Uhlenbeck compactness and removable singularities for self-dual connections

Fix finitely many positive coordinate-ball charts with compactly contained working subcharts covering $X$. A positive linear change makes the metric Euclidean at each chart center; shrink the charts so its coefficient matrix is within [F2]'s fixed small $C^0$ neighborhood throughout. For each sequence member, a frame at a chart center extends over that ball by radial parallel transport. This exists intrinsically: the ordered matrix integral series constructs transport in each bundle chart; finite subdivision of each compact path and uniqueness patch those solutions, and smooth parameter dependence makes the endpoint frame smooth. It is smooth at the center by the same local radial equation there. Thus the coordinate-ball trivializations can be chosen even for a sequence of different bundles; no common topological trivialization is being assumed. Metric energy and Euclidean coefficient energy are uniformly comparable on this finite list. List the coordinate balls with rational centers and radii whose quadrupled closures are in a chart. By successive bounded real subsequence extraction and one diagonal, arrange that the energy integral on every quadrupled listed ball has a limit. Choose $\theta >0$ so energy less than $2\theta$ on such a ball implies the Euclidean curvature smallness in [F1] and the resulting potential smallness in [F2], after dilation. Let $S$ be the points not lying in any listed ball whose quadrupled-ball limiting energy is less than $\theta$. Its complement is open. If $N$ distinct points lie in $S$, choose $N$ listed balls containing them with pairwise disjoint quadrupled closures; this is possible by the coordinate basis and finitely many positive distances. Each limiting energy is at least $\theta$, so $N\theta \le M$. Thus $S$ is finite. For every compact set in its complement finitely many good balls suffice, and all their required smallness bounds hold eventually.

On each good ball apply [F1] on its quadruple. The pulled-back form is Euclidean Coulomb, while its curvature satisfies the actual curved self-duality equation by gauge covariance. Apply [F2] on smaller subballs: the potentials have uniform bounds in every fixed derivative order on the working ball. The constructed gauges are smooth there: their weak equation $du=ua-A_{n}u$, with smooth $A_n,a$, first gives every local first Sobolev order and then every higher order by differentiation. On an overlap, its transformed transition $h_{ij}^n$ is unitary and satisfies $d{h}_{ij}^n=h_{ij}^n{a}_j^n-a_i^n{h}_{ij}^n$, with the fixed coordinate pullback on $a_j^n$ understood. This identity and the uniform local coefficient bounds give uniform derivative bounds on every smaller overlap, inductively; a bound on the transition itself is supplied by compact $SU(2)$. There is no assumption that two different coordinate charts define the same Euclidean Coulomb operator. Apply [F6] on countably many smaller compact balls, to all coefficient and transition derivatives, and diagonalize. This gives smooth limits $a_i,h_{ij}$, convergence in every local derivative norm, the transition equation and the exact cocycle identities. The latter pass pointwise by uniform convergence on triple-overlap compacts. By [F4] they define one smooth bundle $P_{\mathrm{\infty }}$ and connection $A_{\mathrm{\infty }}$ on $X\setminus S$. The equation passes to the smooth limits, so $F_{A_{\mathrm{\infty }}}^{-}=0$. On each compact subset its energy is the limit of the original energy and is at most $M$.

