Celebratio Mathematica

Kar­en Uh­len­beck’s math­em­at­ic­al con­tri­bu­tions and math­em­at­ic­al work in the fields of geo­metry and ana­lys­is laid the found­a­tion for a broad swath of con­tinu­ing re­search in dif­fer­en­tial geo­metry over the past four dec­ades. Think of her work with Jonath­an Sacks on min­im­al sur­faces, her work with Richard Schoen on har­mon­ic maps, and her work with S-T. Yau on Her­mitian Yang–Mills con­nec­tions on stable, holo­morph­ic bundles. And then there is her found­a­tion­al work on gauge the­ory. But, the full scope of Kar­en Uh­len­beck’s con­tri­bu­tions is not the sub­ject of this es­say; this es­say is only about Kar­en Uh­len­beck’s bril­liant gauge the­ory ana­lys­is. (Kar­en Uh­len­beck’s ICM talks in 1983 and 1990 and her Com­ment­ary on “ana­lys­is in the large” (for the centen­ni­al of the AMS) sum­mar­ize much of her early work. However, these pa­pers ser­i­ously down-play her in­flu­ence — she is ex­cep­tion­ally mod­est. And, of course, they say noth­ing about the re­mark­able sub­sequent de­vel­op­ments and ap­plic­a­tions. The ICM talks, her AMS com­ment­ary, and se­lec­ted pa­pers by Kar­en Uh­len­beck on min­im­al sur­faces, har­mon­ic maps, Her­mitian Yang–Mills con­nec­tions and gauge the­ory are lis­ted at the end of this es­say.)

Uh­len­beck’s pa­pers on con­nec­tions with bounded, in­teg­ral curvature norms (see [4], [17], [7], [8]) form the found­a­tion­al ana­lys­is be­hind all of the sub­sequent ap­plic­a­tions of gauge the­ory to to­po­logy and geo­metry. Here is some his­tory: Si­mon Don­ald­son had the bril­liant idea to con­sider cer­tain ca­non­ic­al fam­il­ies of equi­val­ence classes (equi­val­ence by bundle auto­morph­ism) of con­nec­tions on a giv­en man­i­fold as a ho­mo­logy cycle in the space of equi­val­ence classes of all con­nec­tions (see [e1], [e2], [e4] and then [e5]). By vir­tue of the nature of the fam­ily (the con­nec­tions are solu­tions to a ca­non­ic­al par­tial dif­fer­en­tial equa­tion with Fred­holm lin­ear­iz­a­tion), the pair­ing of this cycle with ho­mo­logy classes would give an in­vari­ant of the smooth struc­ture of the am­bi­ent man­i­fold if the cycle is ef­fect­ively com­pact (com­pact, or com­pac­ti­fi­able in a nice way). In hind­sight, com­pact­ness is the truly dif­fi­cult ana­lys­is is­sue. Uh­len­beck’s ana­lys­is is needed to prove that the cycle does in­deed have a ca­non­ic­al com­pac­ti­fic­a­tion. Don­ald­son bril­liantly saw how to ex­ploit her the­or­em to de­vel­op his gauge the­or­et­ic 4-man­i­fold in­vari­ants. Like­wise, her ana­lys­is lies be­hind An­dreas Flo­er’s de­vel­op­ment of Flo­er ho­mo­logy for gauge the­ory [e3] (see also [e6]) be­cause that also re­quires her com­pact­ness the­or­em to prove that the Flo­er dif­fer­en­tial has square zero. All sub­sequent work on gauge the­or­et­ic in­vari­ants in low di­men­sion­al to­po­logy can trace an­ces­try back to Uh­len­beck’s work. (It is still ex­cit­ing; see the ICM 2018 plen­ary talk by Peter Kron­heimer and Tom Mrowka.)

By way of back­ground for this com­pact­ness is­sue: The Hodge dual op­er­at­or from a met­ric on an ori­ented, Rieman­ni­an 4-man­i­fold maps 2-forms to 2-forms with square 1. Its \( +1 \) and \( -1 \) ei­gen­spaces are called (re­spect­ively) the bundles of self-dual and anti-self dual 2-forms. Each is a rank 3, real vec­tor bundle. Mean­while, the curvature of a giv­en con­nec­tion on some prin­cip­al bundle over the man­i­fold is a 2-form with val­ues in an as­so­ci­ated vec­tor bundle (the fiber is the Lie al­gebra of the group), and so it has a self-dual part and an anti-self dual part. Don­ald­son’s space of con­nec­tions are those with van­ish­ing self-dual curvature; they are con­nec­tions with anti-self dual curvature.

