Karen Uhlenbeck

Connections with

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

L^{p}

Complete, conformally flat metrics of constant positive scalar cur- vature on the complement of k points in the n-sphere, k ≥ 2, n ≥ 3, were constructed by R. Schoen in 1988. We consider the problem of determining the moduli space of all such metrics. All such metrics are asymptotically peri- odic, and we develop the linear analysis necessary to understand the nonlinear problem. This includes a Fredholm theory and asymptotic regularity theory for the Laplacian on asymptotically periodic manifolds, which is of indepen- dent interest. The main result is that the moduli space is a locally real analytic variety of dimension k. For a generic set of nearby conformal classes the mod- uli space is shown to be a k-dimensional real analytic manifold. The structure as a real analytic variety is obtained by writing the space as an intersection of a Fredholm pair of infinite dimensional real analytic manifolds. Department of Mathematics, Stanford University, Stanford, California 94305 E-mail address: mazzeo@math.stanford.edu Department of Mathematics, University of Chicago, Chicago, Illinois 60637 E-mail address: pollack@math.uchicago.edu Department of Mathematics, University of Texas, Austin, Texas 78712 E-mail address: uhlen@math.utexas.edu License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use

bounds on curvature

This article is a collection of letters solicited by the editors of the Bulletin in response to a previous article by Jaffe and Quinn [math.HO/9307227]. The authors discuss the role of rigor in mathematics and the relation between mathematics and theoretical physics.

Moduli Spaces of Singular Yamabe Metrics

AbstractAssumeF is the curvature (field) of a connection (potential) onR4 with finiteL2 norm

THE CHERN CLASSES OF SOBOLEV CONNECTIONS

\left( {\int\limits_{R^4 } {\left| F \right|^2 dx< \infty } } \right)

Morse theory on Banach manifolds

. We show the chern number

On the connection between harmonic maps and the self-dual Yang-Mills and the sine-Gordon equations

c_2= {1 \mathord{\left/ {\vphantom {1 8}} \right. \kern-\nulldelimiterspace} 8}\pi ^2 \int\limits_{R^4 } {F \wedge} F