Hostname: page-component-78c5997874-s2hrs Total loading time: 0 Render date: 2024-11-16T17:02:57.033Z Has data issue: false hasContentIssue false
  • English
  • Français

The second variation formula for exponentially harmonic maps

Published online by Cambridge University Press:  17 April 2009

Leung-Fu Cheung  and
Pui-Fai Leung
Leung-Fu Cheung
Affiliation:
Department of Applied MathematicsHong Kong Polytechnic UniversityHung Hom, Kowloon, Hong Kong e-mail: [email protected]
Pui-Fai Leung
Affiliation:
Department of MathematicsNational University of SingaporeKent Ridge, Singapore 119260 e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We derive the formula in the title and deduce some consequences. For example we show that the identity map from any compact manifold to itself is always stable as an exponentially harmonic map. This is in sharp contrast to the harmonic or p-harmonic cases where many such identity maps are unstable. We also prove that an isometric and totally geodesic immersion of Sm into Sn is an unstable exponentially harmonic map if m ≠ n and is a stable exponentially harmonic map if m = n.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 59 , Issue 3 , June 1999 , pp. 509 - 514
Copyright
Copyright © Australian Mathematical Society 1999

References

[1]Hong, M.C., ‘On the conformal equivalence of harmonic maps and exponentially harmonic maps’, Bull. London Math. Soc. 24 (1992), 488–492.Google Scholar
[2]Lawson, H.B., Minimal varieties in real and complex geometry, Séminaire de Mathématiques Supérleures 57 (Les Presses de l'universite de Montreal, Montreal, Que., 1974).Google Scholar
[3]Leung, P.F., ‘A note on stable harmonic maps’, J. London Math. Soc. (2) 29 (1984), 380–384.CrossRefGoogle Scholar
[4]Smith, R.T., ‘The second variation formula for harmonic mappings’, Proc. Amer. Math. Soc. 47 (1975), 229–236.CrossRefGoogle Scholar
[5]Takeuchi, H., ‘Stability and Liouville theorems of p-harmonic maps’, Japan J. Math. 17 (1991), 317–332.CrossRefGoogle Scholar
[6]Yano, K., Integral formulas in Riemannian geometry, Pure and Applied Mathematics 1 (Marcel Dekker, New York, 1970).Google Scholar