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Eigenvalue comparison theorems and its geometric applications. (English) Zbl 0329.53035

MSC:
53C20 Global Riemannian geometry, including pinching
35P15 Estimates of eigenvalues in context of PDEs
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References:
[1] Berger, M., Gauduchon, P., Mazet, E.: Le spectre d’une vari?t? Riemannienne. Lecture Notes in Mathematics 194. Berlin-Heidelberg-New York: Springer 1971 · Zbl 0223.53034
[2] Bishop, R., Crittenden, R.: Geometry of Manifolds. New York-London: Academic Press 1964 · Zbl 0132.16003
[3] Chavel, I., Feldman, E.: The first eigenvalue of the Laplacian on manifolds of non-negative curvature. CUNY preprint · Zbl 0291.53021
[4] Cheeger, J.: The relation between the Laplacian and the diameter for manifolds of non-negative curvature. Arch. der Math.19, 558-560 (1968) · Zbl 0177.50201
[5] Cheng, S. Y.: Eigenvalue and eigenfunctions of the Laplacian. To appear in the Proceedings of the Symposium on Differential Geometry. (Stanford University, 1973)
[6] Hobson, E.W.: Spherical and Ellipsoidal Harmonics. New York: Chelsea 1955 · Zbl 0004.21001
[7] Klingenberg, W.: Contribution to Riemannian geometry in the large. Ann. of Math., II. Ser.69, 654-666 (1959) · Zbl 0133.15003
[8] Mazet, E.: Une majoration de ?1 du type de Cheeger. C.r. Acad. Sci., Paris, S?r. A,277, 171-174 (1973) · Zbl 0264.53021
[9] McKean, H.P. Jr.: An upper bound to the spectrum on a manifold of negative curvature. J. diff. Geometry4, 359-366 (1970) · Zbl 0197.18003
[10] Payne, L.E.: Isoperimetric inequalities and their applications. SIAM Review9, 453-488 (1967) · Zbl 0154.12602
[11] Watson, G.N.: A treatise on the theory of Bessel functions. Cambridge: Cambridge University Press 1944; New York: MacMillan 1944 · Zbl 0063.08184
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