×

D’un résultat Hilbertien à un principe de comparaison entre spectres. Applications. (On a Hilbert type result on a comparison principle between spectra. Applications). (French) Zbl 0722.53037

The authors give several results on estimates of the eigenvalues and Betti numbers for manifolds with or without boundary. They first establish some abstract results in Hilbert space, which enable them to compare the eigenvalues of an operator acting on the sections of a vector bundle with those of an operator acting only on the base space. As applications, they establish upper bounds for Betti numbers and lower bounds for the eigenvalues of the Hodge Laplacian (in the case of manifolds with boundary, the Neumann boundary condition is used) in terms of geometrical informations on the manifold. They give also an example to show that these geometrical informations on the manifold are necessary to obtain a lower bound on the Neumann eigenvalues.
Reviewer: X.-P.Wang (Nantes)

MSC:

53C20 Global Riemannian geometry, including pinching
58J50 Spectral problems; spectral geometry; scattering theory on manifolds
PDF BibTeX XML Cite
Full Text: DOI Numdam EuDML

References:

[1] M. BERGER et P. BÉRARD , Bibliographie: Le spectre d’une variété riemannienne (Spectra of Riemannian Manifolds, Kaigai, 1984 , accessible également en appendice du livre de Pierre Bérard, Lecture Notes in Math., n^\circ 1207, Springer, 1986 ).
[2] R. BISHOP et R. CRITTENDEN , Geometry of Manifolds , Acad. Press, New York, 1964 . MR 29 #6401 | Zbl 0132.16003 · Zbl 0132.16003
[3] P. BÉRARD et S. GALLOT , Inégalités isopérimétriques pour l’équation de la chaleur et applications à l’estimation de quelques invariants géométriques (Séminaire Goulaouic-Meyer-Schwartz, École Polytechnique, Palaiseau, 1983 - 1984 ). Numdam | Zbl 0542.53025 · Zbl 0542.53025
[4] G. BESSON , Appendice au livre de P. Bérard, Spectral Geometry : Direct and Inverse problems (Lecture Notes in Math., n^\circ 1207, Springer, 1986 ). MR 88f:58146 | Zbl 0608.58001 · Zbl 0608.58001
[5] S. Y. CHENG et P. LI , Heat Kernel Estimates and Lower Bound of Eigenvalues (Comment. Math. Helv., vol. 56, 1981 , p. 327-338). MR 83b:58076 | Zbl 0484.53034 · Zbl 0484.53034
[6] G. DUFF et D. C. SPENCER , Harmonic Tensors on Riemannian Manifolds with Boundary (Ann. of Math. Stud., vol. 56, n^\circ 1, 1952 , p. 115-127). MR 13,987a · Zbl 0049.18901
[7] S. GALLOT et D. MEYER , Opérateur de courbure et laplacien des formes différentielles d’une variété riemannienne (J. Math. Pures Appl., vol. 54, 1975 , p. 259-284). MR 56 #13128 | Zbl 0316.53036 · Zbl 0316.53036
[8] S. GALLOT , Inégalités isopérimétriques, courbure de Ricci et invariants géométriques, I (C.R. Acad. Sci. Paris, 296, série I, 1983 , p. 333-336). MR 85c:53069a | Zbl 0535.53035 · Zbl 0535.53035
[9] S. GALLOT , Inégalités isopérimétriques, courbure de Ricci et invariants géométriques, II (C.R. Acad. Sci. Paris, 296, série I, 1983 , p. 365-368). MR 85c:53069b | Zbl 0535.53034 · Zbl 0535.53034
[10] S. GALLOT , Minorations sur le \lambda 1 des variétés riemanniennes [Sém. Bourbaki, 1980 - 1981 (Lecture Notes in Math., Springer, n^\circ 901, 1981 , p. 132-148)]. Numdam | MR 83g:58071 | Zbl 0493.53034 · Zbl 0493.53034
