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

Du, Shi-Zhong; Fan, Xu-Qian

Differ. Integral Equ. 29, No. 7-8, 791-799 (2016).

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.

Reviewer: Gábor Tóth (Camden)

MSC:

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

Keywords:

punctured Euclidean space; punctured hyperbolic space; rotationally symmetric; Riemannian metric