Communications in Mathematical Physics

, Volume 365, Issue 2, pp 685–739 | Cite as

The Hyperbolic Yang–Mills Equation for Connections in an Arbitrary Topological Class

  • Sung-Jin Oh
  • Daniel TataruEmail author


This is the third part of a four-paper sequence, which establishes the Threshold Conjecture and the Soliton-Bubbling versus Scattering Dichotomy for the energy critical hyperbolic Yang–Mills equation in the (4 + 1)-dimensional Minkowski space-time. This paper provides basic tools for considering the dynamics of the hyperbolic Yang–Mills equation in an arbitrary topological class at an optimal regularity. We generalize the standard notion of a topological class of connections on \({\mathbb{R}^{d}}\), defined via a pullback to the one-point compactification \({\mathbb{S}^{d} = \mathbb{R}^{d} \cup \{\infty}\}\), to rough connections with curvature in the critical space \({L^{\frac{d}{2}}(\mathbb{R}^{d})}\). Moreover, we provide excision and extension techniques for the Yang–Mills constraint (or Gauss) equation, which allow us to efficiently localize Yang–Mills initial data sets. Combined with the results in the previous paper (Oh and Tataru in The hyperbolic Yang–Mills equation in the caloric gauge. Local well-posedness and control of energy dispersed solutions, 2017. arXiv:1709.09332), we obtain local well-posedness of the hyperbolic Yang–Mills equation on \({\mathbb{R}^{1+d}}\)\({(d \geq 4)}\) in an arbitrary topological class at optimal regularity in the temporal gauge (where finite speed of propagation holds). In addition, in the energy subcritical case d =  3, our techniques provide an alternative proof of the classical finite energy global well-posedness theorem of Klainerman–Machedon (Ann. Math. (2) 142(1):39–119, 1995., while also removing the smallness assumption in the temporal-gauge local well-posedness theorem of Tao (J. Differ. Equ. 189(2):366–382, 2003. Although this paper is a part of a larger sequence, the materials presented in this paper may be of independent and general interest. For this reason, we have organized the paper so that it may be read separately from the sequence.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.



S.-J. Oh was supported by the Miller Research Fellowship from the Miller Institute, UC Berkeley and the TJ Park Science Fellowship from the POSCO TJ Park Foundation. D. Tataru was partially supported by the NSF Grant DMS-1266182 as well as by a Simons Investigator Grant (Grant No. 291820) from the Simons Foundation.


