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  "body": "\n## Statement\n\nLet $X$ be a compact [[Riemannian manifold|7CBT4SM83Y]] of dimension $n$, let $G$ be a compact Lie group, and let $P \\to X$ be a [[principal $G$-bundle|8E37E74XVP]].\nLet $p \u003e n/2$, and let $(A_i)$ be a sequence of [[connections|NJ21PN5ZGJ]] on $P$ of [[Sobolev class|M4CEZ5K9HP]] $L^p_1$ whose [[curvatures|ZAJFW1S078]] are bounded in $L^p$:\n$$\\sup_i \\norm{F_{A_i}}_{L^p} \u003c \\infty.$$\nThen there are a subsequence $(A_{i_j})$, [[gauge transformations|707DEWCCFX]] $u_j$ of class $L^p_2$, and a connection $A$ of class $L^p_1$ with this property.\nThe connections $u_j^* A_{i_j}$ converge to $A$ weakly in $L^p_1$.\n\n## Remarks\n\nThe 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 $\\norm{A}_{L^p_1} \\le C \\norm{F_A}_{L^p}$.\nFor $p \u003e n/2$, Hölder's inequality turns the $L^p$ bound into a small $L^{n/2}$ bound on balls of a fixed small radius.\nSo one can fix the gauge on each ball of a finite cover, extract weakly convergent subsequences, and patch the local gauges together.\nThe condition $p \u003e n/2$ is needed.\nIn 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.\nThen a sequence may have no subsequence that converges, after gauge, on the whole of $X$.\nFor 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.\nBy Uhlenbeck's removable singularity theorem the limit is a connection on a possibly different bundle; this is the bubbling picture.\n\n## Explanations\n\n### L1 human\n\nA connection can be described in many ways, one for each choice of gauge, and in a bad gauge a harmless sequence can look as if it diverges.\nThe theorem says that if the curvature is bounded in a strong enough sense, then one can choose good gauges in which the sequence has a convergent subsequence.\nSo a bound on curvature, which is gauge invariant, controls the connections themselves.\n\n### L2 human\n\nThe theorem plays for connections the role that the Rellich and Arzelà–Ascoli compactness theorems play for functions.\nIn dimension $4$ and at the critical exponent it underlies the Uhlenbeck compactification of instanton moduli spaces.\nThere, limits of instantons are instantons on a bundle with smaller charge, together with points where curvature has concentrated.\n\nThis compactification is a central tool in Donaldson theory.\n",
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