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Convergence of Riemannian 4-manifolds with L 2 L^{2} -curvature bounds
Advances in Calculus of Variations ( IF 1.7 ) Pub Date : 2019-01-20 , DOI: 10.1515/acv-2017-0058
Norman Zergänge 1
Affiliation  

Abstract In this work we prove convergence results of sequences of Riemannian 4-manifolds with almost vanishing L 2 {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 {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 {L^{2}} -norm of the Riemannian curvature tensor is uniformly bounded from above, and whose L 2 {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 {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 {L^{2}} -curvature flow, which only depend on significant geometric bounds. This is the content of Theorem 1.3.

中文翻译:

具有 L 2 L^{2} -曲率边界的黎曼四流形的收敛

摘要 在这项工作中,我们证明了黎曼 4 流形序列的收敛结果,其中 L 2 {L^{2}} 几乎为零的曲率张量范数和小球体积上的非塌陷边界。在定理 1.1 中,我们考虑了一系列封闭的黎曼 4-流形,其黎曼曲率张量的 L 2 {L^{2}} -范数趋于零。在均匀非塌陷边界和均匀直径边界的假设下,我们证明存在一个关于 Gromov-Hausdorff 拓扑收敛到平面流形的子序列。在定理 1.2 中,我们考虑了一系列封闭的黎曼 4-流形,其黎曼曲率张量的 L 2 {L^{2}} -范数从上方一致有界,其 ​​L 2 {L^{2}} -范数无迹 Ricci 张量趋于零。这里,在均匀非塌陷边界的假设下,这非常接近欧几里得情况,并且具有均匀的直径界限,我们证明存在一个在 Gromov-Hausdorff 意义上收敛到爱因斯坦流形的子序列。为了证明定理 1.1 和定理 1.2,我们使用了一种平滑技术,称为 L 2 {L^{2}} -曲率流。这种方法是由 Jeffrey Streets 引入的。特别是,我们使用他的“管状平均技术”来证明 L 2 {L^{2}} -曲率流的距离估计,这仅取决于重要的几何边界。这就是定理 1.3 的内容。这称为 L 2 {L^{2}} -曲率流。这种方法是由 Jeffrey Streets 引入的。特别是,我们使用他的“管状平均技术”来证明 L 2 {L^{2}} -曲率流的距离估计,这仅取决于重要的几何边界。这就是定理 1.3 的内容。这称为 L 2 {L^{2}} -曲率流。这种方法是由 Jeffrey Streets 引入的。特别是,我们使用他的“管状平均技术”来证明 L 2 {L^{2}} -曲率流的距离估计,这仅取决于重要的几何边界。这就是定理 1.3 的内容。
更新日期:2019-01-20
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