  1. 1.
    Atiyah M.F., Hitchin N.J., Drinfeld V.G., Manin Y.I.: Construction of instantons. Phys. Lett. A 65(3), 185–187 (1978) ADSMathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Atiyah M.F., Hitchin N.J., Singer I.M.: Self-duality in four-dimensional Riemannian geometry. Proc. R. Soc. Lond. Ser. A 362(1711), 425–461 (1978) ADSMathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bor G.: Yang–Mills fields which are not self-dual. Commun. Math. Phys. 145(2), 393–410 (1992) ADSMathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bott R.: An application of the Morse theory to the topology of Lie-groups. Bull. Soc. Math. Fr. 84, 251–281 (1956) MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Chruściel, P.T., Delay, E.: On mapping properties of the general relativistic constraints operator in weighted function spaces, with applications. Mém. Soc. Math. Fr. (N.S.) 94, vi+103 (2003)Google Scholar
  6. 6.
    Corvino J.: Scalar curvature deformation and a gluing construction for the Einstein constraint equations. Commun. Math. Phys. 214(1), 137–189 (2000) ADSMathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Corvino J., Schoen R.M.: On the asymptotics for the vacuum Einstein constraint equations. J. Differ. Geom. 73(2), 185–217 (2006) MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Czimek, S.: An extension procedure for the constraint equations, preprint (2018) PDE 4:2. arXiv:1609.08814
  9. 9.
    Czimek, S.: Boundary harmonic coordinates and the localised bounded L 2 curvature theorem, preprint (2017). arXiv:1708.01667
  10. 10.
    Gursky, M., Kelleher, C., Streets, J.: A conformally invariant gap theorem in Yang–Mills theory, preprint (2017). arXiv:1708.01157
  11. 11.
    Klainerman S., Machedon M.: Finite energy solutions of the Yang-Mills equations in \({{\mathbb{R}}^{3+1}}\). Ann. Math. (2) 142(1), 39–119 (1995) MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Knapp, A.W.: Lie Groups Beyond an Introduction, Progress in Mathematics, 2nd edn., vol. 140. Birkhäuser, Basel (2002)Google Scholar
  13. 13.
    Kobayashi, S., Nomizu, K.: Foundations of Differential Geometry. vol. I. Wiley Classics Library, Wiley, New York, (1996). Reprint of the 1963 original, A Wiley-Interscience PublicationGoogle Scholar
  14. 14.
    Kobayashi, S., Nomizu, K.: Foundations of Differential Geometry. vol. II. Wiley Classics Library, Wiley, New York (1996). Reprint of the 1969 original, A Wiley-Interscience PublicationGoogle Scholar
  15. 15.
    Krieger J., Tataru D.: Global well-posedness for the Yang–Mills equation in 4 + 1 dimensions. Small energy. Ann. Math. (2) 185(3), 831–893 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Milnor, J.W., Stasheff, J.D.: Characteristic Classes, Annals of Mathematics Studies, No. 76. Princeton University Press, Princeton; University of Tokyo Press, Tokyo (1974)Google Scholar
  17. 17.
    Oh S.-J.: Gauge choice for the Yang–Mills equations using the Yang–Mills heat flow and local well-posedness in H 1. J. Hyperbolic Differ. Equ. 11(1), 1–108 (2014) MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Oh S.-J.: Finite energy global well-posedness of the Yang-Mills equations on \({{\mathbb{R}}^{1+3}}\): an approach using the Yang–Mills heat flow. Duke Math. J. 164(9), 1669–1732 (2015) MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Oh, S.-J., Tataru, D.: Local well-posedness of the (4 +  1)-dimensional Maxwell–Klein–Gordon equation at energy regularity. Ann. PDE 2(1), 70, Art. 2 (2016). arXiv:1503.01560,
  20. 20.
    Oh, S.-J., Tataru, D.: The Yang–Mills heat flow and the caloric gauge, preprint (2017). arXiv:1709.08599
  21. 21.
    Oh, S.-J., Tataru, D.: The hyperbolic Yang–Mills equation in the caloric gauge. Local well-posedness and control of energy dispersed solutions, preprint (2017). arXiv:1709.09332
  22. 22.
    Oh, S.-J., Tataru, D.: The Threshold Conjecture for the energy critical hyperbolic Yang–Mills equation, preprint (2017). arXiv:1709.08606
  23. 23.
    Oh, S.-J., Tataru, D.: The threshold theorem for the (4 + 1)-dimensional Yang–Mills equation: an overview of the proof, preprint (2017). arXiv:1709.09088
  24. 24.
    Parker T.H.: A Morse theory for equivariant Yang–Mills. Duke Math. J. 66(2), 337–356 (1992) MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Sadun L., Segert J.: Non-self-dual Yang–Mills connections with nonzero Chern number. Bull. Am. Math. Soc. (N.S.) 24(1), 163–170 (1991) MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Schoen R., Uhlenbeck K.: Boundary regularity and the Dirichlet problem for harmonic maps. J. Differ. Geom. 18(2), 253–268 (1983) MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Sibner L.M., Sibner R.J., Uhlenbeck K.: Solutions to Yang–Mills equations that are not self-dual. Proc. Natl. Acad. Sci. USA 86(22), 8610–8613 (1989) ADSMathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Stein, E.M.: Singular Integrals and Differentiability Properties of Functions. Princeton Mathematical Series, No. 30. Princeton University Press, Princeton (1970)Google Scholar
  29. 29.
    Tao T.: Local well-posedness of the Yang–Mills equation in the temporal gauge below the energy norm. J. Differ. Equ. 189(2), 366–382 (2003) ADSMathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Uhlenbeck K.K.: Connections with L p bounds on curvature. Commun. Math. Phys. 83(1), 31–42 (1982) ADSCrossRefzbMATHGoogle Scholar
  31. 31.
    Uhlenbeck K.K.: The Chern classes of Sobolev connections. Commun. Math. Phys. 101(4), 449–457 (1985) ADSMathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.KIASSeoulKorea
  2. 2.Department of MathematicsUC BerkeleyBerkeleyUSA

Personalised recommendations