An L² curvature pinching result for the Euclidean 3-disk

When studying the Cauchy problem of general relativity we typically obtain L² bounds on the (Ricci) curvature tensor of spacelike hypersurfaces and its derivatives. In many situations it is useful to deduce from these H^{k} bounds that there exists coordinates on the spacelike hypersurface with (optimal) H^{k+2} bounds on the components of the induced Riemannian metric. The general idea is that this can be achieved using harmonic coordinates -- in which the principal terms of the Ricci curvature tensor are the Laplace-Beltrami operators of the metric components -- and standard elliptic regularity results. In this talk, I will make this idea concrete in the case of Riemannian 3-manifolds with boundary, with curvature in L² and second fundamental form of the boundary in H^{1/2} both close to their respective Euclidean unit 3-disk values. The crux of the proof is a refined Bochner identity with boundary for harmonic functions. The cherry on the cake is that this result does not require any topology assumption on the Riemannian 3-manifold, and that we obtain -- as a conclusion -- that it must be diffeomorphic to the 3-disk. This talk is based on a result that I obtained in [Global nonlinear stability of Minkowski space for spacelike-characteristic initial data, Appendix A].