Non-local Curvature and Topology of Locally Conformally Flat Manifolds
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 $(\widetilde{M^n},\tilde{g})$ is conformally embedded in $\mathbb{S}^n$ such that $M^n$ is a Kleinian manifold. Moreover, the limit set of the Kleinian group has Hausdorff dimension $<\frac{n-2}{2}$. If additionally we assume that the non-local curvature $Q_{2\gamma}\geq 0$ for some $1<\gamma<2$, the Hausdorff dimension of the limit set is less than or equal to $\frac{n-2\gamma}{2}$. If $Q_{2\gamma}>0$, 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.
- Publication:
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arXiv e-prints
- Pub Date:
- October 2015
- DOI:
- 10.48550/arXiv.1510.00957
- arXiv:
- arXiv:1510.00957
- Bibcode:
- 2015arXiv151000957Z
- Keywords:
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- Mathematics - Differential Geometry;
- Mathematics - Analysis of PDEs
- E-Print:
- We added some new rigidity results for nonnegative fractional curvature (Theorem 1.3, Theorem 1.4, Theorem 1.5). Also we obtained a sharp upper bound in the nonnegative fractional curvature case (see Theorem 1.1)