Existence Of Rotationally Symmetric Harmonic Diffeomorphisms Between Planes

We study that the existence of rotationally symmetric harmonic diffeomorphisms between the surfaces with hyperbolic metric or Euclidean metric. This thesis is divided into four chapters:In the first chapter, we mainly introduce some notations and the elementary knowledge of harmonic maps.In the second chapter, we are concerned about the existence, or nonexistence, of a rotationally symmetric harmonic diffeomorphism from the complex plane without a disc onto itself, or to annuli with Poincare metric.In the third chapter, we briefly discuss that the existence, or nonexistence, of a rotationally symmetric harmonic diffeomorphism between annuli with Euclidean metric.In the last chapter, we can get the necessary and sufficient condition for existence of rotationally symmetric harmonic diffeomorphism between the unit annuli using the result of the general annuli under the circumstance of Euclidean metric.