Establish removal for the possibly nontrivial limiting bundle near a point of $S$. The needed small punctured curvature gauge has an intrinsic construction. On a dyadic coordinate annulus, choose a frame at one basepoint and transport it along the two spherical path families of [F3]'s annular proof. Transport is defined in finitely many bundle charts along each path, and its variation identity is intrinsic: the covariant derivative of the endpoint frame is the integral of transported curvature over the path variation. The two caps' transports differ by the holonomy of the same explicit spherical lune of area at most $2\pi$. Thus their quaternionic average supplies a global annular frame with potential at most $Cr\mathrm{\parallel }F{\mathrm{\parallel }}_{\mathrm{\infty }}$, without assuming that the punctured bundle already has a global trivialization. On a sufficiently small punctured ball, finite energy and [F2] give, for $r_j=2^{-j}$ and $E_j$ the energy on $\{r_j\mathrm{/}4<\mathrm{\mid }x\mathrm{\mid }<4r_j\}$, the bounds $\mathrm{\parallel }F{\mathrm{\parallel }}_{L^{\mathrm{\infty }}(\{r_j\mathrm{/}2<\mathrm{\mid }x\mathrm{\mid }<2r_j\})}\le C{r}_j^{-2}\sqrt{E_j}$ and ${\sum }_j{E}_j\le 5E$. These are applications on ordinary balls away from the puncture; all rescaled metric bounds are uniform. Make the total energy small enough for the annular threshold. Align neighboring frames by their relative value at one fixed point of each overlap, as in [F3]. The relative frame then differs from identity by at most $C(\sqrt{E_j}+\sqrt{E_{j+1}})$ on that overlap, by integrating its gauge equation along bounded-length angular/radial paths. Average these aligned frames with a dyadic partition having only adjacent overlaps and normalize their quaternionic norm. This gives ONE smooth frame on the whole punctured smaller ball. Its potential $B$ obeys $\mathrm{\mid }B\mathrm{\mid }\le C{r}_j^{-1}(\sqrt{E_j}+\sqrt{E_{j+1}})$ on the $j$th band, hence $\int \mathrm{\mid }B{\mathrm{\mid }}^4\le C{\sum }_j(E_j^2+E_{j+1}^2)\le C{E}^2$. This reconstructs the punctured $L^4$ argument for the curved metric; no Euclidean-ASD theorem is applied to $A_{\mathrm{\infty }}$.

In this actual punctured frame choose smooth cutoffs ${\chi }_j=0$ on $B_{r_j}$ and $1$ outside $B_{2r_j}$, with $\mathrm{\mid }d{\chi }_j\mathrm{\mid }\le C\mathrm{/}{r}_j$. The potentials $B_j={\chi }_{j}B$ extended by zero are smooth on the full fixed ball. If $\mathcal{F}$ is the punctured curvature extended as an $L^2$ class, their exact curvatures are $F_{B_j}={\chi }_j\mathcal{F}+d{\chi }_j\wedge B+({\chi }_j^2-{\chi }_j)B\wedge B$. Holder and absolute continuity show $F_{B_j}\to \mathcal{F}$ in $L^2$: the cutoff derivative term is bounded by $C\mathrm{\parallel }B{\mathrm{\parallel }}_{L^4(\{r_j<\mathrm{\mid }x\mathrm{\mid }<2r_j\})}$ and the quadratic term by its square. Choose the punctured ball initially small enough that all these curvatures are below [F1]'s threshold. Construct full-ball Coulomb gauges $u_j$ and potentials $b_j$ with bounded $W^{1,2}$ and small $L^4$ norms. Weak Hilbert extraction and Rellich give weak $W^{1,2}$ and strong $L^p$, $p<4$, limits; the actual argument in [F3]'s point-removal proof passes $b_j\wedge b_j$ first in $L^1$ and then the bounded curvature in weak $L^2$, giving $F_b$. The $g$-anti-self-dual projections of $F_{B_j}$ tend to zero in $L^2$ because ${\pi }_{-,g}$ is a fixed smooth bounded coefficient operator. Unitary covariance gives the same assertion for $b_j$, so $d_e^{\ast }b=0$ and $F_b^{-,g}=0$ weakly on the ENTIRE ball. By [F2], $b$ is smooth across the point. On punctured compact sets the gauge equations $d{u}_j=u_j{b}_j-B{u}_j$ give bounded $W^{1,4}$ norms; Rellich and a diagonal yield a unitary limit $u$ with $du=ub-Bu$. Passing the products uses strong $L^3$ convergence of $b_j$ and strong $L^{12}$ convergence of the bounded matrices. Smooth $b,B$ bootstrap this equation to a smooth $SU(2)$ map on the puncture. Thus $B^u=b$ there. Glue a new trivial ball using this actual gauge; it extends the limiting bundle and connection smoothly. Repeat at the finitely many points of $S$, obtaining a smooth self-dual $A_{\mathrm{\infty }}$ on a bundle over ALL of $X$. This is the full curved point removal argument, including its actual limiting gauge.

