The second variational formula of the $$k$$-energy and $$k$$-harmonic curves.(English)Zbl 1273.58008

Eells and Lemaire proposed the notion of $$k$$-harmonic maps in [J. Eells and L. Lemaire, Selected topics in harmonic maps. Reg. Conf. Ser. Math. 50 (1983; Zbl 0515.58011)]. These are mappings between manifolds which are stationary for a $$k$$-th order energy generalizing the Dirichlet energy for $$k=1$$. The case $$k=2$$ has been studied a lot recently, and $$2$$-harmonic maps are also known as intrinsically biharmonic maps.
Here, the author calculates the second variation of $$k$$-energy and defines notions for weak stability. Another result is that a product map is $$k$$-harmonic if both of the factors are. Finally, $$k$$-harmonic curves in Riemannian manifolds with constant sectional curvature are studied in some detail.

MSC:

 58E20 Harmonic maps, etc. 53C43 Differential geometric aspects of harmonic maps

Zbl 0515.58011
Full Text:

References:

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