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Harmonic maps from surfaces to R-trees

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References

  • [A] Al’ber, S.I.: Onn-dimensional problems in the calculus of variations in the large. Sov. Math. Dokl5: 700–704 (1964)

    MATH  Google Scholar 

  • [B] Bestvina, M.: Degenerations of the Hyperbolic Space Duke. Math. J.56: 143–161 (1988)

    MATH  MathSciNet  Google Scholar 

  • [C] Corlette, K.: Archimedian Superrigidity and Hyperbolic Geometry. Ann. of Math.135: 165–182 (1992)

    Article  MathSciNet  Google Scholar 

  • [EE] Earle, C. and Eells, J.: A Fibre Bundle Description of Teichmüller Space. J. Differential Geometry3: 19–43 (1969)

    MATH  MathSciNet  Google Scholar 

  • [ES] Eells, J. and Sampson, J.H.: Harmonic Mappings of Riemannian Manifolds. Amer. J. Math.86: 109–160 (1964)

    Article  MATH  MathSciNet  Google Scholar 

  • [FLP] Fathi, A., Laudenbach, F., and Poenaru, V.: Traveaux de Thurston sur les Surfaces Asterisque. pp. 66–67 (1979).

  • [FT] Fischer, A.E. and Tromba, A.J.: On a Purely Riemannian proof of the structure and dimension of the unramified moduli space of a compact Riemann Surface. Math. Ann.267: 311–345 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  • [Gar] Gardiner, F.P.: Teichmüller Theory and Quadratic Differentials. West Sessex, England: John Wiley and Sons 1988

    Google Scholar 

  • [G] Gromov, M.: Métriques pour les Variétés Riemanniennes. J. Lafontaine and P. Pansu (eds.) Paris: Cedic/Fernad Nathan 1981

    Google Scholar 

  • [GS] Gromov, M. and Schoen, R.: Harmonic Maps into Singular Spaces andp-adic Superrigidity for Lattices in Groups of Rank One. Publ. I.H.E.S. (to appear)

  • [H] Hartman, P.: On Homotopic Harmonic Maps. Canad. J. Math.19: 673–687 (1967)

    MATH  MathSciNet  Google Scholar 

  • [Hu-Ma] Hubbard, J. and Masur, H.: Quadratic Differentials and Foliations. Acta Math.142: 221–224 (1979).

    Article  MATH  MathSciNet  Google Scholar 

  • [I] Ishihara, T.: A Mapping of Riemannian Manifolds which preserves Harmonic Functions. J. Math. Kyoto Univ.19: 215–229 (1979)

    MATH  MathSciNet  Google Scholar 

  • [JK] Jäger, W. and Kaul, H.: Uniqueness and Stability of Harmonic Maps and their Jacobi Fields. Man. Math.28: 269–291 (1979)

    Article  MATH  Google Scholar 

  • [J] Jost, J.: Two Dimensional Geometric Variations Problems West Sussex, England: John Wiley and Sons 1991

    Google Scholar 

  • [Ke] Kerchoff, S.: The Asymptotic Geometry of Teichmüller Space. Topology19: 23–41 (1980)

    Article  MathSciNet  Google Scholar 

  • [M1] Morgan, J.: Group Actions on Trees and the Compactification of the Spaces of Classes ofSO(n, 1)representations. Topology25: 1–33 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  • [M2] Morgan, J.:A-Trees and their Applications. Bull. Amer. Math. Soc.26: 87–112 (1992)

    MATH  MathSciNet  Google Scholar 

  • [MS1] Morgan, J. and Shalen, P.B.: Degenerations of Hyperbolic Structures, I: Valuations, Trees and Surfaces. Ann. of Math.120: 401–476 (1984)

    Article  MathSciNet  Google Scholar 

  • [MS2] Morgan, J. and Shalen, P.B.: Free Actions of Surface Groups onR-trees. Topology30: 143–154 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  • [P1] Paulin, F.: Topologie de Gromov équivariante, structures hyperboliques et arbres réels. Invent. Math.94: 53–80 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  • [P2] Paulin, F.: The Gromov Topology onR-trees. Topology and its Applications32: 197–221 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  • [Sa] Sampson, J.H.: Some Properties and Applications of Harmonic Mappings. Ann. Scient. Ec. Norm. Sup4: 265–278 (1978)

    MathSciNet  Google Scholar 

  • [SY] Schoen, R. and Yu, S.-T.: On Univalent Harmonic Maps Between Surfaces. Invent. Math.44: 265–278 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  • [Sk] Skora, R.: Splittings of Surfaces. Preprint

  • [St] Strebel, K.: Quadratic Differentials. Berlin: Springer-Veralg 1984

    MATH  Google Scholar 

  • [Th] Thurston, W.P.: On the Geometry and Dynamics of Diffeomorphisms of Surfaces. Bull. Amer. Math. Soc.19: 417–431 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  • [T] Tromba, A.J.: Teichmüller Theory in Riemannian Geometry. Basel: Birkhauser 1992

    MATH  Google Scholar 

  • [W1] Wolf M.: The Teichmüller Theory of Harmonic Maps. J. Differential Geom.29: 449–479 (1989)

    MATH  MathSciNet  Google Scholar 

  • [W2] Wolf, M.: High Energy Degeneration of Harmonic Maps Between Sufaces and Rays in Teichmüller Space. Topology30: 517–540 (1991)

    Article  MATH  MathSciNet  Google Scholar 

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Partially supported by the National Science Foundation: Alfred P. Sloan Reasearch Fellow

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Wolf, M. Harmonic maps from surfaces to R-trees. Math Z 218, 577–593 (1995). https://doi.org/10.1007/BF02571924

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