Celebratio Mathematica — Uhlenbeck

Karen Uhlenbeck's contributions to gauge theoretic analysis

Works connected to Lesley Millman Sibner

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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 article

The Yang–Mills functional for connections on principle \( SU(2) \) bundles over \( S^4 \) is studied. Critical points of the functional satisfy a system of second-order partial differential equations, the Yang–Mills equations. If, in particular, the critical point is a minimum, it satisfies a first-order system, the self-dual or anti-self-dual equations. Here, we exhibit an infinite number of finite-action nonminimal unstable critical points. They are obtained by constructing a topologically nontrivial loop of connections to which min-max theory is applied. The construction exploits the fundamental relationship between certain invariant instantons on \( S^4 \) and magnetic monopoles on \( H^3 \). This result settles a question in gauge field theory that has been open for many years.

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