The uniqueness of tangent cones for Yang-Mills connections with isolated singularities.

*(English)*Zbl 1049.53021Let \(M\) be an \(n\)-dimensional manifold and \(E\) a vector bundle associated to a principal bundle \(P\) on \(M\) with compact structure group \(G\). Assuming that \((M, g)\) is an \(n\)-dimensional Riemannian manifold, \(n\geq 5\), \(x_0\in M\) and \((E,h)\) a bundle over \(M\)-\(\{x_0\}\), \(h\) being a \(G\)-invariant metric, let \(A\) be a Yang-Mills connection on \(E\) with an isolated singularity at \(x_0\). The author proves the uniqueness, up to gauge transformations, of the tangent cone (or tangent connection) of \(A\) at \(x_0\), under a quadratic growth assumption on the curvature of \(A\) in a neighbourhood of \(x_0\).

He also gives an estimate of the rate of the asymptotic convergence of \(A\) to its cone when \(A\) is stationary, since in this case the a priori estimates of K. K. Uhlenbeck [Commun. Math. Phys. 83, 31–42 (1982; Zbl 0499.58019)] and H. Nakajima [J. Math. Soc. Japan 40, No. 3, 383–392 (1988; Zbl 0647.53030)] ensure the growth condition on the curvature of \(A\). Furthermore, such a convergence rate becomes faster when the tangent cone is assumed to be integrable, even for non-stationary \(A\).

Technically, since the Yang-Mills equation has a degenerate elliptic nature, the author looks for a suitable gauge (using Coulomb gauge) having a long time existence so that he can adapt L. Simon’s method [Ann. Math. (2) 118, 525–571 (1983; Zbl 0549.35071)]. This is achieved by means of a partition of the existence interval of the solution in three parts corresponding to different growth behaviours of the norm of the solution. Such behaviours are modelled on those of the solutions to the linearized equations, and this allows to control their norms on each interval using techniques differing from the variational inequality by Simon and the property that, under the constructed gauge, the time derivative of the connection is uniformly small on the existence interval.

Finally, the author considers Yang-Mills (Y-M) flows which start from a connection sufficiently close, in norm, to a smooth local minimizer of the Y-M functional and he proves the asymptotic convergence to a suitable Y-M connection near the minimizer. Again he constructs a suitable gauge and uses the Simon result for parabolic evolution equations.

He also gives an estimate of the rate of the asymptotic convergence of \(A\) to its cone when \(A\) is stationary, since in this case the a priori estimates of K. K. Uhlenbeck [Commun. Math. Phys. 83, 31–42 (1982; Zbl 0499.58019)] and H. Nakajima [J. Math. Soc. Japan 40, No. 3, 383–392 (1988; Zbl 0647.53030)] ensure the growth condition on the curvature of \(A\). Furthermore, such a convergence rate becomes faster when the tangent cone is assumed to be integrable, even for non-stationary \(A\).

Technically, since the Yang-Mills equation has a degenerate elliptic nature, the author looks for a suitable gauge (using Coulomb gauge) having a long time existence so that he can adapt L. Simon’s method [Ann. Math. (2) 118, 525–571 (1983; Zbl 0549.35071)]. This is achieved by means of a partition of the existence interval of the solution in three parts corresponding to different growth behaviours of the norm of the solution. Such behaviours are modelled on those of the solutions to the linearized equations, and this allows to control their norms on each interval using techniques differing from the variational inequality by Simon and the property that, under the constructed gauge, the time derivative of the connection is uniformly small on the existence interval.

Finally, the author considers Yang-Mills (Y-M) flows which start from a connection sufficiently close, in norm, to a smooth local minimizer of the Y-M functional and he proves the asymptotic convergence to a suitable Y-M connection near the minimizer. Again he constructs a suitable gauge and uses the Simon result for parabolic evolution equations.

