi:10.1023/A:1024750713006
[3] Bessa, G.P., Montenegro, J.F.: An extension of Barta’s theorem and geometric applications. Ann. Global Anal. Geom. 31(4), 345–362 (2007) · Zbl 1116.58006 · doi:10.1007/s10455-007-9058-8
[4] Bessa, G.P., Jorge, L., Montenegro, J.F.: The spectrum of the Martin-Morales-Nadirashvili minimal surfaces is discrete. J. Geom. Anal. 1, 63–71 (2010) · Zbl 1187.58009 · doi:10.1007/s12220-009-9101-z
[5] Bordoni, M.: Spectral estimates for submersions with fibers of basic mean curvature. An. Univ. Vest Timiş. Ser. Mat.-Inform. 44(1), 23–36 (2006) · Zbl 1119.58301
[6] 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
[7] Davies, E.B.: Spectral Theory and Differential Operators. Cambridge University Press, Cambridge (1995) · Zbl 0893.47004
[8] Donnelly, H.: On the essential spectrum of a complete Riemannian manifold. Topology 20(1), 1–14 (1981) · Zbl 0463.53027 · doi:10.1016/0040-9383(81)90012-4
[9] Donnelly, H.: Negative curvature and embedded eigenvalues. Math. Z. 203, 301–308 (1990) · Zbl 0699.53052 · doi:10.1007/BF02570738
[10] Donnelly, H., Garofalo, N.: Riemannian manifolds whose Laplacian have purely continuous spectrum. Math. Ann. 293, 143–161 (1992) · Zbl 0780.58044 · doi:10.1007/BF01444709
[11] Donnelly, H., Li, P.: Pure point spectrum and negative curvature for noncompact manifolds. Duke Math. J. 46, 497–503 (1979) · Zbl 0416.58025 · doi:10.1215/S0012-7094-79-04624-6
[12] Escobar, E.: On the spectrum of the Laplacian on complete Riemannian manifolds. Commun. Partial Differ. Equ. 11, 63–85 (1985) · Zbl 0585.58046 · doi:10.1080/03605308608820418
[13] Gray, A.: Pseudo-Riemannian almost product manifolds and submersions. J. Math. Mech. 16, 715–737 (1967) · Zbl 0147.21201
[14] Greene, R.E., Wu, H.: Function theory on manifolds which possess a pole. In: Lecture Notes in Mathematics, vol. 699. Springer, Berlin (1979) · Zbl 0414.53043
[15] Harmer, M.: Discreteness of the spectrum of the Laplacian and stochastic incompleteness. J. Geom. Anal. 19, 358–372 (2009) · Zbl 1175.58012 · doi:10.1007/s12220-008-9055-6
[16] Karp, L.: Noncompact manifolds with purely continuous spectrum. Mich. Math. J. 31, 339–347 (1984) · Zbl 0578.53034 · doi:10.1307/mmj/1029003078
[17] Kleine, R.: Discreteness conditions for the Laplacian on complete noncompact Riemannian manifolds. Math. Z. 198, 127–141 (1988) · Zbl 0622.53024 · doi:10.1007/BF01183044
[18] Kleine, R.: Warped products with discrete spectra. Results Math. 15, 81–103 (1989) · Zbl 0669.58030 · doi:10.1007/BF03322449
[19] O’Neill, B.: The fundamental equations of a submersion. Mich. Math. J. 13, 459–469 (1966) · Zbl 0145.18602 · doi:10.1307/mmj/1028999604
[20] O’Neill, B.: Submersions and geodesics. Duke Math. J. 34, 363–373 (1967) · Zbl 0147.40605 · doi:10.1215/S0012-7094-67-03440-0
[21] Rellich, F.: Über das asymptotische Verhalten der Lösungen von +{\(\lambda\)}u=0 in unendlichen Gebieten. Jahresber. Dtsch. Math.-Ver. 53, 57–65 (1943) · Zbl 0028.16401
[22] Scott, P.: The geometries of 3-manifolds. Bull. Lond. Math. Soc. 15(5), 401–487 (1983) · Zbl 0561.57001 · doi:10.1112/blms/15.5.401
[23] Tayoshi, T.: On the spectrum of the Laplace–Beltrami operator on noncompact surface. Proc. Jpn. Acad. 47, 579–585 (1971) · Zbl 0225.35080
[24] Torralbo, F.: Rotationally invariant constant mean curvature surfaces in homogeneous 3-manifolds. Preprint 2009, arXiv:0911.5128v1 · Zbl 1196.53040
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.