A counterexample to uniqueness and regularity for harmonic maps between hyperbolic spaces
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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 Classification58E20 35C20 35L70 31B25 32A40 35k20
Key Words and PhrasesAsymptotics Cayley transform conformal contraction exponential growth harmonic map Hölder regularity hyperbolic space
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- [L-T2]Li, Peter, and Tam, Luen-Fai. Uniqueness and regularity of proper harmonic maps. Annals of Mathematics, to appear.Google Scholar