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Harmonic maps between ideal 2-dimensional simplicial complexes

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Abstract

We prove existence and regularity results for energy minimizing maps between ideal hyperbolic 2-dimensional simplicial complexes. The spaces in question were introduced by Charitos–Papadopoulos, who describe their Teichmüller spaces and some compactifications. This work is a first step in introducing harmonic map theory into the Teichmüller theory of these spaces.

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References

  1. Charitos, C., Papadopoulos, A.: The Geometry of ideal 2-dimensional simplicial complexes. Glasgow Math. J. 43, 39–66 (2001)

    Article  MathSciNet  Google Scholar 

  2. Daskalopoulos, G., Mese, C.: Harmonic maps from 2-complexes. Commun. Anal. Geom. 14, 497–549 (2006)

    Article  MathSciNet  Google Scholar 

  3. Daskalopoulos, G., Mese, C.: Harmonic maps from a simplicial complex and geometric rigidity. J. Differ. Geom. 78, 269–293 (2008)

    Article  MathSciNet  Google Scholar 

  4. Daskalopoulos, G., Mese, C.: Harmonic maps between singular spaces I. Commun. Anal. Geom. 18, 257–337 (2010)

    Article  MathSciNet  Google Scholar 

  5. Dubrovin, B.A., Novikov, S.P., Fomenko, A.T.: Modern Geometry Methods and Applications: Part II: The Geometry and Topology of Manifolds. Springer, New York (1985)

    Book  Google Scholar 

  6. Eells, J., Fuglede, B.: Harmonic Maps Between Riemannian Polyhedra, Cambridge Tracts Mathematics, vol. 142. Cambridge University Press, Cambridge (2001)

    MATH  Google Scholar 

  7. Eells, J., Sampson, J.: Harmonic mappings of Riemannian manifolds. Am. J. Math. 86(1), 109–160 (1964)

    Article  MathSciNet  Google Scholar 

  8. Ghys, E., de la Harpe, P. (eds.): Sur les Groupes Hyperboliques d’après Mikhael Gromov, Progress in Mathematics, vol. 83. Birkhäuser Boston Inc, Boston (1990)

    MATH  Google Scholar 

  9. Gerstenhaber, M., Rauch, H.E.: On extremal quasiconformal mappints I, II. Proc. Nat. Acad. Sci. U.S.U. 40, 808–812 (1954)

    Article  MathSciNet  Google Scholar 

  10. Gromov, M., Schoen, R.: Harmonic maps into singular spaces and p-adic superrigidity for lattices in groups of rank one. Publ. Maths. IHES 76, 165–246 (1992)

    Article  MathSciNet  Google Scholar 

  11. Iwaniec, T., Kovalev, L., Onninen, J.: Hopf differentials and smoothing Sobolev homeomorphisms. Int. Math. Res. Not. IMRN 14, 3256–3277 (2012)

    MathSciNet  MATH  Google Scholar 

  12. Jost, J.: Two-Dimensional Geometric Variational Problems. Wiley, Chichester (1991)

    MATH  Google Scholar 

  13. Jost, J.: Equilibrium maps between metric spaces. Calc. Var. Partial Differ. Equ. 2, 173–204 (1994)

    Article  MathSciNet  Google Scholar 

  14. Jost, J., Schoen, R.: On the existence of harmonic diffeomorphisms between surfaces. Invent. Math. 66, 353–359 (1982)

    Article  MathSciNet  Google Scholar 

  15. Korevaar, N., Schoen, R.: Sobolev spaces and harmonic maps for metric space targets. Commun. Anal. Geom. 1, 561–659 (1993)

    Article  MathSciNet  Google Scholar 

  16. Kuwert, E.: Harmonic maps between flat surfaces with conical singularities. Math. Z. 221, 421–436 (1996)

    Article  MathSciNet  Google Scholar 

  17. Mese, C.: A variational charicterization of the Teichmüller map. Calc. Var. PDE 21, 15–46 (2004)

    Article  Google Scholar 

  18. Morrey, C.: Multiple Integrals in the Calculus of Variations. Springer, New York (1966)

    Book  Google Scholar 

  19. Mese, C., Yamada, S.: Local uniformization and free boundary regularity of minimal singular surfaces. J. Differ. Geom. 21, 743–766 (2001)

    MathSciNet  MATH  Google Scholar 

  20. Sampson, J.H.: Some properties and applications of harmonic mappings. Ann. Sci. Ècole Norm. Sup. 4, 211–228 (1978)

    Article  MathSciNet  Google Scholar 

  21. Schoen, R., Yau, S.: On Univalent harmonic maps between surfaces. Invent. Math. 44, 265–278 (1978)

    Article  MathSciNet  Google Scholar 

  22. Thurston, W.P.: Three-Dimensional Geometry and Topology, Vol. 1, Edited by Silvio Levy, Princeton Mathematical Series, vol. 35. Princeton University Press, Princeton (1997)

    Book  Google Scholar 

  23. Wolf, M.: The Teichmüller theory of harmonic maps. J. Differ. Geom. 29, 449–479 (1989)

    Article  Google Scholar 

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Acknowledgements

The authors would like to thank both George Daskalopoulos and Athanase Papadopoulos for suggesting the problem and broader context of the problem, as well as many helpful conversations and suggestions. They also thank Chikako Mese for helpful discussions.

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Authors and Affiliations

  1. Pacific Institute for the Mathematical Sciences, University of British Columbia, Vancouver, BC, Canada

    Brian Freidin

  2. Brown University, Providence, USA

    Victòria Gras Andreu

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  1. Brian Freidin
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  2. Victòria Gras Andreu
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Correspondence to Brian Freidin.

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Freidin, B., Gras Andreu, V. Harmonic maps between ideal 2-dimensional simplicial complexes. Geom Dedicata 208, 129–155 (2020). https://doi.org/10.1007/s10711-020-00514-w

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  • DOI: https://doi.org/10.1007/s10711-020-00514-w

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