Non-local Curvature and Topology of Locally Conformally Flat Manifolds

TL;DR

This paper investigates the geometry and topology of compact conformally flat manifolds with positive scalar curvature, establishing bounds on the Hausdorff dimension of their limit sets under non-local curvature conditions, leading to rigidity and classification results.

Contribution

It introduces new bounds on the Hausdorff dimension of limit sets for conformally flat manifolds with non-local curvature conditions, extending previous results and providing sharp inequalities.

Findings

Hausdorff dimension of limit sets is less than (n-2)/2 for conformally flat manifolds.

Stricter bounds apply when non-local curvature Q_{2γ} > 0, with sharp inequalities.

Derived topological rigidity and classification theorems in lower dimensions.

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.