Huber Theorem Revisited in Dimension 4

We study the second Huber theorem in dimension 4. Given a metric having a pointwise singularity with $L^p$ -bounds on the Bach tensor, we construct a conformal metric which is regular across the singularity. To do so, we introduce another Coulomb-type condition, similar to the case of Yang-Mills connections. This enables us to obtain a conformal metric satisfying an $\epsilon$ -regularity property. We obtain a generalization of the two-dimensional case that can be applied to study the singularities of Bach-flat metrics and immersions with second fundamental forms in $W^{2,\frac{4}{3}+\epsilon }$ .