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Geometry of \(F\)-harmonic maps. (English) Zbl 0941.58010

\(F\)-harmonic maps are a generalization of harmonic maps, \(p\)-harmonic maps or exponential harmonic maps; they are defined as follows. Let \(F:[0,\infty)\to [0,\infty)\) be a \(C^2\)-function such that \(F'> 0\) on \([0,\infty)\). For a smooth map \(\phi: (M,g)\to (N,h)\) between Riemannian manifolds \((M,g)\) and \((N,h)\), the \(F\)-energy \(E_F(\phi)\) of \(\phi\) is defined as \(E_F(\phi)= \int_M F(|d\phi|^2/2) v_g\), where \(|d\phi|\) denotes the Hilbert-Schmidt norm of the differential \(d\phi\in \Gamma(T^*M\otimes\phi^{-1}TN)\) with respect to \(g\) and \(h\), and \(v_g\) is the volume element of \((M,g)\). Thus, \(\phi\) is called an \(F\)-harmonic map if it is a critical point of the \(F\)-energy functional.
In this paper, the author studies \(F\)-harmonic maps by discussing the following issues: (1) the first variation formula for \(F\)-harmonic maps, and the relation between \(F\)-harmonic maps and harmonic maps through conformal deformations; (2) given a smooth map \(\psi: M\to N\), the existence of \(F\)-harmonic maps in the homotopy class of \(\psi\); (3) the stress energy tensor for the \(F\)-energy functional, and weakly conformal \(F\)-harmonic maps; (4) horizontally conformal \(F\)-harmonic maps; (5) the second variation formula and stability for \(F\)-harmonic maps.
One of the main results in this paper is as follows. Let \(\phi:M\to S^n\) be an \(F\)-harmonic map from a compact Riemannian manifold \(M\) to the unit \(n\)-sphere \(S^n\). If \[ \int_M|d\phi|^2\left\{|d\phi|^2 F''\left({|d\phi|^2\over 2}\right)+ (2- n)F'\left({|d\phi|^2\over 2}\right)\right\} v_g< 0, \] then \(\phi\) is unstable. Hence, if either (i) \(F''\leq 0\) and \(n\geq 3\); or (ii) \(F''< 0\) and \(n=2\), then any stable \(F\)-harmonic map from a compact Riemannian manifold \(M\) to \(S^n\) is constant.

MSC:

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