On a lower bound for the first eigenvalue of the Laplace operator on a riemannian manifold

Atsushi Kasue

Annales scientifiques de l'École Normale Supérieure (1984)

  • Volume: 17, Issue: 1, page 31-44
  • ISSN: 0012-9593

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Kasue, Atsushi. "On a lower bound for the first eigenvalue of the Laplace operator on a riemannian manifold." Annales scientifiques de l'École Normale Supérieure 17.1 (1984): 31-44. <http://eudml.org/doc/82135>.

@article{Kasue1984,
author = {Kasue, Atsushi},
journal = {Annales scientifiques de l'École Normale Supérieure},
keywords = {eigenvalue estimates; Busemann functions; Ricci curvature; mean curvature; eigenvalue of the Laplacian; Jacobi differential equation},
language = {eng},
number = {1},
pages = {31-44},
publisher = {Elsevier},
title = {On a lower bound for the first eigenvalue of the Laplace operator on a riemannian manifold},
url = {http://eudml.org/doc/82135},
volume = {17},
year = {1984},
}

TY - JOUR
AU - Kasue, Atsushi
TI - On a lower bound for the first eigenvalue of the Laplace operator on a riemannian manifold
JO - Annales scientifiques de l'École Normale Supérieure
PY - 1984
PB - Elsevier
VL - 17
IS - 1
SP - 31
EP - 44
LA - eng
KW - eigenvalue estimates; Busemann functions; Ricci curvature; mean curvature; eigenvalue of the Laplacian; Jacobi differential equation
UR - http://eudml.org/doc/82135
ER -

References

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  1. [1] J. BARTA, Sur la vibration fondamentale d'une membrane (Comptes Rendus de l'Acad. des Sci., Paris, Vol. 204, 1937, pp. 472-473). Zbl63.0762.02JFM63.0762.02
  2. [2] P. BÉRARD et D. MEYER, Inégalités isopérimétriques et applications (Ann. scient. Éc. Norm. Sup., Paris, 4e série, t. 15, 1982, pp. 513-542). Zbl0527.35020MR84h:58147
  3. [3] R. L. BISHOP and B. O'NEILL, Manifolds of negative curvature (Trans. Amer. Math. Soc., Vol. 145, 1969, pp. 1-49). Zbl0191.52002MR40 #4891
  4. [4] J. CHEEGER and D. GROMOLL, The splitting theorem for manifolds of nonnegative curvature (J. Differential Geometry, Vol. 6, 1971, pp. 119-128). Zbl0223.53033MR46 #2597
  5. [5] R. COURANT and D. HILBERT, Methods of Mathematical Physics, Interscience Pub., Inc., New York, 1970. Zbl57.0245.01
  6. [6] P. EBERLEIN and B. O'NEILL, Visibility manifolds (Pacific J. Math., Vol. 46, 1973, pp. 45-109). Zbl0264.53026MR49 #1421
  7. [7] S. FRIEDLAND and W. K. HAYMAN, Eigenvalue inequalities for the Dirichlet problem on spheres and the growth of subharmonic functions (Comm. Math. Helvetici, Vol. 51, 1976, pp. 133-161). Zbl0339.31003MR54 #568
  8. [8] S. GALLOT, Minorations sur le λ1 des variétés riemanniennes, Séminaire Bourbaki 33e année, 1980-1981, n° 569. Zbl0493.53034
  9. [9] S. GALLOT, A Sobolev inequality and some geometric applications, preprint (séminaire Franco-Japonais de Kyoto). 
  10. [10] R. E. GREENE and H. WU, On the subharmonicity and plurisubharmonicity of geodesically convex function, (Indiana Univ., Math. J., Vol. 22, 1973, pp. 641-653). Zbl0235.53039MR54 #10672
  11. [11] R. E. GREENE and H. WU, D∞ approximations of convex, subharmonic and plurisubharmonic functions (Ann. scient. Éc. Norm. Sup., 4e série, t. 12. 1979, pp. 47-84). Zbl0415.31001MR80m:53055
  12. [12] M. GROMOV, Paul Levy's isoperimetric inequalities, Publications I.H.E.S., 1981. 
  13. [13] E. HEINTZE and H. KARCHER, A general comparison theorem with applications to volume estimates for submanifolds (Ann. scient. Éc. Norm. Sup., Paris, Vol. 11, 1978, pp. 451-470). Zbl0416.53027MR80i:53026
  14. [14] A. KASUE, On a Riemannian manifold with a pole (Osaka J. Math., Vol. 18, 1981, pp. 109-113). Zbl0477.53042MR83f:53030
  15. [15] A. KASUE, A Laplacian comparison theorem and function theoretic properties of a complete Riemannian manifold (Japan J. Math., Vol. 18, 1982, pp. 309-341). Zbl0518.53048MR85h:53031
  16. [16] A. KASUE, Ricci curvature, geodesics and some geometric properties of Riemannian manifolds with boundary (J. Math. Soc. Japan, Vol. 35, 1983, pp. 117-131). Zbl0494.53039MR84g:53068
  17. [17] M. G. KREIN, On certain problems of the maximum and minimum of characteristic values and on the Lyapunov zones of stability (A.M.S. Trans., Vol. 2, n° 1, 1955, pp. 163-187). Zbl0066.33404MR17,484e
  18. [18] A. LICHNEROWICZ, Géométrie des groupes de transformations, Dunod, 1958. Zbl0096.16001MR23 #A1329
  19. [19] P. LI and S. T. YAU, Estimates of eigenvalues of a compact Riemannian manifold (Proc. Symp. Pure Math. A.M.S., Vol. 36, 1980, pp. 205-239). Zbl0441.58014MR81i:58050
  20. [20] H. P. MCKEAN, An upper bound to the spectrum on a manifold of negative curvature (J. Differential Geometry, Vol. 4, 1970, pp. 359-366). Zbl0197.18003MR42 #1009
  21. [21] M. OBATA, Certain conditions for a Riemannian manifold to be isometric with a sphere (J. Math. Soc. Japan, Vol. 14, 1962, pp. 333-340). Zbl0115.39302MR25 #5479
  22. [22] W. A. POOR Jr., Some results on nonnegatively curved manifolds (J. Differential Geometry, Vol. 9, 1974, pp. 583-600). Zbl0292.53037MR51 #11351
  23. [23] R. C. REILLY, Applications of the Hessian operator in a Riemannian manifold, (Indiana Univ. Math. J., Vol. 26, 1977, pp. 459-472). Zbl0391.53019MR57 #13799
  24. [24] H. WU, An elementary method in the study of nonnegative curvature (Acta. Math., Vol. 142, 1979, pp. 57-78). Zbl0403.53022MR80c:53054

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