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

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Abstract

We construct a harmonic diffeomorphism from the Poincaré ballH n=1 to itself, whose boundary value is the identity on the sphereS n, 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.

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References

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Economakis, M. A counterexample to uniqueness and regularity for harmonic maps between hyperbolic spaces. J Geom Anal 3, 27–36 (1993). https://doi.org/10.1007/BF02921328

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  • DOI: https://doi.org/10.1007/BF02921328

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