Uhlenbeck compactness theorem

Remarks

The key step is Uhlenbeck's gauge fixing lemma: on a ball, a connection with small enough curvature in $L^{n/2}$ is gauge equivalent to one in Coulomb gauge, $d^* A = 0$, with $\norm{A}_{L^p_1} \le C \norm{F_A}_{L^p}$. For $p > n/2$, Hölder's inequality turns the $L^p$ bound into a small $L^{n/2}$ bound on balls of a fixed small radius. So one can fix the gauge on each ball of a finite cover, extract weakly convergent subsequences, and patch the local gauges together. The condition $p > n/2$ is needed. In dimension $4$ with only an $L^2$ bound on curvature, the scale-invariant case $p = n/2$, the curvature of instantons can concentrate at points. Then a sequence may have no subsequence that converges, after gauge, on the whole of $X$. For a sequence of instantons on a bundle of fixed charge, a refinement of the argument gives convergence after gauge on the complement of finitely many points. By Uhlenbeck's removable singularity theorem the limit is a connection on a possibly different bundle; this is the bubbling picture.

Going deeper

For a student

The theorem plays for connections the role that the Rellich and Arzelà–Ascoli compactness theorems play for functions. In dimension $4$ and at the critical exponent it underlies the Uhlenbeck compactification of instanton moduli spaces. There, limits of instantons are instantons on a bundle with smaller charge, together with points where curvature has concentrated.

This compactification is a central tool in Donaldson theory.