#### by Clifford Henry Taubes

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], [e5] and then [e4]). 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 the 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
[e4].

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
[21]).

#### Acknowledgement

#### Selected papers by Karen Uhlenbeck on ~~geometric analysis topics~~

##### Commentaries

[2] : “Generic properties of eigenfunctions,” Am. J. Math. 98 : 4 (1976), pp. 1059–1078. MR 464332 Zbl 0355.58017

[16] : “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] : “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] : “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] : “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

##### Selected papers on minimal surfaces

[3] : “The existence of minimal immersions of two-spheres,” Bull. Am. Math. Soc. 83 : 5 (1977), pp. 1033–1036. MR 448408 Zbl 0375.49016

[5] : “The existence of minimal immersions of 2-spheres,” Ann. Math. (2) 113 : 1 (January 1981), pp. 1–24. MR 604040 Zbl 0462.58014

[9] : “Minimal immersions of closed Riemann surfaces,” Trans. Am. Math. Soc. 271 : 2 (1982), pp. 639–652. MR 654854 Zbl 0527.58008

[13] : “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] : “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] : “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

##### Selected papers on harmonic maps

[1] : “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

[6] : “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

[10] : “A regularity theory for harmonic maps,” J. Diff. Geom. 17 : 2 (1982), pp. 307–335. MR 664498 Zbl 0521.58021

[11] : “Boundary regularity and the Dirichlet problem for harmonic maps,” J. Diff. Geom. 18 : 2 (1983), pp. 253–268. MR 710054 Zbl 0547.58020

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

[15] : “Regularity of minimizing harmonic maps into the sphere,” Invent. Math. 78 : 1 (February 1984), pp. 89–100. MR 762354 Zbl 0555.58011

##### Selected papers on Hermitian Yang–Mills connections

[18] : “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

[19] : “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

##### Selected papers on gauge theory

[4] : “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] : “Removable singularities in Yang–Mills fields,” Comm. Math. Phys. 83 : 1 (February 1982), pp. 11–29. MR 648355 Zbl 0491.58032

[8]
:
“Connections with __\( L^p \)__ bounds on curvature,”
Comm. Math. Phys.
83 : 1
(February 1982),
pp. 31–42.
MR
648356
Zbl
0499.58019

[17] : “The Chern classes of Sobolev connections,” Comm. Math. Phys. 101 : 4 (December 1985), pp. 449–457. MR 815194 Zbl 0586.53018

[21] : “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