A sphere theorem for Bach-flat manifolds with positive constant scalar curvature
A sphere theorem for Bach-flat manifolds with positive constant scalar curvature
复制标题
DOI：
10.1016/j.difgeo.2019.01.004
复制
发表时间：
2017-04
影响因子：
0.5
通讯作者：
Yi Fang;Wei Yuan
中科院分区：
文献类型：
--
作者：
Yi Fang;Wei Yuan
完备非紧Ricci平坦流形的一个经典间隙定理。安德森建议，人们可以得到刚性的某些空间简单地通过当地的曲率估计测地球上的全球。利用类似的思想，金证明了一个4维完全非紧巴赫平坦流形与零标量曲率和小的L2曲率张量必须是平坦的。不幸的是，这种方法并没有推广到没有边界的紧致流形。应用不同的方法，我们证明了一个具有正常数量曲率的闭Bach平坦黎曼流形，如果它的Weyl张量和无迹Ricci张量在L∞或Ln 2-范数意义下都很小，则它必须是局部球面的.这些结果推广了M.- A.歌手.作为应用，我们可以部分恢复著名的Chang-Gursky-Yang四维共形球面定理。
A classic gap theorem for complete non-compact Ricci flat manifolds by M. Anderson suggests that one can get the rigidity of certain spaces simply by passing local curvature estimates on geodesic balls to global. Using similar ideas, Kim showed that a 4-dimensional complete non-compact Bach-flat manifold with vanishing scalar curvature and small L 2-curvature tensor has to be flat. Unfortunately, this method does not generalize to compact manifolds without boundary. Applying a different approach, we show that a closed Bach-flat Riemannian manifold with positive constant scalar curvature has to be locally spherical if its Weyl and traceless Ricci tensors are small in the sense of either L∞ or L n 2-norm. 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.