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Bulletin of the American Mathematical Society

The Bulletin publishes expository articles on contemporary mathematical research, written in a way that gives insight to mathematicians who may not be experts in the particular topic. The Bulletin also publishes reviews of selected books in mathematics and short articles in the Mathematical Perspectives section, both by invitation only.

ISSN 1088-9485 (online) ISSN 0273-0979 (print)

The 2020 MCQ for Bulletin of the American Mathematical Society is 0.84.

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From vanishing theorems to estimating theorems: the Bochner technique revisited
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by Pierre H. Bérard PDF
Bull. Amer. Math. Soc. 19 (1988), 371-406
References
  • Toshiaki Adachi and Toshikazu Sunada, Energy spectrum of certain harmonic mappings, Compositio Math. 56 (1985), no. 2, 153–170. MR 809864
  • Thierry Aubin, Nonlinear analysis on manifolds. Monge-Ampère equations, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 252, Springer-Verlag, New York, 1982. MR 681859, DOI 10.1007/978-1-4612-5734-9
  • Thierry Aubin, Équations différentielles non linéaires et problème de Yamabe concernant la courbure scalaire, J. Math. Pures Appl. (9) 55 (1976), no. 3, 269–296. MR 431287
  • Catherine Bandle, Isoperimetric inequalities and applications, Monographs and Studies in Mathematics, vol. 7, Pitman (Advanced Publishing Program), Boston, Mass.-London, 1980. MR 572958
  • Christophe Bavard and Pierre Pansu, Sur le volume minimal de $\textbf {R}^2$, Ann. Sci. École Norm. Sup. (4) 19 (1986), no. 4, 479–490 (French). MR 875084, DOI 10.24033/asens.1514
  • Pierre H. Bérard, Lectures on spectral geometry, 15$^\textrm {o}$ Colóquio Brasileiro de Matemática. [15th Brazilian Mathematics Colloquium], Instituto de Matemática Pura e Aplicada (IMPA), Rio de Janeiro, 1985. Some aspects of direct problems in spectral geometry; With an appendix by Gérard Besson; With a bibliography by Bérard and M. Berger. 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].
  • P. Bérard, G. Besson, and S. Gallot, Sur une inégalité isopérimétrique qui généralise celle de Paul Lévy-Gromov, Invent. Math. 80 (1985), no. 2, 295–308 (French). MR 788412, DOI 10.1007/BF01388608
  • P. Bérard and S. Gallot, Inégalités isopérimétriques pour l’équation de la chaleur et application à l’estimation de quelques invariants, Goulaouic-Meyer-Schwartz seminar, 1983–1984, École Polytech., Palaiseau, 1984, pp. Exp. No. 15, 35 (French). MR 811658
  • Pierre Bérard and Sylvestre Gallot, Remarques sur quelques estimées géométriques explicites, C. R. Acad. Sci. Paris Sér. I Math. 297 (1983), no. 3, 185–188 (French, with English summary). MR 725402
  • Pierre Bérard and Daniel Meyer, Inégalités isopérimétriques et applications, Ann. Sci. École Norm. Sup. (4) 15 (1982), no. 3, 513–541 (French). MR 690651, DOI 10.24033/asens.1435
  • Marcel Berger, Paul Gauduchon, and Edmond Mazet, Le spectre d’une variété riemannienne, Lecture Notes in Mathematics, Vol. 194, Springer-Verlag, Berlin-New York, 1971 (French). MR 0282313, DOI 10.1007/BFb0064643
  • S. Bochner, Vector fields and Ricci curvature, Bull. Amer. Math. Soc. 52 (1946), 776–797. MR 18022, DOI 10.1090/S0002-9904-1946-08647-4
  • K. Yano and S. Bochner, Curvature and Betti numbers, Annals of Mathematics Studies, No. 32, Princeton University Press, Princeton, N. J., 1953. MR 0062505
  • Jean-Pierre Bourguignon, Formules de Weitzenböck en dimension $4$, Riemannian geometry in dimension 4 (Paris, 1978/1979) Textes Math., vol. 3, CEDIC, Paris, 1981, pp. 308–333 (French). MR 769143
    • [Bu-Ka] P. Buser and M. Karcher, Gromov’s almost flat manifolds, Astérisque, 81, Soc. Math. France, Paris, 1981.
  • Isaac Chavel, Riemannian geometry, 2nd ed., Cambridge Studies in Advanced Mathematics, vol. 98, Cambridge University Press, Cambridge, 2006. A modern introduction. MR 2229062, DOI 10.1017/CBO9780511616822
  • Jeff Cheeger and David G. Ebin, Comparison theorems in Riemannian geometry, North-Holland Mathematical Library, Vol. 9, North-Holland Publishing Co., Amsterdam-Oxford; American Elsevier Publishing Co., Inc., New York, 1975. MR 0458335
    • [Co] G. Courtois, in preparation, 1986.
