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Nonexistence of stable harmonic maps to and from certain homogeneous spaces and submanifolds of Euclidean space


Authors: Ralph Howard and S. Walter Wei
Journal: Trans. Amer. Math. Soc. 294 (1986), 319-331
MSC: Primary 58E20
DOI: https://doi.org/10.1090/S0002-9947-1986-0819950-4
MathSciNet review: 819950
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Abstract: Call a compact Riemannian manifold $ M$ a strongly unstable manifold if it is not the range or domain of a nonconstant stable harmonic map and also the homotopy class of any map to or from $ M$ contains elements of arbitrarily small energy. If $ M$ is isometrically immersed in Euclidean space, then a condition on the second fundamental form of $ M$ is given which implies $ M$ is strongly unstable. As compact isotropy irreducible homogeneous spaces have "standard" immersions into Euclidean space this allows a complete list of the strongly unstable compact irreducible symmetric spaces to be made.


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  • [Ch] B.-Y. Chen, Geometry of submanifolds, Dekker, New York, 1973. MR 0353212 (50:5697)
  • [EL] J. Eells and L. Lemaire, Selected topics in harmonic maps, CBMS Regional Conf. Ser. in Math., no. 50, Amer. Math. Soc., Providence, R.I., 1983. MR 703510 (85g:58030)
  • [ES] J. Eells and J. H. Sampson, Harmonic mappings of Riemannian manifolds, Amer. J. Math. 86 (1964), 109-160. MR 0164306 (29:1603)
  • [L] H. B. Lawson, Lectures on minimal submanifolds, Vol. 1, Publish or Perish, Berkeley, Calif., 1980.
  • [Lg] P. F. Leung, On the stability of harmonic maps, Lecture Notes in Math., vol. 949, Springer, Berlin, Heidelberg and New York, 1982, pp. 122-129. MR 673586 (83m:58033)
  • [LS] H. B. Lawson and J. Simons, On stable currents and their application to global problems in real and complex geometry, Ann. of Math. (2) 98 (1973), 427-450. MR 0324529 (48:2881)
  • [M] Min-Oo, Maps of minimum energy from compact simply-connected Lie groups, Annals of Global Analysis and Geometry, Vol. 2, No. 1. MR 755212 (85m:58056)
  • [N] T. Nagano, Stability of harmonic maps between symmetric spaces, Lecture Notes in Math., vol. 949, Springer, Berlin, Heidelberg and New York, 1982, pp. 130-137. MR 673587 (84c:58023)
  • [S] R. T. Smith, The second variation formula for harmonic mappings, Proc. Amer. Math. Soc. 47 (1975), 229-236. MR 0375386 (51:11580)
  • [Sp] M. Spivak, A comprehensive introduction to differential geometry, Publish or Perish, Berkeley, Calif., 1979.
  • [SU] J. Sacks and K. Uhlenbeck, The existence of minimal immersions of $ 2$-spheres, Ann. of Math. (2) 113 (1981), 1-24. MR 604040 (82f:58035)
  • [W] B. White, Infima of energy functionals in homotopy classes of mappings, Preprint. MR 845702 (87m:58039)
  • [X] Y. L. Xin, Some results on stable harmonic maps, Duke Math. J. 47 (1980), 609-613. MR 587168 (81j:58041)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1986-0819950-4
Keywords: Instability of harmonic maps, symmetric spaces
Article copyright: © Copyright 1986 American Mathematical Society

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