The Journal of Geometric Analysis

, Volume 3, Issue 1, pp 27–36

A counterexample to uniqueness and regularity for harmonic maps between hyperbolic spaces

  • Michael Economakis

DOI: 10.1007/BF02921328

Cite this article as:
Economakis, M. J Geom Anal (1993) 3: 27. doi:10.1007/BF02921328


We construct a harmonic diffeomorphism from the Poincaré ballHn=1 to itself, whose boundary value is the identity on the sphereSn, and which is singular at a boundary point, as follows: The harmonic map equations between the corresponding upper-half-space models reduce to a nonlinear o.d.e. in the transverse direction, for which we prove the existence of a solution on the whole R+ that grows exponentially near infinity and has an expansion near zero. A conjugation by the inversion brings the singularity at the origin, and a conjugation by the Cayley transform and an isometry of the ball moves the singularity at any point on the sphere.

Math Subject Classification


Key Words and Phrases

AsymptoticsCayley transformconformalcontractionexponential growthharmonic mapHölder regularityhyperbolic space

Copyright information

© Mathematica Josephina, Inc. 1993

Authors and Affiliations

  • Michael Economakis
    • 1
  1. 1.Department of MathematicsUniversity of WashingtonSeattleUSA