We show the existence of Yang-Mills-Higgs (YMH) fields over a Riemann surface with boundary where a free boundary condition is imposed on the section and a Neumann boundary condition on the connection. In technical terms, we study the convergence and blowup behavior of a sequence of Sacks-Uhlenbeck type α-YMH fields as α → 1. For α > 1, each α-YMH field is shown to be smooth up to the boundary under some gauge transformation. This is achieved by showing a regularity theorem for more general coupled systems, which extends the classical results of Ladyzhenskaya-Ural'ceva and Morrey.
ABSTRACT. We consider the gauge transformations of a metric G-bundle over a compact Riemannian surface with boundary. By employing the heat flow method, the local existence and the long time existence of generalized solution are proved.
For a sequence of approximate Dirac-harmonic maps from a closed spin Riemann surface into a stationary Lorentzian manifold with uniformly bounded energy, we study the blow-up analysis and show that the Lorentzian energy identity holds. Moreover, when the targets are static Lorentzian manifolds, we prove the positive energy identity and the no neck property.
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