Available in electronic format
Available in print format
Bulletin of the American Mathematical Society
Bulletin of the American Mathematical Society
ISSN 1088-9485(e) ISSN 0273-0979(p)
     

From vanishing theorems to estimating theorems: the Bochner technique revisited

Author(s): Pierre H. Bérard
Journal: Bull. Amer. Math. Soc. 19 (1988), 371-406.
MSC (1985): Primary 58G30, 53C20; Secondary 58G11, 58E99
MathSciNet review: 956595
Retrieve article in: PDF

References | Similar articles | Additional information

References:

[Ad-Su] T. Adachi and T. Sunada, Energy spectrum of certain harmonic mappings, Compositio Math. 56 (1985), 153-170. MR 809864

[Au1] Th. Aubin, Non linear analysis on manifolds, Monge-Ampère Equations, Springer-Verlag, 1982. MR 681859

[Au2] Th. Aubin, Equations différentieles non linéaires et problème de Yamabe concernant la courbure scalaire, J. Math. Pures Appl. 55 (1976), 269-296. MR 431287

[Ba] C. Bandle, Isoperimetric inequalities and applications, Pitman Press, 1980. MR 572958

[Ba-Pa] Ch. Bavard and P. Pansu, Sur le volume minimal de R2, Ann. Sci. École Norm. Sup. 19 (1986), 479-490. MR 875084

[Bé] P. Bérard, Lectures on spectral geometry, 15° Colóquio Brasileiro de Matemática, IMPA 1985 and Lecture Notes in Math., vol. 1207, Springer-Verlag, Berlin and New York, 1986. MR 812648

[Bé-Be] P. Bérard and G. Besson, Integral bounds for some geometric invariants via the Bochner technique, Prépublication de l'Institut Fourier n°78, Grenoble, 1987.

[Bé-Ber] P. Bérard and M. Berger, Le spectre d'une variété riemannienne en 1982, Spectra of Riemannian Manifolds, Kaigai Publ., 1983, pp. 139-194, or [Bé, Appendix B].

[Bé-Be-Gal] P. Bérard, G. Besson and S. Gallot, Sur une inégalité isopérimétrique qui généralise celle de P. Lévy-Gromov, Invent. Math. 80 (1985), 295-308. MR 788412

[Bé-Ga] P. Bérard and S. Gallot, Inégalités isopérimétriques pour l'équation de la chaleur et application à l'estimation de quelques invariants, Exp. XV, Séminaire Goulaouic-Meyer-Schwartz 83-84, École Polytechnique, Palaiseau, 1984. MR 811658

[Bé-Ga] P. Bérard and S. Gallot, Remarques sur quelques estimées géométriques explicites, C. R. Acad. Sci. Sér. I Math 297 (1983), 185-188. MR 725402

[Bé-Me] P. Bérard and D. Meyer, Inégalités isopérimétriques et applications, Ann. Sci. École Norm. Sup 15 (1982), 513-542. MR 690651

[B-G-M] M. Berger, P. Gauduchon and E. Mazet, Le spectre d'une variété riemannienne, Lecture Notes in Math., vol. 194, Springer-Verlag, Berlin and New York, 1971. MR 282313

[Bo] S. Bochner, Vector fields and Ricci curvature, Bull. Amer. Math. Soc. 52 (1946), 776-797. MR 18022

[Bo-Ya] S. Bochner and K. Yano Curvature and Betti numbers, Ann. of Math. Studies, no. 32, Princeton Univ. Press, 1953. MR 62505

[Bou] J.-P. Bourguignon, Formules de Weitzenböck en dimension 4, Exposé XVI, A. Besse, Géométrie riemannienne de dimension 4, Cédic-Nathan, 1981. MR 769143

[Bu-Ka] P. Buser and M. Karcher, Gromov's almost flat manifolds, Astérisque, 81, Soc. Math. France, Paris, 1981.

[Ch] I. Chavel, Eigenvalues in Riemannian geometry, Academic Press, New York, 1984. MR 2229062

[Che-Eb] J. Cheeger and D. Ebin, Comparison theorems in Riemannian geometry, American Elsevier, 1975. MR 458335

[Co] G. Courtois, in preparation, 1986.

