Convergence of Riemannian 4-manifolds with L2L^{2}-curvature bounds

Abstract

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 the series of works [Str08], [Str12b], [Str12a], [Str13a], [Str13b] and [Str16]. In particular, we use his "tubular averaging technique", which he has introduced in [Str16, Section 3], 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.