Inventiones mathematicae

, Volume 103, Issue 1, pp 471–495

Manifolds with wells of negative curvature

with an Appendix by Daniel Ruberman
  • K. D. Elworthy
  • Steven Rosenberg
Article

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References

  1. [A] Arnold, L.: A formula connecting sample and moment stability of linear stochastic systems. SIAM J. Appl. Math.44, 793–802 (1984)Google Scholar
  2. [Be] Bérard, P.: From vanishing theorems to estimating theorems: the Bochner technique revisited. Bull. Am. Math. Soc.19, 371–406 (1988)Google Scholar
  3. [BB] Bérard, P., Besson, G.: Number of bound states and estimates on some geometric invariants. (Preprint) 1990Google Scholar
  4. [Br] Brown, E.M.: Proper homotopy theory in simplicial complexes. In: Topology Conference, Virginia Polytechnic Institute and State University (Lect. Notes Math. Vol.375, pp 41–46) Berlin Heidelberg New York: Springer, 1974Google Scholar
  5. [Cha] Chavel, I.: Eigenvalues in Riemannian Geometry. Orlando, San Diego: Academic Press, 1984Google Scholar
  6. [Che] Cheeger, J.: Finiteness theorems for Riemannian manifolds. Am J. Math.92, 61–74 (1970)Google Scholar
  7. [C] Croke, C.: Some isoperimetric inequalities and eigenvalue estimates. Ann. Sci. Ec. Norm. Super.13, 419–435 (1980)Google Scholar
  8. [Dod I] Dodziuk, J.: De Rham-Hodge theory forL 2-cohomology of infinite coverings. Topology16, 157–165 (1977)Google Scholar
  9. [Dod II] Dodziuk, J.: Sobolev spaces of differential forms and de Rham-Hodge isomorphisms. J. Differ. Geom.16, 63–73 (1981)Google Scholar
  10. [Don] Donnelly, H.: The differential form spectrum of hyperbolic space. Manuscr. Math.33, 365–385 (1981)Google Scholar
  11. [E] Elworthy, K.D.: Stochastic Differential Equations on Manifolds. Cambridge University Press, Cambridge: 1982Google Scholar
  12. [ER I] Elworthy, K.D., Rosenberg, S.: Generalized Bochner theorems and the spectrum of complete manifolds. Acta Appl. Math.12, 1–33 (1988)Google Scholar
  13. [ER II] Elworthy, K.D., Rosenberg, S.: Compact manifolds with a little negative curvature. Bull. Am. Maths Soc.20, 41–44 (1989)Google Scholar
  14. [ER III] Elworthy, K.D., Rosenberg, S.: Spectral bounds and the shape of manifolds near infinity. In: Simon B. et al. (eds.), IXth International Congress on Mathematical Physics Bristol: Adam Hilger, 1989, pp. 369–373Google Scholar
  15. [GM] Gallot, S., Meyer, D.: Opérateur de courbure et Laplacien des formes différentielles d'une variété Riemannienne. J. Math. Pures Appl.54, 259–284 (1975)Google Scholar
  16. [GW] Greene, R., Wu, H.: Lipschitz convergence of Riemannian manifolds. Pac. J. Math.131, 119–141 (1988)Google Scholar
  17. [G I] Gromov, M.: Groups of polynomial growth and expanding maps. Publ. Math. I.H.E.S.53, 53–78 (1981)Google Scholar
  18. [G II] Gromov, M.: Structures Métriques pour les Variétés Riemanniennes. Paris: Cedic, 1981Google Scholar
  19. [Ka] Kato, T.: Perturbation Theory for Linear Operators. New York Berlin Heidelberg: Springer, 1966Google Scholar
  20. [Kl] Klingenberg, W.: Contributions of differential geometry in the large. Ann. Math.69, 654–666 (1959)Google Scholar
  21. [L] Li, P.: On the Sobolev constant and thep-spectrum of a compact Riemannian manifold. Ann. Sci. Ec. Norm. Super.13, 451–469 (1980)Google Scholar
  22. [Ma] Malliavin, P.: Annulation de cohomologies et calcul des perturbations dansL 2. Bull. Sc. Math.100, 331–336 (1976)Google Scholar
  23. [MM] Micallef, M., Moore, J.D.: Minimal two-spheres and the topology of manifolds with positive curvature on totally isotropic two-planes.Ann. Math. 87, 199–227 (1988)Google Scholar
  24. [MW] Micallef, M., Wolfson, J.: The second variation of area of minimal surfaces in four manifolds. (Preprint)Google Scholar
  25. [Mi] Milnor, J.: A note on curvature and fundamental group. J. Differ. Geom.2, 1–7 (1968)Google Scholar
  26. [R] Rosenberg, S.: Semigroup domination and vanishing theorems. In. Durrett, R., Pinsky, M. (eds.) Contemporary Mathematics, vol.73, American Mathematical Society, Providence, 1988, pp. 287–302Google Scholar
  27. [W] Wu, J.-Y.: Complete manifolds with a little negative curvature. (Preprint)Google Scholar
  28. [Y] Yau, S.-T.: Some function-theoretic properties of complete Riemannian manifolds and their applications of geometry. Indiana Univ. Math. J.25, 659–670 (1976)Google Scholar

References

  1. [ER1] Elworthy, K.D., Rosenberg, S.: Generalized Bochner theorems and the spectrum of complete manifolds. Acta. Appl. Math.12, 1–33 (1988)Google Scholar
  2. [ER2] Elworthy, K.D., Rosenberg, S.: Manifolds with wells of negative curvature Invent. Math.103, 471–491Google Scholar
  3. [F] Fox, R.H.: A quick trip through knot theory. In: Fort, M.K. (ed.) Topology of 3-Manifolds. Englewood Cliffs, NJ: Prentice-Hall 1962Google Scholar
  4. [GKP] Graham, R.L., Knuth, D.E., Patashnik, O.: Concrete Mathematics: A Foundation for Computer Science New York: Addison Wesley, 1989Google Scholar
  5. [H] Higman, G.: Finitely presented infinite simple groups (Notes Pure Math. vol. 8, Australia National University, Canberra 1974Google Scholar
  6. [I] Lang, S.: Algebra. London: Addison Wesley, 1971Google Scholar
  7. [R] Rolfsen, D.: Knots and Links. Berkeley: Publish or Perish, 1976Google Scholar
  8. [RS] Rourke, C., Sanderson, B.: An Introduction to Piecewise-Linear Topology. Berlin Heidelberg New York: Springer 1972Google Scholar
  9. [SW] Scott, P., Wall, C.T.C.: Topological methods in group theory. In: Homological Methods in Group Theory. LMS Lecture Note Series36, Cambridge: Cambridge University Press, 1979Google Scholar
  10. [S] Stallings, J.: On torsion-free groups with infinitely many ends. Ann. Math.88, 312–334 (1968)Google Scholar
  11. [T] Thompson, R.J. unpublishedGoogle Scholar

Copyright information

© Springer-Verlag 1991

Authors and Affiliations

  • K. D. Elworthy
    • 1
  • Steven Rosenberg
    • 2
  1. 1.Mathematics InstituteUniversity of WarwickCoventryUK
  2. 2.Mathematics DepartmentBoston UniversityBostonUSA

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