This space of con­nec­tions with anti-self dual curvature can not be com­pact by vir­tue of the fact that the ac­tion by pull-back of the group of auto­morph­isms of the prin­cip­al bundle (which is in­fin­ite di­men­sion­al) pre­serves anti-self du­al­ity. But, the space of or­bits of con­nec­tions with anti-self dual curvature un­der this group ac­tion has a chance to be com­pact be­cause it is loc­ally com­pact. In fact, this or­bit space is loc­ally homeo­morph­ic to the or­bit space (via a fi­nite group) of the zero locus of a smooth map between fi­nite di­men­sion­al Eu­c­lidean spaces. To elab­or­ate: Sup­pose that \( A_0 \) is a con­nec­tion on the prin­cip­al bundle. Any nearby con­nec­tion (in the \( C^\infty \) to­po­logy) can be writ­ten as \( A_0+a \) where \( a \) is a 1-form in that as­so­ci­ated bundle with Lie al­gebra fiber. If \( A_0 \) has anti-self dual curvature and like­wise \( A_0+a \); and if \( A_0+a \) is in a slice trans­vers­al to the auto­morph­ism group’s or­bit through \( A_0 \) (and suf­fi­ciently close to \( A_0 \)), then \( a \) will obey a dif­fer­en­tial equa­tion (by vir­tue of \( A_0+a \) hav­ing anti-self dual curvature) that takes the schem­at­ic form \[ D_0 a+\Pi(a\wedge a)=0 \] where \( D_0 \) is a first or­der dif­fer­en­tial op­er­at­or with el­lipt­ic sym­bol that is in­de­pend­ent of \( A_0 \). Mean­while, \( \Pi(a\wedge a) \) de­notes the anti-self dual pro­jec­tion of \( a\wedge a \). The key point in this re­gard is that \( D_0 \) is Fred­holm when its range and do­main are com­pleted us­ing suit­able Banach space norms.

With re­gards to com­pact­ness of the or­bit space: The equa­tion \( D_0 a+\Pi(a\wedge a)=0 \) sug­gests an it­er­at­ive scheme to prove com­pact­ness for the space of auto­morph­ism group or­bits of anti-self dual con­nec­tions; it goes something like this: Sup­pose that some min­im­al bound is known for \( a \), and that this bound leads to a suit­able bound for the quad­rat­ic term \( \Pi(a\wedge a) \). If so, then the equa­tion \( D_0 a+\Pi(a\wedge a) \) says that some lin­ear com­bin­a­tion of first de­riv­at­ives of \( a \) obeys the bound that \( \Pi(a\wedge a) \) obeys; which sug­gests in turn (be­cause \( D_0 \) is el­lipt­ic) that all of the de­riv­at­ives of \( a \) might also obey that same bound. If that were the case, then the equa­tion \( D_0 a+\Pi(a\wedge a) \) could be dif­fer­en­ti­ated ter­m­wise to ob­tain an equa­tion for the first de­riv­at­ives of \( a \) which would be an equa­tion hav­ing the schem­at­ic form \[ D_0\partial a+\Pi(\partial a\wedge a)+\Pi(a\wedge \partial a)+r\cdot a=0 .\] (What is de­noted by \( r \) here comes from the met­ric’s curvature tensor.) This last equa­tion sug­gests that some lin­ear com­bin­a­tion of the second de­riv­at­ives of \( a \) obeys a suit­able bound, and if this is the case for all of the second de­riv­at­ives (which is reas­on­able be­cause \( D_0 \) is el­lipt­ic), then one can dif­fer­en­ti­ate the equa­tion again to ob­tain third de­riv­at­ive bounds, and so on. Con­tinu­ing in this vein would get uni­form bounds for all high­er de­riv­at­ives from an ini­tial bound for just \( a \). Then, by vir­tue of the Ar­zelà–Ascoli the­or­em, any bounded se­quence of \( a \)’s would have a con­ver­gent sub­sequence (which is the de­sired com­pact­ness con­clu­sion).

