A nonexistence result on harmonic diffeomorphisms between punctured spaces

Abstract

In this paper, we will prove a result of nonexistence on harmonic diffeomorphisms between punctured spaces. In particular, we will given an elementary proof to the nonexistence of rotationally symmetric harmonic diffeomorphisms from the punctured Euclidean space onto the punctured hyperbolic space.

An existence and uniqueness result for orientation-reversing harmonic diffeomorphism from $\mathbb{H}_*^n$ to $\mathbb{R}_*^n$

“…Proof. The idea of the proof is similar to that used in Lemma 2.1 [6]. Assume on the contrary, there exists r > 0 such that x(r) < R 1 (r).…”

“…From the results in [16,14,4,2], we know that there is no rotationally symmetric harmonic diffeomorphism between the model spaces R n and H n . Even from R n * to H n * , this is also true [6]. But conversely, from D * to C * , it does not hold [3], although Heinz [8] obtained the nonexistence of harmonic diffeomorphism from the unit disc onto the complex plane.…”