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

Norman Zergänge
• Corresponding author
• Institut für Differentialgeometrie, Leibniz Universität Hannover, Welfengarten 1, 30167 Hannover, Germany
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Published Online: 2019-01-20 | DOI: https://doi.org/10.1515/acv-2017-0058

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.

MSC 2010: 53C25; 53C44

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Revised: 2018-12-11

Accepted: 2018-12-17

Published Online: 2019-01-20

Citation Information: Advances in Calculus of Variations, ISSN (Online) 1864-8266, ISSN (Print) 1864-8258,

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