The prob­lem with im­ple­ment­ing the pre­ced­ing is this: The equa­tion \( D_0 a+\Pi(a\wedge a) \) is an equa­tion for the dif­fer­ence between two anti-self dual con­nec­tions, one be­ing the ori­gin­al \( A_0 \) and the oth­er be­ing \( A_0+a \); and it can be ex­ploited (and, in fact, de­rived in the first place) only in the event that their re­spect­ive auto­morph­ism or­bits are suf­fi­ciently close (which guar­an­tees that \( a \) is small to be­gin with). In par­tic­u­lar, if the auto­morph­ism group or­bits of a se­quence of con­nec­tions with anti-self dual curvature is not con­ver­ging, then (this is es­sen­tially the non­con­ver­gence as­sump­tion) no in­fin­ite sub­set of the se­quence can stay uni­formly close to any giv­en auto­morph­ism group or­bit in the space of con­nec­tions.

More by way of geo­met­ric in­put is needed to pro­ceed. Here is the in­put: If \( A \) is a con­nec­tion on a prin­cip­al bundle with com­pact struc­ture group, then the point­wise norm of its curvature 2-form is in­vari­ant un­der the ac­tion of the auto­morph­ism group. This is be­cause the curvature 2-form at any giv­en point can be viewed as a 2-form with val­ues in the Lie al­gebra of the struc­ture group. Viewed in this light, an auto­morph­ism of the prin­cip­al bundle acts on the curvature 2-form at that point by con­jug­a­tion of the Lie al­gebra factors with an ele­ment in the group. It is the ad­joint ac­tion on the Lie al­gebra. There­fore, if an ad­joint-in­vari­ant norm on the Lie al­gebra (for ex­ample, the trace norm in the case of a mat­rix group) is used to define the norm of the curvature 2-form, then that norm is in­vari­ant with re­spect to the ac­tion of the auto­morph­ism group.

It fol­lows from the pre­ced­ing ob­ser­va­tions that if some func­tion­al of the point­wise norms of the curvature di­verges along a se­quence of con­nec­tions, then the cor­res­pond­ing se­quence of auto­morph­ism or­bits does not con­verge. Kar­en Uh­len­beck’s bril­liant res­ults (from [7] and [8]) were a con­verse which is (in part) this: Sup­pose that \( \{A_n\}_{n=1,2,\dots} \) is a se­quence of anti-self dual con­nec­tions with an a pri­ori bound only on the in­teg­ral of the square of the norm of the curvature (thus, an a pri­ori bound for the \( L^2 \) norm of the curvatures along the se­quence). Then there ex­ists the fol­low­ing data: First, a new prin­cip­al bundle over the man­i­fold and a con­nec­tion on it with anti-self dual curvature 2-form. (The new con­nec­tion is de­noted by \( A \).) Second, a fi­nite set of points in the man­i­fold (an up­per bound for the num­ber can be giv­en a pri­ori in terms of the bound for the in­teg­rals of the squares of the curvature 2-forms). This fi­nite set is de­noted by \( \Omega \). Third, a se­quence (to be de­noted by \( \{g_n\}_{n=1,2,\dots} \)) of iso­morph­isms on the com­ple­ment of \( \Omega \) from the new bundle to the ori­gin­al bundle. These are such that the se­quence \( \{g_n^* A_n\}_{n=1,2,\dots} \) has a sub­sequence that con­verges to the con­nec­tion \( A \) on com­pact sets in the com­ple­ment of \( \Omega \). Kar­en Uh­len­beck’s ana­lys­is in [7] and [8] can also say pre­cisely what hap­pens to the se­quence near the points of \( \Omega \). Some words about the proof of Kar­en Uh­len­beck’s res­ults ap­pear mo­ment­ar­ily.

As it turns out, the \( L^2 \) bound is more im­port­ant than the anti-self dual con­di­tion: Here is what Kar­en Uh­len­beck’s the­or­ems from [7] and [8] im­ply when there is noth­ing but an \( L^2 \) bound: Sup­pose that \( \{A_n\}_{n=1,2,\dots} \) is a se­quence of con­nec­tions with only an a pri­ori bound on the in­teg­ral of the square of the norm of the curvature. Then there ex­ists a new prin­cip­al bundle and now a So­bolev class \( L^2_1 \) con­nec­tion on that bundle (to be de­noted by \( A \)). There is again a fi­nite set of points (de­noted here also by \( \Omega \)) and a se­quence of iso­morph­isms from the new bundle to the old bundle on the com­ple­ment of \( \Omega \). This is \( \{g_n\}_{n=1,2,\dots} \). This data is such that \( \{g_n^* A_n\}_{n=1,2,\dots} \) con­verges in the weak \( L^2_1 \)-to­po­logy to \( A \) on open sets with com­pact clos­ure in the com­ple­ment of \( \Omega \). (A con­nec­tion on a prin­cip­al bundle in the So­bolev class \( L^2_1 \) dif­fers from a smooth con­nec­tion by a meas­ur­able Lie al­gebra val­ued 1-form with meas­ur­able first de­riv­at­ives whose norm is square in­teg­rable and whose first de­riv­at­ives have square in­teg­rable norm.)

