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
Article

Abstract

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

58E20 35C20 35L70 31B25 32A40 35k20 

Key Words and Phrases

Asymptotics Cayley transform conformal contraction exponential growth harmonic map Hölder regularity hyperbolic space 

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References

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    Li, Peter, and Tam, Luen-Fai. The heat equation and harmonic maps of complete manifolds. Invent. Math.105, 1–46 (1991).MATHCrossRefMathSciNetGoogle Scholar
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    Li, Peter, and Tam, Luen-Fai. Uniqueness and regularity of proper harmonic maps. Annals of Mathematics, to appear.Google Scholar
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    Wolf, Michael. Infinite energy harmonic maps and degeneration of hyperbolic surfaces in moduli space. J. Diff. Geom.33, 487–539 (1991).MATHGoogle Scholar

Copyright information

© Mathematica Josephina, Inc. 1993

Authors and Affiliations

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

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