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Curvature and Bubble Convergence of Harmonic Maps

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Abstract

We explore geometric aspects of bubble convergence for harmonic maps. More precisely, we show that the formation of bubbles is characterized by the local excess of curvature on the target manifold. We give a universal estimate for curvature concentration masses at each bubble point and show that there is no curvature loss in the necks. Our principal hypothesis is that the target manifold is Kähler.

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References

  1. Chen, J., Tian, G.: Compactification of moduli space of harmonic mappings. Comment. Math. Helv. 74, 201–237 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  2. Chern, S.S.: On holomorphic mappings of Hermitian manifolds of the same dimension. Proc. Symp. Pure Math. 11, 157–170 (1968)

    Article  MathSciNet  Google Scholar 

  3. Eells, J., Lemaire, L.: Selected Topics in Harmonic Maps. CBMS Regional Conference Series in Mathematics, vol. 50. AMS, Providence (1983). v+85 pp.

    MATH  Google Scholar 

  4. Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Classics in Mathematics. Springer, Berlin (2001). xiv+517 pp. Reprint of the 1998 edition

    MATH  Google Scholar 

  5. Gromov, M.: Pseudoholomorphic curves in symplectic manifolds. Invent. Math. 82, 307–347 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  6. Jost, J.: Two-Dimensional Geometric Variational Problems. Pure and Applied Mathematics. Wiley, Chichester (1991). x+236 pp.

    MATH  Google Scholar 

  7. Kozul, J.L., Malgrange, B.: Sur certaines structures fibrées complexes. Arch. Math. 9, 102–109 (1958)

    Article  Google Scholar 

  8. Lieb, E.H., Loss, M.: Analysis, 2nd edn. Graduate Studies in Mathematics, vol. 14. American Mathematical Society, Providence (2001). xxii+346 pp.

    MATH  Google Scholar 

  9. Lions, P.-L.: The concentration-compactness principle in the calculus of variations. The limit case. I. Rev. Mat. Iberoam. 1(1), 145–201 (1985)

    Article  MATH  Google Scholar 

  10. Lions, P.-L.: The concentration-compactness principle in the calculus of variations. The limit case. II. Rev. Mat. Iberoam. 1(2), 45–121 (1985)

    Article  MATH  Google Scholar 

  11. Lu, Y.C.: Holomorphic mappings of complex manifolds. J. Differ. Geom. 2, 299–312 (1968)

    MATH  Google Scholar 

  12. Parker, T.H.: Bubble tree convergence for harmonic maps. J. Differ. Geom. 44, 595–633 (1996)

    MATH  Google Scholar 

  13. Reshetnyak, Y.G.: Isothermal coordinates on manifolds of bounded curvature. I. Sib. Math. J. 1, 88–116 (1960) (Russian)

    Google Scholar 

  14. Reshetnyak, Y.G.: Isothermal coordinates on manifolds of bounded curvature. II. Sib. Math. J. 1, 248–276 (1960) (Russian)

    Google Scholar 

  15. Toledo, D.: Harmonic maps from surfaces to certain Kähler manifolds. Math. Scand. 45, 13–26 (1979)

    MathSciNet  MATH  Google Scholar 

  16. Troyanov, M.: Un principe de concentration-compacité pour les suites de surfaces Riemanniennes. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 8, 419–441 (1991)

    MathSciNet  MATH  Google Scholar 

  17. Sacks, J., Uhlenbeck, K.: The existence of minimal immersions of 2-spheres. Ann. Math. (2) 113, 1–24 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  18. Schoen, R.: Analytic aspects of the harmonic map problem. In: Seminar on Nonlinear Partial Differential Equations, Berkeley, CA, 1983. Math. Sci. Res. Inst. Publ., vol. 2, pp. 321–358. Springer, New York (1984)

    Chapter  Google Scholar 

  19. Wood, J.C.: Holomorphicity of certain harmonic maps from a surface to complex projective n-space. J. Lond. Math. Soc. (2) 20, 137–142 (1979)

    Article  MATH  Google Scholar 

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Correspondence to Gerasim Kokarev.

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Communicated by Jiaping Wang.

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Kokarev, G. Curvature and Bubble Convergence of Harmonic Maps. J Geom Anal 23, 1058–1077 (2013). https://doi.org/10.1007/s12220-011-9273-1

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  • DOI: https://doi.org/10.1007/s12220-011-9273-1

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