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A finiteness theorem for harmonic maps into Hilbert Grassmannians

Author: Rodrigo P. Gomez
Journal: Trans. Amer. Math. Soc. 353 (2001), 1741-1753
MSC (2000): Primary 58E20; Secondary 53C07
Published electronically: January 10, 2001
MathSciNet review: 1637074
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Abstract | References | Similar Articles | Additional Information


In this article we demonstrate that every harmonic map from a closed Riemannian manifold into a Hilbert Grassmannian has image contained within a finite-dimensional Grassmannian.

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  • 1. M. F. Atiyah and D. W. Anderson, $K$-Theory, Harvard University, 1964. MR 90m:18011
  • 2. Theodor Bröcker and Tammo tom Dieck, Representations of Compact Lie Groups, Springer-Verlag, 1985.
  • 3. D. Burns, Harmonic Maps from $\mathbb{C}P^1$ to $\mathbb{C}P^n$, Proc. Conf. on Harmonic Maps, Springer-Verlag, p. 217-263, 1980.
  • 4. J. Eells and J. C. Wood, Harmonic Maps from Surfaces to Complex Projective Spaces, Advances in Math. (1983), no. 49. MR 85f:58029
  • 5. James Eells and Andrea Ratto, Harmonic Maps and Minimal Immersions with Symmetries, Methods of Ordinary Differential Equations Applied to Elliptic Variational Problems, Anals of Mathematics Studies, no. 130, Princeton University Press, Princeton, New Jersey, 1993. MR 94k:58033
  • 6. D. S. Freed, Flag Manifolds and Infinite Dimensional Kähler Geometry, Infinite Dimensional Groups with Applications, Publ. Math. Sci. Res. Inst. 4, p. 83-124, 1985. MR 87k:58020
  • 7. Rodrigo P. Gomez, A finiteness theorem of harmonic maps from compact Lie groups to $O_{HS}(H)$, Manuscripta Mathematica, no. 93, p. 325-335, 1997. MR 98d:58043
  • 8. S. Kobayashi and K. Nomizu, Foundations of Differential Geometry, vol. I, II, Wiley, 1963, 1969. MR 97c:53001a; MR 97c:53001b
  • 9. Andrew Pressley and Graeme Segal, Loop Groups, Oxford University Press, New York, 1988. MR 88i:22049
  • 10. Graeme Segal, Loop Groups and Harmonic Maps, London Math. Soc. Lecture Note Ser. 139, Cambridge Univ. Press, Cambridge, p. 153-164, 1989. MR 91m:58043
  • 11. Karen Uhlenbeck, Harmonic Maps into Lie Groups (Classical solutions of the Chiral Model), J. Differential Geom. 30, no. 1, p. 1-50, 1989. MR 90g:58028
  • 12. Frank W. Warner, Foundations of Differentiable Manifolds and Lie Groups, Springer-Verlag, 1987. MR 84k:58001
  • 13. R. O. Wells, Differential Analysis on Complex Manifolds, Springer-Verlag, 1980. MR 83f:58001

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Additional Information

Rodrigo P. Gomez
Affiliation: Comprehensive Studies Program, University of Michigan, Ann Arbor, Michigan 48109
Address at time of publication: 8838 Tides Ebb Ct., Columbia, Maryland 21045

Keywords: Harmonic maps, closed Riemannian manifolds
Received by editor(s): May 22, 1997
Received by editor(s) in revised form: July 15, 1998
Published electronically: January 10, 2001
Additional Notes: I would like to thank D. Burns for suggesting this problem to me.
Dedicated: This article is dedicated to my beloved daughter Katherine
Article copyright: © Copyright 2001 American Mathematical Society

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