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A lower bound for the spectrum of the Laplacian in terms of sectional and Ricci curvature. (English) Zbl 0726.58049

The author proves the following result: Let M be a smooth, n-dimensional, complete, simply connected Riemannian manifold. If the sectional curvature of M is bounded above by -k\(\leq 0\) and the Ricci curvature is bounded above by -\(\alpha\leq 0\), then the spectrum of the Laplacian is bounded below by \([\alpha +(n-1)(n-2)k]/4.\) His result improves a previous result due to H. P. McKean, J. Differ. Geom. 4, 359-366 (1970; Zbl 0197.180).

MSC:

58J50 Spectral problems; spectral geometry; scattering theory on manifolds
35P15 Estimates of eigenvalues in context of PDEs

Citations:

Zbl 0197.180
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References:

[1] Isaac Chavel, Eigenvalues in Riemannian geometry, Pure and Applied Mathematics, vol. 115, Academic Press, Inc., Orlando, FL, 1984. Including a chapter by Burton Randol; With an appendix by Jozef Dodziuk. · Zbl 0551.53001
[2] Jeff Cheeger, A lower bound for the smallest eigenvalue of the Laplacian, Problems in analysis (Papers dedicated to Salomon Bochner, 1969) Princeton Univ. Press, Princeton, N. J., 1970, pp. 195 – 199.
[3] Shiu Yuen Cheng, Eigenfunctions and eigenvalues of Laplacian, Differential geometry (Proc. Sympos. Pure Math., Vol. XXVII, Stanford Univ., Stanford, Calif., 1973) Amer. Math. Soc., Providence, R.I., 1975, pp. 185 – 193.
[4] H. P. McKean, An upper bound to the spectrum of \Delta on a manifold of negative curvature, J. Differential Geometry 4 (1970), 359 – 366. · Zbl 0197.18003
[5] Robert S. Strichartz, Analysis of the Laplacian on the complete Riemannian manifold, J. Funct. Anal. 52 (1983), no. 1, 48 – 79. · Zbl 0515.58037 · doi:10.1016/0022-1236(83)90090-3
[6] Shing Tung Yau, Isoperimetric constants and the first eigenvalue of a compact Riemannian manifold, Ann. Sci. École Norm. Sup. (4) 8 (1975), no. 4, 487 – 507. · Zbl 0325.53039
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