By way of a par­en­thet­ic­al re­mark: The \( L^2 \) norm bound is the bor­der­line case for con­ver­gence in two re­spects. First, Kar­en Uh­len­beck’s the­or­ems in [8] also prove this: If \( \{A_n\}_{n=1,2,\dots} \) is a se­quence of anti-self dual con­nec­tions such that the in­teg­rals of any power great­er than 2 of the norms of the curvature 2-forms along the se­quence is bounded, then the cor­res­pond­ing se­quence of auto­morph­ism group or­bits has a con­ver­gent sub­sequence. Second, there are known se­quences of anti-self dual con­nec­tions with a pri­ori \( L^2 \) norm bound that do not con­verge without re­mov­ing some fi­nite set of points. The next para­graph de­scribes an ex­ample.

The Levi-Civ­ita con­nec­tion on the round 4-di­men­sion­al sphere in­duces a cor­res­pond­ing con­nec­tion on the prin­cip­al \( \mathrm{S}\mathrm{O}(3) \) bundle of ori­ented or­thonor­mal frames in the bundle of self dual 2-forms for this round met­ric. And, this \( \mathrm{S}\mathrm{O}(3) \) con­nec­tion has nonzero, anti-self dual curvature 2-form. Since the curvature is nonzero, the in­teg­ral of the square of its norm is like­wise not zero. Keep­ing the pre­ced­ing in mind, note that the anti-self du­al­ity con­di­tion is con­form­ally in­vari­ant, as is the in­teg­ral of the square of the norm of the curvature. (A con­form­al change of met­ric changes the met­ric by mul­tiply­ing it by a pos­it­ive func­tion.) There­fore, if the 4-sphere ad­mits a con­form­al dif­feo­morph­ism (the pull-back of the met­ric is con­form­al to the ori­gin­al), then the pull-back of the con­nec­tion by this dif­feo­morph­ism has anti-self dual curvature whose \( L^2 \) norm is the same as the ori­gin­al. As it turns out, the four di­men­sion­al sphere ad­mits a 1-para­met­er group of con­form­al dif­feo­morph­isms that act by push­ing points to­wards the south pole along the great circles. This fam­ily of con­form­al dif­feo­morph­ism is the im­age via the in­verse of ste­reo­graph­ic pro­jec­tion (from the south pole) of the Eu­c­lidean space res­cal­ing map \( x\to\lambda x \) with \( \lambda \) be­ing a num­ber great­er than 1. Since this fam­ily pushes points to the south pole, the cor­res­pond­ing se­quence of anti-self dual con­nec­tions (all with the same curvature \( L^2 \) norm) can­not have a con­ver­gent sub­sequence on the whole of \( \mathbb{S}^4 \). (But, it does con­verge on the com­ple­ment of the north pole; and the lim­it is a flat con­nec­tion.)

It is for­tu­it­ous (and cru­cial for Don­ald­son’s ap­plic­a­tions) that anti-self dual con­nec­tions on any giv­en prin­cip­al bundle over any giv­en com­pact 4-di­men­sion­al man­i­fold have an a pri­ori \( L^2 \) bound on their curvature. In fact, their curvatures have identic­al \( L^2 \) norms, and here is why: For any con­nec­tion, the in­teg­ral of the square of the norm of the self-dual part of the curvature minus that of the anti-self dual part is a char­ac­ter­ist­ic num­ber that clas­si­fies (in part) the prin­cip­al bundle (it is an in­teger up to a ca­non­ic­al factor). There­fore, all con­nec­tions on the same prin­cip­al bundle with anti-self dual curvature 2-form have the fol­low­ing prop­erty: The in­teg­ral of the square of the norm of their curvature 2-form is in­de­pend­ent of the con­nec­tion; it is the afore­men­tioned char­ac­ter­ist­ic num­ber which is pre­de­ter­mined by the prin­cip­al bundle.