[Cr] Ch. Croke, Some isoperimetric inequalities and eigenvalue estimates, Ann. Sci. École Norm. Sup. Paris 13 (1981), 419-436. MR 608287

[De] J.-P. Demailly, Majoration asymptotique de la cohomologie d'un fibré linéaire hermitien, Prépublication de l'Institut Fourier 25, Grenoble, 1985. Champs magnétiques et inégalités de Morse pour la d" -cohomologie, Prépublication de l'Institut Fourier 37, Grenoble, 1985. MR 799607

[Do] J. Dodziuk, Vanishing theorems for square integrable harmonie forms, Proc. Indian Acad. Sci. Math. Sci. 90 (1981), 21-27. MR 653943

[Ee] J. Eells, Elliptic operators on manifolds, Proc. Summer Course Complex Analysis (I.C.T.P. Trieste 1975), vol. 1, IAEA, 1976, pp. 95-152. MR 482861

[Ee-Le] J. Eells and L. Lemaire, Selected topics in harmonic maps, CBMS Regional Conf. Ser. in Math., no. 50, Amer. Math. Soc. Providence, R. I., 1983. MR 703510

[El] A. El Soufi, Thèse de 3e cycle, Université de Grenoble, 1982.

[Ga1] S. Gallot, Inégalités isopérimétriques, courbure de Ricci et invariants géométriques. I and II, C. R. Acad. Sci. Sér. I Math. 296 (1983), 333-336 and 365-368. MR 697966

[Ga2] S. Gallot, A Sobolev inequality and some geometric applications, Spectra of Riemannian Manifolds, Kaigai Publ., 1983, pp. 45-55.

[Ga3] S. Gallot, Bornes universelles pour des invariants géométriques, Séminaire Théorie Spectrale et Géométrie (Chambéry-Grenoble 1982-1983), Publications de l'Institut Fourier, Grenoble.

[Ga-Me] S. Gallot and D. Meyer, Opérateur de courbure et laplacien des formes différentielles d'une variété riemannienne, J. Math. Pures Appl. 54 (1975), 259-284. MR 454884

[Gam1] N. Gamara, Thèse de 3e cycle, Université de Savoie, Chambéry, 1983.

[Gam2] N. Abdelmoula-Gamara, Symétrisation d'inéquations elliptiques et applications géométriques, Math. Z. (to appear). MR 958647

[Gi-Tr] D. Gilbarg and N. S. Trudinger Elliptic partial differential equations of second order, Grundlehren Math. Wiss. vol 224, Springer-Verlag, Berlin and New York, 1977. MR 473443

[Gr1] M. Gromov, Curvature and Betti numbers, Comment. Math. Helv 56 (1981), 179-195. MR 630949

[Gr2] M. Gromov, P. Levy's isoperimetric inequality, Preprint, Inst. Hautes Etudes Sci., 1980.

[Gr3] M. Gromov, Structure métriques pour les variétés riemanniennes, rédigé par J. Lafontaine et P. Pansu, Cédic-Nathan, 1981. MR 682063

[Gr-La] M. Gromov and B. H. Lawson Positive scalar curvature and the Dirac operator on complete Riemannian manifolds, Inst. Hautes Etudes Sci. Publ. Math. 58 (1983), 83-196. MR 720933

[H-S-U] H. Hess, R. Schrader and D. A. Uhlenbrock, Kato's inequality and the spectral distribution of Laplacians on compact Riemannian manifolds, J. Differential Geom. 15 (1980), 27-38. MR 602436

[Hi] N. Hitchin, Harmonic spinors, Adv. in Math. 14 (1974), 1-55. MR 358873

[Hu] D. Hulin, Pinching and Betti numbers, Ann. Global Anal. Geom. 3 (1985), 85-93. MR 812314

[Ka] J.-L. Kazdan, Positive energy in general relativity, Séminaire Bourbaki, Exp. 593, Astérisque 92-93, 1982. MR 689537

[Ko-No] S. Kobayashi and K. Nomizu, Foundations of differential geometry, vol. II, Interscience, 1969. MR 238225

[La] B.-H. Lawson, Theory of gauge fields in four-manifolds, CBMS Regional Conf. Ser. in Math., no. 58, Amer. Math. Soc. Providence, R. I., 1984.

[Li] P. Li, On the Sobolev constant and the p-spectrum of a compact Riemannian manifold, Ann. Sci. École Norm. Sup. 13 (1980), 451-469. MR 608289

[Lz] A. Lichnerowicz, Spineurs harmoniques, C. R. Acad. Sci. Sér. I Math. 257 (1963), 7-9. MR 156292

[Me1] D. Meyer, Sur les variétés riemanniennes à opérateurs de courbure positif, C. R. Acad. Sci. Sér. I Math. 272 (1971), 482-485. MR 279736

[Me2] D. Meyer, Une inégalité de géométrie hilbertienne et ses applications à la géométrie riemannienne, Preprint 1984, Univ. Paris 7 and C. R. Acad. Sci. Sér. I Math. 295 (1982), 467-469. MR 683666

[Me3] D. Meyer, Work in progress, 1986.

