Some Kind of Stabilities and Instabilities of Energies of Maps Between Kähler Manifolds

  • Tetsuya Taniguchi
  • Seiichi Udagawa
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 111)


We treat the variational problem of the energy of the map between two Riemannian manifolds. It is known that any holomorphic or anti-holomorphic map f: M → N between compact Kähler manifolds is stable for the variation f t of f with fixed Kähler metrics compatible with the holomorphic structures. Is this also stable for the variation g t of the metric g of M with fixed volume of M and fixed isometric map f? In this paper, we show that the answer is no if the dimension of M is no less than 3. This paper is a expositary note of Taniguchi and Udagawa (Characterizations of Ricci flat metrics and Lagrangian submanifolds in terms of the variational problem. To appear in Glasgow Math. J).


Riemannian Manifold Variational Problem Variational Formula Riemannian Metrics Lagrangian Submanifolds 
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  1. 1.
    Ohnita, Y., Udagawa, S.: Stable harmonic maps from Riemann surfaces to compact Hermitian symmetric spaces. Tokyo J. Math. 10, 385–390 (1987)CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    Ohnita, Y., Udagawa, S.: Stability, complex-analyticity and constancy of pluriharmonic maps from compact Kaehler manifolds. Math. Z. 205 629–644 (1990)CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Ohnita, Y., Udagawa, S.: Complex-analyticity of pluriharmonic maps and their constructions. In: Noguchi, J., Ohsawa, T. (eds.) Prospects in Complex Geometry: Proceedings at Katata/Kyoto, 1989. Lecture Notes in Mathematics, vol. 1468, pp. 371–407. Springer, Heidelberg (1991)CrossRefGoogle Scholar
  4. 4.
    Udagawa, S.: Holomorphicity of certain stable harmonic maps and minimal immersions. Proc. Lond. Math. Soc. 57, 577–598 (1988)CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    Taniguchi, T., Udagawa, S.: Characterizations of Ricci flat metrics and Lagrangian submanifolds in terms of the variational problem. To appear in Glasgow Math. J.Google Scholar

Copyright information

© Springer Japan 2014

Authors and Affiliations

  1. 1.College of Liberal Arts and SciencesKitasato UniversitySagamiharaJapan

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