Gap properties of harmonic maps and submanifolds

Qun Chen; Zhen Rong Zhou

Archivum Mathematicum (2005)

  • Volume: 041, Issue: 1, page 59-69
  • ISSN: 0044-8753

Abstract

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In this article, we obtain a gap property of energy densities of harmonic maps from a closed Riemannian manifold to a Grassmannian and then, use it to Gaussian maps of some submanifolds to get a gap property of the second fundamental forms.

How to cite

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Chen, Qun, and Zhou, Zhen Rong. "Gap properties of harmonic maps and submanifolds." Archivum Mathematicum 041.1 (2005): 59-69. <http://eudml.org/doc/249476>.

@article{Chen2005,
abstract = {In this article, we obtain a gap property of energy densities of harmonic maps from a closed Riemannian manifold to a Grassmannian and then, use it to Gaussian maps of some submanifolds to get a gap property of the second fundamental forms.},
author = {Chen, Qun, Zhou, Zhen Rong},
journal = {Archivum Mathematicum},
keywords = {Grassmannian; Gaussian map; mean curvature; the second fundamental form; Gaussian map; mean curvature; the second fundamental form},
language = {eng},
number = {1},
pages = {59-69},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {Gap properties of harmonic maps and submanifolds},
url = {http://eudml.org/doc/249476},
volume = {041},
year = {2005},
}

TY - JOUR
AU - Chen, Qun
AU - Zhou, Zhen Rong
TI - Gap properties of harmonic maps and submanifolds
JO - Archivum Mathematicum
PY - 2005
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 041
IS - 1
SP - 59
EP - 69
AB - In this article, we obtain a gap property of energy densities of harmonic maps from a closed Riemannian manifold to a Grassmannian and then, use it to Gaussian maps of some submanifolds to get a gap property of the second fundamental forms.
LA - eng
KW - Grassmannian; Gaussian map; mean curvature; the second fundamental form; Gaussian map; mean curvature; the second fundamental form
UR - http://eudml.org/doc/249476
ER -

References

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  1. Chen W. H., Geometry of Grassmannian manifolds as submanifolds, (in Chinese), Acta Math. Sinica 31(1) (1998), 46–53. (1998) MR0951473
  2. Chen X. P., Harmonic maps and Gaussian maps, (in Chinese), Chin. Ann. Math. 4A(4) (1983), 449–456. (1983) 
  3. Chern S. S., Goldberg S. I., On the volume decreasing property of a class of real harmonic mappings, Amer. J. Math. 97(1) (1975), 133–147. (1975) Zbl0303.53049MR0367860
  4. Chern S. S., doCarmo M., Kobayashi S., Minimal submanifolds of a sphere with second fundamental form of constant length, Funct. Anal. Rel. Fields (1970), 59–75. (1970) MR0273546
  5. Eells J., Lemaire L., Selected topics on harmonic maps, Expository Lectures from the CBMS Regional Conf. held at Tulane Univ., Dec. 15–19, 1980. (1980) 
  6. Ruh E. A. Vilms J., The tension field of the Gauss map, Trans. Amer. Math. Soc. 149 (1970), 569–573. (1970) MR0259768
  7. Sealey H. C. J., Harmonic maps of small energy, Bull. London Math. Soc. 13 (1981), 405–408. (1981) Zbl0444.58009MR0631097
  8. Takahashi T., Minimal immersions of Riemannian manifolds, J. Math. Soc. Japan. 18 (1966), 380–385. (1966) Zbl0145.18601MR0198393
  9. Wu G. R., Chen W. H., An inequality on matrix and its geometrical application, (in Chinese), Acta Math. Sinica 31(3) (1988), 348–355. (1988) MR0963085
  10. Yano K., Kon M., Structures on Manifolds, Series in Pure Math. 3 (1984), World Scientific. (1984) Zbl0557.53001MR0794310

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