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 Q2γ≥0 for some 1<γ<2, the Hausdorff dimension of the limit set is less than or equal to [Formula presented]. 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.
|Original language||English (US)|
|Number of pages||40|
|Journal||Advances in Mathematics|
|State||Published - Sep 7 2018|
All Science Journal Classification (ASJC) codes
- Nonlocal curvature
- Sharp Hausdorff dimension estimate