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Minimizers of higher order gauge invariant functionals. (English) Zbl 1353.58007

The authors find two higher-order analogs of the Yang-Mills gauge functional for which the analytic theory of Uhlenbeck-Sedláček can be reproduced most closely. One of them is a generalization of the bi-Yang-Mills functional of Bejan-Urakawa, and is given by \[ Y_n(A):=\begin{cases}\int_M|(d_Ad_A^*)^{\frac{n-2}2}F_A|^2+|F_A|^ndx, n\mathrm{ even}\\ \int_M|d_A^* (d_Ad_A^*)^{\frac{n-3}2}F_A|^2+|F_A|^ndx, n\mathrm{ odd}.\end{cases} \] The other involves the full covariant derivative \(D_A\) rather than just the differential: \[ Z_n(A):=\int_M|(D_A)^{n-2}F_A|^2+|F_A|^2dx\,. \] Both functionals are gauge invariant, and scale invariant in the critical dimension \(\dim M=2n\). It turns out that a natural generalization of the classical space \(W^{1,2}\) for minimizers is \(W^{n-1,2}\) rather than \(W^{1,n}\), as previously expected. Either of the two functionals controls the \(W^{n-2,2}\) norms of \(F_A\) (“global” coercivity), and the \(W^{n-1,2}\) norms of \(A\) (in the Uhlenbeck gauge) in dimensions \(\leq 2n\) (“local” coercivity). A recent result of Petrache-Rivière also generalizes: a point singularity of a bundle can be removed if there is a \(W^{n-1,2}\) connection around the point. Weak solutions to the Euler-Lagrange equations in dimensions \(\leq 2n\) are smooth. In the subcritical dimensions \(<2n\) the minimizers exist even on the original bundle (assuming the compact structure group), and are smooth. In the critical dimension \(2n\) the Uhlenbeck gauge can be chosen only away from finitely many points, where removable singularities may develop. After their removal the minimizers turn out to live on a new bundle, which however shares all the Chern classes with the original one, and the minimizers are still smooth.

MSC:

58E15 Variational problems concerning extremal problems in several variables; Yang-Mills functionals
81T13 Yang-Mills and other gauge theories in quantum field theory
53C07 Special connections and metrics on vector bundles (Hermite-Einstein, Yang-Mills)
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References:

[1] Bejan, C.-L., Urakawa, H.: Yang-Mills Fields Analogue of Biharmonic Maps. Topics in Almost Hermitian Geometry and Related Fields. World Scientific Publishing, Hackensack (2005)
[2] Douglis, A., Nirenberg, L.: Interior estimates for elliptic systems of partial differential equations. Comm. Pure Appl. Math. 8, 503-538 (1955) · Zbl 0066.08002 · doi:10.1002/cpa.3160080406
[3] Gastel, A., Scheven, C.: Regularity of polyharmonic maps in the critical dimension. Comm. Anal. Geom. 17(2), 185-226 (2009) · Zbl 1183.58010 · doi:10.4310/CAG.2009.v17.n2.a2
[4] Giaquinta, M.: Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems. Annals of Mathematics Studies, vol. 105, Princeton University Press, Princeton (1983) · Zbl 0516.49003
[5] Ichiyama, T., Inoguchi, J., Urakawa, H.: Biharmonic maps and Bi-Yang-Mills fields. Note Mat. 28(suppl. 1), 233-275 (2008) · Zbl 1201.58012
[6] Ichiyama, T., Inoguchi, J., Urakawa, H.: Classifications and isolation phenomena of bi-harmonic maps and Bi-Yang-Mills fields. Note Mat. 30, 15-48 (2010) · Zbl 1244.58006
[7] Isobe, T.: A regularity result for a class of degenerate Yang-Mills connections in critical dimensions. Forum Math. 20, 1109-1139 (2008) · Zbl 1160.58009 · doi:10.1515/FORUM.2008.051
[8] Iwaniec, T., Scott, C., Stroffolini, B.: Nonlinear Hodge theory on manifolds with boundary. Ann. Mat. Pura Appl. (4) 177, 37-115 (1999) · Zbl 0963.58003 · doi:10.1007/BF02505905
[9] Meyer, Y., Rivière, T.: A partial regularity result for a class of stationary Yang-Mills fields in high dimension. Rev. Mat. Iberoamericana 19(1), 195-219 (2003) · Zbl 1127.35317 · doi:10.4171/RMI/343
[10] Naber, G.L.: Topology, Geometry, and Gauge Fields: Foundations. Texts in Applied Mathematics, 2nd edn. Springer, New York (2011) · Zbl 1231.53002
[11] Nirenberg, L.: An extended interpolation inequality. Ann. Scuola Norm. Sup. Pisa (3) 20, 733-737 (1966) · Zbl 0163.29905
[12] Petrache, M., Rivière, T.: The Resolution of the Yang-Mills Plateau Problem in Super-Critical Dimensions (2013) Preprint · Zbl 1371.58014
[13] Rivière, T.: Interpolation spaces and energy quantization for Yang-Mills fields. Comm. Anal. Geom. 10, 683-708 (2002) · Zbl 1018.58006 · doi:10.4310/CAG.2002.v10.n4.a2
[14] Sedlacek, S.: A direct method for minimizing the Yang-Mills functional. Comm. Math. Phys. 86, 515-527 (1982) · Zbl 0506.53016 · doi:10.1007/BF01214887
[15] Tao, T., Tian, G.: A singularity removal theorem for Yang-Mills fields in higher dimensions. J. Am. Math. Soc. 17, 557-593 (2004) · Zbl 1086.53043 · doi:10.1090/S0894-0347-04-00457-6
[16] Uhlenbeck, K.: Removable singularities in Yang-Mills fields. Comm. Math. Phys. 83, 11-29 (1982) · Zbl 0491.58032 · doi:10.1007/BF01947068
[17] Uhlenbeck, K.: Connections with \[L^p\] Lp bounds on curvature. Comm. Math. Phys. 83, 31-42 (1982) · Zbl 0499.58019 · doi:10.1007/BF01947069
[18] Uhlenbeck, K.: The Chern classes of Sobolev connections. Comm. Math. Phys. 101, 449-457 (1985) · Zbl 0586.53018 · doi:10.1007/BF01210739
[19] Wehrheim, K.: Uhlenbeck Compactness. Series of Lectures in Mathematics. EMS, Zürich (2004) · Zbl 1055.53027 · doi:10.4171/004
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