[Mi-Mo] M. Micallef and J.-D. Moore, Minimal two-spheres and the topology of manifolds with positive curvature on totally isotropic two-planes, Preprint, Univ. of California Santa Barbara, 1986. MR 924677

[Mo] J.-D. Moore, On stability of minimal spheres and a two-dimensional version of Synge's theorem, Arch. Math. 44 (1985), 278-281. MR 784099

[Pa] P. Pansu, Dégénérescence des variétés riemanniennes, Séminaire Bourbaki, Exposé 618, Novembre, 1983.

[Re-Si] M. Reed and B. Simon, Methods of modem mathematical physics, Academic Press, New York, 1978. MR 493421

[Ro] S. Rosenberg, Semigroup domination and vanishing theorems (Proceedings 1987 Cornell Conf. on Geometry of Random Motion) Contemp. Math., vol. 73, Amer. Math. Soc. Providence, R. I., 1988, pp. 287-302. MR 954646

[Sa] T. Sakai, Comparison and finiteness theorems in Riemannian geometry, Geometry of geodesics and related topics, Adv. Stud. Pure Math. 3 (1984), 125-181. MR 758652

[Sc-Yau] R. Schoen and S.-T. Yau, Complete three-dimensional manifolds with positive Ricci curvature and scalar curvature, Seminar on Differential Geometry, Ann. of Math. Studies, no. 102, Princeton Univ. Press, 1982, pp. 209-227. MR 645740

[Sh-So] B. Shiffman and A.-J. Sommese, Vanishing theorems in complex manifolds, Progress in Math., vol 56, Birkhäuser, Boston, 1985. MR 782484

[Si1] Y.-T. Siu, Some recent results in complex manifold theory related to vanishing theorems for the semi-positive case, Arbeitstagung Bonn (1984), Lecture Notes in Math., vol 1111, Springer-Verlag, Berlin and New York, 1985, pp. 169-192. MR 797421

[Si2] Y.-T. Siu, Asymptotic Morse inequalities for analytic sheaf cohomology, Séminaire Bourbaki, Exposé n° 666, Juin, 1986. MR 880038

[Ta] G. Talenti, Best constants in Sobolev inequality, Ann. Math. Pura Appl. 110 (1976), 353-372. MR 463908

[Ur] H. Urakawa, Stability of harmonie maps and eigenvalues of the laplacian, Trans. Amer. Math. Soc. 301 (1987), 557-590. MR 882704

[Wu] H. Wu, The Bochner technique, Proceedings 1980 Beijing symposium on Differential Geometry and Differential Equations. (S. S. Chern and Wu Wen-Tsün, eds.), Science Press and Gordon-Breach 1982, and The Bochner technique in differential geometry, Math. Rep., 1987. MR 714349

[Yau1] S.-T. Yau, Seminar on differential geometry (S. T. Yau, ed.), Ann. of Math. Studies, no. 102, Princeton Univ. Press, 1982. MR 645728

[Yau2] S.-T. Yau, Differential geometry and complex analysis: a volume dedicated to the memory of H. E. Rauch (I. Chavel and H. M. Farkas, eds.), Springer, 1985. MR 780048

S. Gallot, Isoperimetric inequalities based on integral norms of Ricci curvature, Astérisque (to appear). MR 976219

S. Gallot, Théorèmes de comparaison entre variétés et entre spectres. Applications, Astérisque (to appear).

S. Gallot and D. Meyer, D'un résultat hilbertien à un principe de comparaison entre spectres. Applications, Ann. Sci. École Norm. Sup. (to appear).


Similar Articles:

Retrieve articles in Bulletin of the American Mathematical Society with MSC (1985): 58G30, 53C20, 58G11, 58E99

Retrieve articles in all Journals with MSC (1985): 58G30, 53C20, 58G11, 58E99


Additional Information:

DOI: 10.1090/S0273-0979-1988-15679-0
PII: S 0273-0979(1988)15679-0


  AMS Website Logo Small Comments: webmaster@ams.org
© Copyright 2009, American Mathematical Society
Privacy Statement
Search the AMSPowered by Google