The boundary value problem for Yang--Mills--Higgs fields

TL;DR

This paper proves the existence and analyzes the boundary behavior of Yang--Mills--Higgs fields on Riemann surfaces with boundary, extending regularity results for coupled systems and studying the convergence of $ ext{α}$-YMH fields.

Contribution

It establishes existence, boundary regularity, and convergence properties of Yang--Mills--Higgs fields with free boundary conditions on Riemann surfaces.

Findings

Existence of YMH fields with boundary conditions

Boundary regularity of $ ext{α}$-YMH fields for $ ext{α}>1$

Extension of regularity theorems to coupled systems

Abstract

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 blow-up behavior of a sequence of Sacks-Uhlenbeck type $\alpha$-YMH fields as $\alpha\to 1$. For $\alpha>1$, each $\alpha$-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.