Geodesic
Singular Bundles with Bounded $L^2$-Curvatures
Bollettino della Unione matematica italiana, Série 9, Tome 1 (2008) no. 3, pp. 881-901.

Voir la notice de l'article provenant de la source Biblioteca Digitale Italiana di Matematica

We consider calculus of variations of the Yang-Mills functional in dimensions larger than the critical dimension 4. We explain how this naturally leads to a class of - a priori not well-defined - singular bundles including possibly "almost everywhere singular bundles". In order to overcome this difficulty, we suggest a suitable new framework, namely the notion of singular bundles with bounded $L^2$-curvatures. 
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Kessel, Thiemo; Rivière, Tristan. Singular Bundles with Bounded $L^2$-Curvatures. Bollettino della Unione matematica italiana, Série 9, Tome 1 (2008) no. 3, pp. 881-901. http://geodesic.mathdoc.fr/item/BUMI_2008_9_1_3_a17/

[Be1] F. Bethuel, The approximation problem for Sobolev maps between two manifolds. Acta Math., 167, no. 3-4 (1991), 153-206. | DOI | MR | Zbl

[Be2] F. Bethuel, A characterization of maps in $H^1(B^3, S^2)$ which can be approximated by smooth maps. Ann. Inst. H. Poincaré Anal. Non Linéaire, 7, no. 4 (1990), 269-286. | fulltext EuDML | DOI | MR | Zbl

[BCDH] F. Bethuel - J. M. Coron - F. Demengel - F. Hélein, A cohomological criterion for density of smooth maps in Sobolev spaces between two manifolds. Nematic (Orsay, 1990), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 332, Kluwer Acad. Publ., Dordrecht, (1991), 15-23. | MR | Zbl

[BBC] F. Bethuel - H. Brezis - J.-M. Coron, Relaxed energies for harmonic maps. Variational methods (Paris, 1988), 37-52, Progr. Nonlinear Differential Equations Appl., 4, Birkauser Boston, Boston, MA, 1990. | DOI | MR | Zbl

[BCL] H. Brezis - J.M. Coron - E. Lieb, Harmonic maps with defects. Comm. Math. Phys., 107, no. 4 (1986), 649-705. | MR | Zbl

[DK] S. K. Donaldson - P. B. Kronheimer, `The geometry of four-manifolds'. Oxford, (1990). | MR | Zbl

[DT] S. K. Donaldson - R. P. Thomas, Gauge theory in higher dimensions. The geometric universe (Oxford, 1996), 31-47, Oxford Univ. Press, Oxford, 1998. | MR | Zbl

[DV] M. Dubois-Violette, Équation de Yang et Mills, modèles $\sigma$ à deux dimensions et généralisation. Progr. Math., 37, Birkhauser Boston, Boston, MA, 1983. | MR | Zbl

[FrU] D. Freed - K. Uhlenbeck, Instantons and four-manifolds. MSRI Pub. 1, Springer, (1991). | DOI | MR

[GMS] M. Giaquinta - G. Modica - J. Souček, Cartesian currents in the calculus of variations I and II. Springer-Verlag, Berlin, 1998. | DOI | MR

[HL] R. Hardt - F.H. Lin, A remark on $H^1$ mappings of Riemannian manifolds. Manuscripta Math., 56 (1986), 1-10. | fulltext EuDML | DOI | MR

[He] F. Hélein, Harmonic maps, conservation laws and moving frames. Diderot, (1996). | DOI | MR

[Is1] T. Isobe, Energy estimate, energy gap phenomenon and relaxed energy for Yang-Mills functional. J. Geom. Anal., 8 (1998), 43-64. | DOI | MR | Zbl

[Is2] T. Isobe, Relaxed Yang-Mills functional over 4 manifolds. Topological Methods in Non Linear Analysis, 6 (1995), 235-253. | DOI | MR | Zbl

[KR] T. Kessel - T. Rivieáre, Approximation results for singular bundles with bounded $L^p$-curvatures. in preparation (2008).

[MR] Y. Meyer - T. Rivieáre, A partial regularity result for a class of stationary Yang-Mills fields in higher dimensions. Rev. Mat. Iberoamericana, 19, no. 1 (2003), 195-219. | fulltext EuDML | DOI | MR

[NR1] M. S. Narasimhan - S. Ramanan, Existence of universal connections. Amer. J. Math., 83 (1961), 563-572. | DOI | MR | Zbl

[NR2] M. S. Narasimhan - S. Ramanan, Existence of universal connections II. Amer. J. Math., 85 (1963), 223-231. | DOI | MR | Zbl

[Ri1] T. Rivieáre, Everywhere discontinuous harmonic maps into spheres. Acta Math., 175, no. 2 (1995), 197-226. | DOI | MR

[SaU] J. Sacks - K. Uhlenbeck, The existence of minimal immersions of 2-spheres. Ann. of Math., 113 (1981), 1-24. | DOI | MR | Zbl

[ScU] R. Schoen - K. Uhlenbeck, A regularity theory for harmonic maps. J. Diff. Geom., 17 (1982), 307-335. | MR | Zbl

[ScU2] R. Schoen - K. Uhlenbeck, Approximation theorems for Sobolev mappings preprint (1984).

[Se] S. Sedlacek, A Direct Method for Minimizing the Yang-Mills Functional over 4-Manifolds, Commun. Math. Phys., 86 (1982), 515-527. | MR | Zbl

[Ti] G. Tian, Gauge theory and calibrated geometry I. Ann. of Math. (2) 151, no. 1 (2000), 193-268. | fulltext EuDML | DOI | MR | Zbl

[TT] T. Tao - G. Tian, A singularity removal theorem for Yang-Mills fields in higher dimensions. J. Amer. Math. Soc., 17, no. 3 (2004), 557-593 | DOI | MR | Zbl

[Uh1] K. Uhlenbeck, Connections with $L^p$ bounds on curvature. Comm. Math. Phys., 83, (1982), 31-42. | MR | Zbl

[Uh2] K. Uhlenbeck, Removable singularities in Yang-Mills fields. Comm. Math. Phys., 83, (1982), 11-29. | MR | Zbl

[Uh3] K. Uhlenbeck, Variational problems for gauge fields. Seminar on Differential Geometry, Princeton University Press, 1982. | MR | Zbl

[Wh] B. White, Homotopy classes in Sobolev spaces and the existence of energy minimizing maps. Acta. Math., 160 (1988), 1-17. | DOI | MR | Zbl