Construct comparison isomorphisms, not merely unrelated local gauges. On a compact core $K⋐X\setminus S$, select finitely many working charts from step2.1 and smaller charts covering a neighborhood of $K$. Choose smooth weights ${\varphi }_j$ with compact support in the larger charts, summing to one on that neighborhood. In chart $i$ put $T_i^n={\sum }_j{\varphi }_j{h}_{ij}^n{h}_{ji}$. Since the two transition cocycles converge smoothly on the finitely many compact supports, $T_i^n\to I$ in every derivative norm. For large $n$ its quaternionic norm is bounded below by one half. Set ${\sigma }_i^n=T_i^n\mathrm{/}(\frac{1}{2}\mathrm{tr}((T_i^n{)}^{\ast }{T}_i^n){)}^{1\mathrm{/}2}$. The cocycle laws give $T_i^n=h_{ik}^n{T}_k^n{h}_{ki}$; their scalar norms agree, so $h_{ik}=({\sigma }_i^n{)}^{-1}{h}_{ik}^n{\sigma }_k^n$. Therefore these are actual smooth bundle isomorphisms from the limiting bundle to the original one on the core neighborhood. The transferred connections $(a_i^n{)}^{{\sigma }_i^n}$ converge smoothly to $a_i$. This is [F4]'s finite averaging proof with smooth manifold weights, not its Euclidean-cover Statement outside scope.

Choose increasing cores by deleting shrinking disjoint balls about $S$. Apply step5.1 and diagonal selection so that the $n$th chosen comparison is within $1\mathrm{/}n$ in every derivative order at most $n$ on the $n$th core. Extend each comparison to the WHOLE punctured manifold. Indeed near a deleted sphere both bundles have the smooth ball trivializations, one original and one from step4.1. On the collar the comparison is an $SU(2)$ function $h(\vartheta ,r)$. Choose a smooth radial function $f(r)$ equal to $r$ on its outer collar and equal to one interior collar radius for all sufficiently small positive $r$. Then $h(\vartheta ,f(r))$ extends the comparison across the remaining punctured end and agrees with it near its original domain boundary. No extension across the center is requested. Perform this on each disjoint punctured ball. These are actual bundle isomorphisms on $X\setminus S$ for each sequence member, and the extensions change nothing on the exhausting cores. Consequently the transported connections converge in $C_{\mathrm{l}\mathrm{o}\mathrm{c}}^{\mathrm{\infty }}(X\setminus S)$, in particular in the asserted local Sobolev gauges. No assertion that $P_{\mathrm{\infty }}$ and $P$ extend isomorphically across $S$ has been made.

Identify the lost energy before calling it an integer. Choose disjoint coordinate balls $B_r(p)$ about the finite points of $S$ and one decreasing radius sequence tending to zero at each point. Extract further so the integrals of $\mathrm{\mid }{F}_{A_n}{\mathrm{\mid }}^2$ on all these balls converge. Smooth convergence on every compact annulus shows ${\lim }_n{\int }_{B_r(p)}\mathrm{\mid }{F}_{A_n}{\mathrm{\mid }}^2-{\int }_{B_r(p)}\mathrm{\mid }{F}_{A_{\mathrm{\infty }}}{\mathrm{\mid }}^2$ is independent of these sufficiently small radii: subtract two radii and use smooth annular convergence. Denote it ${\lambda }_p$. It is positive. In fact, because $p$ was bad in step1.1, every neighborhood contains a listed ball containing $p$ whose limiting energy on its quadruple is at least $\theta$; choose that quadruple inside $B_r(p)$. Hence the first limiting integral is at least $\theta$, while the smooth extended connection's ball energy tends to zero as $r\to 0$. Thus ${\lambda }_p\ge \theta >0$. For a continuous test function $f$, split its integral over these balls and their complement. On the complement use smooth convergence; on each ball replacing $f$ by $f(p)$ costs at most its oscillation there times the bounded total energy. Let their radii tend to zero. This proves actual weak energy-measure convergence to $\mathrm{\mid }{F}_{A_{\mathrm{\infty }}}{\mathrm{\mid }}^{2}d\mathrm{v}\mathrm{o}\mathrm{l}+{\sum }_p{\lambda }_p{\delta }_p$ and the identity ${\lim }_{n}E(A_n)=E(A_{\mathrm{\infty }})+{\sum }_p{\lambda }_p$. No bubble-tree energy identity has been assumed.

