Karen Uhlenbeck is being awarded the 2019 Abel Prize. It is a remarkable award for a remarkable mathematician. Uhlenbeck did fundamental work in a quickly developing area of mathematics at an early stage of its development. I was a graduate student when some of her significant papers were coming out. There were quite a few people trying to understand her results and her techniques, diving deep into the difficult analysis she was unleashing on geometric problems. Geometric analysts have long recognized Uhlenbeck’s contributions. It is nice to see Uhlenbeck and her work recognized more widely with the Abel Prize.
For an insightful description of some of Uhlenbeck’s work, the recent article by Simon Donaldson in the Notices of the AMS is tremendous. Below are the texts of some reviews of her work in MathSciNet.
MR0264714
Uhlenbeck, K.
Harmonic maps; a direct method in the calculus of variations.
Bull. Amer. Math. Soc. 76 1970 1082–1087.
It was shown by J. H. Sampson and the reviewer [Amer. J. Math. 86 (1964), 109–160; MR0164306] that in every homotopy class of maps of one compact Riemannian manifold M into another N of negative curvature, there is a harmonic map. Furthermore, S. I. Alʹber [Dokl. Akad. Nauk SSSR 178 (1968), 13–16; MR0230254] and P. Hartman [Canad. J. Math. 19 (1967), 673–687;MR0214004] have established certain uniqueness results if N has strictly negative curvature. Our technique was to follow gradient lines (of the tension field of the energy function E) in a suitable space of maps, since limit points are harmonic maps; that involved a rather delicate and explicit study of the appropriate elliptic and parabolic systems. We went on [the reviewer and Sampson, Proc. U.S.-Japan Sem. Differential Geometry (Kyoto, 1965), pp. 22–33, Nippon Hyoronsha, Tokyo, 1966; MR0216519] to show that without any curvature restrictions, there exists in every homotopy class a polyharmonic map, where the degree of polyharmonicity depends on the dimension of M. The proof in that case was based on Morse theory on Hilbertian manifolds of maps, and in particular on verification of the Palais-Smale condition (C) for the poly-energy function.
In the paper under review (which is an announcement, with sketch of proofs. Let us hope that a full account will appear in the near future, as these ideas deserve) a new method of proof is described for the existence of harmonic maps. It involves verification of (C) for perturbations (e.g., adding pth powers of the differential, or of the Laplacian) of the energy E, in the context of Morse theory on Finsler manifolds of maps. (A special case of these perturbations was discovered, for the same purpose, by H. I. Eliasson [Global analysis (Proc. Sympos. Pure Math., Vol. XVI, Berkeley, Calif., 1968), pp. 67–89, Amer. Math. Soc., Providence, R.I., 1970;MR0267605].) The method displays how the curvature restriction on N forces convergence of critical points of the perturbed energies to critical points of the energy itself. In addition to establishing existence, the method provides unique results (in the spirit of the proof of Alʹber). It would be interesting to see whether these methods permit extension to the Plateau problem (where M has a boundary and harmonic maps have prescribed image on the boundary).
Reviewed by J. Eells
MR0464332
Uhlenbeck, K.
Generic properties of eigenfunctions.
Amer. J. Math. 98 (1976), no. 4, 1059–1078.
Let $M^n$ be a compact $n$-manifold and let $L_b$ be a family of self-adjoint elliptic operators on $M^n$ with the parameter $b\in U$ an open subset of a Banach space $B$. The author shows that under reasonable hypotheses the following properties are generic with respect to $B$, i.e., for almost all $b\in U$, (a) $L_b$ has one-dimensional eigenspaces; (b) zero is not a critical value of the eigenfunctions, restricted to the interior of the domain of the operator; (c) the eigenfunctions are Morse functions on the interior of $M$; (d) if $\partial M\neq\varnothing$ and Dirichlet boundary conditions have been imposed, then the normal derivative of the eigenfunctions has zero as a regular value. The author gives several applications. For example, let $\Delta_g$ be the Laplace operator for a metric $g\in\scr M_k=${$C^k$-metrics on $M^n$} for $k>n+3$. Then {$g\in\scr M_k\colon\Delta_g$satisfies (a), (b), (c) and (d) on nonconstant eigenfunctions} is residual in $\scr M_k$.
