Nonlocal curvature and topology of locally conformally flat manifolds

In this paper, we focus on the geometry of compact conformally flat manifolds (Mⁿ,g) with positive scalar curvature. Schoen–Yau proved that its universal cover (Mⁿ˜,g˜) is conformally embedded in Sⁿ such that Mⁿ 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.