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)

 
 

 

Stability of harmonic maps and minimal immersions


Authors: Y. L. Pan and Y. B. Shen
Journal: Proc. Amer. Math. Soc. 93 (1985), 111-117
MSC: Primary 58E20; Secondary 53C42
DOI: https://doi.org/10.1090/S0002-9939-1985-0766539-6
MathSciNet review: 766539
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: It was proved by J. Simons [10] that there does not exist any stable minimal submanifold in the Euclidean sphere $ {S^n}$, and P. F. Leung proved that any stable harmonic map from any Riemannian manifold to $ {S^n}$, where $ n \geqslant 3$, is a constant. In this paper, we generalize their results and indicate that there are many manifolds having such properties as $ {S^n}$.


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

  • [1] S. S. Chern, Minimal submanifolds in Riemannian manifolds, Mimeographed Lecture Notes, Univ. of Kansas, 1968. MR 0248648 (40:1899)
  • [2] S. S. Chern and S. I. Goldberg, On the volume decreasing property of a class of real harmonic mappings, Amer. J. Math. 97 (1975), 133-147. MR 0367860 (51:4102)
  • [3] J. Eells and J. H. Sampson, Harmonic mappings of Riemannian manifolds, Amer. J. Math. 86 (1964), 109-160. MR 0164306 (29:1603)
  • [4] P. F. Leung, On the stability of harmonic maps, Lecture Notes in Math., vol. 949, Springer, New York and Berlin, 1982, pp. 122-129. MR 673586 (83m:58033)
  • [5] M. Obata, The Gauss map of immersions of Riemannian manifolds in a space of constant curvature, J. Differential Geom. 2 (1968), 217-223. MR 0234388 (38:2705)
  • [6] Y. L. Pan, Some nonexistence theorems on stable harmonic mappings, Chinese Ann. Math. 3 (1982), 515-518. MR 681763 (84e:58024)
  • [7] Y. L. Pan and Y. B. Shen, Stable harmonic maps from submanifolds in space form, preprint.
  • [8] C. K. Peng, Some relations between the minimal submanifold and the harmonic mapping, Chinese Ann. Math. Ser. A 5 (1984), 85-89. MR 743786 (85k:58024)
  • [9] R. T. Smith, The second variation formula for harmonic mappings, Proc. Amer. Math. Soc. 47 (1975), 229-236. MR 0375386 (51:11580)
  • [10] J. Simons, Minimal varieties in Riemannian manifolds, Ann. of Math. (2) 88 (1968), 62-105. MR 0233295 (38:1617)
  • [11] Y. L. Xin, Topology of certain submanifolds in the Euclidean sphere, Proc. Amer. Math. Soc. 82 (1981), 643-648. MR 614895 (82e:58036)
  • [12] -, Some results on stable harmonic maps, Duke Math. J. 47 (1980), 609-613. MR 587168 (81j:58041)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 58E20, 53C42

Retrieve articles in all journals with MSC: 58E20, 53C42


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1985-0766539-6
Article copyright: © Copyright 1985 American Mathematical Society

American Mathematical Society