Reviewed by A. J. Tromba
MR0604040
Sacks, J.; Uhlenbeck, K.
The existence of minimal immersions of $2$-spheres.
Ann. of Math. (2) 113 (1981), no. 1, 1–24.
Let $M_p$ be a closed Riemann surface of genus $p\geq 0$ and $N$ a compact Riemannian manifold. In this interesting paper, the authors establish the existence of harmonic maps in three cases: (i) If $\pi_2(N)=0$, every homotopy class of maps $M_p\rightarrow N$ contains a harmonic map of minimum energy—this was established by L. Lemaire [J. Differential Geom. 13(1978), no. 1, 51–78; MR0520601]; see also the article by R. Schoen and S. T. Yau [Ann. of Math. (2) 110 (1979), no. 1, 127–142; MR0541332]. (ii) If $\pi_2(N)\neq 0$, then a generating set for $\pi_2(N)$ modulo the action of $\pi_1(N)$ can be represented by harmonic maps $M_0\rightarrow N$ of minimum energy—such a map is automatically a conformal branched immersion of minimum area. (iii) If $N$ has noncontractible universal covering space, there exists a nonconstant conformal minimal branched immersion $M_0\rightarrow N$ (which may be a saddle point for the energy). The method is to perturb the energy functional $E$ to give a functional $E_\alpha$ ($\alpha\geq 1,E_1=E+$ constant) which satisfies the condition (C) of Palais and Smale and then to study the complicated convergence of critical maps of $E_\alpha$ as $\alpha\rightarrow 1$. Note that a nonexistence theorem contrasting with (ii) was given by A. Futaki [Proc. Japan Acad. Ser. A Math. Sci. 56 (1980), no. 6, 291–293;MR0581474].
Reviewed by John C. Wood
MR0664498
Schoen, Richard; Uhlenbeck, Karen
A regularity theory for harmonic maps.
J. Differential Geom. 17 (1982), no. 2, 307–335.
Let $M^n$ and $N^k$ be Riemannian manifolds, with $M$ compact. Let $u\colon M\rightarrow N$ be an $L_1^2$-map minimizing the energy functional $E(u)=\int\langle du(x),du(x)\rangle dV$, possibly modified to include lower order terms. Theorem: If $u(M)$is in a compact subset of $N$ a.e., then $u$ is smooth on $M-\scr S_u$ for a suitable closed set $\scr S_u$ of Hausdorff dimension $\leq n-3$. If $n=3$, then $\scr S_u$ is discrete. In fact, $\scr S_u=\{a\in M\colon\liminf_{l\rightarrow 0}E_{D_l(a)}(n)/l^{n-2}>0\}$. (An important special case of that result was obtained simultaneously by M. Giaquinta and E. Giusti [MR0648066 above; Analysis, to appear]; there $u(M)$ is required to lie in a coordinate chart.) The case $n=2$ is due to C. B. Morrey; for arbitrary $n$a key idea from potential theory is Morrey’s Dirichlet growth lemma [C. B. Morrey, Multiple integrals in the calculus of variations, Springer, New York, 1966; MR0202511]. Another basic idea (used also by Giaquinta-Giusti), this time from geometric measure theory, is H. Federer’s reduction theorem [Bull. Amer. Math. Soc. 76 (1970), 767–771; MR0260981]. A major technical difficulty overcome in this paper is to find comparison maps (as used by Morrey) which have images in $N$. By refining their arguments the authors obtain the theorem: In addition to the hypotheses of the preceding theorem, suppose that every harmonic map $\theta\colon S^j\rightarrow N$ is constant for $2\leq j\leq n-1$, where $S^j$ is the Euclidean $j$-sphere; (more generally and importantly; if $p\colon{\bf R}^{j+1}-\{0\}\rightarrow S^j$ denotes radial projection and $\theta\colon S^j\rightarrow N$ is a harmonic map such that $\theta\circ p$ minimizes energy on compact subsets of ${\bf R}^{j+1}$, then $\theta\circ p$ is constant $(2\leq j\leq n-1)$). Then $\scr S_u=\varnothing$; i.e., $u$ is smooth on $M$. This is a very important contribution to the theory of harmonic maps. For instance, these hypotheses are satisfied if the universal cover of $N$ supports a strictly convex smooth function. That case provides a new proof and generalization of the existence theorem of the reviewer and J. H. Sampson [Amer. J. Math. 86 (1964), 109–160; MR0164306] —and much else, as well.
