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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

Citations:

Zbl 0515.58011
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Full Text: arXiv Euclid

References:

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