Norman Zergänge

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.

In this work we give a detailed description of Matthias Günther's proof of the Isometric Embedding Theorem of Riemannian manifolds. Subsequently we will use this method to show that it is possible to construct an isometric embedding of a geometric flow, for instance the Ricci-flow, into some Euclidean space.