Energy-minimal diffeomorphisms between doubly connected Riemann surfaces

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Abstract

Let \(M\) and \(N\) be doubly connected Riemann surfaces with boundaries and with nonvanishing conformal metrics \(\sigma \) and \(\rho \) respectively, and assume that \(\rho \) is a smooth metric with bounded Gauss curvature \({\mathcal {K}}\) and finite area. The paper establishes the existence of homeomorphisms between \(M\) and \(N\) that minimize the Dirichlet energy. Among all homeomorphisms \(f :M{\overset{{}_{ \tiny {\mathrm{onto}} }}{\longrightarrow }} N\) between doubly connected Riemann surfaces such that \({{\mathrm{Mod\,}}}M \leqslant {{\mathrm{Mod\,}}}N\) there exists, unique up to conformal automorphisms of M, an energy-minimal diffeomorphism which is a harmonic diffeomorphism. The results improve and extend some recent results of Iwaniec et al. (Invent Math 186(3):667–707, 2011), where the authors considered bounded doubly connected domains in the complex plane w.r. to Euclidean metric.

Mathematics Subject Classification (2000)

Primary 58E20 Secondary 30C62 31A05 
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Notes

Acknowledgments

I thank Professor Leonid Kovalev for very useful discussion about the subject of this paper.