It is also for­tu­it­ous (and im­port­ant for Don­ald­son’s ap­plic­a­tions) that Kar­en Uh­len­beck’s \( L^2 \) com­pact­ness the­or­ems can be turned around (so to speak) and used to con­struct con­nec­tions with anti-self dual curvature 2-forms. The point here is that Kar­en Uh­len­beck’s the­or­ems also de­scribe the be­ha­vi­or of the con­nec­tions in neigh­bor­hoods of the points of the set \( \Omega \) that ap­pears in the state­ment of her the­or­em. What hap­pens is this (to a first ap­prox­im­a­tion): A se­quence of co­ordin­ate res­cal­ings in small balls about any giv­en point in \( \Omega \) (use Gaus­si­an nor­mal co­ordin­ates) pro­duces a con­nec­tion on 4-di­men­sion­al Eu­c­lidean space with anti-self dual curvature which comes from the 4-di­men­sion­al round sphere via the in­verse of the ste­reo­graph­ic pro­jec­tion map. (Just as in the ex­ample two para­graphs back). Know­ing this, one can take a con­nec­tion on the round sphere with anti-self dual curvature, pull it back to Eu­c­lidean space, res­cale it suf­fi­ciently (us­ing the \( x\to \lambda x \) res­cal­ing) and then graft the res­ult in­to any ori­ented 4-man­i­fold us­ing Gaus­si­an nor­mal co­ordin­ates. This gives a con­nec­tion with curvature con­cen­trated around the giv­en point whose curvature is al­most anti-self dual. If this is done clev­erly at suf­fi­ciently many points (tak­ing in­to ac­count the free­dom to choose the points and the Gaus­si­an nor­mal co­ordin­ates and the scal­ing para­met­er), then a small per­turb­a­tion can be made so that the res­ult is a con­nec­tion on a prin­cip­al bundle with identic­ally anti-self dual curvature. The full de­tails of how this is done can be found in [e5].

What fol­lows dir­ectly are the prom­ised words about the proof of Kar­en Uh­len­beck’s res­ults. (What is said does not ne­ces­sar­ily re­flect Kar­en Uh­len­beck’s think­ing.) There are two es­sen­tial ob­ser­va­tions, one geo­met­ric and straight for­ward. The oth­er is bril­liant via ana­lys­is in­sight from Kar­en Uh­len­beck. The geo­met­ric in­sight is this: Any giv­en prin­cip­al Lie group bundle over a ball is iso­morph­ic to the product bundle (the product of the ball with the group); and any giv­en con­nec­tion on the product bundle over a ball with zero curvature 2-form is equi­val­ent via an auto­morph­ism to the product con­nec­tion. Thus, mod­ulo auto­morph­isms and iso­morph­isms, there is just one prin­cip­al bundle over a ball, and there is just one con­nec­tion with zero curvature. This con­nec­tion will be the con­nec­tion \( A_0 \) to use when writ­ing oth­er con­nec­tions over the ball as \( A_0 +a \). Now sup­pose that \( A \) is an­oth­er con­nec­tion on the giv­en prin­cip­al bundle over the giv­en ball. If its curvature 2-form is nearly zero, then it stands to reas­on that there is an auto­morph­ism/iso­morph­ism that writes it as \( A_0+a \) with \( a \) be­ing small. Kar­en Uh­len­beck proved that nearly zero means the fol­low­ing when the di­men­sion is four (she proved ana­log­ous res­ults for oth­er di­men­sions): There ex­ist \( \varepsilon_0 > 0 \) and \( \kappa > 0 \) (that are in­de­pend­ent of the ra­di­us of the ball sup­pos­ing it is small) such that if \( A \) is a con­nec­tion on the ball with the in­teg­ral of the square of the norm its curvature 2-form be­ing less than \( \varepsilon_0^2 \), then there is an iso­morph­ism between the product prin­cip­al bundle over the ball and the giv­en bundle (to be de­noted by \( g \)) such that \( g^* A=A_0 +a \) with \( a \) be­ing small and or­tho­gon­al to the or­bit through \( A_0 \) of the auto­morph­ism group of the product prin­cip­al bundle. (The no­tion of “small” here means that the in­teg­ral over the ball of the sum of the square of \( a \)’s norm and the squares of the norms of \( a \)’s first de­riv­at­ives is at most \( \kappa \) times the in­teg­ral over the ball of the square of the norm of \( A \)’s curvature.) If it is also the case that \( A \) has anti-self dual curvature, then the cor­res­pond­ing Lie al­gebra val­ued 1-form \( a \) obeys the equa­tion \( D_0 a+\Pi(a\wedge a)=0 \) on the giv­en ball; and, as a con­sequence, there are a pri­ori bounds for its point­wise norm and those of its de­riv­at­ives to any giv­en or­der on a slightly smal­ler ra­di­us, con­cent­ric ball. (The lat­ter are de­rived in an it­er­at­ive man­ner by tak­ing de­riv­at­ives of the equa­tion \( D_0 a+\Pi(a\wedge a)=0 \) in the man­ner out­lined pre­vi­ously.)