{In the correction, it is noted that “harmonic” should be deleted from the statement of Lemma 2.5. The other results are not affected by the change.}
Reviewed by J. Eells
MR0710054
Schoen, Richard; Uhlenbeck, Karen
Boundary regularity and the Dirichlet problem for harmonic maps.
J. Differential Geom. 18 (1983), no. 2, 253–268.
This paper follows an earlier one by the authors [same journal 17 (1982), no. 2, 307–335; MR0664498]. Both papers are based on the facts that a harmonic map $u\;(u\in L^2_{1\,{\rm loc}}({\bf R}^n,N))$ which is constant along the rays from 0 a.e. defines a new map $w:S^{n-1} \rightarrow N$ such that $u(x)=w(x/\vert x\vert )$ which is also harmonic and conversely; furthermore $u$ has a singularity at 0 if and only if $w$ is not a constant map (Theorems III and IV). This result supplies an estimate of the Hausdorff dimension of the set of singularities. The first paper dealt with inner regularity for harmonic maps, this one deals with boundary regularity for harmonic maps satisfying a Dirichlet problem. The boundary regularity is actually stronger. This is due to the fact that there are no nontrivial smooth harmonic maps from hemispheres $S^{n-j}_+$ which map the boundary $S^{n-j-1}$to a point $(1\leq j\leq n-2)$. In fact, the Euler-Lagrange equation is deduced by minimizing a functional $\tilde E$ slightly more general than the energy $E,\tilde E(u)=E(u) +V(u)$, where $V(u)$ is the integral over $M$ of $\Sigma_i\Sigma_\alpha\,\gamma ^\alpha_i(x,u(x))\partial u^i/\partial x^\alpha+\Gamma(x,u(x))$. The main regularity theorem is the following: Let $M$ be a compact manifold with $C^{2,\alpha}$ boundary. Suppose $u\in L^2_1(M,N)$ is $E$-minimizing and satisfies $u(x)\in N_0$ a.e. for a compact subset $N_0\subset N$. Suppose $v\in C^{2,\alpha}(\partial M,N_0)$ and $u=v$ on $\partial M$. Then the singular set $S$ of $u$ is a compact subset of the interior of $M$; in particular, $u$ is $C^{2,\alpha}$ in a full neighborhood of $\partial M$.
An application is an amusing proof of a theorem of Sacks and Uhlenbeck on the existence of minimal 2-spheres representing the second homotopy group of a manifold: “If $N$ is compact with convex or empty boundary, any smooth map $v:S^2\rightarrow N$which does not extend continuously to $B^3$ is homotopic to a sum of smooth harmonic (hence minimal) maps$u_j:S^2\rightarrow N$, $j=1,2,\cdots,p$”. The last section deals with approximation of $L^2_1$ maps by smooth maps. The authors give a simple example of a map $u\in L^2_1(B^3_1,S^2)$ such that $u(x)=x/\vert x\vert $ cannot be an $L^2_1$ limit of continuous maps; this result is still true for $L^2_1(M,N)$ with $\dim M\geq 3$. On the contrary if $\dim M=2$, an $L^2_1$ map is a limit of smooth maps.
