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Energy identity for approximations of harmonic maps from surfaces

Author: Tobias Lamm
Journal: Trans. Amer. Math. Soc. 362 (2010), 4077-4097
MSC (2010): Primary 58E20; Secondary 35J60, 53C43
Published electronically: March 23, 2010
MathSciNet review: 2608396
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Abstract | References | Similar Articles | Additional Information

Abstract: We prove the energy identity for min-max sequences of the Sacks-Uhlenbeck and the biharmonic approximation of harmonic maps from surfaces into general target manifolds. The proof relies on Hopf-differential type estimates for the two approximations and on estimates for the concentration radius of bubbles.

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  • 1. J. Chen and G. Tian.
    Compactification of moduli space of harmonic mappings.
    Comm. Math. Helv., 74:201-237, 1999. MR 1691947 (2001k:58024)
  • 2. T. Colding and W. Minicozzi.
    Width and finite extinction time of Ricci flow.
    Preprint, 2007.
  • 3. W. Y. Ding and G. Tian.
    Energy identity for a class of approximate harmonic maps from surfaces.
    Comm. Anal. Geom., 3:543-554, 1995. MR 1371209 (97e:58055)
  • 4. F. Duzaar and E. Kuwert.
    Minimization of conformally invariant energies in homotopy classes.
    Calc. Var. Partial Differ. Equ., 6:285-313, 1998. MR 1624288 (99d:58045)
  • 5. G.Y. Jiang.
    The conservation law for $ 2$-harmonic maps between Riemannian manifolds.
    Acta Math. Sinica, 30:220-225, 1987. MR 891928 (88k:58028)
  • 6. J. Jost.
    Two-dimensional geometric variational problems.
    John Wiley and Sons, Chichester, 1991. MR 1100926 (92h:58045)
  • 7. T. Lamm.
    Fourth order approximation of harmonic maps from surfaces.
    Calc. Var. Partial Differ. Equation s, 27:125-157, 2006. MR 2251990 (2007k:58023)
  • 8. Y. Li and Y. Wang.
    A weak energy identity and the length of necks for a Sacks-Uhlenbeck $ \alpha$-harmonic map sequence.
    Preprint, 2008.
  • 9. F. Lin and T. Rivière.
    Energy quantization for harmonic maps.
    Duke Math. J., 111:177-193, 2002. MR 1876445 (2002k:58036)
  • 10. F. Lin and C. Wang.
    Energy identity of harmonic map flows from surfaces at finite singular time.
    Calc. Var. Partial Differ. Equ., 6:369-380, 1998. MR 1624304 (99k:58047)
  • 11. F. Lin and C. Wang.
    Harmonic and quasi-harmonic spheres.
    Comm. Anal. Geom., 7:397-429, 1999. MR 1685578 (2000b:58028)
  • 12. F. Lin and C. Wang.
    Harmonic and quasi-harmonic spheres II.
    Comm. Anal. Geom., 10:341-375, 2002. MR 1900755 (2003d:58029)
  • 13. E. Loubeau, S. Montaldo and C. Oniciuc.
    The stress-energy tensor for biharmonic maps.
    Math. Z., 259:503-524, 2008. MR 2395125
  • 14. J.D. Moore.
    Energy growth in minimal surface bubbles.
    Preprint, 2007.
  • 15. R.S. Palais.
    Critical point theory and the minimax principle.
    Proc. Symp. Pure Math., 15:185-212, 1970. MR 0264712 (41:9303)
  • 16. T. Parker.
    Bubble tree convergence for harmonic maps.
    J. Differ. Geom., 44:595-633, 1996. MR 1431008 (98k:58069)
  • 17. J. Qing.
    On singularities of the heat flow for harmonic maps from surfaces into spheres.
    Comm. Anal. Geom., 3:297-315, 1995. MR 1362654 (97c:58154)
  • 18. J. Qing and G. Tian.
    Bubbling of the heat flow for harmonic maps from surfaces.
    Comm. Pure Appl. Math., 50:295-310, 1997. MR 1438148 (98k:58070)
  • 19. T. Rivière.
    Interpolation spaces and energy quantization for Yang-Mills fields.
    Comm. Anal. Geom., 10:683-708, 2002. MR 1925499 (2004a:58018)
  • 20. J. Sacks and K. Uhlenbeck.
    The existence of minimal immersions of $ 2$-spheres.
    Annals of Math., 113:1-24, 1981. MR 604040 (82f:58035)
  • 21. M. Struwe.
    The existence of surfaces of constant mean curvature with free boundaries.
    Acta Math., 160:19-64, 1988. MR 926524 (89a:53012)
  • 22. M. Struwe.
    Critical points of embeddings of $ H^{1,n}_0$ into Orlicz spaces.
    Ann. Inst. H. Poincaré, Analyse Non Linéaire, 5:425-464, 1988. MR 970849 (90c:35084)
  • 23. M. Struwe.
    Positive solutions of critical semilinear elliptic equations on non-contractible planar domains.
    J. Eur. Math. Soc., 2:329-388, 2000. MR 1796963 (2001h:35070)
  • 24. M. Struwe.
    Variational Methods, volume 34 of Ergebnisse der Mathematik und ihrer Grenzgebiete,
    Springer-Verlag, Berlin, third edition, 2000. MR 1736116 (2000i:49001)
  • 25. P. Topping.
    Repulsion and quantization in almost-harmonic maps, and asymptotics of the harmonic map flow.
    Annals of Math. (2), 159:465-534, 2004. MR 2081434 (2005g:58029)
  • 26. H. Urakawa.
    Calculus of variations and harmonic maps.
    volume 132 of Translations of Mathematical Monographs, American Mathematical Society, Providence, RI, 1993. MR 1252178 (95c:58050)
  • 27. C. Wang.
    Bubble phenomena of certain Palais-Smale sequences from surfaces to general targets.
    Houston J. Math., 22:559-590, 1996. MR 1417632 (98h:58053)
  • 28. C. Wang.
    Remarks on biharmonic maps into spheres.
    Calc. Var. Partial Differ. Equ., 21:221-242, 2004. MR 2094320 (2005e:58026)

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Additional Information

Tobias Lamm
Affiliation: Max-Planck-Institute for Gravitational Physics, Am Mühlenberg 1, 14476 Golm, Germany
Address at time of publication: Department of Mathematics, University of British Columbia, 1984 Mathematics Road, Vancouver, British Columbia, Canada V6T 1Z2

Keywords: Geometric analysis, harmonic maps, energy identity
Received by editor(s): December 17, 2007
Published electronically: March 23, 2010
Additional Notes: The author would like to thank Yuxiang Li for pointing out an error in an earlier version of the paper.
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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