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  "body": "\n## Statement\n\nLet $B \\subseteq \\R^4$ be the open unit ball with a smooth Riemannian metric, and let $G$ be a compact Lie group.\nLet $P \\to B \\setminus \\set{0}$ be a [[principal $G$-bundle|8E37E74XVP]].\nLet $A$ be a smooth [[connection|NJ21PN5ZGJ]] on $P$ that satisfies the [[Yang–Mills equations|JJWETW5AT8]] $d_A^* F_A = 0$ on $B \\setminus \\set{0}$.\nAssume that $A$ has finite energy:\n$$\\int_{B \\setminus \\set{0}} \\abs{F_A}^2 \u003c \\infty,$$\nwhere $F_A$ is the [[curvature|ZAJFW1S078]] of $A$.\nThen $P$ extends to a principal $G$-bundle $\\bar{P}$ over $B$.\nThere is a [[gauge transformation|707DEWCCFX]] $u$ on $B \\setminus \\set{0}$ such that $u^* A$ extends to a smooth Yang–Mills connection on $\\bar{P}$.\n\n## Remarks\n\nThe proof uses the conformal invariance of the energy in dimension $4$.\nOn dyadic annuli around $0$ the energy tends to $0$, so Uhlenbeck's gauge fixing gives good gauges on each annulus.\nA decay estimate shows that $\\abs{F_A}$ is in fact bounded near $0$.\nThen a Coulomb gauge on the whole ball and elliptic regularity give the smooth extension.\nThe finite energy hypothesis cannot be dropped.\nA connection that extends smoothly over $0$ has bounded curvature near $0$, and the energy is gauge invariant.\nSo a Yang–Mills connection with infinite energy on every punctured ball around $0$ never extends over $0$ in any gauge.\nExample: a finite-energy anti-self-dual connection on $\\R^4$ pulls back, via stereographic projection, to an instanton on $S^4$ minus a point.\nThe theorem extends it over that point, so it comes from a bundle over $S^4$.\nIts energy is a fixed constant times the absolute value of the second Chern number of that bundle.\n\n## Explanations\n\n### L1 human\n\nA Yang–Mills field defined everywhere except at one point of four-dimensional space might be badly singular there.\nThe theorem says that if the total energy of the field is finite, then the singularity is only apparent.\nAfter a change of gauge, the field extends smoothly across the point.\n\n### L2 human\n\nThe 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.\nIt is also the second ingredient, next to the Uhlenbeck compactness theorem, in the description of limits of sequences of instantons.\nThe limit connection away from the bubble points has finite energy, so it extends over those points to a connection on a new bundle.\n",
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  "msc": [
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  "name": "Uhlenbeck's removable singularity theorem",
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