A nonexistence result on harmonic diffeomorphisms between punctured spaces. (English) Zbl 1389.58007

The authors’ main result is the non-existence of rotationally symmetric harmonic diffeomorphisms from \(\mathbb R_\ast^n\) to \(\mathbb H_\ast^n\) for \(n\geq 2\). More precisely, using spherical coordinates, they endow \(\mathbb R^n\) with the metrics \(r^2d\theta^2+dr^2\) and \(g(r)^2d\theta^2+dr^2\), and show that under suitable conditions on the function \(g\), there is no harmonic diffeomorphism from the first metric to the second in the respective punctured spaces. Rotational symmetry reduces this problem to the study of solutions of a second-order differential equation of Abel type.


58E20 Harmonic maps, etc.
34B15 Nonlinear boundary value problems for ordinary differential equations
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