In differential geometry and in particular YangâÂÂMills theory, Uhlenbeck's compactness theorem is a result about sequences of (weak YangâÂÂMills) connections with uniformly bounded curvature having weakly or uniformly convergent subsequences up to gauge. It is an important theorem used in the compactification of the anti self-dual YangâÂÂMills moduli space (ASDYM moduli space), which is central to the construction of Donaldson invariants on four-dimensional manifolds (short 4-manifold) or monopole Floer homology on three-dimensional manifolds (short 3-manifold). The theorem is named after Karen Uhlenbeck, who first described it in 1982. In 2019, Uhlenbeck became the first woman to be awarded the Abel Prize, in part for her contributions to partial differential equations and gauge theory. Uhlenbeck's compactness theorem was generalized to YangâÂÂMills flows by Alex Waldron in 2018.
Let be a -dimensional compact Riemannian manifold and be a principal -bundle with a compact Lie group . Let with and let be a sequence of Sobolev connections with uniform bound for , the norm of their curvatures. Then there exists a sequence of gauge transformations, so that converges weakly. In other words, any -bounded subset of is weakly compact.
Let be a -dimensional compact Riemannian manifold and be a principal -bundle with a compact Lie group . Let with and if . Let be a sequence of weak YangâÂÂMills connections, hence so that:
for all and , with uniform bound for . Then there exists a subsequence, also denoted , and a sequence of gauge transformations, so that converges uniformly to a smooth connection . (Uhlenbeck's strong compactness theorem is not stated explicitly in Uhlenbeck's 1982 paper, but follows from the results within.)