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.
This method was introduced by Jeffrey Streets.
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.
On the $k$-th nullity space of the Riemannian curvature tensor