  • Christopher B. Croke, Some isoperimetric inequalities and eigenvalue estimates, Ann. Sci. École Norm. Sup. (4) 13 (1980), no. 4, 419–435. MR 608287, DOI 10.24033/asens.1390
  • Jean-Pierre Demailly, Champs magnétiques et inégalités de Morse pour la $d''$-cohomologie, C. R. Acad. Sci. Paris Sér. I Math. 301 (1985), no. 4, 119–122 (French, with English summary). MR 799607, DOI 10.5802/aif.1034
  • Jozef Dodziuk, Vanishing theorems for square-integrable harmonic forms, Proc. Indian Acad. Sci. Math. Sci. 90 (1981), no. 1, 21–27. MR 653943, DOI 10.1007/BF02867014
  • J. Eells, Elliptic operators on manifolds, Complex analysis and its applications (Lectures, Internat. Sem., Trieste, 1975) Internat. Atomic Energy Agency, Vienna, 1976, pp. 95–152. MR 0482861
  • James Eells and Luc Lemaire, Selected topics in harmonic maps, CBMS Regional Conference Series in Mathematics, vol. 50, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1983. MR 703510, DOI 10.1090/cbms/050
    • [El] A. El Soufi, Thèse de 3e cycle, Université de Grenoble, 1982.
  • Sylvestre Gallot, Inégalités isopérimétriques, courbure de Ricci et invariants géométriques. I, C. R. Acad. Sci. Paris Sér. I Math. 296 (1983), no. 7, 333–336 (French, with English summary). 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.
  • S. Gallot and D. Meyer, Opérateur de courbure et laplacien des formes différentielles d’une variété riemannienne, J. Math. Pures Appl. (9) 54 (1975), no. 3, 259–284 (French). MR 454884
    • [Gam1] N. Gamara, Thèse de 3e cycle, Université de Savoie, Chambéry, 1983.
  • Najoua Abdelmoula, Symétrisation d’inéquations élliptiques et applications géométriques, Math. Z. 199 (1988), no. 2, 181–190 (French). MR 958647, DOI 10.1007/BF01159651
  • David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, Grundlehren der Mathematischen Wissenschaften, Vol. 224, Springer-Verlag, Berlin-New York, 1977. MR 0473443, DOI 10.1007/978-3-642-96379-7
  • Michael Gromov, Curvature, diameter and Betti numbers, Comment. Math. Helv. 56 (1981), no. 2, 179–195. MR 630949, DOI 10.1007/BF02566208
    • [Gr2] M. Gromov, P. Levy’s isoperimetric inequality, Preprint, Inst. Hautes Etudes Sci., 1980.
  • Mikhael Gromov, Structures métriques pour les variétés riemanniennes, Textes Mathématiques [Mathematical Texts], vol. 1, CEDIC, Paris, 1981 (French). Edited by J. Lafontaine and P. Pansu. MR 682063
  • Mikhael Gromov and H. Blaine Lawson Jr., Positive scalar curvature and the Dirac operator on complete Riemannian manifolds, Inst. Hautes Études Sci. Publ. Math. 58 (1983), 83–196 (1984). MR 720933, DOI 10.1007/BF02953774
  • H. Hess, R. Schrader, and D. A. Uhlenbrock, Kato’s inequality and the spectral distribution of Laplacians on compact Riemannian manifolds, J. Differential Geometry 15 (1980), no. 1, 27–37 (1981). MR 602436, DOI 10.4310/jdg/1214435380
  • Nigel Hitchin, Harmonic spinors, Advances in Math. 14 (1974), 1–55. MR 358873, DOI 10.1016/0001-8708(74)90021-8
  • Dominique Hulin, Pinching and Betti numbers, Ann. Global Anal. Geom. 3 (1985), no. 1, 85–93. MR 812314, DOI 10.1007/BF00054492
  • Jerry L. Kazdan, Positive energy in general relativity, Bourbaki Seminar, Vol. 1981/1982, Astérisque, vol. 92, Soc. Math. France, Paris, 1982, pp. 315–330. MR 689537
  • Shoshichi Kobayashi and Katsumi Nomizu, Foundations of differential geometry. Vol. II, Interscience Tracts in Pure and Applied Mathematics, No. 15 Vol. II, Interscience Publishers John Wiley & Sons, Inc., New York-London-Sydney, 1969. MR 0238225
    • [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.
