Nonlocal curvature and topology of locally conformally flat manifolds

Abstract

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 <n−22. If additionally we assume that the non-local curvature Q2γ≥0 for some 1<γ<2, the Hausdorff dimension of the limit set is less than or equal to n−2γ2. If Q2γ>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.