Eigenvalue estimate on complete noncompact Riemannian manifolds and applications

do Carmo, Manfredo; Zhou, Detang

doi:10.1090/S0002-9947-99-02061-9

Eigenvalue estimate on complete noncompact Riemannian manifolds and applications
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by Manfredo P. do Carmo and Detang Zhou

Trans. Amer. Math. Soc. 351 (1999), 1391-1401

DOI: https://doi.org/10.1090/S0002-9947-99-02061-9

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Abstract:

We obtain some sharp estimates on the first eigenvalues of complete noncompact Riemannian manifolds under assumptions of volume growth. Using these estimates we study hypersurfaces with constant mean curvature and give some estimates on the mean curvatures.

References

H. Alencar and M. do Carmo, Hypersurfaces of constant mean curvature with finite index and volume of polynomial growth, Arch. Math. (Basel) 60 (1993), no. 5, 489–493. MR 1213521, DOI 10.1007/BF01202317
Jeff Cheeger, Mikhail Gromov, and Michael Taylor, Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds, J. Differential Geometry 17 (1982), no. 1, 15–53. MR 658471
Jeff Cheeger and Shing Tung Yau, A lower bound for the heat kernel, Comm. Pure Appl. Math. 34 (1981), no. 4, 465–480. MR 615626, DOI 10.1002/cpa.3160340404
S. Y. Cheng and S. T. Yau, Differential equations on Riemannian manifolds and their geometric applications, Comm. Pure Appl. Math. 28 (1975), no. 3, 333–354. MR 385749, DOI 10.1002/cpa.3160280303
Eells, J. Jr., Kobayashi, S., Problems in differential geometry, In:, Proc. of US-Japan Seminar on differential geometry. Kyoto 1965, 167–177.
Fite, W.B., Concerning the zeros of the solutions of certain differential equations, Trans. Amer. Math. Soc. 19 (1918), 341–352.
D. Fischer-Colbrie, On complete minimal surfaces with finite Morse index in three-manifolds, Invent. Math. 82 (1985), no. 1, 121–132. MR 808112, DOI 10.1007/BF01394782
Katia Rosenvald Frensel, Stable complete surfaces with constant mean curvature, Bol. Soc. Brasil. Mat. (N.S.) 27 (1996), no. 2, 129–144. MR 1418929, DOI 10.1007/BF01259356
Michael E. Gage, Upper bounds for the first eigenvalue of the Laplace-Beltrami operator, Indiana Univ. Math. J. 29 (1980), no. 6, 897–912. MR 589652, DOI 10.1512/iumj.1980.29.29061
Ernst Heintze and Hermann Karcher, A general comparison theorem with applications to volume estimates for submanifolds, Ann. Sci. École Norm. Sup. (4) 11 (1978), no. 4, 451–470. MR 533065, DOI 10.24033/asens.1354
Robert Osserman, Bonnesen-style isoperimetric inequalities, Amer. Math. Monthly 86 (1979), no. 1, 1–29. MR 519520, DOI 10.2307/2320297
Mark A. Pinsky, The spectrum of the Laplacian on a manifold of negative curvature. I, J. Differential Geometry 13 (1978), no. 1, 87–91. MR 520603
Michael E. Taylor, $L^p$-estimates on functions of the Laplace operator, Duke Math. J. 58 (1989), no. 3, 773–793. MR 1016445, DOI 10.1215/S0012-7094-89-05836-5
James S. W. Wong, Oscillation and nonoscillation of solutions of second order linear differential equations with integrable coefficients, Trans. Amer. Math. Soc. 144 (1969), 197–215. MR 251305, DOI 10.1090/S0002-9947-1969-0251305-6
Detang Zhou, Laplace inequalities with geometric applications, Arch. Math. (Basel) 67 (1996), no. 1, 50–58. MR 1392052, DOI 10.1007/BF01196166