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Annals of Global Analysis and Geometry

, Volume 42, Issue 4, pp 565–584 | Cite as

Existence and Liouville theorems for V -harmonic maps from complete manifolds

  • Qun Chen
  • Jürgen JostEmail author
  • Hongbing Qiu
Article

Abstract

We establish existence and uniqueness theorems for V-harmonic maps from complete noncompact manifolds. This class of maps includes Hermitian harmonic maps, Weyl harmonic maps, affine harmonic maps, and Finsler harmonic maps from a Finsler manifold into a Riemannian manifold. We also obtain a Liouville type theorem for V-harmonic maps. In addition, we prove a V-Laplacian comparison theorem under the Bakry-Emery Ricci condition.

Keywords

V-Harmonic map Noncompact manifold Existence Liouville theorem V-Laplacian comparison theorem 

Mathematics Subject Classification

58E20 53C27 

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References

  1. 1.
    Cao H.D.: Recent progress on Ricci solitons. In: Ji, L., Li, P., Schoen, R., Simon, L. (eds.) Recent Advances in Geometric Analysis, Advanced Lectures in Mathematics, vol. 11, pp. 1–38. International Press, Somerville, MA (2010)Google Scholar
  2. 2.
    Centore P.: Finsler Laplacians and minimal-energy maps. Internat. J. Math. 11(1), 1–13 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Chen Q., Jost J., Wang G.: A maximum principle for generalizations of harmonic maps in Hermitian, affine, Weyl and Finsler geometry, Preprint (2011)Google Scholar
  4. 4.
    Cheng S.Y.: Liouville theorem for harmonic maps, geometry of the Laplace operator. In: Proceedings of the Symposium in Pure Mathematics. University of Hawaii, Honolulu, Hawaii, 1979, pp. 147–151, Proceedings of the Symposium in Pure Mathematics, XXXVI, American Mathematical Society, Providence, RI (1980)Google Scholar
  5. 5.
    Choi H.: On the Liouville theorem for harmonic maps. Proc. Amer. Math. Soc. 85, 91–94 (1982)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Colding T.H., Minicozzi W.P.: II: Generic mean curvature flow I: generic singularities. Ann. Math. 175, 755–833 (2012)zbMATHCrossRefGoogle Scholar
  7. 7.
    Ding W.Y., Wang Y.D.: Harmonic maps of complete noncompact Riemannian manifolds. Internat. J. Math. 2(6), 617–633 (1991)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Ding Q., Xin Y.L.: Volume growth, eigenvalue and comapctness for self-shrinkers. arXiv:1101.1411v1 (2011)Google Scholar
  9. 9.
    Dong T.: Hermitian harmonic maps from complete manifolds into convex balls. Nonlinear Anal. 72, 3457–3462 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Gilbarg D., Trudinger N.S.: Elliptic Partial Differential Equations of Second Order, 2nd edn. Springer, Berlin (1983)CrossRefGoogle Scholar
  11. 11.
    Grunau H.-C., Kühnel M.: On the existence of Hermitian-harmonic maps from complete Hermitian to complete Riemannian manifolds. Math. Z. 249, 297–327 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Han J.W., Shen Y.B.: Harmonic maps from complex Finsler manifolds. Pacific J. Math. 236(2), 341–356 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Jäger W., Kaul H.: Uniqueness and stability of harmonic maps and their Jacobi fields. Manuscripta Math. 28, 269–291 (1979)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Jost J.: Harmonic Mappings Between Riemannian Manifolds. ANU-Press, Canberra (1984)Google Scholar
  15. 15.
    Jost J., Simsir F.M.: Affine harmonic maps. Analysis (Munich) 29, 185–197 (2009)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Jost J., Simsir F.M.: Non-divergence harmonic maps. In: Loubeau, E., Montaldo, S., (eds.) Harmonic maps and differential geometry. Contemp. Math, vol. 542, pp. 231–238, Amer. Math. Soc., Providence, RI (2011)Google Scholar
  17. 17.
