$$F$$ -harmonic maps as global maxima
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F-harmonic maps as global maxima Mohammed Benalili · Hafida Benallal
Received: 18 November 2011 / Revised: 26 November 2012 / Published online: 13 December 2012 © Università degli Studi di Napoli "Federico II" 2012
Abstract In this note, we show that some F-harmonic maps into spheres are global maxima of the variations of their energy functional on the conformal group of the sphere. Our result extends partially those obtained in El Soufi and Lejune [C.R.A.S. 315(Serie I):1189–1192, 1992] and El Soufi [Compositio Math 95:343–362,1995] for harmonic and p-harmonic maps. Keywords
F-harmonic maps · Stress-energy tensor · F-tension
Mathematics Subject Classification
Primary 35B53 · 58Z05 · 53C43
1 Introduction Harmonic maps have been studied first by Eells and Sampson in the sixties and since then many articles have appeared (see [6,13,17,20,21,25]) to cite a few of them. Extensions to the notions of p-harmonic, biharmonic, F-harmonic and f -harmonic maps were introduced and similar research has been carried out (see [1–3,7,16,19, 22,24]). Harmonic maps were applied to broad areas in sciences and engineering including the robot mechanics (see [5,8,9]). In this paper for a C 2 -function F : [0, +∞[ → [0, +∞[ such that F (t) > 0 on t ∈ ]0, +∞[, we look for sufficient conditions which present F-harmonic maps into spheres as global maxima of the energy functional. Our result extends similar results obtained in [18,19] for harmonic and p-harmonic maps.
Communicated by editor in chief. M. Benalili (B) · H. Benallal Department of Mathematics, Faculty of Sciences, University Abou-Bekr Belkaid, Tlemcen, Algeria e-mail: [email protected]; [email protected]
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M. Benalili, H. Benallal
Let (M, g) and S n be, respectively, a compact Riemannian manifold of dimension m ≥ 2 and the unit n-dimensional Euclidean sphere with n ≥ 2 endowed with the canonical metric can induced by the inner product of R n+1 . For a C 1 - åapplication φ : (M, g) −→ (S n, can), we define the F-energy functional by
E F (φ) =
F
|dφ|2 2
dvg ,
M
where
|dφ|2 2
denotes the energy density given by |dφ|2 1 |dφ(ei )|2 = 2 2 m
i=1
and where {ei } is an orthonormal basis on the tangent space Tx M and dvg is the Riemannian measure associated to g on M. Let φ −1 T S n and φ −1 T S n be, respectively, the pullback vector fiber bundle of n T S n and the space of sections on φ −1 T S n . Denote by ∇ M, ∇ S and ∇, respectively, n −1 n the Levi-Civita connections on: T M, T S and φ T S . Recall that ∇ is defined by n
∇ X Y = ∇φS∗ X Y where X ∈ T M and Y ∈ φ −1 T S n . Let v be a vector field on S n and denote by γtv t the flow of diffeomorphisms induced by v on S n i.e. d v γt t=0 = v γtv . dt
γ0v = id,
Denote by φt = γtv oφ the flow generated by v along the map φ. The first variation formula of E F (φ) is given by d E F (φt ) |t=0 = dt
M
F
|dφt |2 2
∇∂t dφt , dφt |t=0 dvg
v, τ F (φ) dvg
=− M
2 dφ is the F-tension. where τ F (φ) = traceg ∇ F |dφ| 2 Definition 1 φ is said F-harmonic if
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