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arXiv · 1704.06633

A sphere theorem for Bach-flat manifolds with positive constant scalar curvature

Yi Fang·Wei Yuan
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Abstract

We show a closed Bach-flat Riemannian manifold with a fixed positive constant scalar curvature has to be locally spherical if its Weyl and traceless Ricci tensors are small in the sense of either $L^\infty$ or $L^{\frac{n}{2}}$-norm. Compared with the complete non-compact case done by Kim, we apply a different method to achieve these results. These results generalize a rigidity theorem of positive Einstein manifolds due to M.-A.Singer. As an application, we can partially recover the well-known Chang-Gursky-Yang's $4$-dimensional conformal sphere theorem.

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Yi Fang, Wei Yuan. 2017-04-21. A sphere theorem for Bach-flat manifolds with positive constant scalar curvature. https://arxiv.org/abs/1704.06633

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