    Jost J., Yau S.T.: A nonlinear elliptic system for maps from Hermitian to Riemannian manifolds and rigidity theorem in Hermitian geometry. Acta Math. 170, 221–254 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Kokarev G.: On pseudo-harmonic maps in conformal geometry. Proc. London Math. Soc. 99(3), 168–194 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Li J.Y.: The heat flows and harmonic maps of complete noncompact Riemannian manifolds. Math. Z. 212(2), 161–173 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Li P.: Lecture notes on geometric analysis, Research Institute of Mathematics, Global Analysis Center Seoul National University (1993)Google Scholar
  21. 21.
    Li X.D.: Liouville theorems for symmetric diffusion operators on complete Riemannian manifolds. J. Math. Pures Appl. 84, 1295–1361 (2005)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Li P., Tam L.F.: The heat equation and harmonic maps of complete manifolds. Invent. Math. 105(1), 1–46 (1991)MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Li Y.X., Wang Y.D.: Bubbling location for F-harmonic maps and inhomogeneous Landau-Lifshitz equations. Comment. Math. Helv. 81, 433–448 (2006)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Li Z.Y., Zhang X.: Hermitian harmonic maps into convex balls. Canad. Math. Bull. 50, 113–122 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Li J.Y., Wang M.: Liouville theorems for self-similar solutions of heat flows. J. Eur. Math. Soc. 11, 207–221 (2009)zbMATHCrossRefGoogle Scholar
  26. 26.
    Li J.Y., Zhu X.R.: Non existence of quasi-harmonic spheres. Calc. Var. 37, 441–460 (2010)zbMATHCrossRefGoogle Scholar
  27. 27.
    Lichnerowicz A.: Applications harmoniques et variétés Kähleriennes. Rend. Sem. Mat. Fis. Milano 39, 186–195 (1969)MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    Lin F.H., Wang C.Y.: Harmonic and quasi-harmonic spheres. Comm. Anal. Geom. 7, 397–429 (1999)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Lott J.: Some geometric properties of the Bakry-Emery-Ricci tensor. Comment. Math. Helv. 78, 865–883 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    Mo X.: Harmonic maps from Finsler manifolds. Illinois J. Math. 45, 1331–1345 (2001)MathSciNetzbMATHGoogle Scholar
  31. 31.
    Mo X.H., Yang Y.Y.: The existence of harmonic maps from Finsler manifolds to Riemannian manifolds. Sci. China Ser. 48, 115–130 (2005)MathSciNetzbMATHGoogle Scholar
  32. 32.
    Munteanu O., Wang J.P.: Smooth metric measure spaces with nonnegative curvature. arXiv:1103.0746v2 (2011)Google Scholar
  33. 33.
    Ni L.: Hermitian harmonic maps from complete Hermitian manifolds to complete Riemannian manifolds. Math. Z. 232, 331–355 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  34. 34.
    von der Mosel H., Winklmann S.: On weakly harmonic maps from Finsler to Riemannian manifolds. Ann. Inst. H. Poincaré Anal. Non Linéaire 26, 39–57 (2009)zbMATHCrossRefGoogle Scholar
  35. 35.
    Wang G., Xu Deliang: Harmonic maps from smooth measure spaces. Preprint (2011)Google Scholar
  36. 36.
    Wei G., Wylie W.: Comparison goemetry for the Bakry-Emery Ricci tensor. J. Differential Geom. 83, 377–405 (2009)MathSciNetzbMATHGoogle Scholar
  37. 37.
    Yau S.T.: Harmonic functions on complete Riemannian manifolds. Comm. Pure Appl. Math. 28, 201–228 (1975)MathSciNetzbMATHCrossRefGoogle Scholar
  38. 38.
    Yau S.T.: Some function-theoretic properties of complete Riemannian manifolds and their applications to geometry. Indiana Univ. Math. J. 25, 659–670 (1976)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2012

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsWuhan UniversityWuhanChina
  2. 2.Max Planck Institute for Mathematics in the SciencesLeipzigGermany
  3. 3.Department of MathematicsLeipzig UniversityLeipzigGermany

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