Giv­en the res­ult above on balls, then uni­form ana­lyt­ic con­trol on con­nec­tions with a pri­ori \( L^2 \) bounds on curvature on the whole 4-di­men­sion­al man­i­fold can be had by cov­er­ing the giv­en man­i­fold by balls with radii chosen so that the \( L^2 \) norm of the curvature on each is less than the num­ber \( \varepsilon_0 \) from the pre­ced­ing para­graph. If \( \{A_n\}_{n=1,2,\dots} \) is a se­quence of con­nec­tions with a pri­ori bound on the in­teg­ral of the square of the cor­res­pond­ing curvature 2-forms, then con­ver­gence of a sub­sequence will fol­low auto­mat­ic­ally un­less ever smal­ler ra­di­us balls are needed to ob­tain the de­sired \( \varepsilon_0^2 \) bound on all balls along the se­quence. But note in this re­gard that if the \( L^2 \) norm of the curvature is bounded along the se­quence by a giv­en num­ber \( \mathcal{E} \), then at most \( \mathcal{O}(\mathcal{E}^2/\varepsilon_0^2) \) very small ra­di­us balls are needed for any giv­en con­nec­tion in the se­quence. This bound on the num­ber of small balls im­plies that there is curvature con­cen­tra­tion along a suit­able sub­sequence at just a fi­nite set of points (this is the set \( \Omega \), it has at most \( \mathcal{O}(\mathcal{E}^2/\varepsilon_0^2) \) ele­ments); and con­sequen­tial there is con­ver­gence for the sub­sequence on the com­ple­ment of this same fi­nite set. Kar­en Uh­len­beck’s pa­per Re­mov­able sin­gu­lar­it­ies in Yang–Mills fields [7] ex­plains why the lim­it con­nec­tion that is ob­tained on the com­ple­ment of the fi­nite set \( \Omega \) comes from a con­nec­tion on a prin­cip­al bundle over the whole man­i­fold.

By way of a par­en­thet­ic­al re­mark: The pa­per Re­mov­able sin­gu­lar­it­ies in Yang–Mills fields [7] just noted had some in­flu­ence on the de­vel­op­ment of high en­ergy particle phys­ics in the early 1980s. By way of back­ground: Elec­trons and sub­atom­ic particles called quarks are the fun­da­ment­al con­stitu­ents of the mat­ter around us. Al­most all of the mass of the mat­ter is from the quarks and the force that holds them to­geth­er. Con­nec­tions on the product \( \mathrm{S}\mathrm{O}(3) \) prin­cip­al bundle over \( \mathbb{R}^4 \) sup­ply the force. Now for the math­em­at­ics: A the­or­em in that pa­per said in ef­fect that any com­pact Lie group con­nec­tion over 4-di­men­sion­al Eu­c­lidean space \( (\mathbb{R}^4) \) with the in­teg­ral of the square of the norm of the curvature be­ing fi­nite comes via ste­reo­graph­ic pro­jec­tion from a con­nec­tion on the 4-di­men­sion­al sphere. Kar­en showed in par­tic­u­lar that this has the fol­low­ing im­plic­a­tion: Sup­pose that \( A \) is a con­nec­tion that solves the Yang–Mills equa­tion on \( \mathbb{R}^4 \) or on just the com­ple­ment of a ball in \( \mathbb{R}^4 \). As­sume also that \( A \) has \( L^2 \) curvature 2-form. Then, the norm of \( A \)’s curvature 2-form falls off at large dis­tance on \( \mathbb{R}^4 \) as fast or faster than \( \lvert x\rvert^{-4} \). (The Yang–Mills equa­tions ask that the co­v­ari­ant ex­ter­i­or de­riv­at­ive defined by \( A \) of the Hodge dual of the \( A \)’s curvature 2-form is zero. Con­nec­tions with anti-self dual or self-dual curvature 2-form al­ways solve the Yang–Mills equa­tions.) For a phys­i­cist, this \( \lvert x\rvert^{-4} \) fall off of the curvature im­plies that the force gen­er­ated by the con­nec­tion on sub­atom­ic particles is very small when the particles are far apart. And, the lat­ter fact killed a pro­posed ex­plan­a­tion for the fact (ex­per­i­ment­ally veri­fied) that the force between the fun­da­ment­al quarks grows as quarks try to sep­ar­ate.