[11] M. GROMOV , Paul Levy’s Isoperimetric Inequality , Preprint I.H.E.S., 1980 .
[12] M. GROMOV , Structures métriques pour les variétés riemanniennes , Textes Math., n^\circ 1, Cedic-Nathan, 1981 . MR 85e:53051 | Zbl 0509.53034 · Zbl 0509.53034
[13] M. GROMOV , Curvature, Diameter and Betti Numbers (Comment. Math. Helv., vol. 56, 1981 , p. 179-197). MR 82k:53062 | Zbl 0467.53021 · Zbl 0467.53021
[14] E. HEINTZE et H. KARCHER , A General Comparison Theorem with Applications to Volume Estimates for Submanifolds (Ann. scient. Éc. Norm. Sup., vol. 11, 1978 , p. 451-470). Numdam | MR 80i:53026 | Zbl 0416.53027 · Zbl 0416.53027
[15] C. C. HSIUNG , Curvature and Homology of Riemannian Manifolds with Boundary (Math. Zeit., vol. 82, 1963 , p. 67-81). MR 27 #2943 | Zbl 0119.17202 · Zbl 0119.17202
[16] H. HESS , R. SCHRADER et D. A. UHLENBROCK , Kato’s Inequality and the Spectral Distribution of Laplacian on Compact Riem . Manifolds (J. Differential Geom., vol. 15, 1980 , p. 27-37). MR 82g:58090 | Zbl 0442.58032 · Zbl 0442.58032
[17] H. KARCHER , Pinching Implies Strong Pinching (Comment. Math. Helv., vol. 46, 1971 , p. 124-126). MR 45 #1076 | Zbl 0216.43803 · Zbl 0216.43803
[18] P. LI , On the Sobolev Constant and the p-Spectrum of a Compact Riemannian Manifold (Ann. scient. Éc. Norm. Sup., vol. 13, 1981 , p. 451-457). Numdam | MR 82h:58054 | Zbl 0466.53023 · Zbl 0466.53023
[19] B. LAWSON et M. L. MICHELSON , Imbeddings and Surrounding with Positive Mean Curvature (Invent. Math., vol. 77, 1984 , p. 399-419). Zbl 0555.53027 · Zbl 0555.53027
[20] B. LAWSON , The Unknottedness of Minimal Imbeddings (Invent. Math., vol. 11, 1970 , p. 183-187). Zbl 0205.52002 · Zbl 0205.52002
[21] P. LI et S. T. YAU , Estimates of Eigenvalues of a Compact Riemannian Manifold (Proc. Sympos. Pure Math., vol. 36, 1980 , p. 205-239). MR 81i:58050 | Zbl 0441.58014 · Zbl 0441.58014
[22] P. LI et S. T. YAU , On the Parabolic Kernel of the Schrödinger Operator (Acta Math., vol. 156, 1986 , p. 153-201). MR 87f:58156 | Zbl 0611.58045 · Zbl 0611.58045
[23] D. MEYER , Un lemme de géométrie hilbertienne et des applications à la géométrie riemannienne (C.R. Acad. Sci. Paris, 295, série I, 1982 , p. 467-469). MR 85f:58121 | Zbl 0515.53039 · Zbl 0515.53039
[24] D. MEYER , Un lemme de géométrie hilbertienne et des applications à la géométrie riemannienne, II , preprint. · Zbl 0515.53039
[25] D. MEYER , Une minoration de la première valeur propre non nulle du problème de Neumann pour les variétés riemanniennes à bord (Ann. Inst. Fourier, Grenoble, vol. 36, n^\circ 2, 1986 , p. 113-125). Numdam | MR 88d:58128 | Zbl 0582.58036 · Zbl 0582.58036
[26] D. MEYER , Sur les hypersurfaces minimales des variétés riemanniennes à courbure de Ricci positive ou nulle (Bull. Soc. Math. France, vol. 111, 1983 , p. 359-366). Numdam | MR 86g:53068 | Zbl 0545.53045 · Zbl 0545.53045
[27] B. MAUREY , Communications écrites et orales adressées à D. Meyer , 1982 .
[28] M. REED et B. SIMON , Methods of Modern Mathematical Physics , vol. I à IV, Acad. Press, 1978 . · Zbl 0401.47001
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.