References

  1. 1.
    Ahlfors, L.V.: Lectures on quasiconformal mappings, 2nd edn. University Lecture Series, vol. 38. American Mathematical Society, Providence (2006). With supplemental chapters by C.J. Earle, I. Kra, M. Shishikura and J.H. HubbardGoogle Scholar
  2. 2.
    Astala, K., Iwaniec, T., Martin, G.J.: Deformations of annuli with smallest mean distortion. Arch. Ration. Mech. Anal. 195(3), 899–921 (2010)MATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    Astala, K., Iwaniec, T., Martin, G.: Elliptic Partial Differential Equations and Quasiconformal Mappings in the Plane. Princeton University Press, Princeton (2009)MATHGoogle Scholar
  4. 4.
    Choi, H., Treibergs, A.: Gauss maps of spacelike constant mean curvature hypersurfaces of Minkowski space. J. Differ. Geom. 32, 775–817 (1990)MATHMathSciNetGoogle Scholar
  5. 5.
    Hamilton, R.: Harmonic maps of manifolds with boundary. Lecture Notes in Mathematics, vol. 471, i+168 pp. Springer, Berlin (1975)Google Scholar
  6. 6.
    Hencl, S., Koskela, P.: Regularity of the inverse of a planar Sobolev homeomorphism. Arch. Ration. Mech. Anal. 180(1), 75–95 (2006)MATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    Hencl, S., Malý, J.: Jacobians of Sobolev homeomorphisms. Calc. Var. 38, 233–242 (2010)MATHCrossRefGoogle Scholar
  8. 8.
    Heinz, E.: On certain nonlinear elliptic differential equations and univalent mappings. J. d’ Anal. Math. 5 197–272 (1956/1957)Google Scholar
  9. 9.
    Heinz, E.: Existence theorems for one-to-one mappings associated with elliptic systems of second order. I. J. d’ Anal. Math. 5, 15(1), 325–352 (1965)Google Scholar
  10. 10.
    Hildebrandt, S., Kaul, H., Widman, K.-O.: An existence theorem for harmonic mappings of Riemannian manifolds. Acta Math. 138(1–2), 1–16 (1977)MATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    Hildebrandt, S., Nitsche, J.C.C.: A uniqueness theorem for surfaces of least area with partially free boundaries on obstacles. Arch. Ration. Mech. Anal. 79(3), 189–218 (1982)MATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    Iwaniec, T., Kovalev, L.V., Onninen, J.: The Nitsche conjecture. J. Am. Math. Soc. 24(2), 345–373 (2011)MATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    Iwaniec, T., Kovalev, L.V., Onninen, J.: Harmonic mapping problem and affine capacity. Proc. R. Soc. Edinb. Sect. A 141(5), 1017–1030 (2011)MATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    Iwaniec, T., Kovalev, L.V., Onninen, J.: Hopf differentials and smoothing Sobolev homeomorphisms. Int. Math. Res. Not. IMRN 2012(14), 3256–3277 (2012)MATHMathSciNetGoogle Scholar
  15. 15.
    Iwaniec, T., Koh, N.-T., Kovalev, L.V., Onninen, J.: Existence of energy-minimal diffeomorphisms between doubly connected domains. Invent. Math. 186(3), 667–707 (2011)MATHMathSciNetCrossRefGoogle Scholar
  16. 16.
    Jost, J.: Harmonic maps between surfaces. Lecture Notes in Mathematics, vol. 1062, x+133 pp. Springer, Berlin (1984)Google Scholar
  17. 17.
    Jost, J.: Minimal surfaces and Teichmüller theory. In: Yau, S.-T. (ed.), Tsing Hua Lectures on Geometry and Analysis, Taiwan, 1990–91, pp. 149–211. International Press, Cambridge (1997)Google Scholar
  18. 18.
    Jost, J.: Univalency of harmonic mappings between surfaces. J. Reine Angew. Math. 324, 141–153 (1981)MATHMathSciNetGoogle Scholar
  19. 19.
    Jost, J.: Two-dimensional geometric variational problems. Wiley, Chichester (1991)MATHGoogle Scholar
  20. 20.
    Jost, J., Schoen, R.: On the existence of harmonic diffeomorphisms. Invent. Math. 66(2), 353–359 (1982)MATHMathSciNetCrossRefGoogle Scholar
  21. 21.
    Kalaj, D.: Harmonic maps between annuli on Riemann surfaces. Israel J. Math. 182, 123–147 (2011)MATHMathSciNetCrossRefGoogle Scholar
  22. 22.
    Kalaj, D.: Deformations of annuli on Riemann surfaces with smallest mean distortion. ArXiv:1005.5269Google Scholar
  23. 23.
    Kalaj, D., Mateljević, M.: Inner estimate and quasiconformal harmonic maps between smooth domains. Journal d’Analyse Math. 100, 117–132 (2006)MATHCrossRefGoogle Scholar
  24. 24.
    Kalaj, D.: Quasiconformal harmonic mapping between Jordan domains. Math. Z. 260(2), 237–252 (2008)MATHMathSciNetCrossRefGoogle Scholar
  25. 25.
    Kalaj, D.: Harmonic mappings and distance function. Ann. Scuola Norm. Sup. Pisa Cl. Sci X(5), 1–13 (2011)MathSciNetGoogle Scholar
  26. 26.
    Lehto, O., Virtanen, K.: Quasiconformal mappings in the plane. Springer, New York (1973)MATHCrossRefGoogle Scholar
  27. 27.
    Lloyd, N.G.: Degree theory. Cambridge University Press, Cambridge (1978)MATHGoogle Scholar
  28. 28.
    Malý, J., Martio, O.: Lusin’s condition (N) and mappings of the class \(W^{1, n}\). J. Reine Angew. Math. 458, 19–36 (1995)MATHMathSciNetGoogle Scholar
  29. 29.
    Marković, V., Mateljević, M.: A new version of the main inequality and the uniqueness of harmonic maps. J. Anal. Math. 79, 315–334 (1999)MATHMathSciNetCrossRefGoogle Scholar
  30. 30.
    Mateljević, M., Vuorinen, M.: On harmonic quasiconformal quasi-isometries. J. Inequal. Appl. 2010, 19 (2010), Article ID 178732. doi: 10.1155/2010/178732
  31. 31.
    Nitsche, J.C.C.: On the modulus of doubly connected regions under harmonic mappings. Am. Math. Mon. 69, 781–782 (1962)MATHMathSciNetCrossRefGoogle Scholar
  32. 32.
    Pavlović, M.: Boundary correspondence under harmonic quasiconformal homeomorfisms of the unit disc. Ann. Acad. Sci. Fenn. 27, 365–372 (2002)MATHGoogle Scholar
  33. 33.
    Ruh, E., Vilms, J.: The tension field of the Gauss map. Trans. Am. Math. Soc. 149, 569–573 (1970)MATHMathSciNetCrossRefGoogle Scholar
  34. 34.
    Schoen, R., Yau, S.T.: Lectures on Harmonic Maps. International Press, Cambridge (1997)MATHGoogle Scholar
  35. 35.
    Whyburn, G.T.: Analytic Topology. American Mathematical Society, New York (1942)MATHGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Faculty of Natural Sciences and MathematicsUniversity of MontenegroPodgoricaMontenegro

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