Surveys in Differential Geometry

Volume 24 (2019)

Removeability of a codimension four singular set for solutions of a Yang–Mills–Higgs equation with small energy

Pages: 257 – 291



Penny Smith (Department of Mathematics, Lehigh University, Bethlehem, Pennsylvania, U.S.A.)

Karen Uhlenbeck (School of Mathematics Institute for Advanced Study, Princeton, New Jersey, U.S.A.)


The Yang–Mills–Higgs equations, while they arose in theoretical physics, have become a central object of study in geometric analysis. The moduli spaces of solutions are used to study problems in algebraic geometry as well as topology. To understand these moduli spaces, it is natural to ask what sort of objects are limits in a weak sense of sequences of solutions. In many examples, it can be shown that subsequences of solutions converge to a solution off a set of Hausdorff codimension $4$. In four dimensions, these point singularities are removable. The question arises of when this singular set of dimension $n-4$ is removable in higher dimensions. In a classic paper, Tao and Tian [11] prove an important step in the removability theorem. What they show is that, in general, independent of the equation, if the case the curvature of a connection defined on a set of Hausdorff codimension $4$ is sufficiently small in a certain Morrey space, that there exists a smaller ball and a gauge in which the connection is $d+A$, where $A$ is co-closed and bounded by norms on the curvature. One can now treat the Yang–Mills–Higgs equations as a standard elliptic system and obtain further regularity. This choice of Morrey space is natural, because monotonicity of solutions provides estimates exactly in this space. However, since the solution is only defined off a set of Hausdorff codimension $4$, it is not known that the monotonicity theorem is true for limiting singular solutions. Only by adding the assumption of stationary with respect to diffeomorphisms in the entire domain can one apply the theorem to limiting singular connections. Unfortunately, we cannot fill in this gap. We provide a new proof of the Tao–Tian result. Our proof uses the equation and a maximum principle for a differential inequality derived from the equation via averaging. In this manner, we can avoid the geometric construction of a good gauge in a weak setting via averaging exponential gauges, as we first obtain further estimates on the curvature from the inequality. Our main result Corollary 4 is that, if the curvature is sufficiently small in a Morrey space, and the solution is defined and smooth off a set of Hausdorff codimension $4$ with the covariant derivative of the Higgs field in the same Morrey space, then the solution is smoothly gauge equivalent to one which extends smoothly over the singular set. In the case that the solution has small energy, and is a critical point of the integral, or even just critical with respect to diffeomorphisms in the entire domain, not just off the singular set, the smallness of the curvature in the Morrey space in a smaller domain can be verified and leads to Theorem 11 as a corollary of Corollary 4. There may be other ways to obtain the bounds on the curvature in the Morrey space; hence we state the main result in terms of small curvature in the Morrey space. The results apply to most of the coupled Yang–Mills–Higgs equations, and we do not go into the details in this paper. We do assume that the Higgs field is bounded, and explain why this follows from $L^2$ bounds on it. Many interesting equations, such as the Kapustin–Witten equations and the equations for a complex flat connection do not come with such a bound on sequences. Moreover, the singular set of a renormalized limit is Hausdorff codimension $2$, and we sadly have nothing to say about these singularities.

We develop a new method for proving regularity for small energy stationary solutions of coupled gauge field equations. Our results duplicate those of [10] for the pure Yang–Mills equations, but our proof is simpler, and obtains bounded curvature without the use of Coulomb gauges. It relies instead on the Weitzenblock formulae, and an improved Kato inequality. Our results also extend and simplify those of [3].

2010 Mathematics Subject Classification

05A15, 15A18, 05C38, 15A15

Published 29 December 2021