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Topological and analytical properties of Sobolev bundles, I: The critical case

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We study the interrelationship between topological and analytical properties of Sobolev bundles and describe some of their applications to variational problems on principal bundles. We in particular show that the category of Sobolev principal G-bundles of class W 2,m/2 defined over M m is equivalent to the category of smooth principal G-bundles on M and give a characterization of the weak sequential closure of smooth principal G-bundles with prescribed isomorphism class. We also prove a topological compactness result for minimizing sequences of a conformally invariant Yang-Mills functional.

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References

  1. Bethuel F.: The approximation problem for Sobolev maps between two manifolds. Acta Math. 167, 153–206 (1992)

    Article  MathSciNet  Google Scholar 

  2. Bredon, G.: Topology and Geometry. G.T.M. 139, Springer (1993)

  3. Brezis H.: The interplay between analysis and topology in some nonlinear PDE problems. Bull. Amer. Math. Soc. 40, 179–201 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  4. Brezis H., Li Y.Y.: Topology and Sobolev spaces. J. Funct. Anal. 183, 321–369 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  5. Donaldson S.K., Kronheimer P.B.: The Geometry of Four-Manifolds. Oxford Mathematical Monographs, Oxford (1990)

    MATH  Google Scholar 

  6. Druet, O., Hebey, H., Robert, F.: Blow-up Theory for Elliptic Pdes in Riemannian Geometry. Princeton University Press (2004)

  7. Duzaar F., Kuwert E.: Minimization of conformally invariant energies in homotopy classes. Calc. Var. 6, 285–313 (1997)

    Article  MathSciNet  Google Scholar 

  8. Giaquinta, M., Modica, G., Souček, J.: Cartesian Currents in the Calculus of Variations I & II. Ergebnisse der Mathematik und ihrer Grenzgebiete, Springer-Verlag, Berlin, Heidelberg (1998)

  9. Hang F., Lin F.H.: Topology of Sobolev mappings, II. Acta Math. 191, 55–107 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  10. Hang F., Lin F.H.: Topology of Sobolev bundles, III. Comm. Pure Appl. Math. 56, 1383–1415 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  11. Hardt, R., Rivière, T.: Connecting rational homotopy type singularities (Preprint)

  12. Husemoller, D.: Fibre Bundles, 3rd edn. G.T.M 20, Springer (1994)

  13. Isobe T.: Obstruction theory for the approximation and the deformation problems for Sobolev mappings. Ann. Global Anal. Geom. 27, 299–332 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  14. Isobe T.: On global singularities of Sobolev mappings. Math. Z. 252, 691–730 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  15. Isobe, T.: Topological and analytical properties of Sobolev bundles, II: higher dimensional cases (Preprint)

  16. Isobe, T.: A regularity result for a class of degenerate Yang-Mills connections in critical dimensions (Preprint)

  17. Kobayashi, S., Nomizu, K.: Foundations of Differential Geometry, vol. 1 & 2. Wiley (1963, 1969)

  18. Lions P.L.: The concentration compactness principle in the calculus of variations. The limit case. Part I. Rev. Math. Iberoamericano 1(1), 145–201 (1985)

    MATH  Google Scholar 

  19. Lions P.L.: The concentration compactness principle in the calculus of variations. The limit case. Part II. Rev. Math. Iberoamericano 1(2), 45–121 (1985)

    MATH  Google Scholar 

  20. Pakzad M.R., Rivière T.: Weak density of smooth maps for the Dirichlet energy between manifolds. GAFA 13, 223–257 (2003)

    Article  MATH  Google Scholar 

  21. Parker T.H., Wolfson J.G.: Pseudoholomorphic maps and bubble trees. J. Geom. Anal. 3, 63–98 (1993)

    MATH  MathSciNet  Google Scholar 

  22. Rivière T.: Interpolation spaces and energy quantization for Yang-Mills fields. Comm. Anal. Geom. 10, 683–708 (2002)

    MATH  MathSciNet  Google Scholar 

  23. Sacks P., Uhlenbeck K.: On the existence of minimal immersions of 2-spheres. Ann. Math. 113, 1–24 (1981)

    Article  MathSciNet  Google Scholar 

  24. Schoen R., Uhlenbeck K.: Boundary regularity and the Dirichlet problem for harmonic maps. J. Differ. Geom. 18, 253–268 (1983)

    MATH  MathSciNet  Google Scholar 

  25. Sedlacek S.: A direct method for minimizing the Yang-Mills functional over 4-manifolds. Comm. Math. Phys. 86, 515–527 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  26. Shevchishin V.: Limit holonomy and extension properties of Sobolev and Yang-Mills bundles. J. Geom. Anal. 12, 493–528 (2002)

    MATH  MathSciNet  Google Scholar 

  27. Simon L.: Lectures on Geometric Measure Theory. The Center for Mathematical Analysis, Canberra (1983)

    MATH  Google Scholar 

  28. Struwe M.: A global compactness result for elliptic boundary problems involving limiting nonlinearities. Math. Z. 187, 511–517 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  29. Taubes C.H.: Path-connected Yang-Mills moduli spaces. J. Differ. Geom. 19, 337–392 (1984)

    MATH  MathSciNet  Google Scholar 

  30. Taylor, M.E.: Partial Differential Equations I–III. Applied Mathematical Sciences 115–117, Springer (1996).

  31. Tian G.: Gauge theory and calibrated geometry. Ann. Math. 151, 193–268 (2000)

    Article  MATH  Google Scholar 

  32. Uhlenbeck K.: Connections with L p bounds on curvature. Comm. Math. Phys. 83, 31–42 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  33. Uhlenbeck K.: Removable singularities in Yang-Mills fields. Comm. Math. Phys. 83, 11–29 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  34. Uhlenbeck K.: The Chern classes of Sobolev connections. Comm. Math. Phys. 101, 449–457 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  35. White B.: Infima of energy functionals in homotopy classes of mappings. J. Differ. Geom. 236, 127–142 (1986)

    Google Scholar 

  36. White B.: Homotopy classes Sobolev spaces and the existence of energy minimizing maps. Acta Math. 160, 1–17 (1988)

    Article  MATH  MathSciNet  Google Scholar 

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  1. Department of Mathematics, Graduate School of Science and Engineering, Tokyo Institute of Technology, 2-12-1 Oh-okayama, Meguro-ku, Tokyo, 152-8551, Japan

    Takeshi Isobe

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  1. Takeshi Isobe
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Isobe, T. Topological and analytical properties of Sobolev bundles, I: The critical case. Ann Glob Anal Geom 35, 277–337 (2009). https://doi.org/10.1007/s10455-008-9137-5

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