Here is an­oth­er par­en­thet­ic­al re­mark with re­gards to solu­tions to the Yang–Mills equa­tions on \( \mathbb{R}^4 \): It was con­jec­tured for some time by phys­i­cists that the only solu­tions with the \( L^2 \) norm of the curvature be­ing fi­nite were con­nec­tions with curvature either self-dual or anti-self dual. Kar­en Uh­len­beck with Les­ley Sib­n­er and Bob Sib­n­er con­struc­ted a counter­example to this con­jec­ture (see [19]).

[2] K. Uh­len­beck: “Gen­er­ic prop­er­ties of ei­gen­func­tions,” Am. J. Math. 98 : 4 (1976), pp. 1059–​1078. MR 464332 Zbl 0355.​58017

[15] K. K. Uh­len­beck: “Vari­ation­al prob­lems for gauge fields,” pp. 585–​591 in Pro­ceed­ings of the In­ter­na­tion­al Con­gress of Math­em­aticians, vol. 2. Ed­ited by Z. Ciesiel­ski and C. Olech. PWN (Warsaw), 1984. MR 804715 Zbl 0562.​53059

[20] K. Uh­len­beck: “Com­ment­ary on ‘ana­lys­is in the large’,” pp. 357–​359 in A cen­tury of math­em­at­ics in Amer­ica, part 2. Ed­ited by P. L. Duren, R. As­key, and U. C. Merzbach. His­tory of Math­em­at­ics 2. Amer­ic­an Math­em­at­ic­al So­ci­ety (Provid­ence, RI), 1989. MR 1003144

[22] K. Uh­len­beck: “In­stan­tons and their re­l­at­ives,” pp. 467–​477 in Math­em­at­ics in­to the twenty-first cen­tury (Provid­ence, RI, 8–12 Au­gust 1988). Ed­ited by F. E. Browder. Amer­ic­an Math­em­at­ic­al So­ci­ety Centen­ni­al Pub­lic­a­tions 2. Amer­ic­an Math­em­at­ic­al So­ci­ety (Provid­ence, RI), 1992. MR 1184623 Zbl 1073.​53505

[23] C.-L. Terng and K. Uh­len­beck: “In­tro­duc­tion,” pp. 5–​19 in In­teg­ral sys­tems. Ed­ited by C.-L. Terng and K. Uh­len­beck. Sur­veys in Dif­fer­en­tial Geo­metry 4. In­ter­na­tion­al Press (Cam­bridge, MA), 1998. Zbl 0938.​35182

[3] J. Sacks and K. Uh­len­beck: “The ex­ist­ence of min­im­al im­mer­sions of two-spheres,” Bull. Am. Math. Soc. 83 : 5 (1977), pp. 1033–​1036. MR 448408 Zbl 0375.​49016

[6] J. Sacks and K. Uh­len­beck: “The ex­ist­ence of min­im­al im­mer­sions of 2-spheres,” Ann. Math. (2) 113 : 1 (Janu­ary 1981), pp. 1–​24. MR 604040 Zbl 0462.​58014

[10] J. Sacks and K. Uh­len­beck: “Min­im­al im­mer­sions of closed Riemann sur­faces,” Trans. Am. Math. Soc. 271 : 2 (1982), pp. 639–​652. MR 654854 Zbl 0527.​58008

[13] K. K. Uh­len­beck: “Closed min­im­al sur­faces in hy­per­bol­ic 3-man­i­folds,” pp. 147–​168 in Sem­in­ar on min­im­al sub­man­i­folds. Ed­ited by E. Bom­bieri. An­nals of Math­em­at­ics Stud­ies 103. Prin­ceton Uni­versity Press, 1983. MR 795233 Zbl 0529.​53007

[14] K. K. Uh­len­beck: “Min­im­al spheres and oth­er con­form­al vari­ation­al prob­lems,” pp. 169–​176 in Sem­in­ar on min­im­al sub­man­i­folds. Ed­ited by E. Bom­bieri. An­nals of Math­em­at­ics Stud­ies 103. Prin­ceton Uni­versity Press, 1983. MR 795234 Zbl 0535.​53050

