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Some conditions ensuring the vanishing of harmonic differential forms with applications to harmonic maps and Yang-Mills theory

  • H. C. J. Sealey (a1)
Extract

In (5) it is shown that if m ≽ 3 then there is no non-constant harmonic map φ: ℝm → Sn with finite energy. The method of proof makes use of the fact that the energy functional is not invariant under conformal transformations. This fact has also allowed Xin(9), to show that any non-constant-harmonic map φ:Sm → (N, h), m ≽ 3, is not stable in the sense of having non-negative second variation.

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COPYRIGHT: © Cambridge Philosophical Society 1982
References
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(1)Aronszajn, N.A unique continuation theorem for solutions of elliptic partial differential equations or inequalities. J. Math. Pure Appl. 36 (1957), 235–249.
(2)Bourguignon, J-P. and Lawson, H. B.Stability and isolation phenomena for Yang-Mills fields. Comm. Math. Phys. (to appear).
(3)Cheeger, J. and Gromoll, D.The splitting theorem for manifolds of non-negative Ricci curvature. J. Diff. Geom. 3 (1969), 119–128.
(4)Eells, J. and Lemaire, L.A report on harmonic maps. Bull. London Math. Soc. 10 (1978), 1–68.
(5)Garber, W. D., Ruijsenaas, S. H. H., Seiler, E. and Burns, D.On finite action solutions of the non-linear σ-model. Annals of Physics, 119 (1979), 305–325.
(6)Kobayashi, S. and Nomizu, K.Foundations of differential geometry, I, II (Interscience, 1963, 1969).
(7)Sealey, H. C. J. Some properties of harmonic mappings. Thesis, University of Warwick, 1980.
(8)Wood, J. C. Non-existence of solutions to certain Dirichlet problems for harmonic maps. (Manuscript.)
(9)Xin, Y. L.Some results on stable harmonic maps. Duke Math. J. 47 (1980), 609–613.
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Mathematical Proceedings of the Cambridge Philosophical Society
  • ISSN: 0305-0041
  • EISSN: 1469-8064
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