Karen Uhlenbeck

Bio: Karen Uhlenbeck is an academic researcher from University of Texas at Austin. The author has contributed to research in topics: Harmonic map & Morse theory. The author has an hindex of 35, co-authored 76 publications receiving 8546 citations. Previous affiliations of Karen Uhlenbeck include University of Illinois at Chicago & University of Illinois at Urbana–Champaign.

A regularity theory for harmonic maps

Abstract: 0. Introduction In this paper we develop a regularity theory for energy minimizing harmonic maps into Riemannian manifolds. Let u: M -> N be a map between Riemannian manifolds of dimension n and k. It was shown by C. B. Morrey [17] in 1948 that if n — 2, then an energy minimizing harmonic map is Holder continuous (and smooth if M and N are smooth). Since that time results have been found under special assumptions on N. Eells and Sampson [5] proved in 1963 that if N is compact and has nonpositive curvature, then every homotopy class of maps from a closed manifold M into N has a smooth harmonic representative. In the case where the image of the map is contained in a convex ball of N9 there is a complete existence and regularity theory due to Hildebrandt and Widman [15] as well as Hildebrandt, Kaul and Widman [13]. Recently Giaquinta and Giusti obtained results for the case in which the image lies in a coordinate chart [9], [10]. In this paper we show that a bounded, energy minimizing map u: M -» N is regular (in the interior) except for a closed set S of Hausdorff dimension at most n — 3. We also show S is discrete for n = 3. Moreover, we derive techniques (see Theorem IV) for lowering the dimension of S under the condition that certain smooth harmonic maps of spheres into N are trivial. This can be checked in some interesting cases, for example if N has nonpositive curvature or if the image of the map lies in a convex ball of N, we show S = 0 and any minimizing harmonic map into such a manifold is smooth. Using our methods, it is possible to reduce the dimension of S if N is a sphere or Lie group by studying harmonic spheres in N. Our methods work for functional which are the energy plus lower order terms, and thus have direct bearing on the question of the existence of global Coulomb gauges in nonabelian gauge theories. We point out that there is a strong historical precedent for partial regularity results in problems involving elliptic systems (see Almgren [1], De Giorgi [3],

Connections with l**p bounds on curvature

Abstract: We show by means of the implicit function theorem that Coulomb gauges exist for fields over a ball inR n when the integralL n/2 field norm is sufficiently small. We then are able to prove a weak compactness theorem for fields on compact manifolds withL p integral norms bounded,p>n/2.

Regularity for a class of non-linear elliptic systems

Abstract: where ~ is a smooth positive function satisfying the ellipticity condition o(Q) + 2~'(Q)q > 0, V denotes the gradient, and [Vs[2 = ~ = 1 ]Vskl 2. This type of system arises as the EulerLagrange equations for the stat ionary points of an energy integral which has an intrinsic definition on maps between two Riemannian manifolds; the equations are therefore of geometric interest. However, the method of proof also applies to the equations of non-linear Hodge theory, which have been studied by L. M. and R. B. Sibner 19]. These are systems of equations for a closed p-form so, deo = 0 and

Positive solutions of nonlinear elliptic equations involving critical sobolev exponents

Abstract: Soit Ω un domaine borne dans R n avec n≥3 On etudie l'existence d'une fonction u satisfaisant l'equation elliptique non lineaire -Δu=u P +f(x,u) sur Ω, u>0 sur Ω, u=0 sur ∂Ω, ou p=(n+2)/(n−2), f(x,0)=0 et f(x,u) est une perturbation de u P de bas ordre au sens ou lim u→+α f(x,u)/u P =0

Topological quantum field theory

Abstract: A twisted version of four dimensional supersymmetric gauge theory is formulated. The model, which refines a nonrelativistic treatment by Atiyah, appears to underlie many recent developments in topology of low dimensional manifolds; the Donaldson polynomial invariants of four manifolds and the Floer groups of three manifolds appear naturally. The model may also be interesting from a physical viewpoint; it is in a sense a generally covariant quantum field theory, albeit one in which general covariance is unbroken, there are no gravitons, and the only excitations are topological.

Pseudo holomorphic curves in symplectic manifolds

Abstract: Definitions. A parametrized (pseudo holomorphic) J-curve in an almost complex manifold (IS, J) is a holomorphic map of a Riemann surface into Is, say f : (S, J3 ~(V, J). The image C=f(S)C V is called a (non-parametrized) J-curve in V. A curve C C V is called closed if it can be (holomorphically !) parametrized by a closed surface S. We call C regular if there is a parametrization f : S ~ V which is a smooth proper embedding. A curve is called rational if one can choose S diffeomorphic to the sphere S 2.

The Yang-Mills equations over Riemann surfaces

Abstract: The Yang-Mills functional over a Riemann surface is studied from the point of view of Morse theory. The main result is that this is a ‘perfect9 functional provided due account is taken of its gauge symmetry. This enables topological conclusions to be drawn about the critical sets and leads eventually to information about the moduli space of algebraic bundles over the Riemann surface. This in turn depends on the interplay between the holomorphic and unitary structures, which is analysed in detail.