## Abstract

In this paper, we focus on the geometry of compact conformally flat manifolds (M^{n},g) with positive scalar curvature. Schoen–Yau proved that its universal cover (M^{n}˜,g˜) is conformally embedded in S^{n} such that M^{n} is a Kleinian manifold. Moreover, the limit set of the Kleinian group has Hausdorff dimension <[Formula presented]. If additionally we assume that the non-local curvature Q_{2γ}≥0 for some 1<γ<2, the Hausdorff dimension of the limit set is less than or equal to [Formula presented]. If Q_{2γ}>0, then the above inequality is strict. Moreover, the above upper bound is sharp. As applications, we obtain some topological rigidity and classification theorems in lower dimensions.

Original language | English (US) |
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Pages (from-to) | 130-169 |

Number of pages | 40 |

Journal | Advances in Mathematics |

Volume | 335 |

DOIs | |

State | Published - Sep 7 2018 |

Externally published | Yes |

## All Science Journal Classification (ASJC) codes

- Mathematics(all)

## Keywords

- Nonlocal curvature
- Sharp Hausdorff dimension estimate