Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Remark about the spectrum of the $p$-form Laplacian under a collapse with curvature bounded below


Author: John Lott
Journal: Proc. Amer. Math. Soc. 132 (2004), 911-918
MSC (2000): Primary 58G25; Secondary 53C23
DOI: https://doi.org/10.1090/S0002-9939-03-07121-1
Published electronically: September 18, 2003
MathSciNet review: 2019973
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We give a lower bound on the number of small positive eigenvalues of the $p$-form Laplacian in a certain type of collapse with curvature bounded below.


References [Enhancements On Off] (What's this?)

  • 1. P. Bérard, From vanishing theorems to estimating theorems: the Bochner technique revisited, Bull. Amer. Math. Soc. 19 (1988), 371-406. MR 89i:58152
  • 2. B. Colbois and G. Courtois, A note on the first nonzero eigenvalue of the Laplacian acting on $p$-forms, Manuscripta Math. 68 (1990), 143-160.MR 91g:58290
  • 3. J. Dodziuk, Eigenvalues of the Laplacian on forms, Proc. Amer. Math. Soc. 85 (1982), 437-443.MR 84k:58223
  • 4. K. Fukaya and T. Yamaguchi, The fundamental groups of almost nonnegatively curved manifolds, Ann. of Math. 136 (1992), 253-333.MR 93h:53041
  • 5. K. Grove, P. Petersen and J. Wu, Geometric finiteness theorems via controlled topology, Invent. Math. 99 (1990), 205-213.MR 90k:53075
  • 6. J. Koszul, Sur certains groupes de transformations de Lie, Géométrie differentielle, Colloques Internationaux du CNRS, Strasbourg (1953), pp. 137-141. MR 15:600g
  • 7. J. Lott, Collapsing and the differential form Laplacian: the case of a smooth limit space, Duke Math. J. 114 (2002), 267-306.MR 2003e:58047
  • 8. J. Lott, Collapsing and the differential form Laplacian: the case of a singular limit space, preprint, http://www.math.lsa.umich.edu/~lott.
  • 9. G. Perelman, Construction of manifolds of positive Ricci curvature with big volume and large Betti numbers, Comparison Geometry, Math. Sci. Res. Inst. Publ. 30, Cambridge Univ. Press, Cambridge (1997), pp. 157-163. MR 98h:53062
  • 10. J. Takahashi, Small eigenvalues on $p$-forms for collapsings of the even-dimensional spheres, Manuscripta Math. 109 (2002), 63-71. MR 2003j:58049
  • 11. A. Verona, A de Rham type theorem for orbit spaces, Proc. Amer. Math. Soc. 104 (1988), 300-302.MR 89j:57033
  • 12. T. Yamaguchi, Collapsing and pinching under a lower curvature bound, Ann. of Math. 133 (1991), 317-357.MR 92b:53067

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 58G25, 53C23

Retrieve articles in all journals with MSC (2000): 58G25, 53C23


Additional Information

John Lott
Affiliation: Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109-1109
Email: lott@umich.edu

DOI: https://doi.org/10.1090/S0002-9939-03-07121-1
Received by editor(s): September 19, 2002
Received by editor(s) in revised form: November 3, 2002
Published electronically: September 18, 2003
Additional Notes: Research supported by NSF grant DMS-0072154
Communicated by: Jozef Dodziuk
Article copyright: © Copyright 2003 American Mathematical Society

American Mathematical Society