Energy identity for approximations of harmonic maps from surfaces
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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.References
- Jingyi Chen and Gang Tian, Compactification of moduli space of harmonic mappings, Comment. Math. Helv. 74 (1999), no. 2, 201–237. MR 1691947, DOI 10.1007/s000140050086
- T. Colding and W. Minicozzi. Width and finite extinction time of Ricci flow. Preprint, 2007.
- Weiyue Ding and Gang Tian, Energy identity for a class of approximate harmonic maps from surfaces, Comm. Anal. Geom. 3 (1995), no. 3-4, 543–554. MR 1371209, DOI 10.4310/CAG.1995.v3.n4.a1
- Frank Duzaar and Ernst Kuwert, Minimization of conformally invariant energies in homotopy classes, Calc. Var. Partial Differential Equations 6 (1998), no. 4, 285–313. MR 1624288, DOI 10.1007/s005260050092
- Guo Ying Jiang, The conservation law for $2$-harmonic maps between Riemannian manifolds, Acta Math. Sinica 30 (1987), no. 2, 220–225 (Chinese). MR 891928
- Jürgen Jost, Two-dimensional geometric variational problems, Pure and Applied Mathematics (New York), John Wiley & Sons, Ltd., Chichester, 1991. A Wiley-Interscience Publication. MR 1100926
- Tobias Lamm, Fourth order approximation of harmonic maps from surfaces, Calc. Var. Partial Differential Equations 27 (2006), no. 2, 125–157. MR 2251990, DOI 10.1007/s00526-005-0001-1
- Y. Li and Y. Wang. A weak energy identity and the length of necks for a Sacks-Uhlenbeck $\alpha$-harmonic map sequence. Preprint, 2008.
- Fang-Hua Lin and Tristan Rivière, Energy quantization for harmonic maps, Duke Math. J. 111 (2002), no. 1, 177–193. MR 1876445, DOI 10.1215/S0012-7094-02-11116-8
- Fanghua Lin and Changyou Wang, Energy identity of harmonic map flows from surfaces at finite singular time, Calc. Var. Partial Differential Equations 6 (1998), no. 4, 369–380. MR 1624304, DOI 10.1007/s005260050095
- FangHua Lin and ChangYou Wang, Harmonic and quasi-harmonic spheres, Comm. Anal. Geom. 7 (1999), no. 2, 397–429. MR 1685578, DOI 10.4310/CAG.1999.v7.n2.a9
- FangHua Lin and ChangYou Wang, Harmonic and quasi-harmonic spheres. II, Comm. Anal. Geom. 10 (2002), no. 2, 341–375. MR 1900755, DOI 10.4310/CAG.2002.v10.n2.a5
- E. Loubeau, S. Montaldo, and C. Oniciuc, The stress-energy tensor for biharmonic maps, Math. Z. 259 (2008), no. 3, 503–524. MR 2395125, DOI 10.1007/s00209-007-0236-y
- J.D. Moore. Energy growth in minimal surface bubbles. Preprint, 2007.
- Richard S. Palais, Critical point theory and the minimax principle, Global Analysis (Proc. Sympos. Pure Math., Vol. XV, Berkeley, Calif., 1968) Amer. Math. Soc., Providence, R.I., 1970, pp. 185–212. MR 0264712
- Thomas H. Parker, Bubble tree convergence for harmonic maps, J. Differential Geom. 44 (1996), no. 3, 595–633. MR 1431008
- Jie Qing, On singularities of the heat flow for harmonic maps from surfaces into spheres, Comm. Anal. Geom. 3 (1995), no. 1-2, 297–315. MR 1362654, DOI 10.4310/CAG.1995.v3.n2.a4
- Jie Qing and Gang Tian, Bubbling of the heat flows for harmonic maps from surfaces, Comm. Pure Appl. Math. 50 (1997), no. 4, 295–310. MR 1438148, DOI 10.1002/(SICI)1097-0312(199704)50:4<295::AID-CPA1>3.0.CO;2-5
- Tristan Rivière, Interpolation spaces and energy quantization for Yang-Mills fields, Comm. Anal. Geom. 10 (2002), no. 4, 683–708. MR 1925499, DOI 10.4310/CAG.2002.v10.n4.a2
- J. Sacks and K. Uhlenbeck, The existence of minimal immersions of $2$-spheres, Ann. of Math. (2) 113 (1981), no. 1, 1–24. MR 604040, DOI 10.2307/1971131
- Michael Struwe, The existence of surfaces of constant mean curvature with free boundaries, Acta Math. 160 (1988), no. 1-2, 19–64. MR 926524, DOI 10.1007/BF02392272
- Michael Struwe, Critical points of embeddings of $H^{1,n}_0$ into Orlicz spaces, Ann. Inst. H. Poincaré Anal. Non Linéaire 5 (1988), no. 5, 425–464 (English, with French summary). MR 970849, DOI 10.1016/S0294-1449(16)30338-9
- Michael Struwe, Positive solutions of critical semilinear elliptic equations on non-contractible planar domains, J. Eur. Math. Soc. (JEMS) 2 (2000), no. 4, 329–388. MR 1796963, DOI 10.1007/s100970000023
- Michael Struwe, Variational methods, 3rd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 34, Springer-Verlag, Berlin, 2000. Applications to nonlinear partial differential equations and Hamiltonian systems. MR 1736116, DOI 10.1007/978-3-662-04194-9
- Peter Topping, Repulsion and quantization in almost-harmonic maps, and asymptotics of the harmonic map flow, Ann. of Math. (2) 159 (2004), no. 2, 465–534. MR 2081434, DOI 10.4007/annals.2004.159.465
- Hajime Urakawa, Calculus of variations and harmonic maps, Translations of Mathematical Monographs, vol. 132, American Mathematical Society, Providence, RI, 1993. Translated from the 1990 Japanese original by the author. MR 1252178, DOI 10.1090/mmono/132
- Changyou Wang, Bubble phenomena of certain Palais-Smale sequences from surfaces to general targets, Houston J. Math. 22 (1996), no. 3, 559–590. MR 1417632
- Changyou Wang, Remarks on biharmonic maps into spheres, Calc. Var. Partial Differential Equations 21 (2004), no. 3, 221–242. MR 2094320, DOI 10.1007/s00526-003-0252-7
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
- MR Author ID: 748462
- Email: tlamm@math.ubc.ca
- 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.
- © Copyright 2010
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 362 (2010), 4077-4097
- MSC (2010): Primary 58E20; Secondary 35J60, 53C43
- DOI: https://doi.org/10.1090/S0002-9947-10-04912-3
- MathSciNet review: 2608396