Prove individual charge quantization by clutching. Fix one sufficiently small ball $B=B_r(p)$ as in step7.1 and use the comparison from step6.1 on its fixed boundary collar. In its limiting smooth frame the transported connection $a_n$ converges smoothly to $A_{\mathrm{\infty }}$ there. Modify $a_n$ by a fixed collar cutoff to equal $A_{\mathrm{\infty }}$ near $\mathrm{\partial }B$, and transport this modification back to the original frame. It defines a smooth connection $A_n^{\mathrm{\prime }}$ on the original bundle over $B$, equal to $A_n$ away from the collar. Since the modification tends to zero with all derivatives in the comparison frame, ${\int }_B{c}_2(A_n^{\mathrm{\prime }})-{\int }_B{c}_2(A_n)\to 0$, by the actual curvature polynomial and gauge invariance. Glue this ball bundle to the oppositely oriented limiting ball bundle using the comparison on the boundary. To make the connections match smoothly, flatten each near its boundary to the product pullback of its common tangential connection. This changes neither Chern integral: for a path of connections, direct differentiation and Bianchi give ${\mathrm{\partial }}_t\mathrm{tr}(F_t\wedge F_t)=2d\mathrm{tr}({\dot{A}}_t\wedge F_t)$; a deformation fixing the tangential boundary connection has zero tangential transgression trace. Explicitly write that three-form as ${\sum }_i(-1{)}^{i-1}{V}_{i}d{x}_1\wedge \cdots \widehat{d{x}_i}\cdots \wedge d{x}_4$; its derivative is $(\mathrm{div}V)dx$ and its zero tangential trace is $V\cdot \nu =0$. The actual ball divergence theorem in [F5] therefore makes its derivative integral zero. Product collars then glue a smooth connection on an actual $SU(2)$ bundle over the doubled ball $S^4$. Its Chern number is the integer $q_n={\int }_B{c}_2(A_n^{\mathrm{\prime }})-{\int }_B{c}_2(A_{\mathrm{\infty }})$. By [F5] and step7.1, $q_n\to -{\lambda }_p\mathrm{/}(8{\pi }^2)$. A convergent integer sequence is eventually constant, so $m_p={\lambda }_p\mathrm{/}(8{\pi }^2)$ is an integer; step7.1 makes it positive. This argument constructs the clutching bundle and uses integrality of its Chern class, not an unproved instanton profile or a classification of all bundles on $S^4$.

Let $k_{\mathrm{\infty }}=-⟨c_2(P_{\mathrm{\infty }}),[X]⟩$. The extended bundle is smooth, so [F5] makes this an integer and $E(A_{\mathrm{\infty }})=8{\pi }^2{k}_{\mathrm{\infty }}\ge 0$. The same calculation gives $E(A_n)=8{\pi }^{2}k$ for every sequence member. Substitute step8.1 into step7.1 to get $k=k_{\mathrm{\infty }}+{\sum }_p{m}_p$. In particular there are at most $k$ bubble points counting multiplicity. If $S$ is empty the comparison in step5.1 is over all of $X$ and the bundles are isomorphic, as expected. A sequence with negative prescribed $k$ cannot exist by the energy formula. All claims about the smooth limit, gauges and integer loss are now established.