Reviewer: Anna Maria Pastore (Bari)

##### MSC:

53C07 | Special connections and metrics on vector bundles (Hermite-Einstein, Yang-Mills) |

58E15 | Variational problems concerning extremal problems in several variables; Yang-Mills functionals |

##### Keywords:

Yang-Mills connection; gauge transformations; tangent cone; asymptotic convergence; variational inequality##### References:

[1] | Adams, D.; Simon, L., Rates of asymptotic convergence near isolated singularities of geometric extrema, Indiana univ. math. J., 37, 225-254, (1988) · Zbl 0669.49023 |

[2] | Allard, W.K.; Almgren, F., On the radial behavior of minimal surfaces and the uniqueness of their tangent cones, Ann. of math., 113, 2, 215-265, (1981) · Zbl 0437.53045 |

[3] | Cheeger, J.; Tian, G., On the cone structure at infinity of Ricci flat manifolds with Euclidean volume growth and quadratic curvature decay, Invent. math., 118, 493-571, (1994) · Zbl 0814.53034 |

[4] | Donaldson, S.K.; Kronheimer, P.B., The geometry of four-manifolds, (1990), Oxford University Press New York · Zbl 0820.57002 |

[5] | Freed, D.S.; Uhlenbeck, K.K., Instantons and four-manifolds, Mathematical science research institute publication, Vol. 1, (1984), Springer New York · Zbl 0559.57001 |

[6] | Gilbarg, D.; Trudinger, N.S., Elliptic partial differential equations of second order, (1983), Springer New York · Zbl 0691.35001 |

[7] | Giusti, E., Minimal surfaces and functions of bounded variation, Monographs in mathematics, Vol. 80, (1984), Birkhäuser Basel · Zbl 0545.49018 |

[8] | Morgan, J.W.; Mrowka, T.; Ruberman, D., The L2-moduli space and a vanishing theorem for Donaldson polynomial invariants, (1994), International Press Cambridge, MA · Zbl 0830.58005 |

[9] | Morrey, C.B., Multiple integrals in the calculus of variations, (1966), Springer New York · Zbl 0142.38701 |

[10] | Nakajima, H., Compactness of the moduli space of Yang-Mills connections in higher dimensions, J. math. soc. Japan, 40, 383-392, (1988) · Zbl 0647.53030 |

[11] | Price, P., A monotonicity formula for Yang-Mills fields, Manuscripta math., 43, 131-166, (1983) · Zbl 0521.58024 |

[12] | Schoen, R.; Uhlenbeck, K., A regularity theory for harmonic maps, J. differential geom., 17, 2, 307-335, (1982) · Zbl 0521.58021 |

[13] | Simon, L., Asymptotics for a class of nonlinear evolution equations, with applications to geometric problems, Ann. of math. (2), 118, 3, 525-571, (1983) · Zbl 0549.35071 |

[14] | L. Simon, Lectures on geometric measure theory, Australian National University, 1983. · Zbl 0546.49019 |

[15] | L. Simon, Theorems on regularity and singularity of energy minimizing maps, Lecture in Mathematics, ETH Zürich, 1996. · Zbl 0864.58015 |

[16] | T. Tao, G. Tian, A singualarity removal theorem for Yang-Mills fields in higher dimensions, preprint, 2001. |

[17] | Tian, G., Gauge theory and calibrated geometry, I, Ann. math., 151, 2, 193-268, (2000) · Zbl 0957.58013 |

[18] | Tian, G.; Yang, B., Compactification of the moduli spaces of vortices and coupled vortices, J. reine angew. math., 553, 17-41, (2002) · Zbl 1022.53025 |

[19] | Uhlenbeck, K.K., Connections with Lp bounds on curvature, Comm. math. phys., 83, 1, 31-42, (1982) · Zbl 0499.58019 |

[20] | B. Yang, Construction of Yang-Mills connections with given asymptotic tangent cones, preprint, 2001, available at http://math.stanford.edu/ byang. |

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