Abstract
We show the nonexistence of rotationally symmetric harmonic diffeomorphism between the unit disk without the origin and a punctured disc with hyperbolic metric on the target.
Highlights
The existence of harmonic diffeomorphisms between complete Riemannian manifolds has been extensively studied, please see, for example, [1–34]
Let us prove the first part of this theorem, that is, show the nonexistence of rotationally symmetric harmonic diffeomorphism from D∗ onto P(a) with its Euclidean metric
Let us prove the second part of this theorem, that is, show the existence of rotationally symmetric harmonic diffeomorphisms from P(a) onto D∗ with its Euclidean metric
Summary
The existence of harmonic diffeomorphisms between complete Riemannian manifolds has been extensively studied, please see, for example, [1–34]. Heinz [17] proved that there is no harmonic diffeomorphism from the unit disc onto C with its flat metric. Schoen [25] mentioned a question about the existence, or nonexistence, of a harmonic diffeomorphism from the complex plane onto the hyperbolic 2-space. In [7, 24, 28, 29], the authors therein studied the rotational symmetry case One of their results is the nonexistence of rotationally symmetric harmonic diffeomorphism from C onto the hyperbolic plane. We will study the existence, or nonexistence, of rotationally symmetric harmonic diffeomorphisms from the unit disk without the origin onto a punctured disc. We will consider the Euclidean case and will prove the following theorem. We will give another proof for the nonexistence of rotationally symmetric harmonic diffeomorphism from C onto the hyperbolic disc
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