Specify the compactification and verify that no extra ideal data are required. An ideal connection is a smooth self-dual gauge orbit on a bundle of charge $j\ge 0$, together with a finite unordered collection of points of total positive integer weight $k-j$. Its topology is the actual convergence of connections modulo smooth local comparison gauges off those points, together with weak convergence of the energy measure $\mathrm{\mid }F{\mathrm{\mid }}^{2}d\mathrm{v}\mathrm{o}\mathrm{l}+8{\pi }^2\sum m{\delta }_p$. Steps1.1–9.1 prove that every original sequence has an ideal subsequential limit in this topology. The same proof works for a sequence of background bundles, since no fixed-bundle hypothesis was used before the comparison construction. For a sequence of ideal connections, first pass to a fixed background charge in the finite range $0,\dots ,k$. The finite point configurations have a convergent subsequence by compactness of $X$: list their finitely many repeated points, extract in $X$ and forget that listing. Apply steps1.1–9.1 to the background connections and add their new bubble weights to the limiting old weights at coincident points. The exact charge equation persists. Thus the ideal space, and in particular the closure of the original moduli, is sequentially compact.

For completeness this topology is second-countable and Hausdorff, so sequential compactness really gives compactness. Here is a countable local-data construction. Use the fixed countable coordinate-ball basis, finite deleted-ball lists and finite chart subcovers of their compact complements. Smooth potentials and smooth transition maps on each such finite arrangement have countable $C^r$ coefficient bases: use rational polynomial approximations on finitely many smaller boxes, smooth fixed cutoffs, rational tolerances and integer $r$. Their subspaces satisfying the unitary and transition identities inherit those bases. Arbitrary changes of local frames act by homeomorphisms, so the open saturations of these countable bases form a countable basis on the corresponding gauge quotients. Combine these tests with the countable uniform-dense family of continuous functions on compact $X$, constructed by the same rational-chart approximation, for the weak energy tests. Require also that all listed ideal points lie in the chosen deleted balls. That condition follows locally from finitely many open energy tests: outside those balls the limiting smooth energy measure has no atom, so cover the compact remainder by finitely many smaller balls with test-function integrals less than $4{\pi }^2$. For sufficiently close energy measures these integrals remain less than $8{\pi }^2$, excluding every positive integer ideal atom where a test function is one. Thus these saturated finite-chart and energy tests give a countable basis for exactly the convergence in step10.1. No classification of bundle isomorphism types or global Coulomb slice is used. For uniqueness of a limit, compare two proposed limits on the complement of their finite exceptional sets. Gauges between their smoothly convergent representatives satisfy $d{h}_n=h_n{b}_n-a_n{h}_n$. Unitarity and the smooth coefficient bounds control all gauge derivatives locally, so a diagonal gives a smooth limiting gauge intertwining the two background connections. Weak energy measures have unique limits against the continuous test functions, so their atomic weights and positions agree; the smooth background measure has no atoms. This gauge extends over every exceptional point: in their two smooth extension frames its derivative is bounded by the same equation, and the length distance on a punctured four-ball has the Euclidean completion, so it has a unique continuous limit at the puncture. The bounded derivative gives a weak Lipschitz extension, and the equation then bootstraps to smoothness. The ideal limits are therefore the same. With the countable neighborhood bases just constructed, failure of Hausdorff separation would select a point in every intersection of two shrinking neighborhood bases, giving a sequence with those two different limits. Thus the topology is Hausdorff. Finally second-countability gives a countable subcover of every open cover under AC. If no finite subcover existed, choose a point outside its first $n$ members; a convergent subsequence would contradict the member containing its limit. The sequential compactness in step10.1 therefore gives compactness of the ideal space and of the closed original-moduli closure. This is precisely the compactification by lower-charge smooth limits and integer point labels in the Statement.