Reviewed by Liane Valere Bouche
MR0861491
Uhlenbeck, K.(1-CHI); Yau, S.-T.
On the existence of Hermitian-Yang-Mills connections in stable vector bundles.
Frontiers of the mathematical sciences: 1985 (New York, 1985).
Comm. Pure Appl. Math. 39 (1986), no. S, suppl., S257–S293.
The Yang-Mills equations, which arise in particle physics, have been applied with great success to study the differential topology of $4$-manifolds. Of particular interest are complex surfaces, in which case the Yang-Mills equations have a holomorphic interpretation. Let $X$ be a compact Kähler manifold (of any dimension) and $E$ a holomorphic bundle over $X$. A Hermitian metric on $E$ determines a canonical unitary connection whose curvature is a $(1,1)$-form $F$ which is a skew-Hermitian transformation of $E$. The inner product of $F$ with the Kähler form is then an endomorphism of $E$, and the connection satisfies the Yang-Mills equations if and only if this endomorphism is a multiple of the identity. Metrics which give rise to such connections are called Hermitian-Einstein metrics, and the resulting connection is termed Hermitian-Yang-Mills. It is natural to ask which bundles admit such metrics. Recall first a definition from algebraic geometry. The slope of a bundle is the ratio of its degree to its rank, and a bundle $E$ is said to be stable if the slope of any coherent subsheaf of lower rank is strictly less than the slope of $E$. The main theorem of the present paper asserts that a stable holomorphic bundle over a compact Kähler manifold admits a unique Hermitian-Einstein metric. For complex curves this is an old result of Narasimhan and Seshadri. S. K. Donaldson [Proc. London Math. Soc. (3) 50 (1985), no. 1, 1–26; MR0765366] gave a proof for projective algebraic surfaces, and (subsequent to the work of the authors) extended his work to cover projective complex manifolds of any dimension [Duke Math. J. 54 (1987), no. 1, 231–247;MR0885784].
The authors make a direct study of the partial differential equation arising from the Hermitian-Yang-Mills condition. The continuity method is used to demonstrate the existence of solutions to a perturbed equation. Of course, this involves a priori estimates for the solutions. Let $\varepsilon$ be the perturbation parameter, and $h_\varepsilon$ the solution to the perturbed equation. Then as $\varepsilon\rightarrow 0$ either the $h_\varepsilon$ converge to a solution, or (after appropriate renormalization) the limit represents an $L_1^2$ holomorphic projection onto a subbundle. A major step in the proof consists in showing that this projection is smooth outside a subvariety of codimension at least $2$ and that the image is a coherent subsheaf of $E$. Then the previous estimates are used, via the Chern-Weil theory, to show that this subsheaf is destabilizing for $E$.
The regularity theorem for the projection is stated in general terms. Let $M$ be an algebraic manifold, which we assume is embedded in some projective space. A map $F$ from the unit ball in $\mathbf{C}$ to $M$ is said to be weakly holomorphic if it is in $L_1^2$ and if its differential maps the holomorphic tangent space of the ball into the holomorphic tangent space of $M$ almost everywhere. For balls of higher dimension a map is weakly holomorphic if for every linear coordinate system $\{z_1,\cdots,z_n\}$ and for almost every value of $z_2,\cdots,z_n$ it is weakly holomorphic as a function of $z_1$. Then the authors prove that any weakly holomorphic map into an algebraic manifold is meromorphic.
Both the result and the techniques of this paper are important. They have already found use in the theses of K. Corlette [“Flat $G$-bundles with canonical metrics”, J. Differential Geom., to appear] and C. Simpson [“Systems of Hodge bundles and uniformization”, Ph.D. Thesis, Harvard Univ., Cambridge, Mass., 1987; per revr.].
Reviewed by Daniel S. Freed
