Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Universal constraints on the range of
eigenmaps and spherical minimal immersions

Author: Gabor Toth
Journal: Trans. Amer. Math. Soc. 351 (1999), 1423-1443
MSC (1991): Primary 53C42
MathSciNet review: 1487632
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The purpose of this paper is to give lower estimates on the range dimension of spherical minimal immersions in various settings. The estimates are obtained by showing that infinitesimal isometric deformations (with respect to a compact Lie group acting transitively on the domain) of spherical minimal immersions give rise to a contraction on the moduli space of the immersions and a suitable power of the contraction brings all boundary points into the interior of the moduli space.

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

  • 1. Calabi E., Minimal immersions of surfaces in euclidean spheres, J. Diff. Geom., 1 (1967) 111-125. MR 38:1616
  • 2. DeTurck D.-Ziller W., Minimal isometric immersions of spherical space forms in spheres, Comment. Math. Helvetici 67 (1992) 428-458. MR 93f:53050
  • 3. DoCarmo M.-Wallach N., Minimal immersions of spheres into spheres, Ann. of Math., 93 (1971) 43-62. MR 43:4048
  • 4. Eells J.-Sampson J.H., Harmonic mappings of Riemannian manifolds, Amer. J. Math. 86 (1964) 109-160. MR 29:1603
  • 5. Ejiri, N., Totally real submanifolds in a $6$-sphere, Proc. Amer. Math. Soc., 83 (1981) 759-763. MR 83a:53033
  • 6. Escher Ch., Minimal isometric immersions of inhomogeneous spherical space forms into spheres - A necessary condition for existence, Trans. Amer. Math.Soc., Vol.348, No.9, (1996) 3713-3732. MR 96m:53068
  • 7. Escher Ch., On inhomogeneous space forms, Preprint, Oregon State University, 1997.
  • 8. Gauchman H.-Toth, G., Fine structure of the space of spherical minimal immersions, Trans. Amer. Math. Soc. Vol.348, No.6 (1996) 2441-2463. MR 96i:53058
  • 9. Mashimo K., Minimal immersions of 3-dimensional sphere into spheres, Osaka J. Math., 21 (1984) 721-732. MR 86d:53040
  • 10. Moore J.D., Isometric immersions of space forms into space forms, Pacific J. Math., 40 (1976) 157-166. MR 46:4442
  • 11. Takahashi T., Minimal immersions of Riemannian manifolds, J. Math. Soc. Japan, 18 (1966) 380-385. MR 33:6551
  • 12. Toth G., Classification of quadratic harmonic maps of $S^3$ into spheres, Indiana U. Math. J., 36 (1987) 231-239. MR 89b:58058
  • 13. Toth G., Harmonic maps and minimal immersions through representation theory, Academic Press, Boston, 1990. MR 91a:58050
  • 14. Toth G., Eigenmaps and the space of minimal immersions between spheres, Indiana Univ. Math. J. 46 (1997) 637-658. MR 98k:53082
  • 15. Toth G.-Ziller W., Spherical minimal immersions of the $3$-sphere, Comment. Math. Helvetici (to appear).
  • 16. Vilenkin N.I., Special Functions and the Theory of Group Representations, AMS Translations of Mathematical Monographs, Vol.22, 1968. MR 37:5429
  • 17. Wallach N, Minimal immersions of symmetric spaces into spheres, in Symmetric Spaces, Dekker, New York (1972) 1-40. MR 53:11545
  • 18. Wolf J., Spaces of Constant Curvature, McGraw-Hill, 1967. MR 36:829

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 53C42

Retrieve articles in all journals with MSC (1991): 53C42

Additional Information

Gabor Toth
Affiliation: Department of Mathematics, Rutgers University, Camden, New Jersey 08102

Received by editor(s): April 20, 1997
Article copyright: © Copyright 1999 American Mathematical Society