Convergence of Riemannian $4$-manifolds with $L^2$-curvature bounds

In this work we prove convergence results of sequences of Riemannian $4$-manifolds with almost vanishing $L^2$-norm of a curvature tensor and a non-collapsing bound on the volume of small balls. In Theorem 1.1, we consider a sequence of closed Riemannian $4$-manifolds, whose $L^2$-norm of the Riemannian curvature tensor tends to zero. Under the assumption of a uniform non-collapsing bound and a uniform diameter bound, we prove that there exists a subsequence that converges with respect to the Gromov-Hausdorff topology to a flat manifold. In Theorem 1.2, we consider a sequence of closed Riemannian $4$-manifolds, whose $L^2$-norm of the Riemannian curvature tensor is uniformly bounded from above, and whose $L^2$-norm of the traceless Ricci-tensor tends to zero. Here, under the assumption of a uniform non-collapsing bound, which is very close to the euclidean situation, and a uniform diameter bound, we show that there exists a subsequence which converges in the Gromov-Hausdorff sense to an Einstein manifold. In order to prove Theorem 1.1 and Theorem 1.2, we use a smoothing technique, which is called $L^2$-curvature flow or $L^2$-flow, introduced by Jeffrey Streets in a series of works. In particular, we use his "tubular averaging technique" in order to prove distance estimates of the $L^2$-curvature flow which only depend on significant geometric bounds. This is the content of Theorem 1.3.