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)



On the holomorphicity of harmonic maps from compact Kähler manifolds to hyperbolic Riemann surfaces

Author: Kaoru Ono
Journal: Proc. Amer. Math. Soc. 102 (1988), 1071-1076
MSC: Primary 58E20; Secondary 32H20, 53C55, 58C10
MathSciNet review: 934892
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We give a sufficient condition in order that a harmonic map from a compact Kähler manifold with negative first Chern class to a compact hyperbolic Riemann surface be $ \pm$ holomorphic. The above condition generalizes that of Eells and Wood concerning harmonic maps between Riemann surfaces. As a corollary we get a generalization of Kneser's theorem.

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

  • [1] S. K. Donaldson, Anti self-dual Yang-Mills connections over complex algebraic surfaces and stable vector bundles, Proc. London Math. Soc. (3) 50 (1985), 1-26. MR 765366 (86h:58038)
  • [2] J. Eells and L. Lemaire, Selected topics in harmonic maps, CBMS Regional Conf. Ser. in Math., Amer. Math. Soc., Providence, R. I., 1983. MR 703510 (85g:58030)
  • [3] J. Eells and J. H. Sampson, Harmonic mappings of Riemannian manifolds, Amer. J. Math. 86 (1964), 109-160. MR 0164306 (29:1603)
  • [4] J. Eells and J. C. Wood, Restrictions on Harmonic maps of surfaces, Topology 15 (1976), 263-266. MR 0420708 (54:8720)
  • [5] S. Kobayashi, Stable vector bundles and curvature, Lecture Notes from University of Tokyo, 1984.
  • [6] A. Lichnerowicz, Applications harmoniques et varieétés Kählériennes, Sympos. Math. III (Bologna), 1970, pp. 341-402. MR 0262993 (41:7598)
  • [7] Y. T. Siu, The complex analyticity of harmonic maps and the strong rigidity of compact Kähler manifolds, Ann. of Math. (2) 112 (1980), 73-111. MR 584075 (81j:53061)
  • [8] -, Some recent results in complex manifold theory related to vanishing theorems for semi-positive case, Lecture Notes in Math., vol. 1111, Springer-Verlag, 1985, pp. 169-192.
  • [9] -, Strong rigidity for Kähler manifolds and the construction of bounded holomorphic functions, Discrete Groups in Geometry and Analysis, Papers in Honor of G. D. Mostow on His Sixtieth Birthday (R. Howe, ed.), Birkhauser, Boston, Mass., 1987, pp. 124-151. MR 900825 (89i:32044)
  • [10] S. T. Yau, Calabi's conjecture and some new results in algebraic geometry, Proc. Nat. Acad. Sci. U.S.A. 74 (1977), 1798-1799. MR 0451180 (56:9467)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 58E20, 32H20, 53C55, 58C10

Retrieve articles in all journals with MSC: 58E20, 32H20, 53C55, 58C10

Additional Information

Keywords: Harmonic map, holomorphic map, Kähler manifold, Einstein-Hermitian vector bundle, semistable vector bundle
Article copyright: © Copyright 1988 American Mathematical Society

American Mathematical Society