Uhlenbeck removable singularity theorem - Imported Knowlpedia Content

Statement

Let $G$ be compact and let $A$ be a smooth Yang–Mills connection on a principal $G$ -bundle over the punctured Riemannian four-ball $B^4\setminus\{0\}$ . If its curvature has finite energy,

\int_{B^4}|F_A|^2\,d\operatorname{vol}<\infty,

then there is a gauge in which $A$ extends across $0$ to a smooth Yang–Mills connection on a principal $G$ -bundle over all of $B^4$ . In particular, the apparent isolated singularity is a gauge artifact. The same conclusion applies to self-dual and anti-self-dual connections, since they satisfy the Yang–Mills equation.

Proof mechanism

Finite total energy implies that the curvature energy on sufficiently small annuli tends to zero. Uhlenbeck’s small-curvature theorem supplies compatible Coulomb gauges with critical Sobolev control. In these gauges the Yang–Mills equation becomes an elliptic system, and decay estimates improve the connection’s regularity until standard elliptic bootstrapping gives a smooth extension Uhlenbeck, main theorem.

The conclusion concerns the connection modulo gauge, not the coefficients in an arbitrary trivialization. A badly chosen gauge may remain singular even when the underlying connection extends smoothly.

Role in compactness

In Uhlenbeck compactness, the limiting connection is first constructed only away from finitely many concentration points. The removable singularity theorem extends it over each point, possibly on a bundle with a different topological class from the original one. The discrepancy is recorded by the instanton bubbles and their concentrated energy.

Applying the theorem at infinity after conformally identifying $\mathbb R^4\cup\{\infty\}$ with $S^4$ shows that a finite-energy Yang–Mills field on $\mathbb R^4$ extends over infinity modulo gauge.

Scope and non-example

References

Karen K. Uhlenbeck, “Removable Singularities in Yang–Mills Fields,” Communications in Mathematical Physics 83 (1982), 11–29. DOI record. Relevant: the main removable-singularity theorem and its application at infinity.
Daniel S. Freed and Karen K. Uhlenbeck, Instantons and Four-Manifolds, 2nd ed., Springer, 1991. DOI record. Relevant: Chapter 4, removable singularities and compactification.