Connections withL P bounds on curvature

Abstract

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

Singularity formation in the Yang-Mills Flow

“…Suppose i is large enough so that t ′ is contained in the interval (−δ/λ 2 i , 0) and that D is contained in the ball B r/λ i (0). Fix a point x 0 in D. We will now use a theorem of Uhlenbeck's [13], which states that if the L n norm of the curvature is small enough, then there exists a local Coulomb gauge, in which the L p 1 norm of the connection matrix is bounded by the L p norm of the curvature, for any p ≥ 2n. Since by assumption the curvature is uniformly bounded by C/(T − t), we can apply Uhlenbeck's theorem in a small enough ball B containing x 0 .…”

“…In section 2, we show that given a bound on the curvature, we can derive bounds on all of the derivatives of the curvature. In section 3, we use these estimates, together with a theorem of Uhlenbeck's [13], to get bounds on the connections and then convergence of a sequence of blow-ups near the singular point. We use Hamilton's monotonicity formula [7] to show that, allowing gauge transformations, the sequence converges to a homothetically shrinking soliton.…”

The Gauss-Bonnet-Chern theorem: A probabilistic perspective

“…We achieve this by gradually reducing the computation of these terms to the special case involving the heat kernel. The estimate (1.14d) is trickier and follows using a method reminiscent to the one employed by K. Uhlenbeck in [46].…”

Boundary regularity via Uhlenbeck–Rivière decomposition

“…We intend to apply the following nonlinear decomposition, which is due to Rivière and adapts Uhlenbeck's technique in [Uhl82]:…”