Global analysis, analysis on manifolds Manifolds of mappings and moduli spaces Theorem

Uhlenbeck's removable singularity theorem

Builds on Principal bundle Connection on a principal bundle Curvature form Yang–Mills equations Gauge group

Statement

Let $B \subseteq \R^4$ be the open unit ball with a smooth Riemannian metric, and let $G$ be a compact Lie group. Let $P \to B \setminus \set{0}$ be a principal $G$ -bundle. Let $A$ be a smooth connection on $P$ that satisfies the Yang–Mills equations $d_A^* F_A = 0$ on $B \setminus \set{0}$ . Assume that $A$ has finite energy: $\int_{B \setminus \set{0}} \abs{F_A}^2 < \infty,$ where $F_A$ is the curvature of $A$ . Then $P$ extends to a principal $G$ -bundle $\bar{P}$ over $B$ . There is a gauge transformation $u$ on $B \setminus \set{0}$ such that $u^* A$ extends to a smooth Yang–Mills connection on $\bar{P}$ .

Remarks

The proof uses the conformal invariance of the energy in dimension $4$ . On dyadic annuli around $0$ the energy tends to $0$ , so Uhlenbeck's gauge fixing gives good gauges on each annulus. A decay estimate shows that $\abs{F_A}$ is in fact bounded near $0$ . Then a Coulomb gauge on the whole ball and elliptic regularity give the smooth extension. The finite energy hypothesis cannot be dropped. A connection that extends smoothly over $0$ has bounded curvature near $0$ , and the energy is gauge invariant. So a Yang–Mills connection with infinite energy on every punctured ball around $0$ never extends over $0$ in any gauge. Example: a finite-energy anti-self-dual connection on $\R^4$ pulls back, via stereographic projection, to an instanton on $S^4$ minus a point. The theorem extends it over that point, so it comes from a bundle over $S^4$ . Its energy is a fixed constant times the absolute value of the second Chern number of that bundle.

Going deeper

For a student

The theorem is the reason why finite-energy instantons on $\R^4$ correspond to instantons on the sphere $S^4$ , and why their energy is quantised. It is also the second ingredient, next to the Uhlenbeck compactness theorem, in the description of limits of sequences of instantons. The limit connection away from the bubble points has finite energy, so it extends over those points to a connection on a new bundle.

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Connection on a principal bundle concept
Curvature form concept
Lie group concept
Principal bundle concept
Smooth manifold concept
Yang–Mills equations concept
Gauge group object
Uhlenbeck's removable singularity theorem statement

Tag	`B30ACDNS76`
URI	`m2://B30ACDNS76`
Kind	statement, theorem
Writing	W0 stub
MSC	58E15, 53C07, 81T13
Status	This copy of the site has no build data. Run `m2 status B30ACDNS76`.
About	Yang–Mills equations
Uses	Principal bundle, Connection on a principal bundle, Curvature form, Yang–Mills equations, Gauge group