[24] A. Gonçalves and K. Uh­len­beck: “Mod­uli space the­ory for con­stant mean curvature sur­faces im­mersed in space-forms,” Comm. Anal. Geom. 15 : 2 (2007), pp. 299–​305. MR 2344325 Zbl 1136.​53048 ArX­iv math/​0611295

[1] K. Uh­len­beck: “Har­mon­ic maps: A dir­ect meth­od in the cal­cu­lus of vari­ations,” Bull. Am. Math. Soc. 76 : 5 (1970), pp. 1082–​1087. MR 264714 Zbl 0208.​12802

[5] K. Uh­len­beck: “Morse the­ory by per­turb­a­tion meth­ods with ap­plic­a­tions to har­mon­ic maps,” Trans. Am. Math. Soc. 267 : 2 (1981), pp. 569–​583. MR 626490 Zbl 0509.​58012

[9] R. Schoen and K. Uh­len­beck: “A reg­u­lar­ity the­ory for har­mon­ic maps,” J. Diff. Geom. 17 : 2 (1982), pp. 307–​335. MR 664498 Zbl 0521.​58021

[11] R. Schoen and K. Uh­len­beck: “Bound­ary reg­u­lar­ity and the Di­rich­let prob­lem for har­mon­ic maps,” J. Diff. Geom. 18 : 2 (1983), pp. 253–​268. MR 710054 Zbl 0547.​58020

[12] R. Schoen and K. Uh­len­beck: “Cor­rec­tion to: ‘A reg­u­lar­ity the­ory for har­mon­ic maps’,” J. Diff. Geom. 18 : 2 (1983), pp. 329. MR 710058

[16] R. Schoen and K. Uh­len­beck: “Reg­u­lar­ity of min­im­iz­ing har­mon­ic maps in­to the sphere,” In­vent. Math. 78 : 1 (Feb­ru­ary 1984), pp. 89–​100. MR 762354 Zbl 0555.​58011

[18] K. Uh­len­beck and S.-T. Yau: “On the ex­ist­ence of Her­mitian-Yang–Mills con­nec­tions in stable vec­tor bundles,” pp. S257–​S293 in Pro­ceed­ings of the Sym­posi­um on Fron­ti­ers of the Math­em­at­ic­al Sci­ences: 1985 (New York, Oc­to­ber 1985), pub­lished as Comm. Pure Ap­pl. Math. 39 : Sup­ple­ment S1. Is­sue ed­ited by C. Mor­awetz. J. Wiley and Sons (New York), 1986. MR 861491 Zbl 0615.​58045

[21] K. Uh­len­beck and S. T. Yau: “A note on our pre­vi­ous pa­per: On the ex­ist­ence of Her­mitian Yang–Mills con­nec­tions in stable vec­tor bundles,” Com­mun. Pure Ap­pl. Math. 42 : 5 (1989), pp. 703–​707. MR 997570 Zbl 0678.​58041

[4] K. K. Uh­len­beck: “Re­mov­able sin­gu­lar­it­ies in Yang–Mills fields,” Bull. Am. Math. Soc. (N.S.) 1 : 3 (May 1979), pp. 579–​581. MR 526970 Zbl 0416.​35026

[7] K. K. Uh­len­beck: “Re­mov­able sin­gu­lar­it­ies in Yang–Mills fields,” Comm. Math. Phys. 83 : 1 (Feb­ru­ary 1982), pp. 11–​29. MR 648355 Zbl 0491.​58032

[8] K. K. Uh­len­beck: “Con­nec­tions with \( L^p \) bounds on curvature,” Comm. Math. Phys. 83 : 1 (Feb­ru­ary 1982), pp. 31–​42. MR 648356 Zbl 0499.​58019

[17] K. K. Uh­len­beck: “The Chern classes of So­bolev con­nec­tions,” Comm. Math. Phys. 101 : 4 (Decem­ber 1985), pp. 449–​457. MR 815194 Zbl 0586.​53018

[19] L. M. Sib­n­er, R. J. Sib­n­er, and K. Uh­len­beck: “Solu­tions to Yang–Mills equa­tions that are not self-dual,” Proc. Natl. Acad. Sci. U.S.A. 86 : 22 (Novem­ber 1989), pp. 8610–​8613. MR 1023811 Zbl 0731.​53031