On the \(f\)-harmonic and \(f\)-biharmonic maps. (English) Zbl 1209.58014

Ouakkas, S.; Nasri, R.; Djaa, M.

JP J. Geom. Topol. 10, No. 1, 11-27 (2010).

The paper studies \(f\)-harmonic maps which are maps \(\phi: (M^m,g)\rightarrow (N^n,h)\) between Riemannian manifolds that are critical points of the \(f\)-energy \(E_f(\phi)=\frac{1}{2}\int_Mf\,|d\phi|^2\,dv_g\) [A. Lichnerowicz, Sympos. Math., Roma 3, Probl. Evolut. Sist. solare, Nov. 1968 e Geometria, Febb. 1969, 341–402 (1970; Zbl 0193.50101); J. Eells and L. Lemaire, Bull. Lond. Math. Soc. 10, 1–68 (1978; Zbl 0401.58003), Section 10.20]. These \(f\)-harmonic maps should not be confused with \(F\)-harmonic maps which are critical points of the \(F\)-energy \(E_F(\varphi)=\int_M F(\frac{|d\varphi|^2}{2})\,dv_g\) [M. Ara, Kodai Math. J. 22, No. 2, 243–263 (1999; Zbl 0941.58010)]. \(F\)-harmonic maps include harmonic, \(p\)-harmonic, and exponential harmonic maps as special cases whilst an \(f\)-harmonic map with \(f=1\) is the usual harmonic map.
The authors of the paper derive the \(f\)-tension field, \(f\)-stress energy impulsion, and the second variation of the \(f\)-energy functional. An application of these is a characterization of the \(f\)-harmonic conformal diffeomorphisms. In the second part of the paper, the authors introduce the notion of \(f\)-biharmonic maps as the critical points of the \(f\)-bienergy defined by the integration of the square of the norm of the \(f\)-tension field. Equations of \(f\)-biharmonic identity maps and \(f\)-biharmonic conformal maps are derived.
It would be more interesting if the paper gives some examples of such maps.

Reviewer: Ye-Lin Ou (Commerce)

MSC:

58E20	Harmonic maps, etc.
53C20	Global Riemannian geometry, including pinching

Keywords:

\(f\)-harmonic maps; conformal maps; \(f\)-biharmonic maps

Citations:

Zbl 0193.50101; Zbl 0401.58003; Zbl 0941.58010