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On the asymptotic behaviour of rotationally symmetric harmonic maps. (English) Zbl 0767.34029
$n$-dimensional Riemann spaces are considered that are the direct product of the $n-1$-dimensional Euclidean unit sphere and the half line and are equipped with a rotationally symmetric metric. Such spaces may be isometric to the Euclidean or to the hyperbolic space of dimension $n$. Rotationally symmetric maps of one space to the other are considered and an energy integral is attached to each map. The Euler-Lagrange equation of this integral is a second order nonlinear scalar differential equation, and a map is called harmonic if its defining scalar function is a positive solution tending to zero at infinity. Conditions are given under which any bounded harmonic map is constant and other properties are studied, too.

34C40ODE on manifolds
34D05Asymptotic stability of ODE
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