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Locally minimizing harmonic maps from noncompact manifolds

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Abstract

By introducing the “relative energy”, we develop a new method for finding harmonic maps from noncompact complete Riemannian manifolds with prescribed asympototic behaviour at infinity. This method is an extension of the well known direct method of energy-minimization for compact domains. As an application of our method, we show that the Dirichlet problem at infinity with Hölder continuous boundary data for harmonic maps from a Cartan-Hadarmard manifold with bounded negative curvature into a compact manifold, has a locally minimizing solution which is smooth near infinity.

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References

  1. Anderson, M.: The Dirichlet problem at infinity for manifolds of negative curveature,J. Diff. Geom. 18 (1983), 701–721.

    MATH  Google Scholar 

  2. Avilés, P., Choi, H. and Micallef, H.: Boundary behavior of harmonic maps on non-smooth domains and complete negatively curved manifolds. J Funct. Anal.99 293–331 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  3. Anderson, M. and Schoen, R.: Positive harmonic functions on complete manifolds of negative curvature. Ann. of Math.121, 429–461 (1985)

    Article  MathSciNet  Google Scholar 

  4. Ding, W.-Y. and Wang, Y.: Harmonic maps of complete noncompact Riemannian manifolds. Intern. J. Math.2, 617–633 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  5. Gilbarg, D. and Trudinger, N.: Elliptic Partial Differential Equations of Second Order. 2nd ed., Berlin New York: Springer 1983

    MATH  Google Scholar 

  6. Jost, J. and Karcher, H.: Geometrische Methoden zur Gewinnung von apriori-Schranken für harmonische Abbildungen. Manuscripta Math.40, 27–77 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  7. Li P. and Tam, L.: The heat equation and harmonic maps of complete manifolds. Invent. Math.105 1–46 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  8. Schoen, R.: Analitic aspects of the harmonic map problem, in Seminar on Partial Differential Equations (S.S. Chern ed.), pp. 321–358, Berlin New York: Springer 1984.

    Google Scholar 

  9. Sullivan, D.: The Dirichlet problem at infinity for a negatively curved manifold. J. Diff. Geom.18, 723–732 (1983)

    MATH  MathSciNet  Google Scholar 

  10. Schoen, R. and Uhlenbeck, K.: A regularity theory for harmonic maps, J. Diff. Geom.17, 307–335 (1982)

    MathSciNet  Google Scholar 

  11. Schoen, R. and Uhlenbeck K.: Regularity of minimizing harmonic maps into the sphere, Invent. Math.78, 89–100 (1984)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Institute of Mathematics, Academia Sinica, 100080, Beijing, China

    Wei-Yue Ding

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  1. Wei-Yue Ding
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This work is supported by the Chinese National Science Foundation

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Ding, WY. Locally minimizing harmonic maps from noncompact manifolds. Manuscripta Math 85, 283–297 (1994). https://doi.org/10.1007/BF02568199

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  • DOI: https://doi.org/10.1007/BF02568199

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