  • Peter Li, On the Sobolev constant and the $p$-spectrum of a compact Riemannian manifold, Ann. Sci. École Norm. Sup. (4) 13 (1980), no. 4, 451–468. MR 608289, DOI 10.24033/asens.1392
  • André Lichnerowicz, Spineurs harmoniques, C. R. Acad. Sci. Paris 257 (1963), 7–9 (French). MR 156292
  • Daniel Meyer, Sur les variétés riemanniennes à opérateur de courbure positif, C. R. Acad. Sci. Paris Sér. A-B 272 (1971), A482–A485 (French). MR 279736
  • Daniel Meyer, Un lemme de géométrie hilbertienne et des applications à la géométrie riemannienne, C. R. Acad. Sci. Paris Sér. I Math. 295 (1982), no. 7, 467–469 (French, with English summary). MR 683666
    • [Me3] D. Meyer, Work in progress, 1986.
  • Mario J. Micallef and John Douglas Moore, Minimal two-spheres and the topology of manifolds with positive curvature on totally isotropic two-planes, Ann. of Math. (2) 127 (1988), no. 1, 199–227. MR 924677, DOI 10.2307/1971420
  • John Douglas Moore, On stability of minimal spheres and a two-dimensional version of Synge’s theorem, Arch. Math. (Basel) 44 (1985), no. 3, 278–281. MR 784099, DOI 10.1007/BF01237864
    • [Pa] P. Pansu, Dégénérescence des variétés riemanniennes, Séminaire Bourbaki, Exposé 618, Novembre, 1983.
  • Michael Reed and Barry Simon, Methods of modern mathematical physics. I. Functional analysis, Academic Press, New York-London, 1972. MR 0493419
  • Steven Rosenberg, Semigroup domination and vanishing theorems, Geometry of random motion (Ithaca, N.Y., 1987) Contemp. Math., vol. 73, Amer. Math. Soc., Providence, RI, 1988, pp. 287–302. MR 954646, DOI 10.1090/conm/073/954646
  • Takashi Sakai, Comparison and finiteness theorems in Riemannian geometry, Geometry of geodesics and related topics (Tokyo, 1982) Adv. Stud. Pure Math., vol. 3, North-Holland, Amsterdam, 1984, pp. 125–181. MR 758652, DOI 10.2969/aspm/00310125
  • Richard Schoen and Shing Tung Yau, Complete three-dimensional manifolds with positive Ricci curvature and scalar curvature, Seminar on Differential Geometry, Ann. of Math. Stud., vol. 102, Princeton Univ. Press, Princeton, N.J., 1982, pp. 209–228. MR 645740
  • Bernard Shiffman and Andrew John Sommese, Vanishing theorems on complex manifolds, Progress in Mathematics, vol. 56, Birkhäuser Boston, Inc., Boston, MA, 1985. MR 782484, DOI 10.1007/978-1-4899-6680-3
  • Yum Tong Siu, Some recent results in complex manifold theory related to vanishing theorems for the semipositive case, Workshop Bonn 1984 (Bonn, 1984) Lecture Notes in Math., vol. 1111, Springer, Berlin, 1985, pp. 169–192. MR 797421, DOI 10.1007/BFb0084590
  • Yum Tong Siu, Asymptotic Morse inequalities for analytic sheaf cohomology, Astérisque 145-146 (1987), 5, 283–297. Séminaire Bourbaki, Vol. 1985/86. MR 880038
  • Giorgio Talenti, Best constant in Sobolev inequality, Ann. Mat. Pura Appl. (4) 110 (1976), 353–372. MR 463908, DOI 10.1007/BF02418013
  • Hajime Urakawa, Stability of harmonic maps and eigenvalues of the Laplacian, Trans. Amer. Math. Soc. 301 (1987), no. 2, 557–589. MR 882704, DOI 10.1090/S0002-9947-1987-0882704-8
  • H. Wu, The Bochner technique, Proceedings of the 1980 Beijing Symposium on Differential Geometry and Differential Equations, Vol. 1, 2, 3 (Beijing, 1980) Sci. Press Beijing, Beijing, 1982, pp. 929–1071. MR 714349
  • Shing Tung Yau (ed.), Seminar on Differential Geometry, Annals of Mathematics Studies, No. 102, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1982. Papers presented at seminars held during the academic year 1979–1980. MR 645728
  • Shing-Tung Yau, On the structure of complete manifolds with positive scalar curvature, Differential geometry and complex analysis, Springer, Berlin, 1985, pp. 219–222. MR 780048
  • Sylvestre Gallot, Isoperimetric inequalities based on integral norms of Ricci curvature, Astérisque 157-158 (1988), 191–216. Colloque Paul Lévy sur les Processus Stochastiques (Palaiseau, 1987). 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).
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Additional Information
  • Journal: Bull. Amer. Math. Soc. 19 (1988), 371-406
  • MSC (1985): Primary 58G30, 53C20; Secondary 58G11, 58E99
  • DOI: https://doi.org/10.1090/S0273-0979-1988-15679-0
  • MathSciNet review: 956595